ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2

More Exercises

Question 1.
(a) In the figure (i) given below if DE || BG, AD = 3 cm, BD = 4 cm and BC = 5 cm. Find (i) AE : EC (ii) DE.
(b) In the figure (ii) given below, PQ || AC, AP = 4 cm, PB = 6 cm and BC = 8 cm. Find CQ and BQ.
(c) In the figure (iii) given below, if XY || QR, PX = 1 cm, QX = 3 cm, YR = 4.5 cm and QR = 9 cm, find PY and XY.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q1.1
Solution:
(a) In the figure (i)
Given : DE || BC, AD = 3 cm, BD = 4 cm and BC = 5 cm.
To find (i) AE : EC and
(ii) DE Since DE || BC of ∆ABC
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q1.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q1.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q1.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q1.5

Question 2.
In the given figure, DE || BC.
(i) If AD = x, DB = x – 2, AE = x + 2 and EC = x – 1, find the value of x.
(ii) If DB = x – 3, AB = 2x, EC = x – 2 and AC = 2x + 3, find the value of x.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q2.1
Solution:
In the given figure, DE || BC
(i) AD = x, DB = x – 2, AE = x + 2, EC = x – 1
In ∆ABC,
∵DE || BC
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q2.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q2.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q2.4

Question 3.
E and F are points on the sides PQ and PR respectively of a ∆PQR. For each of the following cases, state whether EF || QR:
(i) PE = 3.9 cm, EQ = 3 cm, PF = 8 cm and RF = 9 cm.
(ii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm.
Solution:
(i) In ∆PQR, E and F are the points on the sides PQ and PR respectively
PE = 3.9 cm, EQ = 3 cm, PF = 8 cm,
RF = 9 cm
Is EF || QR ?
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q3.1

Question 4.
A and B are respectively the points on the sides PQ and PR of a triangle PQR such that PQ = 12.5 cm, PA = 5 cm, BR = 6 cm and PB = 4 cm. Is AB || QR? Give reasons for your answer.
Solution:
In ∆PQR, A and B are points on the sides PQ and PR such that
PQ = 12.5 cm, PA = 5 cm, BR = 6 cm and PB = 4 cm
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q4.1

Question 5.
(a) In figure (i) given below, DE || BC and BD = CE. Prove that ABC is an isosceles triangle.
(b) In figure (ii) given below, AB || DE and BD || EF. Prove that DC² = CF x AC.
Solution:
(a) Given: In the figure,
DE || BC and BD = CE
To prove: ∆ABC is an isosceles triangle
Proof: In ∆ABC, DE || BC
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q5.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q5.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q5.3

Question 6.
(a) In the figure (i) given below, CD || LA and DE || AC. Find the length of CL if BE = 4 cm and EC = 2 cm.
(b) In the given figure, ∠D = ∠E and \(\frac { AD }{ BD } =\frac { AE }{ EC } \). Prove that BAC is an isosceles triangle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q6.1
Solution:
(a) Given : CD || LA and DE || AC
Length of BE = 4 cm
Length of EC = 2 cm
Now, in ∆BCA
DE || AC
\(\frac { BE }{ BC } =\frac { BD }{ BA } \)
(Corallary of basic proportionality theorem)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q6.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q6.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q6.4

Question 7.
In the figure given below, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. show that BC || QR.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q7.1
Solution:
In the given figure, A, B, C are points on
OP, OQ and OR respectively
and AB || PQ and AC || PR
To prove: BC || QR
Proof: In POQ,
AB || PQ
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q7.2

Question 8.
ABCD is a trapezium in which AB || DC and its diagonals intersect each other at O. Using Basic Proportionality theorem, prove that \(\frac { AO }{ BO } =\frac { CO }{ DO } \)
Solution:
Given : ABCD is a trapezium in which AB || DC
Its diagonals AC and BD intersect each other at O
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q8.1

Question 9.
(a) In the figure (1) given below, AB || CR and LM || QR.
(i) Prove that \(\frac { BM }{ MC } =\frac { AL }{ LQ } \)
(ii) Calculate LM : QR, given that BM : MC = 1 : 2.
(b) In the figure (2) given below AD is bisector of ∠BAC. If AB = 6 cm, AC = 4 cm and BD = 3cm, find BC
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q9.1
Solution:
(a) Given: AB || CR and LM II QR.
Also BM : MC = 1:2
To Prove:
(i) \(\frac { BM }{ MC } =\frac { AL }{ LQ } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q9.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q9.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q9.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 Q9.5

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

CA Foundation Business Laws Study Material Chapter 21 The Companies Act, 2013

CA Foundation Business Laws Study Material Chapter 21 The Companies Act, 2013

21.1 INTRODUCTION

The Companies Act, 2013 was preceded by the Companies Act, 1956. A need was felt to replace the Companies Act, 1956 with a new legislation due to the changes in the economic environment in India as well as abroad. The Companies Act, 2013 was enacted to consolidate and amend the law relating to the companies.
The Companies Act, 2013 contains 470 sections and seven schedules which has been divided into 29 chapters. A substantial part of this Act is in the form of Companies Rules. The Companies Act, 2013 seeks to make our corporate regulations more contemporary. It aims to:

  • Consolidate and amend the law relating to the companies.
  • Meet the changed national and international economic environment.
  • Accelerate the expansion and growth of our economy.
  • Increase accountability in corporate governance.
  • Simplify law and regulations (Reduced Sections).
  • Strengthen the interests of minority investors.
  • Legislate the role of whistle-blowers.
  • Speedy settlement of company disputes (NCLT/NCLAT).
  • Impose stringent punishment for violations and mismanagement.

Applicability of the Companies Act, 2013:
-As per Section 1(4), it applies to:

  • Companies incorporated under this Act or under any previous company law
  • Insurance companies, except if inconsistent with Insurance Act, 1938 or Insurance Regulatory & Development Authority Act, 1999
  • Banking companies, except if inconsistent with the provisions of the Banking Regulation Act, 1949
  • Companies engaged in generation or supply of electricity, except if inconsistent with Electricity Act, 2003
  • Other company governed by any special Act, except if inconsistent with provisions of such special Act
  • Such body corporate, incorporated by any Act, as CG may, by notification specify, subject to exceptions, modifications or adaptation.

-It extends to the whole of India [Section 1(2)]

-It does not apply to:

  • Trusts governed by the Indian Trust Act, 1882
  • Societies, club and professional associations governed by the Societies Registration Act, 1860
  • Co-operative societies.

21.2 COMPANY: MEANING AND ITS FEATURES

21.2- 1A Meaning
As defined in the Companies Act, 2013, a “Company” means a company incorporated under this Act or under any previous company law. [Sec. 2(20)]
Lord Justice Lindley has defined a company as

“An association of many persons who contribute money or money’s worth to a common stock and employed it in some trade or business and who share the profit or loss arising there from. The common stock so contributed is denoted in money and is the capital of the company. The persons who contributed it or to whom it belongs are members. The proportion of capital to which each member is entitled is his share. The shares are always transferable, although the right to transfer them may be restricted.”

According to Chief Justice Marshall, “a corporation is an artificial being, invisible, intangible, existing only in contemplation of law. Being a mere creation of law, it possesses only those properties which the charter of its creation confers upon it, either expressly or as accidental to its very existence.
In the words of professor Haney “A company is an incorporated association, which is an artificial person created by law, having a separate entity, with a perpetual succession and a common seal.” This definition sums up the meaning as well as the features of a company succinctly.

21.2- 1B Features of a Company
I. Separate Legal Entity
When a company is registered, it becomes a separate legal personality. It comes to have almost the same rights and powers as a human being. Its existence is distinct and separate from that of its members. A company can own property, have bank account, raise loans, incur liabilities and enter into contracts.
It is a different person altogether from the subscribers to the memorandum of association. Its personality is distinct and separate from the personality of those who compose it. Even members can contract with company, acquire right against it or incur liability to it. Only the creditors of a company can sue it for the debts, and not its members.

In Lee v. Lee Air Farming Limited (1960): Lee, a qualified pilot held all but one of the shares in the company and by the articles was appointed director of the company and the chief pilot. The life of the employees of the company was insured by an insurer. Lee died while piloting the company’s aircraft and his widow claimed compensation for his death, in the course of his employment. Insurers challenged the claim on the ground that no compensation was due to Lee, as Lee and Lee Air Farming Limited was one and the same person. Held, there was a valid contract of service between Lee and the company and Lee was therefore, an employee. Lee was a separate person from the company and so compensation was due to the widow. The magic of corporate personality enabled Lee to be the master and servant at the same time. Mrs. Lee’s contention was upheld.

II. Perpetual Succession
A company has a continued existence and it can be wound up only as per law. A company being a separate legal entity is not affected by the death, insolvency, lunacy etc. of any or all of its members. In case of the death or insolvency of a member, the shares held by him shall be transmitted to his
nominee/legal representative or official assignee/pfficial receiver. Even if all the members of a company die, the company survives. Thus, “Members may come and go, but the company goes on forever.”

III. Limited Liability
The liability of a member depends upon the kind of company of which he is a member. We know that company is a separate legal entity which is distinct from its members.

  1. Thus, in the case of a limited liability company, the liability of the members of the company is limited to the extent of the nominal value of shares held by them. In no case can the share- holders be asked to pay anything more than the unpaid value of their shares.
  2. In the case of a company limited by guarantee, the members are liable only to the extent of the amount guaranteed by them and that too only when the company goes into liquidation.
  3. However, if it is an unlimited company, the liability of its members is unlimited as well.

IV. Artificial Legal Person
A company is a legal or judicial person as created by law. It is an artificial person as it is created by a process other than natural birth. It is a person since it is clothed with all the rights of an individual.
It can do everything which any natural person can do except be sent to jail, take an oath, marry or practice a learned profession. Hence, it is a legal person in its own sense.
As the company is an artificial person, it can act only through some human agency, viz., directors.
The directors cannot control affairs of the company and act as its agency, but they are not the “agents” of the members of the company. The directors can either on their own or through the common seal (of the company) can authenticate its formal acts.
Thus, a company is called an artificial legal person.

V. Separate Property
The company being a separate legal entity can own property, have banking account, raise loans, incur liabilities and enter into contracts. Even members can contract with company, acquire right against it or incur liability to it. A company is capable of owning, enjoying and disposing of property in its own name. Although the capital and assets are contributed by the shareholders, the company becomes the owner of its capital and assets. The shareholders are not the private or joint owners of the company’s property. A member does not even have an insurable interest in the property of the company.
The leading case on this point is of Macaura v. Northern Assurance Co. Limited (1925): Macaura (M) was the holder of nearly all (except one) shares of a timber company. He was also a major creditor of the company. M Insured the company’s timber in his own name. The timber was lost in a fire. M claimed insurance compensation. Held, the insurance company was not liable to him as no shareholder has any right to any item of property owned by the company, for he has no legal or equitable interest in them.

VI. Common Seal
A company being an artificial person cannot sign a document as a natural person can do. The common seal is a seal used by a company as a substitute for a signature. In legal terms the common seal is the official signature of the company. As per Companies Amendment Act, 2015, the provision of a common seal has been made optional for a company. It is a metal seal on which the name of the company is engraved. [Section 12(3)(b)]
In case a company does not have a common seal, the authorization is made as per the articles. Table F of the articles states that “such authorization shall be by two directors or by a director and the Company Secretary, wherever the company has appointed a Company Secretary.”

The Companies (Amendment) Act, 2015 has made the common seal optional by omitting the words “and a common seal” from Section 9 so as to provide an alternative mode of authorization for companies who opt not to have a common seal. Rational for this amendment is that common seal is seen as a relic of medieval times. Even in the U.K., common seal has been made optional since 2006.

This amendment provides that the documents which need to be authenticated by a common seal will be required to be so done, only if the company opts to have a common seal.

VII. Separation of ownership from management
The shareholders who are the owners of the share capital of the company and they bear risk but they do not actually manage the company. The management is vested in the board of directors who are elected by the shareholders.

VIII. Can Sue and be sued
A company can sue others and it can be sued in its own name.

IX. Transferability of shares
The capital of a company is divided into shares. Shares of a company are movable property, transferable subject to certain conditions which may be provided in the articles.

21.2- 2 Is company a Citizen?
A company has nationality and domicile and residence.
But it is not a citizen and therefore cannot be said to have the fundamental rights expressly conferred upon citizens only. (State Trading Corporation of India Ltd. v. CTO 1963, 33 Comp. Cas. 1057 (SC). However those fundamental rights which are available to all persons, whether citizens or not, like the right to equality, etc. are available to the company.
As per Citizenship Act, 1955, only natural persons can be citizens of India. So a company cannot be a citizen of India.

21.2- 3 Is company the property of shareholders/members?
The company is not the property of its shareholders. All the property in the name of the company is its separate property which is controlled, managed and disposed of by the company in its own name. Thus the company is the owner of its assets and capital. Moreover, the company being a separate legal person, it cannot be construed as property of the shareholders. The decision of the Supreme Court in the case, National Textile Workers Union v. P.R. Ramkrishnan, AIR 1993 (SC), has set at rest at the debate which was going on this issue. According to the verdict given in this case, “a company, according to the new socio-economic thinking is a social institution having duties and responsibilities towards the community in which it functions. Maximization of social welfare should be the legitimate goal of a company and shareholders should be regarded not as proprietors of the company, but merely as suppliers of capital entitled to no more than reasonable return and the company should be responsible not only to shareholders but also to workers, consumers and the other members of the community and should be guided by consideration of national economy and progress. ”

21.3 CORPORATE VEIL THEORY

21.3- 1 Corporate Veil
From the juristic point of view a company is a legal person distinct from its members (Salomon v. Salomon & Co. Ltd.). This principle may be referred to as the veil of incorporation. The effect of his principle is that there is a fictional veil between the company and its members. Corporate Veil refers to a legal concept whereby the company is identified separately from the members of the company. Thus, the shareholders are protected from the acts of the company. The Salomon v. Salomon and Co Ltd. laid down the foundation of the concept of corporate veil or independent corporate personality.
(Veil is a piece of fine material worn by women to protect or conceal the face.)

In Salomon v. Salomon & Co. Ltd. the House of Lords laid down that a company is a person distinct and separate from its members. In this case, Salomon incorporated a company named “Salomon & Co. Ltd.”, with seven subscribers consisting of himself, his wife, four sons and one daughter. This company took over the personal business assets of Salomon for Rs. 38,782 and in turn, Salomon took 20,000 shares of Rs. 1 each, debentures worth Rs. 10,000 of the company with charge on the company’s assets and the balance in cash. His wife, daughter and four sons took up one share each. Subsequently, the company went into liquidation due to general trade depression. The unsecured creditors to the tune of Rs. 7,000 contended that Salomon could not be treated as a secured creditor of the company, in respect of the debentures held by him, as he was the managing director of the company, and the company and Solomon were one and the same person.

It was held by Lord Mac Naughten: “The Company is at law a different person altogether from the subscribers to the memorandum, and though it may be that after incorporation the business is precisely the same as it was before and the same persons are managers, and the same hands receive the profits, the company is not in law the agent of the subscribers or trustees for them. Nor are the subscribers, as members, liable, in any shape or form, except to the extent and in the manner provided by the Act. ”
Thus, this case clearly established that company has its own existence and as a result, a shareholder cannot be held liable for the acts of the company even though he holds virtually the entire share capital. The whole law of corporation is in fact based on this theory of separate corporate entity.

21.3- 2 Lifting of Corporate Veil
‘Lifting the veil’ means looking behind the company as a legal person Le.; disregarding the corporate entity and paying regard instead to the realities behind the legal form. Where the courts ignore the corporate personality and concern themselves directly with the members or directors, the corporate veil may be said to have been lifted. Where the Courts ignore the company and concern themselves directly with the members or managers, the corporate veil may be said to have been lifted. Only in appropriate circumstances, the Courts are willing to lift the corporate veil and that too, when questions of control are involved rather than merely a question of ownership.

The following are the cases where company law disregards the principle of corporate personality or the principle that the company is a legal entity distinct and separate from its shareholders or members:
1. To determine the character of the company i.e. to find out whether company is an enemy or a friend:
The leading case in this point is Daimler Company Ltd. v. Continental Tyre & Rubber Co. (Great Britain) Ltd. [1916] 2 AC 307. In this case the Daimler Company was incorporated in London. Its majority shareholders and directors were Germans. On declaration of war between England and Germany in 1914 it was held that the company was a German company. Accordingly, the suit filed by the company to recover a trade debt was dismissed on the ground that such payment would amount to trading with enemy.
If the public interest is not likely to be in jeopardy, the Court may not be willing to crack the corporate shell. But it may rend the veil for ascertaining whether a company is an enemy company. Of course, unlike a natural person, â company does not have mind or conscience; therefore, it cannot be a friend or foe. It may, however, be characterised as an enemy com-pany, if its affairs are under the control of people of an enemy country. For this purpose, the Court may examine the character of the persons who are really at the helm of affairs of the company.

2. Company is formed to evade taxes:
Where corporate entity is used to evade or circumvent tax, the Court can disregard the corporate entity [Juggilal v. Commissioner of Income Tax AIR (SC)]. In certain matters concerning the law of taxes, duties and stamps particularly where question of the controlling interest is in issue. [S. Berendsen Ltd. v. Commissioner of Inland Revenue].
In Sir Dinshaw Maneckjee Petit, Re AIR 1927 Bom. 371, D formed four private companies and transferred his investments to them. D took pretended loans from the companies, which he never repaid. In a legal proceeding the corporate veils of all the companies were lifted and the incomes of the companies treated as if they were of D. The Court decided that the private companies were a sham and the corporate veil was lifted to decide the real owner of the income. Also, affirmed in CITv. Sri Meenakshi Mills Ltd. AIR 1967 SC 819.

3. Company is formed to avoid a legal obligation/welfare legislation:
If the sole purpose for the formation of the company was to use it as a device to reduce the amount to be paid by way of bonus to workmen, the Supreme Court upheld the piercing of the veil to look at the real transaction (The Workmen Employed in Associated Rubber Industries Limited, Bhavnagar v. The Associated Rubber Industries Ltd., Bhavnagar and another).
Workmen of Associated Rubber Industry Ltd., v. Associated Rubber Industry Ltd.: The
facts of the case are that “A Limited” purchased shares of “B Limited” by investing a sum of Rs. 4,50,000. The dividend in respect of these shares was shown in the profit and loss account of the company, year after year. It was taken into account for the purpose of calculating the bonus payable to workmen of the company. Sometime in 1968, the company transferred the shares of B Limited, to C Limited a subsidiary, wholly owned by it. Thus, the dividend income did not find place in the Profit & Loss Account of A Ltd., with the result that the surplus available for the purpose for payment of bonus to the workmen got reduced.
Here a company created a subsidiary and transferred to it, its investment holdings in a bid to reduce its liability to pay bonus to its workers. Thus, the Supreme Court disregarded the separate existence of the subsidiary company. The new company so formed had no assets of its own except those transferred to it by the principal company, with no business or income of its own except receiving dividends from shares transferred to it by the principal company and serving no purpose except to reduce the gross profit of the principal company so as to reduce the amount paid as bonus to workmen.

4. Formation of subsidiaries to act as agents:
A company may sometimes be regarded as an agent or trustee of its members, or of another company, and may therefore be deemed to have lost its individuality in favour of its principal. Here the principal will be held liable for the acts of that company.
In.the case of Merchandise Transport Limited v. British Transport Commission (1982), a transport company wanted to obtain licences for its vehicles, but could not do so if applied in its own name. It, therefore, formed a subsidiary company, and the application for licence was made in the name of the subsidiary. The vehicles were to be transferred to the subsidiary company. Held, the parent and the subsidiary were one commercial unit and the application for licences was rejected.

5. Company formed for fraud/improper conduct or to defeat law:
The corporate veil may be lifted if the company is formed to-

  1. defeat the law;
  2. defraud creditors;
  3. avoid legal obligations (arising by way of a contract).

Where the device of incor-poration is adopted for some illegal or improper purpose, e.g., to defeat or circumvent law, to defraud creditors or to avoid legal obligations. [Gilford Motor Co. v. Horne]
In Jones v. Lipman [1962] 11 ALL ER 442, the defendant attempted to avoid completing the sale of his house to the plaintiff by transferring to a company formed for the purpose. The court ordered both the defendant and the company specifically to perform the contract with the plaintiff.

6. To determine the technical competence of the company:
In New Horizons Ltd. v. UOI (1997) 89 Comp. Case 849 (SC), a company was formed as joint venture by other companies for purpose of telecom tender. The company was new but its major shareholders had vast experience in the field. However, tender evaluation company rejected the tender on the ground that the company has no experience in the field. Supreme Court held that experience of major shareholders can be considered as experience of the company, for purpose of awarding a tender or contract.

21.4 CLASSES OF COMPANIES UNDER THE ACT

The Companies Act, 2013 has broadly classified the companies into various classes.
On the basis of number of members: A company may be incorporated as a one-person company, private company or a public company, on the basis of the number of members joining it.
On the basis of Liability: Again, on the basis of liability, it may either be an unlimited company, or may be limited by shares or by guarantee or by both.
On the basis of control: Companies can be classified as associate company, holding company and subsidiary company on the basis of control.
On the basis of access to capital: A company may be classified as a Listed company or an Unlisted company.
Other Classifications: Some other forms of classification of companies are Foreign Company, Government Company, Small company, Dormant company, Nidhi Company and Company formed for Charitable Objects.

Companies may be classified into various classes on the following basis:
1. On the basis of liability:
a. Company limited by shares
“Company limited by shares” means a company having the liability of its members limited by x memorandum, to amount, If any, unpaid on the shares respectively held by them; i.e. his personal property cannot be undertaken to meet company’s total debt]. [Sec. 2(22)]
The memorandum of association mentions whether the liability of the members is limited j or not. In these companies there is a share capital divided into shares of fixed amount. The liability of the shareholder is limited to the nominal amount of the shares held by him.
Thus, if a person buys 100 shares of Rs. 10 each, his maximum liability is to the extent of Rs. 1,000 only. He cannot be asked to pay more than this amount. If he has paid Rs. 6 on each share, his remaining liability will be only 14 per share ( i.e. 4 X 100 = Rs. 400). A majority of the companies in India belong to this category.

b. Company limited by guarantee
“Company limited by guarantee” means a company having the liability of its members limited by the memorandum, to such amount as the members may respectively undertake to contribute to the assets of the company in the event of its being wound up. [Sec. 2(21 )]
Such an amount is called the Guarantee. The memorandum of association lays down the guarantee amount. No member is liable to pay more than the amount, which he has guaran- teed to contribute.

These companies may or may not have a share capital In the case of a guarantee company with a share capital, the members are required to purchase shares of fixed amount and also give a guarantee for a further sum in the event of winding up. Generally, guarantee companies are formed for non-trading purposes. Such as promotion of commerce, art, science, sports etc., and do not aim for profit. The Chambers of Commerce, charitable institutions, sport clubs, are generally organized as guarantee companies.

In Narendra Kumar Agarwal v. Saroj Maloo, the Supreme Court has laid down that the right of a guarantee company to refuse to accept the transfer by a member of his interest in the company is on a different footing than that of a company limited by shares. The membership of a guarantee company may carry privileges much different from those of ordinary shareholders.
The common features between a ‘guarantee company’ and ‘the company having share capital’ are legal personality and limited liability. In the latter case, the member’s liability is limited by the amount remaining unpaid on the share, which each member holds. Both of them have to state in their memorandum that the members’ liability is limited.

The point of distinction between these two types of companies is that in the former case the members may be called upon to discharge their liability only after commencement of the winding up and only subject to certain conditions; but in the latter case, they may be called upon to do so at any time, either during the company’s life-time or during its winding up. It is clear from the definition of the guarantee company that it does not raise its initial working funds from its members. Therefore, such a company may be useful only where no working funds are needed or where these funds can be held from other sources like endowment, fees, charges, donations, etc.

POINTS OF DISTINCTION

COMPANY LIMITED BY SHARES

COMPANY LIMITED BY GUARANTEE

Purpose: Profit/non-profit both. Generally not for profit.
Usefulness: When initial funds are required to be raised to commence business. Only where no working funds are needed or where these funds can be held from other sources like endowment, fees, charges, donations, etc.
Transfer of interest May not be restricted. Restricted & different than that of those limited by shares
Liability of members Limited to amount unpaid on shares. Limited to amount guaranteed.
Amount Called Unpaid amount on shares may be called even before winding up. Amount guaranteed can be called only on winding up. If company has a share capital, unpaid amount on shares can be called before winding up.
Share capital Must have share capital May have share capital
To start Raises initial funds from members Does not raise initial funds from members, unless it has a share capital.

c. Unlimited company
An unlimited company is defined as a company not having any limit on the liability of its members. Thus the members of an unlimited company have unlimited liability, but he will be entitled to claim contribution from other members. In such a company liability of member ceases on cessation of membership. If company is running & is not wound up the liability on the shares is the only liability which can be enforced by the company. [Sec. 2(92)].
The liability of each member extends to the whole amount of the company’s debts and liabilities but he will be entitled to claim contribution from other members. In case the company has share capital, the articles of association must state the amount of share capital and the amount of each share. So long as the company is a going concern the liability on the shares is the only liability which can be enforced by the company.
The creditors can institute proceedings for winding up of the company for their claims. The official liquidator may call the members for their contribution towards the liabilities and debts of the company, which can be unlimited.

2. On the basis of number of members
a. One person company
Section 2(62) of the Companies Act, 2013 defines one person company (OPC) as a company which has only one person as a member.
Companies Act, 2013 introduced a new class »of companies which can be incorporated by a single person.
One person company has been introduced to encourage entrepreneurship and corporatization of business. OPC differs from sole proprietary concern in an aspect that OPC is a separate legal entity with a limited liability of the member whereas in the case of sole proprietary, the liability of owner is not restricted and it extends to the owner’s entire assets constituting of official and personal.
The procedural requirements of an OPC are simplified through exemptions provided under the Act in comparison to the other forms of companies.
According to section 3(1 )(c) of the Companies Act, 2013, OPC is a private limited company with the minimum paid up share capital as may be prescribed and has only one member.

Important points relater to a OPC (One Person Company)

  • Only one person as member.
  • Minimum paid up capital – not yet prescribed.
  • The memorandum of OPC shall indicate the name of the other person, who shall, in the event of the subscriber’s death or his incapacity to contract, become the member of the company.
  •  The other person whose name is given in the memorandum shall give his prior written consent in prescribed form and the same shall be filed with Registrar of companies at the time of incorporation.
  • Such other person may be given the right to withdraw his consent.
  • The member of OPC may at any time change the name of such other person by giving notice to the company and the company shall intimate the same to the Registrar.
  • Any such change in the name of the person shall not be deemed to be an alteration of the memorandum.
  • Only a natural person who is an Indian citizen and resident in India (person who has stayed in India for a period of not less than 182 days during the immediately preceding one calendar year)
    • shall be eligible to incorporate a OP
    • shall be a nominee for the sole member of a OPC.
  • No person shall be eligible to incorporate more than one OPC or become nominee in more than one such company.
  • No minor shall become member or nominee of the OPC or can hold share with beneficial interest.
  • Such Company cannot be incorporated or converted into a company under section 8 of the Act. Though it may be converted to private or public companies in certain cases.
  • Such Company cannot carry out Non-Banking Financial Investment activities including investment in securities of any body corporate.
  • OPC cannot convert voluntarily into any kind of company unless two years have expired from the date of incorporation, except where the paid up share capital is increased beyond fifty lakh rupees or its average annual turnover during the relevant period exceeds two crore rupees.
  • If One Person Company or any officer of such company contravenes the provisions, they shall be punishable with fine which may extend to ten thousand rupees and with a further fine which may extend to one thousand rupees for every day after the first during which such contravention continues.
  • Here the member can be the sole member and director.

b. Private Company [Section 2(68)]
“Private Company” means a company having a minimum paid-up share capital as may be prescribed, and which by its articles,—

  1. restricts the right to transfer its shares;
  2. except in case of One Person Company, limits the number of its members to two hundred:
    Provided that where two or more persons hold one or more shares in a company jointly, they shall, for the purposes of this clause, be treated as a single member:
    Provided further that—
    (A) persons who are in the employment of the company; and
    (B) persons who, having been formerly in the employment of the company, were members of the company while in that employment and have continued to be members after the employment ceased, shall not be included in the number of members; and
  3. prohibits any invitation to the public to subscribe for any securities of the company;

Important points related to a Private company:

  • No minimum paid-up capital requirement.
  • Minimum number of members – 2 (except if private company is an OPC, where it will be 1).
  • Maximum number of members – 200, excluding present employee-cum-members and erstwhile employee-cum-members.
  • Right to transfer shares restricted.
  • Prohibition on invitation to subscribe to securities of the company.
  • Small company is a private company.
  • OPC can be formed only as a private company.

Small Company: Small company given under the section 2(85) of the Companies Act, 2013 which means a company, other than a public company-

  1. paid-up share capital of which does not exceed fifty lakh rupees or such higher amount as may be prescribed which shall not be more than five crore rupees; and
  2. turnover of which as per its last profit and loss account does not exceed two crore rupees or such higher amount as may be prescribed which shall not be more than twenty crore rupees:

Exceptions: This section shall not apply to:
(A) a holding company or a subsidiary company;
(B) a company registered under section 8; or
(C) a company or body corporate goverqed by any special Act.

c. Public company [Section 2(71)]
“Public company” means a company which—

  1. is not a private company;
  2. has a minimum paid-up share capital, as may be prescribed:

Provided that a company which is a subsidiary of a company, not being a private company, shall be deemed to be public company for the purposes of this Act even where such subsidiary company continues to be a private company in its articles.

Important points related to a Public company
Is not a private company (Articles do not have the restricting clauses).

  • Shares are freely transferable.
  • No minimum paid up capital requirement.
  • Minimum number of members – 7.
  • Maximum numbers of members – No limit.
  • Subsidiary of a public company is deemed to be a public company.

According to section 3(1 )(a), a company may be formed for any lawful purpose by seven or more persons, where the company to be formed is to be a public company.

3. On the basis of control
a. Holding and subsidiary companies
‘Holding and subsidiary’ companies are relative terms.
A company is a holding company in relation to one or more other companies, means a com-pany of which such companies are subsidiary companies. [Section 2(46)]
Whereas section 2(87) defines “subsidiary company” in relation to any other company (that is to say the holding company), means a company in which the holding company—

  1. controls the composition of the Board of Directors; or
  2. exercises or controls more than one-half of the total share capital either at its own or together with one or more of its subsidiary companies:

Provided that such class or classes of holding companies as may be prescribed shall not have layers of subsidiaries beyond such numbers as may be prescribed. [Layers are yet to be notified]
For the purposes of this section—

  1. a company shall be deemed to be a subsidiary company of the holding company even if the control referred to in sub-clause
    1. or sub-clause
    2. is of another subsidiary company of the holding company;
  2. the composition of a company’s Board of Directors shall be deemed to be controlled by another company if that other company by exercise of some power exercisable by it at its discretion can appoint or remove all or a majority of the directors;
  3. the expression “company” includes anybody corporate;
  4. “layer” in relation to a holding company means its subsidiary or subsidiaries.

The term “Total Share Capital”, means the aggregate of the—

  1. Paid-up equity share capital; and
  2. Convertible preference share capital.

Example 1: A will be subsidiary of B, if B controls the composition of the Board of Directors of A, i.e., if B can, without the consent or approval of any other person, appoint or remove a majority of directors of A.
Example 2: A will be subsidiary of B, if B holds more than 50% of the share capital of A.
Example 3: B is a subsidiary of A and C is a subsidiary of B. In such a case, C will be the subsidiary of A. In the like manner, if D is a subsidiary of C, D will be subsidiary of B as well as of A and so on.
Status of private company, which is subsidiary to public company: In view of Section 2(71) of the Companies Act, 2013 a Private company, which is subsidiary of a public company shall be deemed to be public company for the purpose of this Act, even where such subsidiary company continues to be a private company in its articles.

The Ministry clarified that the shares held, or power exercisable by company in another company in a ‘fiduciary capacity’ shall not be counted for the purpose of determining the holding subsidiary.
Fiduciary capacity: Holding only in the capacity of a trustee. For instance, when a company holds shares or exercise powers on behalf of any individual, wherein the company is just a trustee holding shares Le.; in good faith, trust and confidence for that individual.

b. Associate company [Section 2(6)]
In relation to another company, means a company in which that other company has a signifi-cant influence, but which is not a subsidiary company of the company having such influence and includes a joint venture company.
The term “significant influence” means control of at least 20% of total share capital, or of business decisions under an agreement. [Section 2(6)]
The term “Total Share Capital”, means the aggregate of the—

  1. Paid-up equity share capital; and
  2. Convertible preference share capital.

This is a new definition inserted in the 2013 Act.
Vide General Circular No. 24/2014 dated 25th of June 2014, the Ministry of Corporate Affairs has clarified that the shares held by a company in another company in a ‘fiduciary capacity’ shall not be counted for the purpose of determining the relationship of ‘associate company’ under section 2(6) of the Companies Act, 2013.

4. On the basis of access to capital

  1. Listed company: As per the definition given in section 2(52) of the Companies Act, 2013, it is a company which has any of its securities listed on any recognised stock exchange.
  2. Unlisted company: Whereas the word securities as per section 2(81) of the Companies Act, 2013 has been assigned the same meaning as defined in clause (h) of section 2 of the Securities Contracts (Regulation) Act, 1956.
    Unlisted company: means company other than listed company.

5. Other companies
a. Government company [Section 2(45)]
Government Company means any company in which not less than 51% of the paid-up share capital is held by—

  1. the Central Government, or
  2. by any State Government or Governments, or
  3. partly by the Central Government and partly by one or more State Governments, and the section includes a company which is a subsidiary company of such a Government company.

b. Foreign Company [Section 2(42)] 
It means any company or body corporate incorporated outside India which-

  1. has a place of business in India whether by itself or through an agent, physically or through electronic mode; and
  2. conducts any business activity in India in any other manner

c. Formation of companies with charitable objects etc. (Section 8 company)
Section 8 of the Companies Act, 2013 deals with the formation of companies which are formed to—

  1. Promote the charitable objects of commerce, art, science, sports, education, research, social welfare, religion, charity, protection of environment etc.
  2. Such company intends to apply its profit in promoting its objects and
  3. Prohibiting the payment of any dividend to its members.

Examples of section 8 companies are FICCI, ASSOCHAM, National Sports Club of India, CII, etc.

Power of Central government to issue the license-

  1. Section 8 allows the Central Government to register such person or association of 1 persons as a company with limited liability without the addition of words ‘Limited’ or ‘Private limited’ to its name, by issuing licence on such conditions as it deems fit.
  2. The registrar shall on application register such person or association of persons as a company under this section.
  3. On registration the company shall enjoy same privileges and obligations as of a limited company.

Note: Central Government has delegated the power to grant License to the ROC.

Revocation of license
The Central Government may by order revoke the licence of the company where the company contravenes any of the requirements or the conditions of this sections subject to which a j licence is issued or where the affairs of the company are conducted fraudulently, or violative j of the objects of the company or prejudicial to public interest, and on revocation the Registrar shall put ‘Limited’ or ‘Private Limited’ against the company’s name in the register. But before such revocation, the Central Government must give it a written notice of its intention to revoke the licence and opportunity to be heard in the matter.
Note: Central Government has delegated the power to revoke license to the Regional Directors.

Order of the Central Government
Where a licence is revoked there the Central Government may, in the public interest order that the company registered under this section should be amalgamated with another compa- j ny registered under this section having similar objects, to form a single company with such constitution, properties, powers, rights, interest, authorities and privileges and with such liabilities, duties and obligations as may be specified in the order, or the company be wound up.

Penalty/punishment in contravention
If a company makes any default in complying with any of the requirements laid down in this section, the company shall, be punishable with fine varying from ten lakh rupees to one crore rupees and the directors and every officer of the company who is in default shall be punishable with imprisonment for a term which may extend to three years or with fine varying from twenty-five thousand rupees to twenty-five lakh rupees, or with both and where it is proved that the affairs of the company were conducted fraudulently, every officer in default shall be liable for action under section 447 which deals with Fraud.

Section 8 Company – Significant points

  • Formed for the promotion of commerce, art, science, religion, charity, protection en-vironment, sports, etc.
  • Requirement of minimum share capital does not apply.
  • Uses its profits for the promotion of the objective for which formed.
  • Does not declare dividend to members.
  • Operates under a special licence from Central Government.
  • Need not use the word Ltd. /Pvt. Ltd. in its name and adopt a more suitable name such as club, chambers of commerce etc. ‘
  • Licence revoked if conditions contravened.
  • On revocation, Central Government may direct it to
    •  Converts its status and change its name
    •  Wind-up
    •  Amalgamate with another company having similar object.
  • Can call its general meeting by giving a clear 14 days’ notice instead of 21 days.
  • Requirement of minimum number of directors, independent directors etc. does not apply.
  • Need not constitute Nomination and Remuneration Committee and Shareholders Relationship Committee.
  • A partnership firm can be a member of Section 8 company.

d. Dormant company (Section 455)
Where a company is formed and registered under this Act for a future project or to hold an asset or intellectual property and has no significant accounting transaction, such a company or an inactive company may make an application to the Registrar in such manner as may be prescribed for obtaining the status of a dormant company.
“Inactive company” means a company which has not been carrying on any business or operation, or has not made any significant accounting transaction during the last two financial years, or has not led financial statements and annual returns during the last two financial years.
“Significant accounting transaction” means any transaction other than—

  1. payment of fees by a company to the Registrar;
  2. payments made by it to full the requirements of this Act or any other law;
  3. allotment of shares to full the requirements of this Act; and
  4. payments for maintenance of its office and records.

e. Nidhi Companies
Company which has been incorporated as a Nidhi’with the object of cultivating the habit of thrift (cost cutting) and savings amongst its members, receiving deposits from, and lending to, its members only, for their mutual benefit and which complies with such rules as are pre-scribed by the Central Government for regulation of such class of companies. [Section 406 of the Companies Act, 2013]

Public Financial Institutions (PFI)
By virtue of Section 2(72) of the Companies Act, 2013, the following institutions are to be regarded as public financial institutions:

  1. the Life Insurance Corporation of India, éstablished under the Life Insurance Corporation Act, 1956;
  2. the Infrastructure Development Finance Company Limited;
  3. specified company referred to in the Unit Trust of India (Transfer of Undertaking and Repeal) Act, 2002;
  4. institutions notified by the Central Government under section 4A(2) of the Companies Act, 1956 so repealed under section 465 of this Act.
  5. such other institution as may be notified by the Central Government in consultation with the Reserve Bank of India.

Conditions for an institution to be notified as PFI: Mo institution shall be so notified unless—

  1. it has been established or constituted by or under any Central or State Act; or
  2. not less than fifty-one per cent of the paid-up share capital is held or controlled by the Central
    Government or by any State Government or Governments or partly by the Central Government and partly by one or more State Government.

21.5 MODE OF REGISTRATION/INCORPORATION OF COMPANY PROMOTERS

21.5-1 Promoter [Sec. 2(69)]
The Companies Act, 2013 defines the term “Promoter” under section 2(69) which means a person—

  1. who has been named as such in a prospectus or is identified by the company in the annual return referred to in section 92; or
  2. who has control over the affairs of the company, directly or indirectly whether as a shareholder, director or otherwise; or
  3. in accordance with whose advice, directions or instructions the Board of Directors of the company is accustomed to act.

In simple terms we can say,

  • Persons who form the company are known as promoters.
  • It is they who conceive the idea of forming the company.
  • They take all necessary steps for its registration.
  • It should, however, be noted that persons acting only in a professions capacity e.g., thesolicitor, banker, accountant etc. are not regarded as promoters.

21.5- 2 Formation of Company
Section 3 of the Companies Act, 2013 deals with the basic requirement with respect to the constitution of the company.
In the case of a public company, any 7 or more persons can form a company for any lawful purpose ) by subscribing their names to memorandum and complying with the requirements of this Act in respect of registration.
In exactly the same way, 2 or more persons can form a private company and one person where company to be formed is one person company.
Procedure for reservation of name:

  • An application to the Registrar of Companies (ROC) concerned shall be made electronically in Form with fee.
    (Availability of a name can be checked using the ‘Check Company Name’ Service under company Services option under MCA services tab on homepage of MCA i.e. www.mca.gov.in) Name shall within the parameters prescribed under the Act.
  • 6 company names in order of priority should be submitted to afford flexibility to the Registrar. The ROC shall furnish the information regarding the approval of name/rejection of proposed name within 7 days of the receipt of the application.
  • The approved name shall remain available for adoption by the promoters for a period of 60 days from the date of application. This period may, however, be extended by the ROC.
  • 2 person in case of a private company and 7 in case of public company should be named as promoters/subscribers. Every person who intends to become a Director should obtain DIN. (Director Identification Number).
  • Applicant will get SRN (Service Request Number), which can be used to trace position about approval of name.
  • The Registrar shall give three opportunities for resubmission under one registration form (i.e.; INC 1) if the application is rejected.

21.5- 3 Incorporation of Company
Section 7 of the Companies Act, 2013 provides for the procedure to be followed for incorporation | of a company.
1. Filing of the documents and information with the registrar
For the registration of the company following documents and information are required to be filed with the registrar within whose jurisdiction the registered office of the company is proposed to be situated—

  • the memorandum and articles of the company duly signed by all the subscribers to the memorandum.
  • a declaration by person who is engaged in the formation of the company (an advocate, a chartered accountant, cost accountant or company secretary in practice), and by a person named in the articles (director, manager or secretary of the company), that all the requirements of this Act and the rules made thereunder in respect of registration and matters precedent or incidental thereto have been complied with.
  • an affidavit from each of the subscribers to the memorandum and from persons named as the first directors, if any, in the articles stating that—
    • he is not convicted of any offence in connection with the promotion, formation or management of any company, or
    • he has not been found guilty of any fraud or misfeasance or of any breach of duty to any company under this Act or any previous company law during the last five years,
    • and that all the documents filed with the Registrar for registration of the com-pany contains information that is correct and complete and true to the best of his knowledge and belief;
  • the address for correspondence till its registered office is established;
  • the particulars (names, including surnames or family names, residential address, na-tionality) of every subscriber to the memorandum along with proof of identity, and in the case of a subscriber being a body corporate, such particulars as may be prescribed.
  • the particulars (names, including surnames or family names, the Director Identification Number, residential address, nationality) of the persons mentioned in the articles as the first directors and such other particulars including proof of identity as may be prescribed; and
  • the particulars of the interests of the persons mentioned in the articles as the first di-rectors of the company in other firms ‘or bodies corporate along with their consent to act as directors of the company in such form and manner as may be prescribed.
    Particulars provided in this provision shall be of the individual subscriber and not of the professional engaged in the incorporation of the company [The Companies (Incorporation) Rules, 2014],

2. Issue of certificate of incorporation on registration
The Registrar on the basis of documents and information filed, shall register all the documents and information in the register and issue a certificate of incorporation in the prescribed form to the effect that the proposed company is incorporated under this Act.

3. Corporate Identity Number (CIN)
On and from the date mentioned in the certificate of incorporation, the Registrar shall allot to the company a corporate identity number, which shall be a distinct identity for the company and which shall also be included in the certificate.

4. Maintenance of copies of all documents and information
The company shall maintain and preserve at its registered office copies of all documents and information as originally filed, till its dissolution under this Act.

5. Furnishing of false or incorrect information or suppression of material fact at the time of incorporation
If any person furnishes any false or incorrect particulars of any information or suppresses any material information, of which he is aware in any of the documents filed with the Registrar in relation to the registration of a company, he shall be liable for action for fraud under section 447.

6. Company already incorporated by furnishing any false or incorrect information or represen-tation or by suppressing any material fact (i.e. post Incorporation)
Where, at any time after the incorporation of a company, it is proved that the company has been got incorporated by furnishing any false or incorrect information or representation or by suppressing any material fact or information in any of the documents or declaration filed or made for incorporating such company, or by any fraudulent action, the promoters, the persons named as the first directors of the company and the persons making declaration under this section shall each be liable for action for fraud under section 447.

7. Order of the Tribunal
Where a company has been got incorporated by furnishing false or incorrect information or representation or by suppressing any material fact or information in any of the documents or declaration filed or made for incorporating such company or by any fraudulent action, the Tribunal may, on an application made to it, on being satisfied that the situation so warrants,—

  1. pass such orders, as it may think fit, for regulation of the management of the company including changes, if any, in its memorandum and articles, in public interest or in the interest of the company and its members and creditors; or
  2. direct that liability of the members shall be unlimited; or
  3. direct removal of the name of the company from the register of companies; or
  4. pass an order for the winding up of the company; or
  5. pass such other orders as it may deem fit.

Provided that before making any order,—
the company shall be given a reasonable opportunity of being heard in the matter; and
the Tribunal shall take into consideration the transactions entered into by the company, including the obligations, if any, contracted or payment of any liability.

Simplified Proforma for Incorporating Company Electronically (SPICe):
The Ministry of Corporate Affairs has taken various initiatives for ease of business. In a step towards easy setting up of business, MCA has Simplified the process of filing of forms for incorporation of a company through Simplified Proforma for incorporating company electronically.

21.5-4 Effect of Registration
Section 9 of the Companies Act, 2013 provides for the effect of registration of a company.
When a company is registered and a certificate of incorporation is issued by the Registrar, following important consequences follow:—

  1. The company becomes a distinct legal entity, Its life commences from the date mentioned in the certificate of incorporation.
  2. It becomes a body corporate and it acquires a perpetual succession and a common seal.
  3. It is capable of suing and be sued in its corporate name.
  4. Its property is not the property of the shareholders. The shareholders have a right to share in the profits of the company when realized and divided. Likewise any liability of the company is not the liability of individual shareholders.

From the date of incorporation mentioned in the certificate, the company becomes a legal person separate from the incorporators; and there comes into existence a binding contract between the company and its members as evidenced by the Memorandum and Articles of Association [Hari Nagar Sugar Mills Ltd. v. S.S. Jhunjhunwala]. It has perpetual existence until it is dissolved by liquidation or struck out of the register.
A shareholder who buys shares, does not buy any interest in the property of the company but in certain cases a writ petition will be maintainable by a company or its shareholders.
A legal personality emerges from the moment of registration of a company and from that moment the persons subscribing to the Memorandum of Association and other persons joining as members are regarded as a body corporate or a corporation in aggregate and the legal person begins to function as an entity. A company on registration acquires a separate existence and the law recognises it as a legal person separate and distinct from its members [State Trading Corporation of India v. Commercial Tax Officer],
It may be noted that under the provisions of the Act, a company may purchase shares of another company and thus become a controlling company. However, merely because a company purchases all shares of another company it will not serve as a means of putting an end to the corporate character of another company and each company is a separate juristic entity [Spencer & Co. Ltd. Madras v. CWT Madras].
As has been stated above, the law recognizes such a company as a juristic person separate and distinct from its members. The mere fact that the entire share capital has been contributed by the Central Government and all its shares are held by the President of India and other officers of the Central Government does not make any difference in the position of registered company and it does not make a company an agent either of the President or the Central Government [Heavy Electrical Union v. State of Bihar].

21.5- 5 Effect of memorandum and articles
As per section 10 of the Companies Act, 2013,
The memorandum and articles, when registered, binds the company and its members to the same extent as if they have been signed by the company and by each member.
The company & each of its members are to observe & be bound by all provisions of memorandum & of the articles.
Thus, the company is bound to the member the members are bound to the company; and the members are bound to the other members by whatever is contained in these documents.
But, in relation to articles, neither a company not its members are bound to outsiders.
All monies payable by any member to the company under the memorandum or articles shall be a debt due from him to the company.

21.6 CLASSIFICATION OF CAPITAL

The term Capital has a variety of meanings. It means one thing to economists; another to accountants and still another to businessmen and lawyers. In relation to a company limited by shares, the word capital means share capital, i.e., the capital or figure in terms of so many rupees divided into shares of fixed amount. In other words, the contributions of persons to the common stock of the company form the capital of the company. The proportion of the capital to which each member is entitled, is his share. A share is not a sum of money; it is rather an interest measured by a sum of money and made up of various rights contained in the contract.

In the domain of Company Law, the term ‘capital’ is used in the following senses:
(a) Nominal or authorised or registered capital
This form of capital has been defined in section 2(8) of the Companies Act, 2013. “Authorised capital” or “Nominal capital” means such capital as is authorised by the memorandum of a company to be the maximum amount of share capital of the company. Thus, it is the sum stated in the memorandum as the capital of the company with which it is to be registered being the maximum amount which it is authorised to raise by issuing shares, and upon which it pays the stamp duty. It is usually fixed at the amount, which, it is estimated, the company will need, including the working capital and reserve capital, if any.

(b) Issued capital
Section 2(50) of the Companies Act, 2013 defines “issued capital” which means such capital as the company issues from time to time for subscription. It is that part of authorised capital which is offered by the company for subscription and includes the shares allotted for consideration other than cash.
Schedule III to the Companies Act, 2013, makes it obligatory for a company to disclose its issued capital in the balance sheet.
For e.g. A company may have total authorised share capital of Rs. 10 lacs divided into 1 lac shares of Rs. 10 each. It may decide to issue 80,000 shares of Rs. 10 each. In that case the issued capital shall be X 8,00,000.

(c) Subscribed capital
Section 2(86) of the Companies Act, 2013 defines “subscribed capital” as such part of the capital which is for the time being subscribed by the members of a company.
It is the nominal amount of shares taken up by the public. Where any notice, advertisement or other social communication or any business letter, bill head or letter paper of a company states the authorised capital, the subscribed and paid-up capital must also be stated in equally conspicuous characters. A default in this regard will make the company and every officer who is in default liable to pay penalty extending X 10,000 and X 5,000 respectively. [Section 60],
In the above example out of 80,000 shares issued by the company, if applications are received for only 70,000 shares of Rs. 10 each, the subscribed capital will be X 7,00,000.

(d) Called-up capital
Section 2(15) of the Companies Act, 2013 defines “called-up capital” as such part of the capital, which has been called for payment. It is the total amount called up on the shares issued.
In the above example if the company has called up 5 per share, then it’s called up capital shall be 70,000 X 5 = Rs. 3.5 lacs.
(e) Paid-up capital
Paid-up capital is the total amount paid or credited as paid up on shares issued. It is equal to called up capital less calls in arrears.
In the example given above, if only X 3,00,000 is actually received by the company, then the paid up capital shall be to X 3,00,000.

21.7 SHARES

(I) Nature of shares
Section 2(84) of the Companies Act, 2013 defines the term ‘share’ which means a share in the share capital of a company and includes stock.
A share thus represents such proportion of the interest of the shareholders as the amount paid up thereon bears to the total capital payable to the company.
It is a measure of the interest in the company’s assets to which a person holding a share is entitled.
Shares are a movable property: According to section 44 of the Companies Act, 2013, share or other interest of any member in a company shall be movable property, transferable in the manner provided by the Articles of the company.
Shares shall be numbered: Section 45 provides, every share in a company having a share capital shall be distinguished by its distinctive number. [Not apply to a share held by a person whose name is entered as holder of beneficial interest in such share in the records of a depository.]
Share is an interest in the company: Farwell Justice, in Borland Trustees v. Steel Bros. & Co. Ltd.
observed that “a share is not a sum of money but is an interest measured by a sum of money and made up of various rights contained in the contract, including the right to a sum of money of a more or less amount”. The rights and obligations attaching to a share are those prescribed by the memorandum and the articles of a company. It must, however, be remembered that a shareholder has not only contractual rights against the company, but also certain other rights which accrue to him according to the provisions of the Companies Act.
Thus, a share of a company in the hand so of a shareholder signifies a bundle of rights and obligations.

(II) Kinds of share capital
Section 43 of the Companies Act, 2013 provides the kinds of share capital. According to the provision «jj the share capital of a company limited by shares shall be of two kinds, namely:—
(i) Equity share capital

  1. with voting rights; or
  2. with differential rights as to dividend, voting or otherwise in accordance with prescribed rules;

Example:
It is to be noted that, Tata Motors in 2008 introduced equity shares with differential voting rights called ‘A’ equity shares in its rights issue. In the issue, every 10 ‘A’ equity shares carried only one voting right but would get 5 percentage points more dividend than that declared on each of the ordinary shares. Since ‘A’ equity share did not carry the similar voting rights, it was being traded at discount to other common shares having full voting. Other companies which have issued equity shares with differential voting rights (popularly called DVRs) are Future Retail, Jain Irrigation among others.
(ii) Preference share capital
However, this Act shall not affect the rights of the preference shareholders who are entitled to participate in the proceeds of winding up before the commencement of this Act.

According to Explanation to section 43:

  1. “Equity share capital”, with reference to any company limited by shares, means all share capital which is not preference share capital;
  2. “Preference share capital”, with reference to any company limited by shares, means that part of the issued share capital of the company which carries or would carry a preferential right with respect to—

(a) payment of dividend, either as a fixed amount or an amount calculated at a fixed rate, which may either be free of or subject to income-tax; and
(b) repayment, in the case of a winding up or repayment of capital, of the amount of the share capital paid-up or deemed to have been paid-up, whether or not, there is a preferential right to the payment of any fixed premium or premium on any fixed scale, specified in the memorandum or articles of the company;

Capital shall be deemed to be preference capital, despite that it is entitled to either or both of the following rights, namely:-

  1. that in respect of dividends, in addition to the preferential rights to the amounts specified as above, it has a right to participate, whether fully or to a limited extent, with capital not entitled to the preferential right aforesaid;
  2. that in respect of capital, in addition to the preferential right to the repayment, on a winding up, of the amounts specified above, it has a right to participate, whether fully for to a limited extent, with capital not entitled to that preferential right in any surplus which may remain after the entire capital has been repaid.

Exception: In case of private company – Section 43 shall not apply where memorandum or articles of association of the private company so provides.

21.8 MEMORANDUM OF ASSOCIATION

21.8- 1 Meaning
The Memorandum of Association is a document of great importance. It contains the basic conditions on the strength of which a company is incorporated, namely, the name of the company, the place of its registered office, the objects within which it can operate, the nature of liability of its members and capital structure. Having regard to these basic conditions, it has also been described as the charter or constitution of the company. It defines as well as confines the powers of the company. It states what the company can do, what are its powers and at the same time sets out the limit outside which the company cannot function. It also regulates the affairs of the company in relation to the outsiders.

Definition: According to sec. 2(56) of the Companies Act, 2013 memorandum of association means “the memorandum of association of a company as originally framed or as altered from time to time in pursuance of any previous company law or of this Act”. This definition is not satisfactory, as it does not tell us what a memorandum of association is. We can define memorandum of association is the basic document of a company. It states positively the range of activities of the company and what the company can do and it also states negatively the limitation of the powers of a company i.e., what the company cannot do.
The memorandum must be printed, divided into paragraphs, numbered consecutively, and signed by at least seven persons (two in the case of a private company and one in the case of One Person Company) in the presence of at least one witness, who will attest the signatures. The particulars about the signatories to the memorandum as well as the witness, as to their address, description, occupation etc., must also be entered.

It is to be noted that a company being a legal person can through its agent, subscribe to the memorandum. However, a minor cannot be a signatory to the memorandum as he is not competent to contract. The guardian of a minor, who subscribes to the memorandum on his behalf, will be deemed to have subscribed in his personal capacity.

The above clauses of the Memorandum are called compulsory clauses, or “Conditions”. In addition to these a memorandum may contain other provisions, for example rights attached to various classes of shares.
The Memorandum of Association of a company cannot contain anything contrary to the provisions of the Companies Act If it does, the same shall be devoid of any legal effect. Similarly, all other documents of the company must comply with the provisions of the Memorandum.

21.8-2 Object of registering a memorandum of association:
It contains the object for which the company is formed and therefore identifies the possible scope of its operations beyond which its actions cannot go.
The purpose of memorandum is two-fold:

  • To enable the prospective investors to know the purpose for which their money is going to be used and what risk they are taking in making the investment.
  • To inform outsiders dealing with company as to what is its permitted range of activities in which it may lawfully engage.

Public document: The memorandum of association is a public document, which can be inspected by anybody at the Office of the Registrar of Companies. Every person dealing with company is presumed to have sufficient knowledge of its contents. Thus, memorandum helps in regulating external affairs 1 of company in relation to outsiders. Outsiders after reading contents of memorandum can know whether contract, which they wish to make, is within object of company.

A company cannot depart from the provisions contained in the memorandum however imperative may be the necessity for the departure. It cannot enter into a contract or engage in any trade or business, which is beyond the power confessed on it by the memorandum. If it does so, it would be ultra vires the company and void.
As per Section 4, Memorandum of a company shall be drawn up in such form as is given in Tables A, B, C, D and E in Schedule I of the Companies Act, 2013.
Section 4(6) of the Companies Act, 2013, provides that the memorandum of association should be in any one of the following model forms specified in Schedule I:
Table A for company limited by shares.
Table B for company limited by guarantee & not having share capital.
Table C for company limited by guarantee & having a share capital.
Table D for an unlimited company and not having share capital.
Table E for an unlimited company and having share capital.
The memorandum and articles of a company must be as closed to model forms, as possible, depending upon the circumstances.

21.8- 3 Content of the memorandum
A. Name Clause
The memorandum of association shall state the name of the company (Name Clause) with the last word “Limited” in the case of a public limited company, or the last words “Private Limited” in the case of a private limited company. This clause is not applicable on the com-panies formed under section 8 of the Act.
The name including phrase ‘Electoral Trust’ may be allowed for Registration of companies to be formed under section 8 of the Act, in accordance with the Electoral Trusts Scheme, 2013 notified by the Central Board of Direct Taxes (CBDT). For the Companies under section 8 of the Act, the name shall include the words foundation, Forum, Association, Federation, Chambers, Confederation, council, Electoral trust and the like etc. [The Companies (Incorporation) Rules, 2014],
As per MCA notification dated 5th June, 2015, a Government company’s name must end with | the word “Limited”. In the case of One Person Company, the words “One Person Company” should be included below its name.

B. Registered Office clause
The memorandum of association shall state the State in which the registered office of the company (Registered Office clause) is to be situated.

C. Object clause
The memorandum of association shall contain objects for which the company is proposed to be incorporated and any matter considered necessary in furtherance thereof (Object clause).
If any company has changed its activities which are not reflected in its name, it shall change its name in line with its activities within a period of six months from the change of activities after complying with all the provisions as applicable to change of name.

D. Liability clause
The memorandum shall also state the liability of members of the company (Liability clause), whether limited or unlimited, and also state,— in the case of a company limited by shares, that the liability of its members is limited to the amount unpaid, if any, on the shares held by them; and in the case of a company limited by guarantee, the amount up to which each member undertakes to contribute- to the assets of the company in the event of its being wound-up while he is a member or within one year after he ceases to be a member, for payment of the debts and liabilities of the company or of such debts and liabilities as may have been contracted before he ceases to be a member, as the case may be; and to the costs, charges and expenses of winding-up and for adjustment of the rights of the contributories among themselves;

E. Capital Clause
The memorandum shall state amount of authorized capital (Capital Clause) divided into share of fixed amounts and the number of shares with the subscribers to the memorandum have agreed to take, indicated opposite their names, which shall not be less than one share.
A company not having share capital need not have this clause.

F. Association clause
It contains the desire of the subscribers to be formed into a company. The Memorandum j shall conclude with the association clause. Every subscriber to the Memorandum shall take  atleast one share, and shall write against his name, the number of shares taken by him. j
In the case of OPC, the name of the person who, in the event of death of the subscriber, shall become [
the member of the company.

21.9 DOCTRINE OF ULTRA VIRES

21.9- 1 Meaning
The meaning of the term ultra vires is simply “beyond (their) powers”. The legal phrase “ultra vires ” is applicable only to acts done in excess of the legal powers of the doers. This presupposes that the powers in their nature are limited.
It is a fundamental rule of Company Law that the objects of a company as stated in its memorandum can be departed from only to the extent permitted by the Act, thus far and no further. In consequence, any act done or a contract made by the company which travels beyond the powers not only of the directors but also of the company is wholly void and inoperative in law and is therefore not binding on the company. On this account, a company can be restrained from employing its fund for purposes other than those sanctioned by the memorandum. Likewise, it can be restrained from carrying on a trade different from the one it is authorised to carry on.
The impact of the doctrine of ultra vires is that a company can neither be sued on an ultra vires transaction, nor can it sue on it. Since the memorandum is a “public document”, it is open to public inspection. Therefore, when one deals with a company one is deemed to know about the powers of the company. If in spite of this you enter into a transaction which is ultra vires the company, you cannot enforce it against the company.

21.9- 2 Example
If you have supplied goods or performed service on such a contract or lent money, you cannot obtain payment or recover the money lent. But if the money advanced to the company has not been expended, the lender may stop the company from parting with it by means of an injunction; this is because the company does not become the owner of the money, which is ultra vires the company. As the lender remains the owner, he can take back the property in specie. If the ultra vires loan has been utilised in meeting lawful debt of the company then the lender steps into the shoes of the debtor paid off and consequently he would be entitled to recover his loan to that extent from the company.
An act which is ultra vires the company being void, cannot be ratified by the shareholders of the company. Sometimes, act which is ultra vires can be regularised by ratifying it subsequently. For instance, if the act is ultra vires the power of the directors, the shareholders can ratify it; if it is ultra vires the articles of the company, the company can alter the articles; if the act is within the power of the company but is done irregularly, shareholder can validate it.
The leading case through which this doctrine was enunciated is that of Ashbury Railway Carriage and Iron Company Limited v. Riche (1875).
The facts of the case are:
The main objects of a company were:

  1. To make, sell or lend on hire, railway carriages and wagons.
  2. To carry on the business of mechanical engineers and general contractors.
  3. To purchase, lease, sell and work mines.
  4. To purchase and sell as merchants or agents, coal, timber, metals etc.

The directors of the company entered into a contract with Riche, for financing the construction of a railway line in Belgium, and the company further ratified this act of the directors by passing a special resolution. The company however, repudiated the contract as being ultra vires. And Riche brought an action for damages for breach of contract. His contention was that the contract was well within the meaning of the word general contractors and hence within its powers. Moreover it had been ratified by a majority of shareholders. However, it was held by the Court that the contract was null and void. It said that the terms general contractors was associated with mechanical engineers, Le. it had to be read in connection with the company’s main business. If, the term general contractor’s was not so interpreted, it would authorize the making of contracts of any kind and every description, for example, marine and re-insurance.
An ultra vires contract can never be made binding on the company. It cannot become “Intra vires” by reasons of estoppel, acquiescence, lapse of time, delay or ratification.

21.9- 3 The whole position regarding the doctrine of ultra vires can be summed up as

  1. When an act is performed, which though leggl in itself, is not authorized by the object clause of the memorandum, or by the statute, it is said to be ultra vires the company, and hence null and void.
  2. An act which is ultra vires, the company cannot be ratified even by the unanimous consent of all the shareholders.
  3. An act which is ultra vires the directors, but intra vires the company can be ratified by the members of the company through a resolution passed at a general meeting.
  4. If an act is ultra vires the Articles, it can be ratified by altering the Articles by a Special Resolution at a general meeting.

However, the disadvantages of this doctrine outweigh its main advantage, namely to provide protection to the shareholders and creditors. Although it may be useful to members in restraining the activities of the directors, it is only a nuisance insofar as it prevents the company from changing its activities in a direction which is agreed by all. Again, the purpose of doctrine of ultra vires has been defeated as now the object clause can be easily altered, by passing just a special resolution of the shareholders.

21.10 ARTICLES OF ASSOCIATION

Meaning
The articles of association of a company are its rules and regulations, which are framed to manage its internal affairs. Just as the memorandum contains the fundamental conditions upon which the company is allowed to be incorporated, so also the articles are the internal regulations of the company (Guiness v. Land Corporation of Ireland). These general functions of the articles have been aptly summed up by Lord Cairns in Ashbury Carriage Co. v. Riche as follows: “The articles play a part subsidiary to memorandum of association. They accept the memorandum as the charter of incorporation, and so accepting it the articles proceed to define the duties, the rights and powers of the governing body as between themselves and the company and the mode and form in which the business of the company is to be carried on, and the mode and form in which changes in the internal regulation of the company may from time to time be made.” The document containing the articles of association of a company (the Magna Carta) is a business document; hence it has to be construed strictly. It regulates domestic management of a company and creates certain rights and obligations between the members and the company [S.S. Rajkumar v. Perfect Castings (P) Ltd.].

The articles of association are in fact the bye-laws of the company according to which directors and other officers are required to perform their functions as regards the management of the company, its accounts and audit. It is important therefore that the auditor should study them and, while doing so he should note the provisions therein in respect of relevant matters.
Section 5 of the Companies Act, 2013 seeks to provide the contents and model of articles of association. The section lays the following law-
(1) Contains regulations
The articles of a company shall contain the regulations for management of the company.

(2) Inclusion of matters
The articles shall also contain such matters, as are prescribed under the rules. However, a company may also include such additional matters in its articles as may be considered necessary for its management.

(3) Contain provisions for entrenchment
The articles may contain provisions for entrenchment (to protect something) to the effect that specified provisions of the articles may be altered only if conditions or procedures as that are more restrictive than those applicable in the case of a special resolution, are met or complied with.

(4) Manner of inclusion of the entrenchment prdvision
The provisions for entrenchment shall only be made either on formation of a company, or by an amendment in the articles agreed to by all the members of the company in the case of a private company and by a special resolution in the case of a public company.

(5) Notice to the registrar of the entrenchment provision
Where the articles contain provisions for entrenchment, whether made on formation or by amendment, the company shall give notice to the Registrar of such provisions in such form and manner as may be prescribed.

(6) Forms of articles
The articles of a company shall be in respective forms specified in Tables F, G, H, I and J in Schedule I as may be applicable to such company.

(7) Model articles
A company may adopt all or any of the regulations contained in the model articles applicable to such company.
(8) Company registered after the commencement of this Act
In case of any company, which is registered after the commencement of this Act, insofar as the registered articles of such company do not exclude or modify the regulations contained in the model articles applicable to such company, those regulations shall, so far as applicable, be the regulations of that company in the same manner and to the extent as if they were contained in the duly registered articles of the company.

21.11 THE FOLLOWING ARE THE KEY DIFFERENCES BETWEEN THE MEMORANDUM OF ASSOCIATION VS. ARTICLES OF ASSOCIATION:

MOA

AOA

Power It is the charter and constitution of the company. The articles are subordinate to memorandum. If there is conflict between the two, memorandum shall prevail.
Ultra vires Acts done by a company beyond the scope of the memorandum are absolutely void (ineffective). They cannot be ratified even by unanimous vote of all the shareholders. Articles of association govern the internal relationship between the company and its members. Acts done by the company beyond its Articles can be ratified by the shareholders.
Registration Every company must have its own memo­randum. It must be compulsorily filed for registration. It must be in the following forms:

Model : A, B, C, D, E of Schedule I

Every company must have its own Articles. It must be compulsorily filed for registration. It must be in the following forms:

Model : F, G, H, I, J of Schedule I

Alteration MOA cannot be altered easily. AOA can be altered if it is desired by 3/4th majority.
Nature Memorandum of association contains the basic conditions on which the company is incorporated. It provides for name, situation objects, capital and liability of the company. Articles of association are the rules governing the internal management of the company. It provides for rules and procedures for the conduct of its business.
Scope It determines the objects, scope and extent of the activities of the company. It governs the ways in which the objects of the company are to be carried out.

 

21.12 DOCTRINE OF INDOOR MANAGEMENT

21.12- 1 Doctrine of Constructive Notice
Section 399 of the Companies Act, 2013 provides that any person can inspect by electronic means any document kept by the Registrar, or make a record of the same, or get a copy or extracts of any document, including certificate of incorporation of any company, on payment of prescribed fees.
Section 399 provides that the memorandum and articles when registered with Registrar of Companies ‘become public documents’ and then they can be inspected by any one on payment of a nominal fee. Therefore, any person who contemplates entering into a contract with the company has the means of ascertaining the powers of the company and is thus, presumed to have read these documents and understood them in their true perspective. This is known as “doctrine of constructive notice”.
Even if the party dealing with the company does not have actual notice of the contents of these

  • documents it is presumed that he has an implied (constructive) notice of them. Consequently,
  • if a person enters into a contract which is beyond the powers of the company, as defined in the
  • memorandum, or outside the limit set on the authority of the directors as per the memorandum or articles, he cannot, as a general rule, acquire any rights under the contract against the company.

By constructive notice is meant

  1. Whether a person reads the documents or not, he is presumed to have knowledge of the contents of the documents. He is not only presumed to have read the documents but also understood them in their true perspective, and
  2. Every person dealing with the company not only has the constructive notice of the memorandum and articles, but also of all the other related documents, such as Special Resolutions etc., which are required to be registered with the Registrar.

Thus, if a person enters into a contract which is beyond the powers of the company as defined in the memorandum, or outside the authority of directors as per memorandum or articles, he cannot acquire any rights under the contract against the company.

21.12- 2 Doctrine of Indoor Management
The Doctrine of Indoor Management is the exception to the doctrine of constructive notice. The aforesaid doctrine of constructive notice does in no sense mean that outsiders are deemed to have notice of the internal affairs of the company. For instance, if an act is authorised by the articles or
memorandum, an outsider is entitled to assume that all the detailed formalities for doing that act
have been observed. This can be explained with the help of a landmark case The Royal British Bank v. Turquand. This is the doctrine of indoor management popularly known as TurquandRule.
FACTS of The Royal British Bank v. Turquand
Mr. Turquand was the social manager (liquidator) of the insolvent Cameron’s Coalbrook Steam, Coal and Swansea and Loughor Railway Company. It was incorporated under the Joint Stock Companies Act, 1844. The company had given a bond for Rs. 2,000 to the Royal British Bank, which secured the company’s drawings on its current account. The bond was under the company’s seal, signed by two directors and the secretary. When the company was sued, it alleged that under its registered deed of settlement (the articles of association), directors only had power to borrow up to an amount authorized by a company resolution. A resolution had been passed but not specifying how much the directors could borrow.
Held, that the bond was valid, so the Royal British Bank could enforce the terms. He said the bank was deemed to be aware that the directors could borrow only up to the amount resolutions allowed.
Articles of association were registered with Companies House, so there was constructive notice. But the bank could not be deemed to know which ordinary resolutions passed, because these were not registerable. The bond was valid because there was no requirement to look into the company’s internal workings. This is the indoor management rule, that the company’s indoor affairs are the company’s problem.

21.12- 3 Exceptions to the doctrine of Indoor Management
Thus, you will notice that the aforementioned rule of Indoor Management is important to persons dealing with a company through its directors or other persons. They are entitled to assume that the acts of the directors or other officers of the company are validly performed, if they are within th scope of their apparent authority. So long as an act is valid under the articles, if done in a particular manner, an outsider dealing with the company is entitled to assume that it has been done in the manner required.

The abovementioned doctrine of Indoor Management or Turquand Rule has limitations of its own. That is to say, it is inapplicable to the following cases, namely:
a. Actual or constructive knowledge of irregularity
The rule does not protect any person when the person dealing with the company has notice, whether actual or constructive, of the irregularity.
In Howard v. Patent Ivory Mfg. Co. (1888)38 Ch. D. 156, the directors of a company could borrow upto £ 1,000 without the sanction of members in General Meeting. The consent of the shareholders was required to borrow in excess of £1,000. The directors themselves lent £3,500 to the company. It was held that the directors had the notice of the internal irregularity and therefore the company was liable to them only for £1,000.
In Morris v, Kansseen, a director could not defend an allotment of shares to him as he par-ticipated in the meeting, which made the allotment. His appointment as a director also fell through because none of the directors appointed him was validly in office.

b. Suspicion of Irregularity
The doctrine is not applicable in case of negligent persons. If an officer of the company acts in a manner, which would not ordinarily be within his powers, the person dealing with him must make proper inquiries and satisfy himself as to the officer’s authority. If he fails to make enquiry, he cannot rely on the rule. Where the transaction is unusual or not in the ordinary course of business, it is the duty of the outsider to make the necessary enquiry.
The protection of the “Turquand Rule” is also not available where the circumstances sur-rounding the contract are suspicious and therefore invite inquiry. Suspicion should arise, for example, from the fact that an officer is purporting to act in matter, which is apparently outside the scope of his authority.
Where, for example, as in the case of Anand Bihari Lai v. Dinshaw & Co., an accountant of a company transferred some property of the company in favour of Anand Bihari, who brought an action for the breach of contract against the company. The transfer was held by the Court to be void, since the power of transferring property could not be considered as within the apparent authority of an accountant.
Similarly, in the case of Haughton & Co. v. Nothard, Lowe & Wills Ltd. where a person holding directorship in two companies agreed to apply the money of one company in payment of the debt to other, the court said that it was something so unusual “that the plaintiff were v put upon inquiry to ascertain whether the persons making the contract had any authority in fact to make it.” Any other rule would “place limited companies without any sufficient reasons for so doing, at the mercy of any servant or agent who should purport to contract on their behalf.”

c. Forgery
The doctrine of indoor management applies only to irregularities which might otherwise affect a transaction but it cannot apply to forgery which must be regarded as nullity. Forgery may in circumstances exclude the ‘Turquand Rule’.
The only clear illustration is found in the Ruben v. Great Fingall Consolidated.
In this case, the Secretary of the company issued a share certificate in favour of Ruben, which apparently complied with company’s articles, as it was purported to be signed by 2 directors & secretary & it had company’s common seal affixed to it. In fact, the secretary had forged the signature of the directors and affixed the seal without any authority. It was held that the certificate was not binding upon the company. Lord Loreburn held : “It is quite true that personal dealing with limited liability companies are not bound to inquire into their indoor management, but this doctrine which is well established, applied to irregularities which otherwise might affect genuine transaction. It cannot apply to a forgery”.
The plaintiff contended that whether the signature were genuine or forged was apart of the internal management, and therefore, the company should be estopped from denying genu-ineness of the document. But it was held, that the rule has never been extended to cover such a complete forgery.

MULTIPLE CHOICE QUESTIONS:

1. In which case the principle of corporate veil was ignored by the courts:
(a) Determination of character of the company
(b) ForprotectionofrevenueoftheGovemment
(c) For prevention of fraud or improper conduct
(d) All of these

2. A company is incorporated in India, but all its members are the Americans. Such company is:
(a) An Indian company
(b) An American company
(c) An illegal association
(d) A government company

3. A company cannot be treated as a citizen, yet it has its:
(a) Nationality
(b) Domicile
(c) Residence
(d) All of these

4. Which of the following is called the charter of a company?
(a) Prospectus
(b) Articles of Association
(c) Memorandum
(d) Certificate of Incorporation

5. Which of the following tables in Schedule of the Act does not contain any prescribed form of Memorandum?
(a) F
(b) B
(c) E
(d) D

6. How many fundamentals clauses are contained in the Memorandum of a company?
(a) 4
(b) 5
(c) 6
(d) 7

7. Under the Registered office clause, a company has to give the name of the:
(a) Town
(b) Village
(c) District
(d) State

8. Acts done by a company ‘beyond its powers’ are termed:
(a) Intra vires
(b) illegal
(c) ultra vires
(d) none of these.

9. The regulations contained in which of the following Tables are also called ‘articles’:
(a) A
(b) B
(c) C
(d) none of these.

10. Articles of Association are the regulator of the behaviour of:
(a) Creditors
(b) Directors and officers
(c) Suppliers
(d) Outside public

11. Articles of Association are subordinate to:
(a) Prospectus
(b) Board of Directors
(c) Memorandum and Companies Act
(d) None of these

12. Which of the following documents contains bye- g laws that govern the internal management of a company:
(a) Memorandum
(b) Articles
(c) Prospectus
(d) All of these

13. Registration of Articles of Association is compulsory for which of the following companies:
(a) Unlimited companies
(b) Companies limited by guarantee
(c) Private companies limited by shares
(d) All of these

14. Which of the following companies may or may not frame (register) their articles of association
(a) Unlimited companies
(b) Private companies limited by shares
(c) Companies limited by guarantee
(d) None, all has to have its own articles.

15. The Articles of association must be signed by:
(a) All the directors
(b) All the promoters
(c) The auditors
(d) All the subscribers of the memorandum

16. Which of the following is the necessary condition in case of the Articles of a company:
(a) Be printed
(b) Be divided into paragraphs numbered consecutively
(c) Be signed by each subscriber of the Memorandum
(d) All of these.

17. Memorandum and Articles of a company are:
(a) Private documents
(b) Public documents
(c) Government documents
(d) All of the above

18. Under the doctrine or rule of ‘constructive notice’ which of the following persons are presumed to have read the Memorandum and the Articles:
(a) Members
(b) Directors
(c) Outsiders
(d) Officers of the company

19. The benefit of the Doctrine of Indoor Management is available only to the:
(a) Directors
(b) Shareholders
(c) Officers of the company
(d) Outsiders

Answers:
CA Foundation Business Laws Study Material Chapter 21 The Companies Act, 2013 1

STATE WHETHER THE FOLLOWING ARE TRUE OR FALSE:

1. ‘Patel & Co.’ is a company in the eyes of the law.
2. A company has its own distinct entity.
3. If the members composing a company die or dissociate themselves from the company, such company gets annihilated alongwith.
4. The creditors of a company can sue its members for the debts of the company.
5. A company being an artificial person cannot own property and cannot sue or be sued.
6. A private company need not have its own Articles.
7. A company limited by guarantee cannot have share capital.
8. A company that does not hold more than 50% of total share capital of another company, may still be a holding company of that another one.
9. A Government company is that company of which all the share are held by the Government.
10. A licenced company can be registered with limited liability without adding the word ‘Limited’ to its name.
11. A ‘one-man company’ is a legal entity in the eyes of the law.
12. Memorandum is the basic document facilitating the incorporation of a company.
13. It is not necessary for a company to mention the actual address of its registered office in the Memorandum.
14. The domicile and nationality of a company are determined on the basis of its registered office.
15. The acts ultra vires the directors are null and void and therefore cannot be ratified by the shareholders.
16. The transactions ultras vires the Memorandum can be ratified by the unanimous consent of all the shareholders.
17. Articles are rules framed for governing the internal affairs of the company.
18. Articles are bye-laws for the general administration of a company.
19. Memorandum is subsidiary to articles.
20. No form has been prescribed for the articles of companies in the Act.
21. Memorandum and articles bind the company to its members, and also each member to the company interse.
22. Memorandum and articles are public documents open to public inspection.
23. An outsider dealing with a company is presumed to have knowledge of the provisions of its memorandum and articles.
24. An outsider dealings with a company has a rights to assume that its internal management has no irregularity.

Answers:
CA Foundation Business Laws Study Material Chapter 21 The Companies Act, 2013 2

CA Foundation Business Laws Study Material Chapter 20 The Limited Liability Partnership Act, 2008

CA Foundation Business Laws Study Material Chapter 20 The Limited Liability Partnership Act, 2008

INTRODUCTION

A need has been felt for a long time for a new corporate form that would provide an alternative to the traditional partnership with unlimited personal liability on the one hand, and, the compliance based structure of the private or unlisted public company on the other hand. With this objective in view, the Limited Liability Partnership Act, 2008 was enacted by the Parliament on 12th December, 2008, which received the assent of the President on 7th January, 2009 and was notified with effect from 31st March, 2009. Most of the sections of Limited Liability Act, 2008 came into force from 31st March, 2009.

LLP is a hybrid form of business organization structure which combines the advantages of the Partnership firm and the Company Structure. LLP offers the flexibility of a partnership firm and reduces the compliances of a company structure.
Though the enactment regarding Limited Liability Partnership (“LLP”) finally came into operation in 2009 in India, the concept of limited liability partnership is not new; in fact is more than a 100 year old concept. It is one of the most renowned forms of business organizations worldwide.
The Ministry of Corporate Affairs (MCA) and the Registrar of Companies (ROC) are entrusted with the task of administrating the LLP Act, 2008. The Central Government has the authority to frame the Rules with regard to the LLP Act, 2008, and can amend them by notifications in the Official Gazette, from time to time.

This Act deals with the formation and regulation of Limited Liability Partnerships and for matters incidental thereto.
It contains 81 sections and divided into 4 schedules.

  1. The First Schedule deals with mutual rights and duties of partners, as well as the rights and duties of limited liability partnership and its partners in case of absence of formal agreement with respect to them.
  2. The Second Schedule deals with conversion of a firm into LLP.
  3. The Third Schedule deals with conversion of a private company into LLP.
  4. The Fourth Schedule deals with conversion of unlisted public company into LLP.

Note: The Indian Partnership Act, 1932 is not applicable to LLPs.

WHY LIMITED LIABILITY PARTNERSHIP?

A need has been felt to make a new legislation related to a new corporate form of business organization in India to meet with the contemporary growth of the Indian economy. It provides an alternative to the traditional partnership with unlimited liability on the one hand and the statute- based governance structure of the limited liability company on the other hand, in order to enable professional expertise and entrepreneurial initiative to combine, organize and operate in flexible, innovative and efficient manner.
Limited Liability Partnership (LLP) is a corporate business organization that provides the benefits of limited liability but also allows its members the flexibility of organizing their internal structure just like in case of a partnership, based on a mutually arrived agreement. The LLP form enables entrepreneurs, professionals and enterprises providing services of any kind or engaged in scientific and technical disciplines, to form commercially efficient vehicles suited to their requirements. Owing to flexibility in its structure and operation, the LLP is a suitable vehicle for small enterprises and for investment by venture capital.

LIMITED LIABILITY PARTNERSHIP – MEANING AND CONCEPT

Meaning: A LLP is a new form of legal business entity with limited liability. It is a separate legal entity where LLP itself is liable to the third parties upto the assets it owns but the liability of the partners is limited. It is an alternative corporate business vehicle that not only gives the benefits of limited liability at low compliance cost but allows its partners the flexibility of organising their internal structure as a traditional partnership. It gives the benefits of limited liability of a company and the flexibility of a partnership.
LLP is also called as a hybrid between a company and a partnership as it contains elements of both, a corporate entity as well as a partnership.
Since LLP contains elements of both ‘a corporate structure’ as well as ‘a partnership firm structure’ LLP is called a hybrid between a company and a partnership.

CHARACTERISTIC/SALIENT FEATURES OF LLP

1. A body corporate
A LLP is a body corporate formed and incorporated under LLP Act and is a legal entity separate from the partners constituting it. [Sec. 3]

2. Separate Legal Entity
The LLP is a separate legal entity. It is liable to the full extent of its assets but liability of the partners is limited to their agreed contribution in the LLP. In other words, creditors of LLP shall be the creditors of LLP alone and not of the partners.

3. Perpetual Succession
Death, insanity, retirement or insolvency of partners has no impact on the existence of LLP. The LLP can continue its existence irrespective of changes in partners. It is can enter into contracts in its own name. It can also hold properties in its own name. It is created by law and law alone can dissolve it.

4. Absence of Mutual Agency
The cardinal principal of mutual agency of partners in a partnership is missing in LLP. In case of LLP, the partners of LLP are agents of LLP alone and not of the other partners. Hence, no partner can be held liable on account of the independent or un-authorized actions of other partners. Thus individual partners cannot be held liable for liability incurred by another partner’s wrongful business decisions or misconduct.

5. LLP Agreement
The partners are free to make rules related to the mutual rights and duties of the partners as per their choice. This is done through an agreement. In the absence of any such agreement, the mutual rights and duties shall be governed by the provisions of the LLP Act, 2008.

6. Artificial Person
A LLP is an Artificial legal person created by law capable of enjoying all the rights of an individual. It can do everything which a natural person can do, except the contracts of very personal nature like, it cannot marry, it cannot go to jail, cannot take an oath, cannot marry or get divorce. Further, it cannot practice a learned profession like CA, Law or Medicine. A LLP is invisible, intangible, immortal but not fictitious because it really exists.

7. Common Seal
Being an artificial person, a LLP work on its own but it has to act through its partners. Hence, it may have a common seal which can be considered as its official signature. [Section 14(c)], It should be noted that it is not mandatory for a LLP to have a common seal. If it decides to have one, then it shall remain under the custody of some responsible official and it shall be a fixed in the presence of at least 2 designated partners of the LLP.

8. Limited Liability
Every partner of a LLP is, for the purpose of the business of LLP, the agent of the LLP, but not of other partners (Section 26). The liability of the partners will be limited to their agreed contribution in the LLP.

9. Management of Business
The partners in the LLP are entitled to manage the business of LLP. However, only the designated partners are responsible for legal compliances.

10. Minimum and Maximum number of Partners
Every LLP shall have least two partners and shall also have at least 2 individuals as designated partners. It is mandatory that at least one of the designated partners shall be resident in India. Further, there is no maximum limit of partners in LLP.

11. Business for profit Only
LLP can be formed only for carrying on any lawful business with a view to earn profit. Thus LLP cannot be formed for charitable or non-for-profit purpose.

12. Investigation
The Central Government shall have powers to investigate the affairs of an LLP by appointment of competence authority.

13. Compromise or Arrangement
Any compromise or arrangement including merger and amalgamation of LLPs shall be in accordance with the provisions of the LLP Act, 2008.

14. Conversion into LLP
A firm, private company or an unlisted public company would be allowed to be converted into LLP in accordance with the provisions of LLP Act, 2008.

15. E-Filling of Documents
Every form or application of document required to be led or delivered under the act and rules made thereunder, shall be led in computer readable electronic form on its website www.mca. gov.in and authenticated by a partner or designated partner of LLP by the use of electronic or digital signature.

16. Foreign LLPs
Section 2(l)(m) defines foreign limited liability partnership “as a limited liability partnership formed, incorporated, or registered outside India which established a place of business within India”. Foreign LLP can become a partner in an Indian LLP.

The following are the advantages of LLP form of business organization:

  1. It is easier to form a LLP as compared to a company.
  2. The partners of a LLP enjoy limited liability.
  3. It operates on the basis of an agreement.
  4. It is not rigid as far as capital structure is concerned.
  5. It provides flexibility without imposing detailed legal and procedural requirements.
  6. It is easy to dissolve an LLP as compared to a Company.

INCORPORATION OF LLP

Essential elements to incorporate LLP
Limited Liability Partnerships are body corporates which must be registered with the Registrar of LLP after following the provisions specified in the LLP Act. The process is quite similar to setting up of a company. Under the LLP Act, 2008, the following elements are very essential to form a LLP in India:

  1. Persons intending to incorporate a LLP shall decide a name for the LLP.
  2. A LLP shall execute a limited liability partnership agreement between the partners inter se or between the LLP and its partners. In the absence of any agreement the provisions as set out in First Schedule of LLP Act, 2008 will be applied.
  3. Then they shall complete and submit the incorporation document in the form prescribed with the Registrar electronically, along with the prescribed fees.
  4. There must be at least two partners for incorporation of LLP [Individual or body corporate],
  5. A LLP shall have a registered office in India so as to send and receive communications;
  6. It should appoint atleast two individuals as designated partners who will be responsible for number of duties including doing of all acts, matters and things as are required to be done by the LLP. At least one of them should be resident in India. Each designated partner shall hold a Designated Partner Identification Number (DPIN) which is allotted by MCA.
  7. As soon as the process is completed, a certificate of registration shall be issued which shall contain a Limited Liability Partnership Identification Number (LLPIN)

STEPS OR PROCESS FOR INCORPORATING AN LLP

Step I: Reservation of name

  • The first step while incorporating a LLP is the reservation of name of LLP.
  • The name of a LLP shall not be similar to that of an existing LLP, Company or a Partnership Firm.
  • The applicant has to file e-form 1, for ascertaining the availability and reservation of name.
    6 names in order of preference can be indicated.
  • The name should contain the suffix “Limited Liability Partnership” or “LLP”.

Step 2: Incorporation

  • In the second step, the applicant has to file e-form 2 for incorporating a new LLP.
  • This form contains the details of the proposed LLP and the Partners and Designated Partners along with their consent to act as such.

Step 3: Execute a LLP Agreement 

  • It is mandatory to execute LLP Agreement. [Sec. 23]
  • LLP agreement shall be filed with the registrar in e-form 3 within 30 days of incorporation of LLP.

The contents of the LLP Agreement are enumerated below:

  1. Name of LLP
  2. Name and address of partners and designated partners
  3. Form of contribution & interest on contribution
  4. Profit sharing ratio
  5. Remuneration of Partners
  6. Rights & Duties of Partners
  7. Proposed Business
  8. Rules for governing LLP.

DIFFERENCES WITH OTHER FORMS OF ORGANISATION

A. Distinction between LLP and Partnership Firm:
The points of distinction between a limited liability partnership and partnership firm are tabulated as follows:

Sr.No:

Basis LLP

Partnership

1

Regulating Act

The Limited Liability Partnership Act, 2008. The Indian Partnership Act, 1932.

2

Body corporate

It is a body corporate. It is not a body corporate.

3

Separate legal entity

It is a legal entity separate from its members. It is a group of persons with no separate legal entity.

4

Creation

It is created by a legal process called registration under the LLP Act, 2008. It is created by an agreement between the partners.

5

Registration

Registration is mandatory. LLP can sue and be sued in its own name. Registration is voluntary. Only the registered partnership firm can sue the third parties.

6

Perpetual
succession

The death, insanity, retirement or insolvency of the partner(s) does not affect its existence of LLP. Members may join or leave but its existence continues forever. The death, insanity retirement or insolvency of the partner(s) may affect its existence. It has no perpetual succession.

7

Name

Name of the LLP to contain the word limited liability partners (LLP) as suffix. No guidelines. The partners can have any name as per their choice.

8

Liability

Liability of each partner limited to the extent to agreed contribution except in case of wilful fraud. Liability of each partner is unlimited. It can be extended upto the personal assets of the partners.

9

Mutual agency

Each partner can bind the LLP by his own acts but not the other partners. ‘ Each partner can bind the firm as well as other partners by his own acts.

10

Designated
partners

At least two designated partners and atleast one of them shall be resident in India. There is no provision for such partners under the Indian Partnership Act, 1932.

11

Common seal

It may have its common seal as its official signatures. There is no such concept in partnership

12

Legal
compliances

Only designated partners are responsible for all the compliances and penalties under this Act. All partners are responsible for all the compliances and penalties under the Act.

13

Annual filing of documents

LLP is required to file:

(i) Annual statement of accounts

(ii) Statement of solvency

(iii) Annual return with the registration of LLP every year.

Partnership firm is not required to file any annual document with the registrar of firms.

14

Foreign
partnership

Foreign nationals can become a partner in a LLP. Foreign nationals cannot become a partner in a partnership firm.

15

Minor as partner

Minor cannot be admitted to the benefits of LLP. Minor can be admitted to the benefits of the partnership with the prior consent of the existing partners.

B. Distinction between LLP and Limited Liability Company (LLC)

Sr.No:

Basis LLP

Limited Liability Company

1

Regulating Act

The LLP Act, 2008. The Companies Act, 2013.

2

Members/
Partners

The persons who contribute to LLP are known as partners of the LLP. The persons who invest the money in the shares are known as members of the company.

3

Internal
governance
structure

The internal governance structure of a LLP is governed by agreement between the partners. The internal governance structure of a company is regulated by statute (i.e., Companies Act, 2013).

4

Name

Name of the LLP to contain the word “Limited Liability partnership” or “LLP” as suffix. Name of the public company to contain the word “limited” and Private company to contain the word “Private limited” as suffix.

5

Number of members/ partners

Minimum – 2 members

Maximum – No such limit on the members in the Act.

The members of the LLP can be individuals/ or body corporate through the nominees.

Private company: Minimum – 2 members Maximum – 200 members

Public company: Minimum – 7 members Maximum – No such limit on the members.

Members can be organizations, trusts, another business form or individuals.

6

Liability of members/ partners

Liability of a partners is limited to the extent of agreed contribution except in case of wilful fraud. Liability of a member is limited to the amount unpaid on the shares held by them.

7

Management

The business of the company managed by the partners including the designated partners authorized in the agreement. The affairs of the company are managed by board of directors elected by the shareholders.

8

Minimum number of directors/ designated partners

Minimum 2 designated partners. Private Co. – 2 directors Public Co. – 3 directors

MULTIPLE CHOICE QUESTIONS:

1. The LLP Act, 2008 came into force from:
(a) 31st March, 2008
(b) 31st March, 2009
(c) 1st April, 2008
(d) 1st April, 2009

2. Maximum number of partners in a LLP can be:
(a) 100
(b) 200
(c) 50
(d) Unlimited

3. In case of a LLP, the partners are agents of
(a) LLP
(b) Other Partners
(c) Both a) and b)
(d) None of the above

4. For reservation of name, e-Form is
required.
(d) E-Form 1
(b) E-Form A
(c) E-Form B
(d) E-Form 2

5. Minimum designated partners required in a LLP are:
(a) 1
(b) 2
(c) 3
(d) 0

6. Common seal is mandatory for
(a) Company
(b) LLP
(c) Both the above
(d) None of the above

7. In case of legal non-compliance and penalties under the LLP Act, are responsible.
(a) Partners
(b) Designated Partners
(c) LLP
(d) All the above

Answers:
CA Foundation Business Laws Study Material Chapter 20 The Limited Liability Partnership Act, 2008 1

STATE WHETHER THE FOLLOWING ARE TRUE OR FALSE:

1. It is mandatory for a LLP to have a common seal.
2. A LLP is not a body corporate.
3. Executing an LLP Agreement is discretionary.
4. In a LLP, all partners have an unlimited liability.
5. Limited Liability Partnership is governed by Partnership Act, 1932.
6. Every partner in a LLP is the agent for the purpose of the business of LLP but not of the other partners.
7. LLP can be incorporated for charitable purpose as well as for business.
8. Foreign Nationals can become a partner in a LLP.

Answers:
CA Foundation Business Laws Study Material Chapter 20 The Limited Liability Partnership Act, 2008 2

CA Foundation Business Laws Study Material Chapter 19 Reconstitution and Dissolution of Firm

CA Foundation Business Laws Study Material Chapter 19 Reconstitution and Dissolution of Firm

RECONSTITUTION OF A FIRM

Incoming and outgoing Partners: The constitution of a firm may be changed by the introduction of a new partner; death, retirement, insolvency and expulsion of a partner; or by the transfer of a partner’s share to an outsider. All these are included within the term reconstitution of a firm. Upon reconstitution, the rights and liabilities of the incoming and outgoing partners have to be determined. The provisions of the Partnership Act regarding such cases are stated below.

Introduction of a New Partner (Sec. 31)
A new partner can be introduced only with the consent of all the partners. The share of profits which a new partner is entitled to get is fixed at the time he becomes a partner. He is liable for all the debts of the firm after the date of his admission but he is not responsible for any act of the firm done before he became a partner, unless otherwise agreed. These rules do not apply to a minor becoming a partner under Sec. 30.
The incoming partner may, however, assume liability for past debts by novation that is, by tripartite agreement between

  1. the creditor
  2. the partners and
  3. the incoming partner.

Retirement of a Partner (Sec. 32)
A partner may retire

  1. with the consent of all the other partners,
  2. in accordance with the terms of the agreement of partnership or
  3. where the partnership is at will, by giving notice in writing to all the other partners of his intention to retire.

A retiring partner remains liable for the partnership debts contracted while he was a partner. He may however be discharged from any liability to any third party for acts of the firm done before his retirement by novation, if it is so agreed with the third party and the partners of the reconstituted firm. Such agreement may be implied from the course of dealing between the firm and the third party after he had knowledge of the retirement.
The retired partner continues to remain liable to third parties for all acts of the firm until public notice is given of the retirement. Such notice may be given either by the retired partner or by any member of the reconstituted firm. A retired partner is not liable for the debts of the firm incurred after public notice of his retirement. A sleeping partner may retire without giving a public notice of his retirement because he is not known to be a partner to third parties.

Expulsion of a partner (Sec. 33)
A partner can be expelled only when the following conditions are fulfilled:

  1. When the contract of partnership contains a provision for expulsion under stated circumstances.
  2. The power to expel is exercised in good faith by the majority of the partners.
  3. The expelled partner has been given notice of the charges against him and has been given an opportunity to answer the charges.

The liabilities of an expelled partner for the debts of the firm are the same as those of retired partner.

Insolvency of a partner (Sec. 34)
When the partner of a firm is adjudicated an insolvent, he ceases to be a partner from the date on which the order of adjudication was passed by the court. Whether the firm is thereby dissolved or not depends on the terms of the agreement between the partners.

Death of Partners (Sec. 35)
Ordinarily the death of partner has the effect of dissolving the firm. But it is competent for the partners to agree that the firm will continue to exist even after the death of partner.

Transfer of interest (Sec. 29)
What are the Rights of Transferee of Partner’s share?
Transferee’s rights: A share in a partnership is transferable like any other property, but as the 1z partnership relationship is based on mutual confidence, the assignee of a partner’s interest by sale, mortgage or otherwise cannot enjoy the same rights and privileges as the original one (section 29 x of the Partnership Act). The Supreme Court in Narayanappa v. Krishnappa( 1966) has held that the assignee will enjoy only the rights to receive the shares of the profits of the assignor and amount of profits agreed to by other partners.

The rights of such a transferee may be noted as follows:
(a) During the continuance of partnership
During the continuance of partnership, such transferee is entitled to receive the share of the
profits of the transferring partner. However, he is bound to accept the profits as agreed to by the partners i.e. he cannot challenge the accounts.
A transferee of a Partner’s share is not entitled:

  1. to interfere with the conduct of the business.
  2. to require accounts or
  3. to inspect books of the firm.

(b) On the dissolution of the firm
On the dissolution of the firm or on the retirement of the transferring partner, the transferee will be entitled against the remaining partners:

  1. to receive the share of the assets of the firm to which the transferring partners was entitled and
  2. for the purpose of ascertaining the share, to an account as from the date of the dissolution.

Rights and liabilities of an outgoing partner
(I) Rights of an outgoing partner
An outgoing partner possesses following rights:
(a) Right to carry on competing business
An outgoing partner has the right to carry on the business competing with that of the firm, and he may advertise such business (Sec. 36). But section 36 imposes some restrictions on his activities in order to prevent unfair competition with the firm. The ‘ restrictions imposed upon outgoing partner are:

  1. he may not use firm’s name,
  2. he may not represent himself as carrying on the business on behalf of the firm or
  3. he may not solicit the customers or the persons who were already dealing with the firm before he left the firm. The above restrictions are subject to a contract to the contrary.

However, the firm may enter into an agreement with the retiring partner not to do competitive business, and then he will not be entitled to carry on competitive business. This agreement will not be void as it will not be treated as an agreement in restraint of trade.

(b) Right to share subsequent profit in certain cases
As per section 37, in case the accounts of the outgoing partners continue to remain unsettled and the remaining partner continues to run the business, such a partner is entitled to receive his share of profit or interest at the rate of 6% p.a. on the amount of his share in the firm.

(II) Liabilities of an outgoing partner
These may be classified into two stages:
(a) Liability for acts done before leaving the firm
A retiring partner is liable for the acts done and debts incurred before his retirement, but he may be exempted from this liability in case on an agreement made by him with the third party and the remaining partners of the reconstituted firm.
(b) Liability for acts done after leaving the firm
In case of retirement of a partner, a public notice is essential to this effect. If it is not given, the retiring partner will continue to be liable to third parties for the acts of the . firm even after his retirement. A public notice is not essential in case of sleeping and deceased partners who is not known to be partner, and so will not be liable for such acts.

DISSOLUTION OF FIRM (SECS. 39 TO 55)
Dissolution of partnership and dissolution of firm

Dissolution of firm

Dissolution of Partnership

It involves closing down of the business as the partnership between all the partners comes to an end. It,involves a change in the relationship amongst the partners due to retirement, expulsion etc., and the business of the firm does not necessary come to an end. It leads to reconstitution of firm.

Modes of dissolution of firm
A firm may be dissolved on any of the following grounds:
1. By agreement (Sec. 40)
A firm may be dissolved any time with the consent of all the partners of the firm. Partnership is created by contract, it can also be terminated by contract.

2. By Compulsory Dissolution (Sec. 41)
A firm is dissolved-

  1. by the adjudication of all the partners or of all the partners but one as insolvent, or
  2. by the happening of any event which makes the business of the firm unlawful.

3. Dissolution on the happening of certain contingencies (Sec. 42)
Subject to contract between the partners, a firm is dissolved-

  1. if constituted for a fixed term, by the expiry of that term,
  2. if constituted to carry out one or more adventures of undertakings, by the completion thereof,
  3. by the death of a partner, and
  4. by the adjudication of a partner as an insolvent.

The partnership agreement may provide that the firm will not be dissolved in any of the aforementioned cases. Such a provision is valid.

4. Dissolution by notice (Sec. 43)
Where the partnership is at will, the firm may be dissolved by any partner giving notice in writing to all other partners of his intention to dissolve the firm. The firm is dissolved as from the date mentioned in the notice as the date of dissolution, or, if no date is mentioned, as from the date of communication of the notice.

5. Dissolution by the Court (Sec. 44)
At the suit of a partner, the court may dissolve a firm on any one of the following grounds:
A. Insanity
If a partner has become of unsound mind. The suit for dissolution in this case can be filed by the next friend of the insane partner or by any other partner.

B. Permanent incapacity
If a partner becomes permanently incapable of performing his duties as a partner. Permanent incapacity may arise from an incurable illness like paralysis. The suit for dissolution in this case must be brought by a partner other than the person who has become incapable.

C. Guilty conduct
If a partner is guilty of conduct which is likely to affect prejudicially the carrying on of the business, regard being had to the nature of the business. The suit for dissolution on the ground mentioned in this clause must be brought by a partner other than the partner who is guilty of misconduct.

D. Persistent breach of agreement
If a partner wilfully and persistently commits breach of the partnership agreement regarding management or otherwise conducts himself in such a way that it is not rea-sonably practicable for the other partriers to carry on business in partnership with him.

E. Transfer of whole interest
If a Partner has transferred the whole of his interest in the firm to an outsider or as allowed his interest to be sold in execution of a decree. Transfer of partner’s interest does not by itself dissolves the firm. But the other partners may ask the court to dissolve the firm if such a transfer occurs.

F. Loss
If the business of the firm cannot be carried on except at a loss. Since the motive, with which partnerships are formed, is acquisition of gain, the courts have been given discretion to dissolve a firm in cases where it is impossible to make profits.

G. Just and Equitable clause
If the court considers it just and equitable to dissolve the firm. This clause gives a discretionary power to the court to dissolve a firm in cases which do not come within j any of the foregoing clauses but which are considered to be fit and proper cases for dissolution.

Consequences of dissolution:
a. Rights & liabilities of partners on dissolution (Secs. 45 to 55)
When the firm is dissolved, the business of the firm is wound up, the assets are realised to pay the debts and the surplus, if any is distributed amongst the partners. For the purposes of winding up of the firm, the partners possess certain rights and are subject to certain liabilities. These are discussed below.
1. Right to have the business wound up (Sec. 46)
On the dissolution of a firm, a partner has the right

  1. to have the business of the firm wound up and the debts of the firm settled out of the property of the firm and
  2. to have the surplus distributed among the partners according to their rights.

2. Continuing authority of partners for purpose of winding up (Sec. 47)
The partners authority to act for the firm and to bind their co-partners continues even after dissolution of the firm for the following two purposes :

  1. to wind up the affairs of the firm (for example recovering money from debtors)
  2. to complete transaction begun but unfinished at the time of the dissolution.

3. Right to share in personal profits earned after dissolution (Sec. 50)
Every partner has a right to share in any secret profits derived by any partner under any transaction carried out in the firm name or by use of the property or business connection of the firm, after the dissolution but before winding up.

4. Right to have premium returned on premature dissolution (Sec. 51)
Where a partner has paid a premium (goodwill) on entering into partnership for a fixed term, and the firm is dissolved before the expiration of that term, he shall be entitled ! to repayment of the whole or a reasonable part of the premium. The amount of repayment will depend upon
(a) the terms upon which he became a partner &
(b) length of the time during which he was a partner.

Example: X entered into a partnership, in a firm for a period of 10 years and paid Rs. 5,00,000 as premium. The firm was dissolved after expiration of 3 years because of the insolvency of a partner. Here, X shall be entitled to Rs. 3,50,000 (5,00,000/10 *3= 1,50,000; 5,00,000 -1,50,000 = 3,50,000) as return of premium.
No such premium shall be paid to the partner if such premature dissolution

  1. is due to death of a partner, or
  2. is due to his own misconduct or
  3. is as per agreement which contains no provision for the return of premium.

5. Rights where partnership contract is rescinded for fraud or misrepresentation (Sec. 52)
Where a partner was induced to join the firm by the fraud or misrepresentation of any other partner, the aggrieved partner has the right to rescind the partnership agreement and is entitled:

  1. to a lien on, or a right of retention of, the surplus or the assets of the firm remaining after the debts of the firm have been paid, for any sum paid by him for the purchase of a share in the firm and for any capital contributed by him;
  2. to rank as a creditor of the firm in respect of any payment made by him towards the debts of the firm; and
  3. to be indemnified by the partner or partners guilty of fraud or misrepresentation against all the debts of the firm.

6. Right to restrain the use of firm name or firm property (Sec. 53)
After a firm is dissolved, every partner, may restrain any other partner:

  1. from carrying on a similar business in the firm name or
  2. from using any of the property of the firm for his own benefit, until the affairs of the firm have been completely wound up. However, it will not affect the right of any partner or his representative who has bought the goodwill of the firm to use the firm name.

b. Liabilities of partners on dissolution
1. Continuing liability until public notice (Sec. 45)
If a public notice is not given of the dissolution of a firm the partners continue to be liable to third parties for any act done by any of them after dissolution.

2. Liability for continuing authority of Partners for purpose of winding up (Sec. 47)
After the dissolution of the firm, the partners continue to be liable for acts done to wind up the affairs of the firm and to complete transactions begun but unfinished at the time of the dissolution.

Mode of settlement of accounts after dissolution
The partners may lay down their own procedure for the settlement of accounts after dissolution. In the absence of a prior agreement between the partners in this regard, the accounts may be settled in accordance with the provisions provided in sections 48, 49 and 55 of the Indian Partnership Act which are discussed below:

  1. Goodwill shall be included in the assets and it might be sold separately or along with other property of the firm.
  2. Losses, including deficiencies of capital, shall be paid first out of profits, next out of capital, and lastly, for the balance, the partners shall individually subscribe in their profit sharing ratio.
  3. Assets of the firm, including partners’ contributions to make deficiencies of capital, shall be applied firstly, for paying the debts of the firm to third parties, secondly if there remains any surplus, it shall be utilized in paying each partner the amount of advances given to the firm. Such payments are made in the ratio of advances made by the partners. For example, if X gives an advance of Rs. 50,000 and Y of Rs. 60,000, then the ratio of payment shall be 5:6. thirdly, if still there remains any surplus, it shall be utilized for paying each partner rateably on account of capital. For example, the capitals of X, Y and Z have been contributed for Rs. 6,00,000, Rs. 7,00,000 and Rs. 8,00,000 respectively. Here, the proportion of capital shall be 6:7:8. and finally, the residue to be divided amongst partners in their profit sharing ratio.
  4. In case one of the partners is insolvent and nothing is recoverable from him, then, the deficiency of such a partner is borne by the solvent partners in the ratio of their capitals in accordance with the rule in Garner v. Murray.
  5. Payment of firm debts and separate debts: According to Section 49 of Partnership Act, where there are debts of the firm as well as individual debts of the partners, then the following rules shall apply:
    1. The property of the firm shall be first utilized in payment of the debts of the firm; and if there remains any surplus, then the share of each partner in such surplus shall be applied in payment of his individual debts, or if there is no such individual debt then his share shall be paid to him.
    2. The individual property of any partner shall be applied first in the payment of his individual debts; and if there remains any surplus, it shall be utilized in the payment of the debts of the firm.

PUBLIC NOTICE (SEC. 72)
1. The Partnership Act requires that a public notice must be given in each of the following cases:

  1. On minor attaining majority
    A minor partner on becoming a major must give public notice of his intention to remain or not to remain a partner. [Sec. 30(5)]
  2. Retirement of a partner
    When a partner retires from the firm, he must give public notice to terminate further liability. [Sec. 32(3)]
  3. Expulsion of a partner
    When a partner is expelled from the partnership business he must give public notice to terminate further liability. [Sec. 33]
  4. Dissolution of the firm
    When a partnership firm is dissolved, the partners of the dissolved firm must give public notice to terminate further liability [Section 45(1)]

2. Mode of the Public Notice
According to Sec. 72 the Public Notice becomes effective when the following steps have been taken:

  1. The notice has been published in the Official Gazette.
  2. The notice has been published in at least one vernacular newspaper (Le. which is published in Indian language) circulating in the district where the concerned firm has its place or principal place of business.
  3. If the firm is registered, the notice has been sent to the Registrar of Firms.

3. Consequences of not giving public notice
(a) On minor attaining majority
If a minor is admitted to the benefits of partnership under Section 30 he has to give public notice within 6 months of his attaining majority or of his obtaining knowledge that he has been admitted to the benefits of partnership, whichever date is later. If he fails to give notice, that he has elected to become or not to become a partner in the firm, he shall become a partner in the firm on the expiry of the said 6 months and is liable as a partner of the firm.

(b) Retirement of a partner
If a retiring partner does not give a public notice of the retirement from the firm under section 32, he and the other partners shall continue to be liable as partners to third parties for any act done by any of them which would have been an act of the firm if done before the retirement.

(c) Expulsion of a partner
If in case of expulsion of a partner from the firm a public notice is not given, the expelled partner and the other partners shall continue to be liable to third parties dealing with the firm as in the case of a retired partner. [Section 33],

(d) Dissolution of the firm
If a public notice is not given on dissolution of a registered firm, the partners shall be liable to third persons of any act done by any of them which would have been an act of the firm if done before the dissolution (section 45). When public notice is given of the dissolution of a firm, no partner shall have authority to bind the firm except for certain specific purposes as given in Section 47. According to this section, after the dissolution of a firm, the authority of each partner to bind the firm and their mutual rights and obligations of the partners shall continue :

  1. so far as may be necessary wind up the affairs of the firm; and
  2. to complete transactions begun but unfinished at the time of the dissolution.

MULTIPLE CHOICE QUESTIONS:

1. A partner may not be expelled from the firm by any majority of partners unless:
(a) The terms of partnership agreement confer the power to expel a partner
(b) The expulsion is made by a majority of the partners of the firm
(c) The decision of expulsion is made by all the partners in good faith
(d) All the above.

2. Agreement in restraint of trade is void. But if an
outgoing partner agrees with the firm that he will not carry on any competing business, such an agreement will be valid if:
(a) Such restraint is in respect of carrying of any business similar to that of the firm
(b) Such agreement is made by the partners beforehand i.e. well in advance
(c) Such agreement is made without any specific reference to time period.
(d) Such agreement is made without reference to local limits.

3. A notice in writing by one partner must be given to all the partners of the firm in case of:
(a) Dissolution on the happening of contingencies
(b) Dissolution of partnership at will
(c) Dissolution by court
(d) Compulsory dissolution

4. A firm is compulsorily dissolved
(a) By adjudication of any partner of the firm as insolvent
(b) By the death of a partner
(c) By adjudication of all the partners or of all the partners but one is insolvent
(d) In any of the above circumstances

5. The partners authority to act for the firm and to bind their co-partners continues even after the dissolution of the firm:
(a) To wind up the affairs of the firm
(b) To complete the unfinished transactions
(c) Both of above
(d) None of the above.

6. Retiring partner continues to remain liable to third parties for acts of the firm :—
(a) Until public notice is given of the retirement.
(b) From the date of retirement
(c) Upto the close of the financial year in which he retires.
(d) So long as the firm uses his name.

7. A partner can be expelled from a firm
(a) If power to expel is conferred by express agreement.
(b) If the power is exercised in good faith.
(c) By majority of partners after giving opportunity of explanation.
(d) All of the above.

8. A retired partner may be liable
(a) For debts incurred before retirement.
(b) For debts incurred after retirement until public notice is given.
(c) Either (a) or (b)
(d) Both (a) and (b)

9. Which of the following conditions is not necessary for expulsion of a partner?
(a) The power of expulsion must be given in the partnership deed.
(b) Such power has been exercised by a majority of the partners.
(c) Such power has been exercised in good faith for the interest of the firm and not used as vengeance against a partner
(d) An FIR has been filed in the Police Station.

10. No public notice is required
(a) On the death of a partner.
( b) On minor attaining majority.
(c) Retirement of partner.
(d) Dissolution of firm.

11. An outgoing partner can carry on a competing business and also advertise such business. For this purpose, in the absence of contract to the contrary —
(a) He can use the firm’s name
(b) He cannot use the firm’s name
(c) He cannot represent himself as carrying on the business of the firm.
(d) Both (b) and (c).

12. If all partners, or all but one partner, of the firm are declared insolvent —
(a) Firm is automatically dissolved
(b) Firm becomes illegal association.
(c) Firm is also declared insolvent.
(d) Firm becomes illegal entity.

13. Dissolution of partnership between all the partners of a firm is called —
(a) Dissolution of partnership.
(.b) Dissolution of partners.
(c) Dissolution of the firm.
(d) Reconstitution of firm.

14. The accounting rule in respect of loss arising due to insolvency of a partner is dealt within
(a) Derry v. Peek
(b) Carlill v. Carbolic Smoke Ball Co.
(c) Garner v. Murray
(d) Chinnaiah v. Ramaiya.

15. While selling goodwill of the firm, the selling partners may agree with the buyer that they will not carry on similar business, within a specified period or within specified local limits. Such agreement in restraint of trade shall be :
(a) Valid, if the restrictions imposed are rea-sonable
(b) Valid (whether restrictions are reasonable or not)
(c) Void
(d) Voidable

16. Public Notice under the Partnership Act, is given in the following manner:
(a) Serving a copy of the Notice to the Registrar of firms
( b) Publishing the Notice in the Official Gazette
(c) Publishing the Notice in one vernacular newspaper circulating in the district where the firm’s principal place of business is situated
(d) All of the above.

Answers:
CA Foundation Business Laws Study Material Chapter 19 Reconstitution and Dissolution of Firm 1

STATE WHETHER THE FOLLOWING ARE TRUE OR FALSE:

1. A partner who has purchased the goodwill of the firm on dissolution of partnership has a right to make use of the firm’s name for earning profits.
2. All partners are not the joint owners of the property of the firm, unless otherwise provided in the agree-ment.
3. The test of existence of partnership is the element of sharing of profits rather than mutual agency.
4. Legal representatives are required to give public notice so as to avoid the liability of the deceased partner.
5. Permanent incapacity of a partner is not a ground for dissolution of partnership firm.
6. A firm can be held liable for all wrongful acts of a partner done in the ordinary course of partnership business.
7. A firm signifies the abstract legal relation of the partners.
8. Dissolution of firm automatically results in dissolution of partnership.
9. Partnership will get dissolved if all the partnership except one are declared insolvent.
10. Public notice is not necessary on a minor admitted to the benefits of partnership opting to become a partner in the firm.
11. Losses including deficiencies of capital are to be paid by the partners in the portion in which they were entitled to share the profits.
12. Losses including deficiencies of capital shall be first paid out of capital.
13. In settling the accounts of a firm after dissolution, the assets are first utilized in paying the debts of the firm to the third parties.
14. The term dissolution of partnership and dissolution of firm are synonymous.
15. Indian Partnership act imposes penalty for non-registration of the firm.
16. The assignee of a partner’s interest, will enjoy the right to receive the share of the profits of the assignor and receive the accounts of profits agreed to by other partners.
17. When a partner of a firm becomes lunatic, the firm dissolves automatically.

Answers:
CA Foundation Business Laws Study Material Chapter 19 Reconstitution and Dissolution of Firm 2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3

More Exercises

Question 1.
If A = \(\begin{bmatrix} 3 & \quad 5 \\ 4 & \quad -2 \end{bmatrix}\) and B = \(\left[ \begin{matrix} 2 \\ 4 \end{matrix} \right] \), is the product AB possible ? Give a reason. If yes, find AB.
Solution:
Yes, the product is possible because of
number of column in A = number of row in B
i.e., (2 x 2). (2 x 1) = (2 x 1) is the order of the matrix.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q1.1

Question 2.
If A = \(\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}\),B = \(\begin{bmatrix} 1 & -1 \\ -3 & 2 \end{bmatrix}\), find AB and BA, Is AB = BA ?
Solution:
A = \(\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}\),
B = \(\begin{bmatrix} 1 & -1 \\ -3 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q2.1

Question 3.
If P = \(\begin{bmatrix} 4 & 6 \\ 2 & -8 \end{bmatrix}\),Q = \(\begin{bmatrix} 2 & -3 \\ -1 & 1 \end{bmatrix}\)
Find 2PQ
Solution:
P = \(\begin{bmatrix} 4 & 6 \\ 2 & -8 \end{bmatrix}\),
Q = \(\begin{bmatrix} 2 & -3 \\ -1 & 1 \end{bmatrix}\)
\(2PQ=2\begin{bmatrix} 4 & \quad 6 \\ 2 & -8 \end{bmatrix}\times \begin{bmatrix} 2\quad & -3 \\ -1 & \quad 1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q3.1

Question 4.
Given A = \(\begin{bmatrix} 1 & 1 \\ 8 & 3 \end{bmatrix}\) , evaluate A² – 4A
Solution:
A = \(\begin{bmatrix} 1 & 1 \\ 8 & 3 \end{bmatrix}\)
A² – 4A = \(\begin{bmatrix} 1 & \quad 1 \\ 8 & \quad 3 \end{bmatrix}\begin{bmatrix} 1\quad & 1 \\ 8\quad & 3 \end{bmatrix}-4\begin{bmatrix} 1\quad & 1 \\ 8\quad & 3 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q4.1

Question 5.
If A = \(\begin{bmatrix} 3 & \quad 7 \\ 2 & \quad 4 \end{bmatrix}\), B = \(\begin{bmatrix} 0 & \quad 2 \\ 5 & \quad 3 \end{bmatrix}\) and C = \(\begin{bmatrix} 1 & \quad -5 \\ -4 & \quad 6 \end{bmatrix}\)
Find AB – 5C
Solution:
A = \(\begin{bmatrix} 3 & \quad 7 \\ 2 & \quad 4 \end{bmatrix}\), B = \(\begin{bmatrix} 0 & \quad 2 \\ 5 & \quad 3 \end{bmatrix}\) and C = \(\begin{bmatrix} 1 & \quad -5 \\ -4 & \quad 6 \end{bmatrix}\)
AB = \(\begin{bmatrix} 3 & \quad 7 \\ 2 & \quad 4 \end{bmatrix}\)\(\begin{bmatrix} 0 & \quad 2 \\ 5 & \quad 3 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q5.1

Question 6.
If A = \(\begin{bmatrix} 1 & \quad 2 \\ 2 & \quad 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 2 & \quad 1 \\ 1 & \quad 2 \end{bmatrix}\), find A(BA)
Solution:
A = \(\begin{bmatrix} 1 & \quad 2 \\ 2 & \quad 1 \end{bmatrix}\)
B = \(\begin{bmatrix} 2 & \quad 1 \\ 1 & \quad 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q6.1

Question 7.
Given matrices:
A = \(\begin{bmatrix} 2 & \quad 1 \\ 4 & \quad 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 3 & \quad 4 \\ -1 & \quad -2 \end{bmatrix}\), C = \(\begin{bmatrix} -3 & \quad 1 \\ 0 & \quad -2 \end{bmatrix}\)
Find the products of (i) ABC (ii) ACB and state whether they are equal.
Solution:
A = \(\begin{bmatrix} 2 & \quad 1 \\ 4 & \quad 2 \end{bmatrix}\)
B = \(\begin{bmatrix} 3 & \quad 4 \\ -1 & \quad -2 \end{bmatrix}\),
C = \(\begin{bmatrix} -3 & \quad 1 \\ 0 & \quad -2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q7.1

Question 8.
Evaluate : \(\begin{bmatrix} 4\sin { { 30 }^{ o } } & \quad 2cos{ 60 }^{ o } \\ sin{ 90 }^{ o } & \quad 2cos{ 0 }^{ o } \end{bmatrix}\begin{bmatrix} 4 & 5 \\ 5 & 4 \end{bmatrix}\)
Solution:
\(\begin{bmatrix} 4\sin { { 30 }^{ o } } & \quad 2cos{ 60 }^{ o } \\ sin{ 90 }^{ o } & \quad 2cos{ 0 }^{ o } \end{bmatrix}\begin{bmatrix} 4 & 5 \\ 5 & 4 \end{bmatrix}\)
\(sin{ 30 }^{ o }=\frac { 1 }{ 2 } ,cos{ 60 }^{ o }=\frac { 1 }{ 2 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q8.1

Question 9.
If A = \(\begin{bmatrix} -1 & \quad 3 \\ 2 & \quad 4 \end{bmatrix}\), B = \(\begin{bmatrix} 2 & \quad -3 \\ -4 & \quad -6 \end{bmatrix}\) find the matrix AB + BA
Solution:
A = \(\begin{bmatrix} -1 & \quad 3 \\ 2 & \quad 4 \end{bmatrix}\),
B = \(\begin{bmatrix} 2 & \quad -3 \\ -4 & \quad -6 \end{bmatrix}\)
\(AB=\begin{bmatrix} -1 & \quad 3 \\ 2 & \quad 4 \end{bmatrix}\times \begin{bmatrix} 2 & \quad -3 \\ -4 & \quad -6 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q9.1

Question 10.
A = \(\begin{bmatrix} 1 & \quad 2 \\ 3 & \quad 4 \end{bmatrix}\) and B = \(\begin{bmatrix} 6 & \quad 1 \\ 1 & \quad 1 \end{bmatrix}\), C = \(\begin{bmatrix} -2 & \quad -3 \\ 0 & \quad 1 \end{bmatrix}\)
find each of the following and state if they are equal.
(i) CA + B
(ii) A + CB
Solution:
(i) CA + B
CA = \(\begin{bmatrix} -2 & \quad -3 \\ 0 & \quad 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & \quad 2 \\ 3 & \quad 4 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q10.1

Question 11.
If A = \(\begin{bmatrix} 1 & -2 \\ 2 & -1 \end{bmatrix}\) and B = \(\begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}\)
Find 2B – A²
Solution:
A = \(\begin{bmatrix} 1 & -2 \\ 2 & -1 \end{bmatrix}\)
B = \(\begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}\)
2B = \(2\begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}\)
= \(\begin{bmatrix} 6 & 4 \\ -4 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q11.1

Question 12.
If A = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) and B = \(\begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix}\), C = \(\begin{bmatrix} 5 & 1 \\ 7 & 4 \end{bmatrix}\), compute
(i) A(B + C)
(ii) (B + C)A
Solution:
(i) A(B + C)
A = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\)
B = \(\begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix}\),
C = \(\begin{bmatrix} 5 & 1 \\ 7 & 4 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q12.1

Question 13.
If A = \(\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}\) and B = \(\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}\), C = \(\begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix}\)
find the matrix C(B – A)
Solution:
A = \(\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}\)
B = \(\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}\),
C = \(\begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q13.1

Question 14.
A = \(\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 2 & 3 \\ -1 & 0 \end{bmatrix}\)
Find A² + AB + B²
Solution:
Given that
A = \(\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}\)
B = \(\begin{bmatrix} 2 & 3 \\ -1 & 0 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q14.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q14.2

Question 15.
If A = \(\begin{bmatrix} 2 & 1 \\ 0 & -2 \end{bmatrix}\) and B = \(\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}\), C = \(\begin{bmatrix} -3 & 2 \\ -1 & 4 \end{bmatrix}\)
Find A² + AC – 5B
Solution:
A = \(\begin{bmatrix} 2 & 1 \\ 0 & -2 \end{bmatrix}\)
B = \(\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}\),
C = \(\begin{bmatrix} -3 & 2 \\ -1 & 4 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q15.1

Question 16.
If A = \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\), find A2 and A3.Also state that which of these is equal to A
Solution:
A = \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
A² = A x A = \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q16.1

Question 17.
If X = \(\begin{bmatrix} 4 & 1 \\ -1 & 2 \end{bmatrix}\), show that 6X – X² = 9I Where I is the unit matrix.
Solution:
Given that
X = \(\begin{bmatrix} 4 & 1 \\ -1 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q17.1

Question 18.
Show that \(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\) is a solution of the matrix equation X² – 2X – 3I = 0,Where I is the unit matrix of order 2
Solution:
Given
X² – 2X – 3I = 0
Solution = \(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\)
or
X = \(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\)
∴ X² = \(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q18.1

Question 19.
Find the matrix X of order 2 × 2 which satisfies the equation
\(\begin{bmatrix} 3 & 7 \\ 2 & 4 \end{bmatrix}\begin{bmatrix} 0 & 2 \\ 5 & 3 \end{bmatrix}+2X=\begin{bmatrix} 1 & -5 \\ -4 & 6 \end{bmatrix}\)
Solution:
Given
\(\begin{bmatrix} 3 & 7 \\ 2 & 4 \end{bmatrix}\begin{bmatrix} 0 & 2 \\ 5 & 3 \end{bmatrix}+2X=\begin{bmatrix} 1 & -5 \\ -4 & 6 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q19.1

Question 20.
If A = \(\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}\), find the value of x, so that A² – 0
Solution:
Given
A = \(\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}\)
A² = \(\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}\)\(\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q20.1

Question 21.
If \(\begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} x \\ 0 \end{matrix} \right] \) Find the value of x
Solution:
\(\begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} x \\ 0 \end{matrix} \right] \)
⇒ \(\begin{bmatrix} 2 & -3 \\ 0 & 0 \end{bmatrix}=\left[ \begin{matrix} x \\ 0 \end{matrix} \right] \)
⇒ \(\left[ \begin{matrix} -1 \\ 0 \end{matrix} \right] =\left[ \begin{matrix} x \\ 0 \end{matrix} \right] \)
Comparing the corresponding elements
x = -1

Question 22.
(i) Find x and y if \(\begin{bmatrix} -3 & 2 \\ 0 & -5 \end{bmatrix}\left[ \begin{matrix} x \\ 2 \end{matrix} \right] =\left[ \begin{matrix} -5 \\ y \end{matrix} \right] \)
(ii) Find x and y if \(\begin{bmatrix} 2x & x \\ y & 3y \end{bmatrix}\left[ \begin{matrix} 3 \\ 2 \end{matrix} \right] =\left[ \begin{matrix} 16 \\ 9 \end{matrix} \right] \)
Solution:
(i) \(\begin{bmatrix} -3 & 2 \\ 0 & -5 \end{bmatrix}\left[ \begin{matrix} x \\ 2 \end{matrix} \right] =\left[ \begin{matrix} -5 \\ y \end{matrix} \right] \)
⇒ \(\begin{bmatrix} -3x & 4 \\ 0 & -10 \end{bmatrix}=\left[ \begin{matrix} -5 \\ y \end{matrix} \right] \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q22.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q22.2
Here x = 2, y = 1

Question 23.
Find x and y if
\(\begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} 3 \\ 2 \end{matrix} \right] \)
Solution:
Given
\(\begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} 3 \\ 2 \end{matrix} \right] \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q23.1

Question 24.
If \(\begin{bmatrix} 1 & 2 \\ 3 & 3 \end{bmatrix}\begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix}=\begin{bmatrix} x & 0 \\ 9 & 0 \end{bmatrix} \) find the values of x and y
Solution:
Given
\(\begin{bmatrix} 1 & 2 \\ 3 & 3 \end{bmatrix}\begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix}=\begin{bmatrix} x & 0 \\ 9 & 0 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q24.1

Question 25.
If \(\begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) write down the values of a,b,c and d
Solution:
Given
\(\begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q25.1
Comparing the corresponding elements
a = 3, b = 4, c = 2, d = 5

Question 26.
Find the value of x given that A² = B
Where A = \(\begin{bmatrix} 2 & 12 \\ 0 & 1 \end{bmatrix}\) and
B = \(\begin{bmatrix} 4 & x \\ 0 & 1 \end{bmatrix}\)
Solution:
A = \(\begin{bmatrix} 2 & 12 \\ 0 & 1 \end{bmatrix}\) and
B = \(\begin{bmatrix} 4 & x \\ 0 & 1 \end{bmatrix}\)
A² = B
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q26.1

Question 27.
If A = \(\begin{bmatrix} 2 & x \\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 4 & 36 \\ 0 & 1 \end{bmatrix}\), find the value of x, given that A² – B
Solution:
Given
A² = \(\begin{bmatrix} 2 & x \\ 0 & 1 \end{bmatrix}\)\(\begin{bmatrix} 2 & x \\ 0 & 1 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q27.1

Question 28.
If A = \(\begin{bmatrix} 3 & x \\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 9 & 16 \\ 0 & -y \end{bmatrix}\) find x and y when A² = B
Solution:
Given
A = \(\begin{bmatrix} 3 & x \\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 9 & 16 \\ 0 & -y \end{bmatrix}\) find x and y when A² = B
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q28.1

Question 29.
Find x, y if \(\begin{bmatrix} -2 & 0 \\ 3 & 1 \end{bmatrix}\left[ \begin{matrix} -1 \\ 2x \end{matrix} \right] +3\left[ \begin{matrix} -2 \\ 1 \end{matrix} \right] =2\left[ \begin{matrix} y \\ 3 \end{matrix} \right] \)
Solution:
Given
\(\begin{bmatrix} -2 & 0 \\ 3 & 1 \end{bmatrix}\left[ \begin{matrix} -1 \\ 2x \end{matrix} \right] +3\left[ \begin{matrix} -2 \\ 1 \end{matrix} \right] =2\left[ \begin{matrix} y \\ 3 \end{matrix} \right] \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q29.1

Question 30.
If \(\begin{bmatrix} a & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 4 & 3 \\ -3 & 2 \end{bmatrix}=\begin{bmatrix} b & 11 \\ 4 & c \end{bmatrix} \) find a,b and c
Solution:
\(\begin{bmatrix} a & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 4 & 3 \\ -3 & 2 \end{bmatrix}=\begin{bmatrix} b & 11 \\ 4 & c \end{bmatrix} \)
⇒ \(\begin{bmatrix} 4a-3 & 3a+2 \\ 4+0 & 3+0 \end{bmatrix}=\begin{bmatrix} b & 11 \\ 4 & c \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q30.1

Question 31.
If A = \(\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}\) ,B = \(\begin{bmatrix} 2 & x \\ 0 & -\frac { 1 }{ 2 } \end{bmatrix} \) find the value of x if AB = BA
Solution:
Given
AB = \(\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}\)\(\begin{bmatrix} 2 & x \\ 0 & -\frac { 1 }{ 2 } \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q31.1

Question 32.
If A = \(\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}\) find x and y so that A² – xA + yI
Solution:
Given
A² = \(\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}\)\(\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q32.1

Question 33.
If P = \(\begin{bmatrix} 2 & 6 \\ 3 & 9 \end{bmatrix}\), Q = \(\begin{bmatrix} 3 & x \\ y & 2 \end{bmatrix}\)
find x and y such that PQ = 0
Solution:
Given
P = \(\begin{bmatrix} 2 & 6 \\ 3 & 9 \end{bmatrix}\),
Q = \(\begin{bmatrix} 3 & x \\ y & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q33.1

Question 34.
Let \(M\times \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix}=\left[ \begin{matrix} 1 & 2 \end{matrix} \right] \) where M is a matrix
(i) State the order of matrix M
(ii) Find the matrix M
Solution:
Given
(i) M is the order of 1 x 2
let M = [x y]
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q34.1

Question 35.
Given \(\begin{bmatrix} 2 & 1 \\ -3 & 4 \end{bmatrix}\) ,X = \(\left[ \begin{matrix} 7 \\ 6 \end{matrix} \right] \)
(i) the order of the matrix X
(ii) the matrix X
Solution:
We have
\(\begin{bmatrix} 2 & 1 \\ -3 & 4 \end{bmatrix}\) , X = \(\left[ \begin{matrix} 7 \\ 6 \end{matrix} \right] \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q35.1

Question 36.
Solve the matrix equation : \(\left[ \begin{matrix} 4 \\ 1 \end{matrix} \right] \) ,X = \(\begin{bmatrix} -4 & 8 \\ -1 & 2 \end{bmatrix}\)
Solution:
\(\left[ \begin{matrix} 4 \\ 1 \end{matrix} \right] \) , X = \(\begin{bmatrix} -4 & 8 \\ -1 & 2 \end{bmatrix}\)
Let matrix X = [x y]
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q36.1

Question 37.
(i) If A = \(\begin{bmatrix} 2 & -1 \\ -4 & 5 \end{bmatrix}\) and B = \(\left[ \begin{matrix} -3 \\ 2 \end{matrix} \right] \) find the matrix C such that AC = B
(ii) If A = \(\begin{bmatrix} 2 & -1 \\ -4 & 5 \end{bmatrix}\) and B = [0 -3] find the matrix C such that CA = B
Solution:
(i) given
A = \(\begin{bmatrix} 2 & -1 \\ -4 & 5 \end{bmatrix}\)
B = \(\left[ \begin{matrix} -3 \\ 2 \end{matrix} \right] \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q37.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q37.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q37.3

Question 38.
If A = \(\begin{bmatrix} 3 & -4 \\ -1 & 2 \end{bmatrix}\) , find matrix B such that BA = I,where I is unity matrix of order 2
Solution:
A = \(\begin{bmatrix} 3 & -4 \\ -1 & 2 \end{bmatrix}\)
BA = I, where I is unity matrix of order 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q38.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q38.2

Question 39.
If B = \(\begin{bmatrix} -4 & 2 \\ 5 & -1 \end{bmatrix}\) and C = \(\begin{bmatrix} 17 & -1 \\ 47 & -13 \end{bmatrix}\)
find the matrix A such that AB = C
Solution:
B = \(\begin{bmatrix} -4 & 2 \\ 5 & -1 \end{bmatrix}\)
C = \(\begin{bmatrix} 17 & -1 \\ 47 & -13 \end{bmatrix}\)
and AB = C
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q39.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 Q39.2

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1

More Exercises

Question 1.
Find the slope of a line whose inclination is
(i) 45°
(ii) 30°
Solution:
(i) tan 45° = 1
(ii) tan 30° = \(\frac { 1 }{ \sqrt { 3 } } \)

Question 2.
Find the inclination of a line whose gradient is
(i) 1
(ii) √3
(iii) \(\frac { 1 }{ \sqrt { 3 } } \)
Solution:
(i) tan θ = 1 ⇒ θ = 45°
(ii) tan θ = √3 ⇒ θ = 60°
(iii) tan θ = \(\frac { 1 }{ \sqrt { 3 } } \) ⇒ θ = 30°

Question 3.
Find the equation of a straight line parallel 1 to x-axis which is at a distance
(i) 2 units above it
(ii) 3 units below it.
Solution:
(i) A line which is parallel to x-axis is y = a
⇒ y = 2
⇒ y – 2 = 0
(ii) A line which is parallel to x-axis is y = a
⇒ y = -3
⇒ y + 3 = 0

Question 4.
Find the equation of a straight line parallel to y-axis which is at a distance of:
(i) 3 units to the right
(ii) 2 units to the left.
Solution:
(i) The equation of line parallel to y-axis is at a distance of 3 units to the right is x = 3 ⇒ x – 3 = 0
(ii) The equation of line parallel to y-axis at a distance of 2 units to the left is x = -2 ⇒ x + 2 = 0

Question 5.
Find the equation of a straight line parallel to y-axis and passing through the point ( – 3, 5).
Solution:
The equation of the line parallel to y-axis passing through ( – 3, 5) to x = -3
⇒ x + 3 = 0

Question 6.
Find the equation of the a line whose
(i) slope = 3, y-intercept = – 5
(ii) slope = \(– \frac { 2 }{ 7 } \), y-intercept = 3
(iii) gradient = √3, y-intercept = \(– \frac { 4 }{ 3 } \)
(iv) inclination = 30°,y-intercept = 2
Solution:
Equation of a line whose slope and y-intercept is given is
y = mx + c
where m is the slope and c is the y-intercept
(i) y = mx + c
⇒ y = 3x + (-5)
⇒ y = 3x – 5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q6.1

Question 7.
Find the slope and y-intercept of the following lines:
(i) x – 2y – 1 = 0
(ii) 4x – 5y – 9 = – 0
(iii) 3x +5y + 7 = 0
(iv) \(\frac { x }{ 3 } +\frac { y }{ 4 } =1\)
(v) y – 3 = 0
(vi) x – 3 = 0
Solution:
We know that in the equation
y = mx + c, m is the slope and c is the y-intercept.
Now using this, we find,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q7.2

Question 8.
The equation of the line PQ is 3y – 3x + 7 = 0
(i) Write down the slope of the line PQ.
(ii) Calculate the angle that the line PQ makes with the positive direction of x-axis.
Solution:
Equation of line PQ is 3y – 3x + 7 = 0
Writing in form of y = mx + c
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q8.1

Question 9.
The given figure represents the line y = x + 1 and y = √3x – 1. Write down the angles which the lines make with the positive direction of the x-axis. Hence determine θ.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q9.1
Solution:
Slope of the line y = x + 1 after comparing
it with y = mx + c, m = 1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q9.2

Question 10.
Find the value of p, given that the line \(\frac { y }{ 2 } =x-p\) passes through the point ( – 4, 4) (1992).
Solution:
Equation of line is \(\frac { y }{ 2 } =x-p\)
It passes through the points (-4, 4)
It will satisfy the equation
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q10.1

Question 11.
Given that (a, 2a) lies on the line \(\frac { y }{ 2 } =3x-6\) find the value of a.
Solution:
∵ Point (a, 2a) lies on the line
\(\frac { y }{ 2 } =3x-6\)
∴this point will satisfy the equation
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q11.1

Question 12.
The graph of the equation y = mx + c passes through the points (1, 4) and ( – 2, – 5). Determine the values of m and c.
Solution:
Equation of the line is y = mx + c
∴ it passes through the points (1, 4)
∴ 4 = m x 1 + c
⇒ 4 = m + c
⇒ m + c = 4 … (i)
Again it passes through the point (-2, -5)
∴ 5 = m (-2) + c
⇒ 5 = -2 m + c
⇒ 2m – c = 5 …(ii)
Adding (i) and (ii)
3m = 9
⇒ m = 3
Substituting the value of m in (i)
3 + c = 4
⇒ c = 4 – 3 = 1
Hence m = 3, c = 1

Question 13.
Find the equation of the line passing through the point (2, – 5) and making an intercept of – 3 on the y-axis.
Solution:
∴ The line intersects y-axis making an intercept of -3
∴ the co-ordinates of point of intersection will be (0, -3)
Now the slope of line (m) = \(\frac { { y }_{ 2 }-{ y }_{ 1 } }{ { x }_{ 2 }-{ x }_{ 1 } } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q13.1

Question 14.
Find the equation of a straight line passing through ( – 1, 2) and whose slope is \(\\ \frac { 2 }{ 5 } \).
Solution:
Equation of the line will be
\(y-{ y }_{ 1 }=m(x-{ x }_{ 1 }) \)
\(y-2=\frac { 2 }{ 5 } (x+1)\)
⇒ 5y – 10 = 2x + 2
⇒ 2x – 5y + 2 + 10 = 0
⇒ 2x – 5y + 12 = 0

Question 15.
Find the equation of a straight line whose inclination is 60° and which passes through the point (0, – 3).
Solution:
The equation of line whose slope is wand passes through a given point is
\(y-{ y }_{ 1 }=m(x-{ x }_{ 1 }) \)
Here m = tan 60° = √3 and point is (0, -3)
∴ y + 3 = √3 (x – 0)
⇒ y + 3 = √3x
⇒ √3x – y – 3 = 0

Question 16.
Find the gradient of a line passing through the following pairs of points.
(i) (0, – 2), (3, 4)
(ii) (3, – 7), ( – 1, 8)
Solution:
m = \(\frac { { y }_{ 2 }-{ y }_{ 1 } }{ { x }_{ 2 }-{ x }_{ 1 } } \)
Given
(i) (0, -2), (3, 4)
(ii) (3, -7), (-1, 8)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q16.1

Question 17.
The coordinates of two points E and F are (0, 4) and (3, 7) respectively. Find :
(i) The gradient of EF
(ii) The equation of EF
(iii) The coordinates of the point where the line EF intersects the x-axis.
Solution:
Co-ordinates of points E (0, 4) and F (3, 7) are given, then
(i) The gradient of EF
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q17.1

Question 18.
Find the intercepts made by the line 2x – 3y + 12 = 0 on the co-ordinate axis.
Solution:
Putting y = 0, we will get the intercept made on x-axis,
2x – 3y + 12 = 0
⇒ 2x – 3 × 0 + 12 = 0
⇒ 2x – 0 + 2 = 0
⇒ 2x = -12
⇒ x = -6
and putting x = 0, we get the intercepts made on y-axis,
2x – 3y + 12 = 0
⇒ 2 × 0 – 3y + 12 = 0
⇒ -3y = -12
⇒ \(y= \frac { -12 }{ -3 } \) = 4

Question 19.
Find the equation of the line passing through the points P (5, 1) and Q (1, – 1). Hence, show that the points P, Q and R (11, 4) are collinear.
Solution:
The two given points are P (5, 1), Q(1, -1).
∴ Slope of the line (m)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q19.1

Question 20.
Find the value of ‘a’ for which the following points A (a, 3), B (2,1) and C (5, a) are collinear. Hence find the equation of the line.
Solution:
Given That
A(a, 3), B (2, 1) and C (5, a) are collinear.
Slope of AB = Slope of BC
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q20.1

Question 21.
Use a graph paper for this question. The graph of a linear equation in x and y, passes through A ( – 1, – 1) and B (2, 5). From your graph, find the values of h and k, if the line passes through (h, 4) and (½, k). (2005)
Solution:
Points (h, 4) and (½, k) lie on the line passing
through A(-1, -1) and B(2, 5)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q21.1

Question 22.
ABCD is a parallelogram where A (x, y), B (5, 8), C (4, 7) and D (2, – 4). Find
(i) the coordinates of A
(ii) the equation of the diagonal BD.
Solution:
Given that
ABCD is a parallelogram where A (x, y), B (5, 8), C (4, 7) and D (2, -4)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q22.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q22.2

Question 23.
In ∆ABC, A (3, 5), B (7, 8) and C (1, – 10). Find the equation of the median through A.
Solution:
AD is median
⇒ D is mid point of BC
∴ D is \(\left( \frac { 7+1 }{ 2 } ,\frac { 8-10 }{ 2 } \right) \)
i.e (4, -1)
slope of AD
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q23.1

Question 24.
Find the equation of a line passing through the point ( – 2, 3) and having x-intercept 4 units. (2002)
Solution:
x-intercept = 4
∴ Co-ordinates of the point will be (4, 0)
Now slope of the line passing through the points (-2, 3) and (4, 0)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q24.1

Question 25.
Find the equation of the line whose x-intercept is 6 and y-intercept is – 4.
Solution:
x-intercept = 6
∴ The line will pass through the point (6, 0)
y -intercept = -4 ⇒ c = -4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q25.1

Question 26.
Write down the equation of the line whose gradient is \(\\ \frac { 1 }{ 2 } \) and which passes through P where P divides the line segment joining A ( – 2, 6) and B (3, – 4) in the ratio 2 : 3. (2001)
Solution:
P divides the line segment joining the points
A (-2, 6) and (3, -4) in the ratio 2 : 3
∴ Co-ordinates of P will be
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q26.1

Question 27.
Find the equation of the line passing through the point (1, 4) and intersecting the line x – 2y – 11 = 0 on the y-axis.
Solution:
line x – 2y – 11 = 0 passes through y-axis
x = 0,
Now substituting the value of x in the equation x – 2y – 11 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q27.1

Question 28.
Find the equation of the straight line containing the point (3, 2) and making positive equal intercepts on axes.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q28.1
Let the line containing the point P (3, 2)
passes through x-axis at A (x, 0) and y-axis at B (0, y)
OA = OB given
∴ x = y
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q28.2

Question 29.
Three vertices of a parallelogram ABCD taken in order are A (3, 6), B (5, 10) and C (3, 2) find:
(i) the coordinates of the fourth vertex D.
(ii) length of diagonal BD.
(iii) equation of side AB of the parallelogram ABCD. (2015)
Solution:
Three vertices of a parallelogram ABCD taken in order are
A (3, 6), B (5, 10) and C (3, 2)
(i) We need to find the co-ordinates of D
We know that the diagonals of a parallelogram bisect each other
Let (x, y) be the co-ordinates of D
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q29.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q29.2

Question 30.
A and B arc two points on the x-axis and y-axis respectively. P (2, – 3) is the mid point of AB. Find the
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q30.1
(i) the co-ordinates of A and B.
(ii) the slope of the line AB.
(iii) the equation of the line AB. (2010)
Solution:
Points A and B are on x-axis and y-axis respectively
Let co-ordinates of A be (X, O) and of B be (O, Y)
P (2, -3) is the midpoint of AB
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q30.2

Question 31.
Find the equations of the diagonals of a rectangle whose sides are x = – 1, x = 2 , y = – 2 and y = 6.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q31.1
Solution:
The equations of sides of a rectangle whose equations are
x1 = -1, x2 = 2, y1 = -2, y2 = 6.
These lines form a rectangle when they intersect at A, B, C, D respectively
Co-ordinates of A, B, C and D will be
(-1, -2), (2, -2), (2, 6) and (-1, 6) respectively.
AC and BD are its diagonals
(i) Slope of the diagonal AC
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q31.2

Question 32.
Find the equation of a straight line passing through the origin and through the point of intersection of the lines 5x + 1y – 3 and 2x – 3y = 7
Solution:
5x + 7y = 3 …(i)
2x – 3 y = 7 …(ii)
Multiply (i) by 3 and (ii) by 7,
15x + 21y = 9
14x – 21y = 49
Adding we get,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q32.1

Question 33.
Point A (3, – 2) on reflection in the x-axis is mapped as A’ and point B on reflection in the y-axis is mapped onto B’ ( – 4, 3).
(i) Write down the co-ordinates of A’ and B.
(ii) Find the slope of the line A’B, hence find its inclination.
Solution:
A’ is the image of A (3, -2) on reflection in the x-axis.
∴ Co-ordinates of A’ will be (3, 2)
Again B’ (- 4, 3) in the image of A’, when reflected in the y-axis
∴ Co-ordinates of B will be (4, 3)
(ii) Slope of the line joining, the points A’ (3, 2) and B (4, 3)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 Q33.1

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.1 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

CA Foundation Business Laws Study Material Chapter 18 Relations of Partners 170

CA Foundation Business Laws Study Material Chapter 18 Relations of Partners 170

RIGHTS & DUTIES OF PARTNERS (SECS. 9 TO 17)

The mutual rights & duties of the partners are usually governed by the agreement between them. Where there is no specific agreement, their relations to one another are governed by Secs. 9 to 17 of the Partnership Act.
a. Rights of partners
Subject to contract between the partners, the Partnership Act confers the following rights upon the partners of a firm:

1. Right to take part in the conduct of the business [Sec.12(a)]
Every partner has a right to take part in the conduct of the business of the firm. The partners among themselves, may agree to entrust the work of management to one or more of them and they may even agree to make payment to such partners by way of an extra remuneration.

2. Right to be consulted[Sec.12(c)]
Every partner has a right to be consulted and heard before any matter is decided. Ordinary matters may be decided by majority opinion but matters of fundamental nature would require unanimity.
The matters which are to be decided by unanimous consent of all the partners are discussed below :

  1. Nature of business [Sec. 12]:
    No change can be made in nature of the business without the consent of all the partners.
  2. Admission of a partner [Sec. 31(1)] :
    A person can be admitted as a partner, only with the consent of all the existing partners.
  3. Transfer by a partner of his interest in the firm. (sec. 29):
    A partner can transfer his share in the firm to a third person with the consent of all other partners.
  4. Admission of a minor to the benefits of partnership [Sec. 30(1)]:
    A minor is incompetent to contract and, therefore a contract of partnership cannot be entered into with a minor. However, he can be admitted to the benefits of an existing partnership firm provided all the partners consent to it.

3. Right to access to books [Sec.12(d)]
Every partner has a right to have access, to inspect and copy any of the records and books of the firm.

4. Right to share the profits [Sec. 13(b)]
Every partner has right to share equally in the profits earned and to contribute equally to the losses sustained by the firm. This provision is irrespective of the amount of capital j contribution made or business expertise offered. However, they may agree to share the profits in some other ratio.
5. Right to interest on Capital [Sec. 13(c)]
Every partner has right to interest on capital, if so agreed, out of profits only.

6. Right to interest on advances [Sec. 13(d)]
A Partner is entitled to receive interest at 6 % p.a. on any advance, in excess of the agreed amount of capital, made for the purposes of the business.

7. Right to indemnity [Sec. 13(e)]
Every partner has a right to claim indemnity from the firm in respect of payments made or liabilities incurred by him

  1. in the ordinary and proper conduct of the business, and
  2. in doing such act, in an emergency, for the purpose of protecting the firm from loss, as would be done by a person of ordinary prudence, in his own case, under similar circumstances.

8. Right to prevent the introduction of new partner [Sec. 31(1)]
Every partner is entitled to prevent the admission of a new partner into the firm.

9. Right to retire [Sec.32(1)]
A partner to retire from the firm

  1. with the consent of all other partners, or
  2. in accordance with the terms of the deed, or
  3. by giving a notice to all other partners, of his intention to retire.

10. Right not to be expelled [Sec.33]
Every partner has right to continue in the partnership and not to be expelled from the firm.

11. Right to carry on competing business after retirement [sec.36(1)]
Every outgoing partner has a right to carry on a competitive business under certain conditions.

12. Right to dissolve the firm (sec. 43)
Where the partnership is at will, the firm may dissolve by any partner giving notice in writing to all the other partners of his intention to dissolve the firm.

b. Duties of Partners
1. General Duties of Partners:
Section 9 of Partnership Act lays down that all the partners are bound:

  1. to carry on the business of the firm to the greatest common advantage,
  2. to be just and faithful to each other, and
  3. to render to any partner or his legal representative the true accounts and
  4. to render full information of all things affecting the firm.

2. Duty to indemnify for loss caused by fraud:
According to Section 10 of Partnership Act, every partner shall indemnify (reimburse or pay back) the firm for any loss caused to it by his fraud in the conduct of the business of the firm.

3. Duty to attend diligently to his duties[Sec. 12 (b)]:
Every partner is bound to attend diligently to his duties in the conduct of the business.

4. Duty to work without remuneration [Sec. 13(a)]:
A partner is normally not entitled to receive any remuneration for taking part in the business of the firm. However if the partnership agreement provides or business custom allows, a partner can be given remuneration.

5. Duty to contribute to the losses [(Sec. 13(b)]:
The partners shall contribute equally to the losses sustained by the firm without regard to the capital contribution made by the firm.

6. Duty to indemnify for wilful neglect [Sec. 13 (f)]:
A partner shall indemnify the firm for any loss caused to it by his wilful neglect in the conduct of the business of the firm.

7. Duty to use firm’s property exclusively by for the firm. (Sec. 15):
Subject to contract between the partners, the property of the firm shall be held exclu¬sively for the purposes of the business of the firm.

8. Duty to account for personal profits derived [Sec. 16(a)]:
If a partner derives any profit for himself from any transaction of the firm, or from the use of the property or business connection of the firm in the firm’s name, he shall account for that profit and pay it to the firm.

9. Duty not to compete with the business of the firm [Sec. 16 (b)]:
No partner can carry on a business which is competing with that of the firm without the consent of the other partners, otherwise the partner carrying on such a business will have to account for and pay to the firm all profits made by him in that business.

10. Not to assign (transfer) his interest in the firm (sec. 29):
It is the duty of a partner not to assign his interest in the firm to a stranger (outsider) without the consent of all other partners.

c. Mutual rights and duties of partners—

  1. after a change in the constitution of the firm,
  2. after the expiry of the term of the firm, and
  3. where additional undertakings or adventures are carried out.

(1) Rights and duties of partners after a change occurs in the constitution of the firm. Subject to contract between the partners, where a change occurs in the constitution of the firm, the mutual rights and duties of the partners in the reconstituted firm remain the same as they were immediately before the change, as far as may be [Sec. 17(a)],
(2) Rights and duties of the partners after the expiry of the term of the firm. Subject
to contract between the partners, where a firm constituted for a fixed term con¬tinues to carry on business after the expiry of that term, the mutual rights and duties of the partners remain the same as they were before the expiry, of the fixed term [Sec. 17(b)]. ’
(3) Rights and duties of partners where additional undertakings or adventures are carried out. Subject to contract between the partners, where a firm constituted to carry out one or more adventures or undertakings carries out other adventures or undertakings, the mutual rights and duties of the partners in respect of the other adventures or undertakings are the same as those in respect of the original adventures or undertakings [Sec. 17(c)].

Relation of Partners with third parties: [Secs. 18 to 30]

PARTNERS AS AGENTS

The law of partnership is an extension of the law of agency. This is evident from the concluding portion of the definition of partnership which says that the business may be carried on “by all or any all of them acting for all.” “This clearly establishes the implied agency, the partner conducting the affairs of the business is considered as agent of the remaining partners. Sec. 18 of the Partnership Act provides, “Subject to the provisions of this Act, a partner is the agent of the firm for the purposes of the business of the firm”.
In carrying on the business of the firm, partners act as agent as well as principals. While the relation between the partners inter se is that of principals, they are agents of one another in relation to third parties for purposes of business of the firm. Every partner has a two-fold character, he is an agent of the other partners (because other partners are bound by his acts) and also he himself is the principal (because he is bound by the acts of other partners). The liability of one partner for the acts of his co-partners is in truth the liability of a principal for the acts of his agent. This concept of mutual agency is, in fact, the true test of the existence of partnership.
The acts of the partner in the usual course of the business, bind the firm unless:

  1. The partner so acting has no authority to act for the firm in that matter; and
  2. The person with whom he is dealing knows that he has no authority; or
  3. Does not know or believe him to be a partner.

IMPLIED AUTHORITY OF PARTNER
a. Meaning of implied authority
Meaning of Implied Authority. The authority of a partner means the capacity of a partner to bind the firm by his act. This authority may be express or implied. Where the authority to a partner to act is expressly conferred by an agreement it is called express authority. It is implied when the law presumes certain powers exercisable by every partner unless negatived by a contract to the contrary. Sections 19(1) and 22 deal with the implied authority of a partner.

According to Sec. 19(1) of the Act, “the act of a partner which is done to carry on, in the usual way business of the kind carried on by the firm. ” is called Implied Authority of partner. It is subject to the following 3 conditions:

  1. The act done by the partner must relate to the normal business of the firm and must be within the scope of the business of the firm. For example, if the partner of a firm dealing in electronic goods, purchases some wine in the name of the firm, the firm would not be liable.
  2. The act must be done in the usual way Le., in the normal course. What is usual way of carrying on the business, will depend on the nature of the business, customs and usages in that kind of business and circumstances of each Particular case.
  3. The act must be done in the name of the firm, or in any other manner expressing or implying an intention to bind the firm (Sec. 22).

Examples of Implied Authority : The implied authority of a Partner shall usually include general powers of partners as agents of the firm. If the partnership is of general nature, the implied authority of a partner shall include the following acts :

  1. Buy or sell goods on account of firm,
  2. to borrow money,
  3. to employ or engage servants,
  4. settle accounts with third parties,
  5. receive payments of debts „ present accounts to creditors
  6. engage lawyer to defend the actions brought against the firm
  7. to draw negotiable instruments & cheques on name of firm. The implied authority of partners may differ from business to business.

b. Limitation on Implied authority or Statutory Restrictions on Implied Authority [Sec. 19(2)]
Sec. 19(2) contains the list of acts regarding which a partner does not have an implied authority unless there is usage or custom or contract to the contrary. Accordingly, a partner cannot:

  1. submit a dispute relating to the business of the firm to arbitration;
  2. open a banking account on behalf of the firm in his own name;
  3. compromise or relinquish any claim or portion of a claim by the firm;
  4. withdraw a suit or proceeding filed on behalf of the firm;
  5. admit any liability in a suit or proceeding against the firm;
  6. acquire immovable property on behalf of the firm;
  7. transfer immovable property belonging to the firm; or
  8. enter into partnership on behalf of the firm.

A partner can do any of the above thing if:

  1. he has specific or express authority of the partners or
  2. the usage or custom of trade permits him.

Illustrations

  1. A being one of a firm of solicitors and attorneys, draws a bill of exchange in the name of the firm without authority. The other partners are not liable on the bill, for it is no part of the ordinary business of a solicitor to draw, accept, or endorse bills of exchange.
  2. A and B carry on business in partnership as bankers. A sum of money is received by A on behalf of the firm. A does not inform B of such receipt, afterwards A appropriates the money to his own use. The partnership is liable to make good the money.
  3. A and B are partners. A with the intention of cheating B goes to a shop and purchases articles on behalf of the firm, such as might be used in the ordinary course of the partnership business, and converts them to his own separate use, there being no collusion between him and the seller. The firm is liable for the price of the goods.

EXTENT OF LIABILITY OF PARTNERS

1. Liability of a partner for acts of the firm. (Sec 25)
Every partner is liable jointly with all the other partners and also severally for all acts of the firm done while he is a. partner. A creditor can sue the partners jointly as well as separately and successively. For example when the partnership firm incurred liability to telephone dept., it became the liability of all the partners. The department is competent to disconnect the personal telephone line of the partner to meet the liability of the firm to the dept.

Every partner is liable, to an unlimited extent, for all debts due to third parties from the firm incurred while he was a partner. As between the partners, the liability is adjustable according to the terms of the partnership agreement. Thus if a partner is entitled to receive !4th share of the losses. The accounts between the partners will be adjusted on this basis. But a third party, who is a creditor of the firm, is entitled to realise the whole of his claim from any one of the partners.

An Act of a firm means any act or omission by all the partners or by any partner or agent of the firm which gives rise to right enforceable by or against the firm. Sec. 2(a). Illustration : Suppose R and S are partners. R enters into contract with X. Now if X commits breach of contract, the firm can sue X. Similarly, if the firm commits a breach of contract, X can sue the firm. The contract between R on behalf of the firm and the third party X would be an act of the firm.

2. Liability of firm for wrongful acts of partners (Sec. 26)
Where, by the wrongful act or omission of a partner (a) acting in the ordinary course of the business of a firm, or (b) with authority of his partner, loss or injury is caused to any third party, or any penalty is incurred, the firm is liable therefore to the same extent as the partner.

3. Liability of firm for misapplication by Partners. (Sec. 27)
Where –

  1. a Partner acting within in his apparent authority receives money or property from a third party and misapplies it or
  2. a firm in the course of its business receives money or property from a third Party, and the money or property is misapplied by any of the partners while it is in the custody of the firm, the firm is liable to make good the loss.

Example: P, Q and R carry on a partnership business. M, a debtor of the firm, repays his debt of ? 50,000 to P who does not inform Q and R about the repayment and purchases a television for the members of his family. Here, M is discharged from the debt after making payment of ? 50,000 to P. ”

4. The law relating to the liability of the estate of a deceased partner
Section 35 of the Indian Partnership Act provides that “where under a contract between the partners, the estate of the deceased partner is not liable for any act of the firm done after his death.” Thus, a deceased partner’s estate is not liable to third parties for what may be done after his death by the surviving partners.
Proviso to Sec. 45 lays down in identical rule applicable to a case where the death of a partner has caused dissolution of the firm.
No public notice is required in the case of death in order to absolve the estate of the deceased from future obligations of the firm.
Suppose, the surviving partners borrow money to pay for and take delivery of the goods ordered by the firm during the life-time of the deceased partner. In such a case, the latter’s estate is not * liable for the debt. The creditor can have only a personal decree against the surviving partners and a decree against the partnership assets in the hands of those partners. A suit for goods sold and delivered would not lie against the representatives of the deceased partner. This is because there was no debt due in respect of the goods during the life time of the deceased.

MINOR AS A PARTNER

Admission of a minor into the benefits of the firm: According to Sec. 11 of the Indian Contract Act, an agreement by or with a minor is void. As such, he is incapable of entering into a contract of partnership. But with consent of all the partner^ for the time being, a minor may be admitted to the benefits of partnership [Sec. 30(1)]. This provision is based on the rule that a minor cannot be a promiser, but he can be a promisee or a beneficiary. It should, however, be noted that a new partnership cannot be formed with a partner who is a minor. Also, there cannot be partnership of minors among themselves as they are incapable of entering into a contract.

The position of a minor partner may be studied, under two heads:
1. Position before attaining majority

  1. Rights:
    1. He has a right to such share of the property and of profits of the firm as may have been agreed upon.
    2. He has also a right to have access to and inspect and copy any of the accounts, but not books of the firm. [Sec. 30(2)]
    3. When he is not given his due share of profit, he has a right to file a suit for his share of property of the firm. But he can do so only, if he wants to sever his connection with the firm. [Sec. 30(4)].
  2. Liabilities: The liability of the minor partner is confined only to the extent of his share in the profits and property of the firm. Over and above this, he is neither personally liable nor is his private estate liable.

2. Position on attaining majority
On attaining majority, the minor partner has to decide within 6 months whether he shall continue in the firm or leave it. Within this period he should give a public notice of his choice:

  1. to become, or
  2. not to become, a partner in the firm.

If he fails to give a public notice he is deemed to have become a partner in the firm on the expiry of the said six months [Sec. 30(5)]
Where he elects to become a partner:

  1. He becomes personally liable to third par ties for all acts of the firm done since he was admitted to the benefits of partnership;
  2. His share in the property and profits oi the firm is the share to which he was entitled as a minor partner [Sec. 30(7)].

Where he elects not to become a partner:

  1. His rights & liabilities continue to be those of a minor upto the date of the public notice;
  2. His share is not liable for any acts of the firm done after the date of the public notice;
  3. He is entitled to sue the partners for his share of the property and profits in the firm [Sec. 30(8)]

MULTIPLE CHOICE QUESTIONS:

1. As per section 18, a partner in a partnership firm functions :
(a) In a dual capacity of principal and agent.
(b) As a principal.
(c) As an agent.
(d) Neither as a principal nor as an agent.

2. If a partner commits fraud in the conduct of the business of the firm:
(a) He shall indemnify the firm for any loss caused to it by his fraud.
(b) He is not liable to the firm.
(c) He is liable to the partners.
(d) He is liable to the third parties.

3. Partners are bound to carry on the business of the firm —
(a) To the greatest common advantage.
(b) For the welfare to the society.
(c) For the advantage of the family members.
(d) For earning’personal profits.

4. Which are the matters that require unanimous consent of all the partners:
(a) Admission of a partner.
(b) Transfer by a partner of his interest in the firm.
(c) Fundamental change in the nature of the business.
(d) All the above.

5. The liability of a minor partner is limited to the extent of :
(a) His share in the firm
(b) His personal assets
(c) His share in the firm as well as his personal assets
(d) He is not liable

6. Subject to contract between the partners, a partner does not have any one of the following rights :
(a) Right to receive remuneration.
( b) Right to share profits.
(c) Right to take part in the business.
(d) Right to claim interest on capital.

7. A partner can bind a firm by his act if he:
(a) Submits a dispute to arbitration.
( b) Withdraws suit or proceeding filed on behalf of the firm.
(c) Transfer immovable property belonging to the firm.
(d) Buys goods on behalf of the firm.

8. For ordinary business matters the decisions in the firm are taken on the basis of :
(a) Decision of majority of partners.
(b) Unanimous decision of partners.
(c) 2/3 majority.
(d) 1/3 majority.

9. For changing the nature of a business
(a) Consent of all the partners is needed.
(b) Consent of majority of partners is needed.
(c) Consent of court is needed.
(d) Consent of Registrar of firm is needed in reference to conduct of the business.

10. Where a partner has advanced any loan to the firm and the agreement provides for interest, but does not specify any rate, the rate shall be —
(a) 6%
(b) 8%
(c) 10%
(d) Nil

11. Property of the firm does not include:
(a) Trademark owned by the firm.
(b) Property acquired by or for the firm.
(c) Goodwill of the business.
(d) Property belonging to the partners.

12. Notice to a partner operates as notice to the firm. For such purpose, notice may be given to :
(a) All the partners jointly.
( b) A partner who habitually acts in the business of the firm.
(c) Any two partners.
(d) Only the dormant partners.

13. When a partner transfers his share, the transferee of partners share has the following rights:
(a) To take part in the conduct of the business
(b) To require accounts
(c) To inspect books of the firm
(d) To receive the share of the profits of the transferring partner

14. A minor may give public notice of his decision to continue or withdraw from the firm on his attaining majority within:
(a) Three months
(b) Six months
(c) Nine months
(d) One year

15. When a minor on attaining the age of majority, has elected to become a partner, he becomes personally liable to third parties for all the acts of the firm from the date of his:
(a) Decision to become a partner
(b) Attaining the age of majority
(c) Admission to the benefits of the firm
(d) Attaining majority or decision to become a partner, whichever is earlier

Answers:
CA Foundation Business Laws Study Material Chapter 18 Relations of Partners 170 1

STATE WHETHER THE FOLLOWING ARE TRUE OR FALSE:

1. It is the duty of the partners to work full time in the business of the firm.
2. Partner has a right to receive remuneration.
3. A partner can be admitted to the partnership with the consent of majority of the partners.
4. A partner is not the agent of the other partners.
5. On becoming major, the liability of a minor admitted to the benefits of partnership becomes unlimited from the fate of majority.
6. Where a partner is entitled to interest on the capital he will be paid interest only if there are profits.
7. If a partner advances money to the firm he will be entitled to interest @ 6% p.a. from the firm only in case of profits.

Answers:
CA Foundation Business Laws Study Material Chapter 18 Relations of Partners 170 2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3

More Exercises

Question 1.
Find the sum of the following A.P.s :
(i) 2, 7, 12, … to 10 terms
(ii) \(\frac { 1 }{ 15 } ,\frac { 1 }{ 12 } ,\frac { 1 }{ 10 } ,… \) t0 11 terms
Solution:
(i) 2, 7, 12, … to 10 terms
Here a = 2, d = 7 – 2 = 5 and n = 10
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q1.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q1.2

Question 2.
How many terms of the A.P. 27, 24, 21, …, should be taken so that their sum is zero?
Solution:
A.P. = 27, 24, 21,…
a = 27
d = 24 – 27 = -3
Sn =0
Let n terms be there in A.P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q2.1

Question 3.
Find the sums given below :
(i) 34 + 32 + 30 + … + 10
(ii) – 5 + ( – 8) + ( – 11) + … + ( – 230)
Solution:
(i) 34 + 32 + 30 + … + 10
Here, a = 34, d = 32 – 34 = -2, l = 10
Tn = a + (n – 1)d
10 = 34 + (n – 1)(-2)
-24 = -2 (n – 1)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q3.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q3.2

Question 4.
In an A.P. (with usual notations) :
(i) given a = 5, d = 3, an = 50, find n and Sn
(ii) given a = 7, a13 = 35, find d and S13
(iii) given d = 5, S9 = 75, find a and a9
(iv) given a = 8, an = 62, Sn = 210, find n and d
(v) given a = 3, n = 8, S = 192, find d.
Solution:
(i) a = 5, d = 3, an = 50
an = a + (n – 1 )d
50 = 5 + (n – 1) x 3
⇒ 50 – 5 = 3(n – 1)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q4.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q4.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q4.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q4.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q4.5

Question 5.
(i) The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
(ii) The sum of first 15 terms of an A.P. is 750 and its first term is 15. Find its 20th term.
Solution:
(i) First term of an A.P. (a) = 5
Last term (l) = 45
Sum = 400
l = a + (n – 1 )d
45 = 5 + (n – 1)d
⇒ (n – 1)d = 45 – 5 = 40 …(i)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q5.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q5.2

Question 6.
The first and the last terms of an A.P. are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?
Solution:
First term of an A.P. (a) = 17
and last term (l) = 350
d= 9
l = Tn = a + (n – 1 )d
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q6.1

Question 7.
Solve for x : 1 + 4 + 7 + 10 + … + x = 287.
Solution:
1 + 4 + 7 + 10 + .. . + x = 287
Here, a = 1, d = 4 – 1 = 3, n = x
l = x = a = (n – 1)d = 1 + (n – 1) x 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q7.2

Question 8.
(i) How many terms of the A.P. 25, 22, 19, … are needed to give the sum 116 ? Also find the last term.
(ii) How many terms of the A.P. 24, 21, 18, … must be taken so that the sum is 78 ? Explain the double answer.
Solution:
(i) A.P. is 25, 22, 19, …
Sum = 116
Here, a = 25, d = 22 – 25 = -3
Let number of terms be n, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q8.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q8.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q8.4

Question 9.
Find the sum of first 22 terms, of an A.P. in which d = 7 and a22 is 149.
Solution:
Sum of first 22 terms of an A.P. whose d = 7
a22 = 149 and n = 22
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q9.1

Question 10.
(i) Find the sum of first 51 terms of the A.P. whose second and third terms are 14 and 18 respectively.
(ii) If the third term of an A.P. is 1 and 6th term is – 11, find the sum of its first 32 terms.
Solution:
Sum of first 51 terms of an A.P. in which
T2 = 14, T3 = 18
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q10.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q10.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q10.3

Question 11.
If the sum of first 6 terms of an A.P. is 36 and that of the first 16 terms is 256, find the sum of first 10 terms.
Solution:
S6 = 36
S16 = 256
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q11.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q11.2

Question 12.
Show that a1, a2, a3, … form an A.P. where an is defined as an = 3 + 4n. Also find the sum of first 15 terms.
Solution:
an = 3 + 4n
a1 = 3 + 4 x 1 = 3 + 4 = 7
a2 = 3 + 4 x 2 = 3 + 8 = 11
a3 = 3 + 4 x 3 = 3 + 12 = 15
a4 = 3 + 4 x 4 = 3 + 16 = 19
and so on Here, a = 1 and d = 11 – 7 = 4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q12.1

Question 13.
(i) If an = 3 – 4n, show that a1, a2, a3, … form an A.P. Also find S20.
(ii) Find the common difference of an A.P. whose first term is 5 and the sum of first four terms is half the sum of next four terms.
Solution:
(i) an = 3 – 4n
a1 = 3 – 4 x 1 = 3 – 4 = -1
a2 = 3 – 4 x 2 = 3 – 8 = -5
a3 = 3 – 4 x 3 = 3 – 12 = -9
a4 = 3 – 4 x 4 = 3 – 16 = -13 and so on
Here, a = -1, d = -5 – ( -1) = -5 + 1 = -4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q13.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q13.2

Question 14.
The sum of first n terms of an A.P. whose first term is 8 and the common difference is 20 equal to the sum of first 2n terms of another A.P. whose first term is – 30 and the common difference is 8. Find n.
Solution:
In an A.P.
Sn = S2n
For the first A.P. a = 8, d = 20
and for second A.P. a = -30, d = 8
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q14.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q14.2

Question 15.
The sum of first six terms of an arithmetic progression is 42. The ratio of the 10th term to the 30th term is \(\\ \frac { 1 }{ 3 } \). Calculate the first and the thirteenth term.
Solution:
T10 : T30 = 1 : 3, S6 = 42
Let a be the first term and d be a common difference, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q15.1

Question 16.
In an A.P., the sum of its first n terms is 6n – n². Find is 25th term.
Solution:
Sn = 6n – n²
T25 = ?
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q16.1

Question 17.
If the sum of first n terms of an A.P. is 4n – n², what is the first term (i. e. S1)? What is the sum of the first two terms? What is the second term? Also, find the 3rd term, the 10th term, and the nth terms?
Solution:
Sn = 4n – n²
Sn – 1 = 4(n – 1) – (n – 1)²
= 4n – 4 – (n² – 2n + 1)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q17.1

Question 18.
If Sn denotes the sum of first n terms of an A.P., prove that S30 = 3(S20 – S10).
Solution:
Sn denotes the sum of first n terms of an A.P.
To prove: S30 = 3(S20 – S10)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q18.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q18.2

Question 19.
(i) Find the sum of first 1000 positive integers.
(ii) Find the sum of first 15 multiples of 8.
Solution:
(i) Sum of first 1000 positive integers
i. e., 1 + 2 + 3+ 4 + … + 1000
Here, a = 1, d = 1, n = 1000
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q19.1

Question 20.
(i) Find the sum of all two digit natural numbers which are divisible by 4.
(ii) Find the sum of all natural numbers between 100 and 200 which are divisible by 4.
(iii) Find the sum of all multiples of 9 lying between 300 and 700.
(iv) Find the sum of all natural numbers less than 100 which are divisible by 6.
Solution:
(i) Sum of two digit natural numbers which are divisible by 4
which are 12, 16, 20, 24, …, 96
Here, a = 12, d = 16 – 12 = 4, l = 96
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q20.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q20.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q20.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q20.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q20.5

Question 21.
(i) Find the sum of all two digit odd positive numbers.
(ii) Find the sum of all 3-digit natural numbers which are divisible by 7.
(iii) Find the sum of all two digit numbers which when divided by 7 yield 1 as the
Solution:
(i) Sum of all two-digit odd positive numbers which are 11, 13, 15, …, 99
Here, a = 11, d = 2, l = 99
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q21.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q21.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q21.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q21.4

Question 22.
A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows : Rs 200 for the first day, Rs 250 for the second day, Rs 300 for the third day, etc., the penalty for each succeeding day being Rs 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work for 30 days ?
Solution:
Penalty for
First day = Rs 200
Second day = Rs 250
Third day = Rs 300
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q22.1

Question 23.
Kanika was given her pocket money on 1st Jan, 2016. She puts Rs 1 on Day 1, Rs 2 on Day 2, Rs 3 on Day 3, and continued on doing so till the end of the month, from this money into her piggy bank. She also spent Rs 204 of her pocket money, and was found that at the end of the month she still has Rs 100 with her. How much money was her pocket money for the month ?
Solution:
Pocket money for Jan. 2016
Out of her pocket money, Kanika puts
Rs 1 on the first day i.e., 1 Jan.
Rs 2 on second Jan
Rs 3 on third Jan
Rs 31 on 31st Jan
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q23.1

Question 24.
Yasmeen saves Rs 32 during the first month, Rs 36 in the second month and Rs 40 in the third month. If she continues to save in this manner, in how many months will she save Rs 2000?
Solution:
Savings for the first month = Rs 32
For the second month = Rs 36
For the third month = Rs 40
Total savings for the period = Rs 2000
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q24.1

Question 25.
The students of a school decided to beautify the school on the Annual Day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2 m. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance did she cover in completing this job and returning back to collect her books ? What is the maximum distance she travelled carrying a flag ?
Solution:
Total number of flags = 27
To fixed after every = 2 m
The flag is stored at the middlemost flag
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 Q25.1

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2

More Exercises

Question 1.
Given that M = \(\begin{bmatrix} 2 & 0 \\ 1 & 2 \end{bmatrix} \) and N = \(\begin{bmatrix} 2 & 0 \\ -1 & 2 \end{bmatrix}\),find M + 2N
Solution:
M = \(\begin{bmatrix} 2 & 0 \\ 1 & 2 \end{bmatrix} \)
N = \(\begin{bmatrix} 2 & 0 \\ -1 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q1.1

Question 2.
If A = \(\begin{bmatrix} 2 & 0 \\ -3 & 1 \end{bmatrix} \) and B = \(\begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} \)
find 2A – 3B
Solution:
A = \(\begin{bmatrix} 2 & 0 \\ -3 & 1 \end{bmatrix} \)
B = \(\begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q2.1

Question 3.
If A = \(\begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \) and B = \(\begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix} \)
Compute 3A + 4B
Solution:
A = \(\begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \)
B = \(\begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q3.1

Question 4.
Given A = \(\begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \) and B = \(\begin{bmatrix} -4 & -1 \\ -3 & -2 \end{bmatrix} \)
(i) find the matrix 2A + B
(ii) find a matrix C such that C + B = \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
Solution:
A = \(\begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \)
B = \(\begin{bmatrix} -4 & -1 \\ -3 & -2 \end{bmatrix} \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q4.1

Question 5.
A = \(\begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix} \) and B = \(\begin{bmatrix} -2 & -1 \\ 1 & 2 \end{bmatrix} \) , C = \(\begin{bmatrix} 0 & 3 \\ 2 & -1 \end{bmatrix} \)
Find A + 2B – 3C
Solution:
A = \(\begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix} \) and B = \(\begin{bmatrix} -2 & -1 \\ 1 & 2 \end{bmatrix} \) , C = \(\begin{bmatrix} 0 & 3 \\ 2 & -1 \end{bmatrix} \)
∴ A + 2B – 3C
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q5.1

Question 6.
If A = \(\begin{bmatrix} 0 & -1 \\ 1 & 2 \end{bmatrix} \) and B = \(\begin{bmatrix} 1 & 2 \\ -1 & 1 \end{bmatrix} \)
Find the matrix X if :
(i) 3A + X = B
(ii) X – 3B = 2A
Solution:
A = \(\begin{bmatrix} 0 & -1 \\ 1 & 2 \end{bmatrix} \)
B = \(\begin{bmatrix} 1 & 2 \\ -1 & 1 \end{bmatrix} \)
(i) 3A + X = B
⇒ X = B – 3A
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q6.1

Question 7.
Solve the matrix equation
\(\begin{bmatrix} 2 & 1 \\ 5 & 0 \end{bmatrix}-3X=\begin{bmatrix} -7 & 4 \\ 2 & 6 \end{bmatrix}\)
Solution:
\(\begin{bmatrix} 2 & 1 \\ 5 & 0 \end{bmatrix}-3X=\begin{bmatrix} -7 & 4 \\ 2 & 6 \end{bmatrix}\)
\(\begin{bmatrix} 2 & 1 \\ 5 & 0 \end{bmatrix}-\begin{bmatrix} -7 & 4 \\ 2 & 6 \end{bmatrix}=3X\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q7.1

Question 8.
If \(\begin{bmatrix} 1 & \quad 4 \\ -2 & \quad 3 \end{bmatrix}+2M=3\begin{bmatrix} 3 & \quad 2 \\ 0 & -3 \end{bmatrix}\), find the matrix M
Solution:
\(\begin{bmatrix} 1 & \quad 4 \\ -2 & \quad 3 \end{bmatrix}+2M=3\begin{bmatrix} 3 & \quad 2 \\ 0 & -3 \end{bmatrix}\)
2M =
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q8.1

Question 9.
A = \(\begin{bmatrix} 2 & -6 \\ 2 & 0 \end{bmatrix} \) and B = \(\begin{bmatrix} -3 & 2 \\ 4 & 0 \end{bmatrix} \) , C = \(\begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix} \)
Find the matrix X such that A + 2X = 2B + C
Solution:
A = \(\begin{bmatrix} 2 & -6 \\ 2 & 0 \end{bmatrix} \) and B = \(\begin{bmatrix} -3 & 2 \\ 4 & 0 \end{bmatrix} \) , C = \(\begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix} \)
let X = \(\begin{bmatrix} x & y \\ z & t \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q9.1

Question 10.
Find X and Y if X + Y = \(\begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}\) and X – Y = \(\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}\)
Solution:
X + Y = \(\begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}\)…..(i)
X – Y = \(\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}\)…….(ii)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q10.1

Question 11.
If \(2\begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 7 & 0 \\ 10 & 5 \end{bmatrix}\) Find the values of x and y
Solution:
\(2\begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 7 & 0 \\ 10 & 5 \end{bmatrix}\)
\(\begin{bmatrix} 6 & 8 \\ 10 & 2x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 7 & 0 \\ 10 & 5 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q11.1

Question 12.
If \(2\begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} z & 0 \\ 10 & 5 \end{bmatrix}\) Find the values of x and y
Solution:
\(2\begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} z & 0 \\ 10 & 5 \end{bmatrix}\)
\(\begin{bmatrix} 6 & 8 \\ 10 & 2x \end{bmatrix}+\begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} z & 0 \\ 10 & 5 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q12.1

Question 13.
If \(\begin{bmatrix} 5 & 2 \\ -1 & \quad y+1 \end{bmatrix}-2\begin{bmatrix} 1 & 2x-1 \\ 3 & -2 \end{bmatrix}=\begin{bmatrix} 3 & -8 \\ -7 & 2 \end{bmatrix}\) Find the values of x and y
Solution:
\(\begin{bmatrix} 5 & 2 \\ -1 & \quad y+1 \end{bmatrix}-2\begin{bmatrix} 1 & 2x-1 \\ 3 & -2 \end{bmatrix}=\begin{bmatrix} 3 & -8 \\ -7 & 2 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q13.1

Question 14.
If \(\begin{bmatrix} a & \quad 3 \\ 4 & \quad 2 \end{bmatrix}+\begin{bmatrix} 2 & \quad b \\ 1 & -2 \end{bmatrix}-\begin{bmatrix} 1\quad & 1 \\ -2\quad & c \end{bmatrix}=\begin{bmatrix} 5 & 0 \\ 7 & 3 \end{bmatrix}\)
Find the value of a,b and c
Solution:
\(\begin{bmatrix} a & \quad 3 \\ 4 & \quad 2 \end{bmatrix}+\begin{bmatrix} 2 & \quad b \\ 1 & -2 \end{bmatrix}-\begin{bmatrix} 1\quad & 1 \\ -2\quad & c \end{bmatrix}=\begin{bmatrix} 5 & 0 \\ 7 & 3 \end{bmatrix}\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q14.1

Question 15.
If A = \(\begin{bmatrix} 2 & a \\ -3 & 5 \end{bmatrix} \) and B = \(\begin{bmatrix} -2 & 3 \\ 7 & b \end{bmatrix} \) , C = \(\begin{bmatrix} c & 9 \\ -1 & -11 \end{bmatrix} \) and 5A + 2B = C, find the values of a,b,c
Solution:
A = \(\begin{bmatrix} 2 & a \\ -3 & 5 \end{bmatrix} \) and B = \(\begin{bmatrix} -2 & 3 \\ 7 & b \end{bmatrix} \) , C = \(\begin{bmatrix} c & 9 \\ -1 & -11 \end{bmatrix} \)
and 5A + 2B = C
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 Q15.1

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.2 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test

More Exercises

 

Question 1.
Write the first four terms of the A.P. when its first term is – 5 and the common difference is – 3.
Solution:
First 4 term of A.P. whose first term (a) = -5
and common difference (d) = -3
= -5, -8, -11, -14

Question 2.
Verify that each of the following lists of numbers is an A.P., and the write its next three terms :
(i) \(0,\frac { 1 }{ 4 } ,\frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,… \)
(ii) \(5,\frac { 14 }{ 3 } ,\frac { 13 }{ 3 } ,4,… \)
Solution:
(i) \(0,\frac { 1 }{ 4 } ,\frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,… \)
Here a = 0, d = \(\\ \frac { 1 }{ 4 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q2.1

Question 3.
The nth term of an A.P. is 6n + 2. Find the common difference.
Solution:
Tn of an A.P. = 6n + 2 .
T1 = 6 x 1 + 2 = 6 + 2 = 8
T2 = 6 x 2 + 2 = 12 + 2 = 14
T3 = 6 x 3 + 2 = 18 + 2 = 20
d = 14 – 8 = 6

Question 4.
Show that the list of numbers 9, 12, 15, 18, … form an A.P. Find its 16th term and the nth.
Solution:
9, 12, 15, 18, …
Here, a = 9, d = 12 – 9 = 3
or 15 – 12 = 3
or 18 – 15 = 3
Yes, it form an A.P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q4.1

Question 5.
Find the 6th term from the end of the A.P. 17, 14, 11, …, – 40.
Solution:
6th term from the end of
A.P. = 17, 14, 11, …… 40
Here, a = 17, d = -3, l = -40
l = a + (n – 1 )d
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q5.1

Question 6.
If the 8th term of an A.P. is 31 and the 15th term is 16 more than its 11th term, then find the A.P.
Solution:
In an A.P.
a8 = 31, a15 = a11 + 16
Let a be the first term and d be a common difference, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q6.1

Question 7.
The 17th term of anA.P. is 5 more than twice its 8th term. If the 11th term of the A.P. is 43, then find the wth term.
Solution:
In an A.P.
a17 = 2 x a8 + 5
a11 = 43, find an
Let a be the first term and d be the common
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q7.2

Question 8.
The 19th term of an A.P. is equal to three times its 6th term. If its 9th term is 19, find the A.P.
Solution:
In an A.P.
a19 = 3 x a6 and a9 = 19
Let a be the first term and d be the common
difference, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q8.2

Question 9.
If the 3rd and the 9th terms of an A.P. are 4 and – 8 respectively, then which term of this A.P. is zero?
Solution:
In an A.P.
a3 = 4, a9 = -8, which term of A.P. will be zero
Let a be the first term and d be a common difference, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q9.1

Question 10.
Which term of the list of numbers 5, 2, – 1, – 4, … is – 55?
Solution:
A.P. is 5, 2, -1, – 4, …
Which term of A.P. is -55
Let it be nth term
Here, a = 5, d = 2 – 5 = -3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q10.1

Question 11.
The 24th term of an A.P. is twice its 10th term. Show that its 72nd term is four times its 15th term.
Solution:
In an A.P.
24th term = 2 x 10th term
To show that 72nd term = 4 x 15th term
Let a be the first term and d be a common difference, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q11.1

Question 12.
Which term of the list of numbers \(20,19\frac { 1 }{ 4 } ,18\frac { 1 }{ 2 } ,17\frac { 3 }{ 4 } ,..\) is the first negative term?
Solution:
In A.P., which is the first negative term
\(20,19\frac { 1 }{ 4 } ,18\frac { 1 }{ 2 } ,17\frac { 3 }{ 4 } ,..\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q12.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q12.2

Question 13.
If the pth term of an A.P. is q and the qth term is p, show that its nth term is (p + q – n)
Solution:
In an A.P.
pth term = q
qth term = p
Show that (p + q – n) is nth term
Let a be the first term and d be the common
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q13.1

Question 14.
How many three digit numbers are divisible by 9?
Solution:
3-digit numbers which are divisible by 9 are 108, 117, 126, 135, …, 999
Here, a = 108, d = 9 and l = 999
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q14.1

Question 15.
The sum of three numbers in A.P. is – 3 and the product is 8. Find the numbers.
Solution:
Sum of three numbers of an A.P. = -3
and their product = 8
Let the numbers be
a – d, a, a + d, then
a – d + a + a + d = -3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q15.1

Question 16.
The angles of a quadrilateral are in A.P. If the greatest angle is double of the smallest angle, find all the four angles.
Solution:
Angles of a quadrilateral are in A.P.
Greatest angle is double of the smallest
Let the smallest angle of the quadrilateral is
a + 3d…..(i)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q16.1

Question 17.
The nth term of an A.P. cannot be n² + n + 1. Justify your answer.
Solution:
nth term of an A.P. can’t be n² + n + 1
Giving some different values to n such as 1, 2, 3, 4, …
we find then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q17.1

Question 18.
Find the sum of first 20 terms of an A.P. whose nth term is 15 – 4n.
Solution:
Giving some different values such as 1 to 20
We get,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q18.1

Question 19.
Find the sum :
\(18+15\frac { 1 }{ 2 } +13+…+\left( -49\frac { 1 }{ 2 } \right) \)
Solution:
Find the sum
\(18+15\frac { 1 }{ 2 } +13+…+\left( -49\frac { 1 }{ 2 } \right) \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q19.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q19.2

Question 20.
(i) How many terms of the A.P. – 6,\(– \frac { 11 }{ 2 } \) – 5,… make the sum – 25?
(ii) Solve the equation 2 + 5 + 8 + … + x = 155.
Solution:
(i) Sum = -25
A.P. = -6, \(– \frac { 11 }{ 2 } \) -5,…
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q20.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q20.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q20.3

Question 21.
If the third term of an A.P. is 5 and the ratio of its 6th term to the 10th term is 7 : 13, then find the sum of first 20 terms of this A.P.
Solution:
3rd term of an A.P. = 5
Ratio in 6th term and 10th term = 7 : 13
Find S20
Let a be the first term and d be the common difference
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q21.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q21.2

Question 22.
In an A.P., the first term is 2 and the last term is 29. If the sum of the terms is 155, then find the common difference of the A.P.
Solution:
In an A.P.
First term (a) = 2
Last term (l) = 29
Sum of terms = 155
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q22.1

Question 23.
The sum of first 14 terms of an A.P. is 1505 and its first term is 10. Find its 25th term.
Solution:
Sum of first 14 terms = 1505
First term (a) = 10
Find 25th term
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q23.1

Question 24.
Find the number of terms of the A.P. – 12, – 9, – 6, …, 21. If 1 is added to each term of this A.P., then find the sum of all terms of the A.P. thus obtained.
Solution:
A.P. -12, -9, -6,…, 21
If 1 is added to each term, find the sum of there terms
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q24.1

Question 25.
The sum of first n term of an A.P. is 3n² + 4n. Find the 25th term of this A.P.
Solution:
S= 3n² + 4n
Sn – 1 = 3(n – 1)² + 4(n – 1)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q25.1

Question 26.
In an A.P., the sum of first 10 terms is – 150 and the sum of next 10 terms is – 550. Find the A.P.
Solution:
In an A.P.
Sum of first 10 terms = -150
Sum of next 10 terms = -550, A.P. = ?
Sum of first 10 terms = -150
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q26.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q26.2

Question 27.
The sum of first m terms of an A.P. is 4m² – m. If its nth term is 107, find the value of n. Also find the 21 st term of this A.P.
Solution:
Sm = 4m² – m
Sn = 4n² – n
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q27.1

Question 28.
If the sum of first p, q and r terms of an A.P. are a, b and c respectively, prove that
\(\frac { a }{ p } (q-r)+\frac { b }{ q } (r-p)+\frac { c }{ r } (p-q)=0 \)
Solution:
Let the first term of A.P. be A and common difference be d.
Sum of the first p terms is
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q28.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q28.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q28.3

Question 29.
A sum of Rs 700 is to be used to give 7 cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.
What is the importance of an academic prise in students life? (Value Based)
Solution:
Total sum = Rs 700
Cash prizes to 7 students = 7 prize
Each prize is Rs 20 less than its preceding prize
d = -20, d = -20, n = 7
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q29.1

Question 30.
Find the geometric progression whose 4th term is 54 and 7th term is 1458.
Solution:
In a G.P.
a4 = 54
a7 = 1458
Let a be the first term and r be the common difference
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q30.1

Question 31.
The fourth term of a G.P. is the square of its second term and the first term is – 3. Find its 7th term.
Solution:
In G.P.
a4 = (a2)², a1 = -3
Let a be the first term and r be the common ratio
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q31.1

Question 32.
If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively, prove that x, y and z are in G.P.
Solution:
In a G.P.
a= x, a10 = y, a16 = z
Show that x, y, z are in G.P.
Let a be the first term and r be the common ratio, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q32.1

Question 33.
The original cost of a machine is Rs 10000. If the annual depreciation is 10%, after how many years will it be valued at Rs 6561 ?
Solution:
Original cost of machine = Rs 10000
Since, machine depreciates at the rate of 10%
on reducing the balance,
Value of machine after one year
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q33.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q33.2

Question 34.
How many terms of the G.P. \(3,\frac { 3 }{ 2 } ,\frac { 3 }{ 4 } \),are needed to give the sum \(\\ \frac { 3069 }{ 512 } \) ?
Solution:
G.P. \(3,\frac { 3 }{ 2 } ,\frac { 3 }{ 4 } \)
Sn = \(\\ \frac { 3069 }{ 512 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q34.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q34.2

Question 35.
Find the sum of first n terms of the series : 3 + 33 + 333 + …
Solution:
Series is
3 + 33 + 333 + … n terms
= 3[1 + 11 + 111 +…n terms]
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q35.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q35.2

Question 36.
Find the sum of the series 7 + 7.7 + 7.77 + 7.777 + … to 50 terms.
Solution:
The given sequence is 7, 7.7, 7.77, 7.777,…
Required sum = S50
= 7 + 7.7 + 7.77 + … 50 terms
= 7(1 + 1.1 + 1. 11 + … 50 terms)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q36.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q36.2

Question 37.
The inventor of chessboard was a very clever man. He asked the king, h reward of one grain of wheat for the first square, 2 grains for the second, 4 grains for the third, and so on, doubling the number of the grains for each subsequent square. How many grains would have to be given?
Solution:
In a chessboard, there are 8 x 8 = 64 squares
If a man put 1 grain in first square,
2 grains in second square,
4 grains in third square
and goes on upto the last square, i.e. 64th square
Therefore, 1 + 2 + 4 + 8 + 16 + … 64 terms
Here, a = 1, r = 2 and n = 64
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test Q37.1

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS

More Exercises

Choose the correct answer from the given four options (1 to 12) :

Question 1.
The points A (9, 0), B (9, 6), C ( – 9, 6) and D ( – 9, 0) are the vertices of a
(a) rectangle
(b) square
(c) rhombus
(d) trapezium
Solution:
A (9, 0), B (9, 6), C (-9, 6), D (-9, 0)
AB² = (x2 – x1)² + (y2 – y1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS Q1.1

Question 2.
The mid-point of the line segment joining the points A ( – 2, 8) and B ( – 6, – 4) is
(a) ( – 4, – 6)
(b) (2, 6)
(c) ( – 4, 2)
(d) (4, 2)
Solution:
Mid-point of the line segment joining the points A (-2, 8), B (-6, -4)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS Q2.1

Question 3.
If \(P\left( \frac { a }{ 3 } ,4 \right) \) segment joining the points Q ( – 6, 5) and R ( – 2, 3), then the value of a is
(a) – 4
(b) – 6
(c) 12
(d) – 12
Solution:
\(P\left( \frac { a }{ 3 } ,4 \right) \) is mid-point of the line segment
joining the points Q (-6, 5) and R (-2, 3)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS Q3.1

Question 4.
If the end points of a diameter of a circle are A ( – 2, 3) and B (4, – 5), then the coordinates of its centre are
(a) (2, – 2)
(b) (1, – 1)
(c) ( – 1, 1)
(d) ( – 2, 2)
Solution:
End points of a diameter of a circle are (-2, 3) and B (4,-5)
then co-ordinates of the centre of the circle
= \(\left( \frac { -2+4 }{ 2 } ,\frac { 3-5 }{ 2 } \right) or\left( \frac { 2 }{ 2 } ,\frac { -2 }{ 2 } \right) \)
= (1, -1) (b)

Question 5.
If one end of a diameter of a circle is (2, 3) and the centre is ( – 2, 5), then the other end is
(a) ( – 6, 7)
(b) (6, – 7)
(c) (0, 8)
(d) (0, 4)
Solution:
One end of a diameter of a circle is (2, 3) and centre is (-2, 5)
Let (x, y) be the other end of the diameter
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS Q5.1

Question 6.
If the mid-point of the line segment joining the points P (a, b – 2) and Q ( – 2, 4) is R (2, – 3), then the values of a and b are
(a) a = 4, b = – 5
(b) a = 6, b = 8
(c) a = 6, b = – 8
(d) a = – 6, b = 8
Solution:
the mid-point of the line segment joining the
points P (a, b – 2) and Q (-2, 4) is R (2, -3)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS Q6.1

Question 7.
The point which lies on the perpendicular bisector of the line segment joining the points A ( – 2, – 5) and B (2, 5) is
(a) (0, 0)
(b) (0, 2)
(c) (2, 0)
(d) ( – 2, 0)
Solution:
the line segment joining the points A (-2, -5) and B (2, -5), has mid-point
= \(\left( \frac { -2+2 }{ 2 } ,\frac { -5+5 }{ 2 } \right) \) = (0, 0)
(0, 0) lies on the perpendicular bisector of AB. (a)

Question 8.
The coordinates of the point which is equidistant from the three vertices of ∆AOB (shown in the given figure) are
(a) (x, y)
(b) (y, x)
(c) \(\left( \frac { x }{ 2 } ,\frac { y }{ 2 } \right) \)
(d) \(\left( \frac { y }{ 2 } ,\frac { x }{ 2 } \right) \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS Q8.1
Solution:
In the given figure, vertices of a ∆OAB are (0, 0), (0, 2y) and (2x, 0)
The point which is equidistant from O, A and B is the mid-point of AB.
∴ Coordinates are \(\left( \frac { 0+2x }{ 2 } ,\frac { 2y+0 }{ 2 } \right) \) or (x, y) (a)

Question 9.
The fourth vertex D of a parallelogram ABCD whose three vertices are A ( – 2, 3), B (6, 7) and C (8, 3) is
(a) (0, 1)
(b) (0, – 1)
(c) ( – 1, 0)
(d) (1, 0)
Solution:
ABCD is a ||gm whose vertices A (-2, 3), B (6, 7) and C (8, 3).
The fourth vertex D will be the point on which diagonals AC and BD
bisect each other at O.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS Q9.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS Q9.2

Question 10.
A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, – 5) is the mid-point of PQ, then the coordinates of P and Q are, respectively
(a) (0, – 5) and (2, 0)
(b) (0, 10) and ( – 4, 0)
(c) (0, 4) and ( – 10, 0)
(d) (0, – 10) and (4, 0)
Solution:
A line intersects y-axis at P and x-axis a Q.
R (2, -5) is the mid-point
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS Q10.1

Question 11.
The points which divides the line segment joining the points (7, – 6) and (3, 4) in the ratio 1 : 2 internally lies in the
(a) Ist quadrant
(b) IInd quadrant
(c) IIIrd quadrant
(d) IVth quadrant
Solution:
A point divides line segment joining the points
A (7, -6) and B (3, 4) in the ratio 1 : 2 internally.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS Q11.1
Let (x, y) divides it in the ratio 1 : 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS Q11.2

Question 12.
The centroid of the triangle whose vertices are (3, – 7), ( – 8, 6) and (5, 10) is
(a) (0, 9)
(b) (0, 3)
(c) (1, 3)
(d) (3, 3)
Solution:
Centroid of the triangle whose Vertices are (3, -7), (-8, 6) and (5, 10) is
\(\left( \frac { 3-8+5 }{ 3 } ,\frac { -7+6+10 }{ 3 } \right) or\left( 0,\frac { 9 }{ 3 } \right) \)
or (0, 3) (b)

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.