ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion 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 7 Ratio and Proportion Chapter Test

More Exercises

Question 1.
Find the compound ratio of:
(a + b)2 : (a – b )2 ,
(a2 – b2) : (a2 + b2),
(a4 – b4) : (a + b)4
Solution:
(a + b)2 : (a – b )2 ,
(a2 – b2) : (a2 + b2),
(a4 – b4) : (a + b)4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q1.1

Question 2.
If (7 p + 3 q) : (3 p – 2 q) = 43 : 2 find p : q
Solution:
(7p + 3q) : (3p – 2q) = 43 : 2
⇒ \(\frac { 7p+3q }{ 3p-2q } =\frac { 43 }{ 2 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q2.1

Question 3.
If a : b = 3 : 5, find (3a + 5b): (7a – 2b).
Solution:
a : b = 3 : 5
⇒ \(\frac { a }{ b } =\frac { 3 }{ 5 } \)
⇒ 3a + 5n : 7a – 2b
Dividing each term by b
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q3.1

Question 4.
The ratio of the shorter sides of a right angled triangle is 5 : 12. If the perimeter of the triangle is 360 cm, find the length of the longest side.
Solution:
Let the two shorter sides of a right-angled triangle be 5x and 12x.
Third (longest side)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q4.1

Question 5.
The ratio of the pocket money saved by Lokesh and his sister is 5 : 6. If the sister saves Rs 30 more, how much more the brother should save in order to keep the ratio of their savings unchanged?
Solution:
Let the savings of Lokesh and his sister are 5x and 6x.
and the Lokesh should save Rs y more Now, according to the problem,
⇒ \(\frac { 5x+y }{ 6x+30 } =\frac { 5 }{ 6 } \)
⇒ 30x + 6y = 30x + 150
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q5.1

Question 6.
In an examination, the number of those who passed and the number of those who failed were in the ratio of 3 : 1. Had 8 more appeared, and 6 less passed, the ratio of passed to failures would have been 2 : 1. Find the number of candidates who appeared.
Solution:
Let number of passed = 3 x
and failed = x
Total candidates appeared = 3x + x = 4x.
In second case
No. of candidates appeared = 4 x + 8
and No. of passed = 3 x – 6
and failed = 4x + 8 – 3x + 6 = x + 14
then ratio will be = 2 : 1
Now according to the condition
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q6.1

Question 7.
What number must be added to each of the numbers 15, 17, 34 and 38 to make them proportional ?
Solution:
Let x be added to each number, then numbers will be
15 + x, 17 + x, 34 + x, and 38 + x.
Now according to the condition
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q7.1

Question 8.
If (a + 2 b + c), (a – c) and (a – 2 b + c) are in continued proportion, prove that b is the mean proportional between a and c.
Solution:
(a + 2 b + c), (a – c) and (a – 2 b + c) are in continued proportion
⇒ \(\frac { a+2b+c }{ a-c } =\frac { a-c }{ a-2b+c } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q8.1

Question 9.
If 2, 6, p, 54 and q are in continued proportion, find the values of p and q.
Solution:
2, 6, p, 54 and q are in continued proportional then
⇒ \(\frac { 2 }{ 6 } =\frac { 6 }{ p } =\frac { p }{ 54 } =\frac { 54 }{ q } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q9.1

Question 10.
If a, b, c, d, e are in continued proportion, prove that: a : e = a4 : b4.
Solution:
a, b, c, d, e are in continued proportion
⇒ \(\frac { a }{ b } =\frac { b }{ c } =\frac { c }{ d } =\frac { d }{ e } \) = k (say)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q10.1

Question 11.
Find two numbers whose mean proportional is 16 and the third proportional is 128.
Solution:
Let x and y be two numbers
Their mean proportion = 16
and third proportion = 128
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q11.1

Question 12.
If q is the mean proportional between p and r, prove that:
\({ p }^{ 2 }-{ 3q }^{ 2 }+{ r }^{ 2 }={ q }^{ 4 }\left( \frac { 1 }{ { p }^{ 2 } } -\frac { 3 }{ { q }^{ 2 } } +\frac { 1 }{ { r }^{ 2 } } \right) \)
Solution:
q is mean proportional between p and r
q² = pr
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q12.1

Question 13.
If \(\frac { a }{ b } = \frac { c }{ d } = \frac { e }{ f } \), prove that each ratio is
(i) \(\sqrt { \frac { { 3a }^{ 2 }-{ 5c }^{ 2 }+{ 7e }^{ 2 } }{ { 3b }^{ 2 }-{ 5d }^{ 2 }+{ 7f }^{ 2 } } } \)
(ii) \({ \left[ \frac { { 2a }^{ 3 }+{ 5c }^{ 3 }+{ 7e }^{ 3 } }{ { 2b }^{ 3 }+{ 5d }^{ 3 }+{ 7f }^{ 3 } } \right] }^{ \frac { 1 }{ 3 } } \)
Solution:
\(\frac { a }{ b } = \frac { c }{ d } = \frac { e }{ f } \) = k(say)
∴ a = k, c = dk, e = fk
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q13.1

Question 14.
If \(\frac { x }{ a } = \frac { y }{ b } = \frac { z }{ c } \), prove that
\(\frac { { 3x }^{ 3 }-{ 5y }^{ 3 }+{ 4z }^{ 3 } }{ { 3a }^{ 3 }-{ 5b }^{ 3 }+{ 4c }^{ 3 } } ={ \left( \frac { 3x-5y+4z }{ 3a-5b+4c } \right) }^{ 3 }\)
Solution:
\(\frac { x }{ a } = \frac { y }{ b } = \frac { z }{ c } \) = k (say)
x = ak, y = bk, z = ck
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q14.1

Question 15.
If x : a = y : b, prove that
\(\frac { { x }^{ 4 }+{ a }^{ 4 } }{ { x }^{ 3 }+{ a }^{ 3 } } +\frac { { y }^{ 4 }+{ b }^{ 4 } }{ { y }^{ 3 }+{ b }^{ 3 } } =\frac { { \left( x+y \right) }^{ 4 }+{ \left( a+b \right) }^{ 4 } }{ { \left( x+y \right) }^{ 3 }+{ \left( a+b \right) }^{ 3 } } \)
Solution:
\(\frac { x }{ a } = \frac { y }{ b } \) = k (say)
x = ak, y = bk
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q15.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q15.2

Question 16.
If \(\frac { x }{ b+c-a } =\frac { y }{ c+a-b } =\frac { z }{ a+b-c } \) prove that each ratio’s equal to :
\(\frac { x+y+z }{ a+b+c } \)
Solution:
\(\frac { x }{ b+c-a } =\frac { y }{ c+a-b } =\frac { z }{ a+b-c } \) = k(say)
x = k(b + c – a),
y = k(c + a – b),
z = k(a + b – c)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q16.1

Question 17.
If a : b = 9 : 10, find the value of
(i) \(\frac { 5a+3b }{ 5a-3b } \)
(ii) \(\frac { { 2a }^{ 2 }-{ 3b }^{ 2 } }{ { 2a }^{ 2 }+{ 3b }^{ 2 } } \)
Solution:
a : b = 9 : 10
⇒ \(\frac { a }{ b } = \frac { 9 }{ 10 }\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q17.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q17.2

Question 18.
If (3x² + 2y²) : (3x² – 2y²) = 11 : 9, find the value of \(\frac { { 3x }^{ 4 }+{ 25y }^{ 4 } }{ { 3x }^{ 4 }-{ 25y }^{ 4 } } \) ;
Solution:
\(\frac { { 3x }^{ 4 }+{ 25y }^{ 4 } }{ { 3x }^{ 4 }-{ 25y }^{ 4 } } =\frac { 11 }{ 9 } \)
Applying componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q18.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q18.2

Question 19.
If \(x=\frac { 2mab }{ a+b } \) , find the value of
\(\frac { x+ma }{ x-ma } +\frac { x+mb }{ x-mb } \)
Solution:
\(x=\frac { 2mab }{ a+b } \)
⇒ \(\frac { x }{ ma } +\frac { 2b }{ a+b } \)
Applying componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q19.1

Question 20.
If \(x=\frac { pab }{ a+b } \) ,prove that \(\frac { x+pa }{ x-pa } -\frac { x+pb }{ x-pb } =\frac { 2\left( { a }^{ 2 }-{ b }^{ 2 } \right) }{ ab } \)
Solution:
\(x=\frac { pab }{ a+b } \)
⇒ \(\frac { x }{ pa } +\frac { b }{ a+b } \)
Applying componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q20.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q20.2

Question 21.
Find x from the equation \(\frac { a+x+\sqrt { { a }^{ 2 }-{ x }^{ 2 } } }{ a+x-\sqrt { { a }^{ 2 }-{ x }^{ 2 } } } =\frac { b }{ x } \)
Solution:
\(\frac { a+x+\sqrt { { a }^{ 2 }-{ x }^{ 2 } } }{ a+x-\sqrt { { a }^{ 2 }-{ x }^{ 2 } } } =\frac { b }{ x } \)
Applying componendo and dividendo,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q21.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q21.2

Question 22.
If \(x=\frac { \sqrt [ 3 ]{ a+1 } +\sqrt [ 3 ]{ a-1 } }{ \sqrt [ 3 ]{ a+1 } -\sqrt [ 3 ]{ a-1 } } \), prove that :
x³ – 3ax² + 3x – a = 0
Solution:
\(x=\frac { \sqrt [ 3 ]{ a+1 } +\sqrt [ 3 ]{ a-1 } }{ \sqrt [ 3 ]{ a+1 } -\sqrt [ 3 ]{ a-1 } } \)
Applying componendo and dividendo,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q22.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q22.2

Question 23.
If \(\frac { by+cz }{ b^{ 2 }+{ c }^{ 2 } } =\frac { cz+ax }{ { c }^{ 2 }+{ a }^{ 2 } } =\frac { ax+by }{ { a }^{ 2 }+{ b }^{ 2 } } \), prove that each of these ratio is equal to \(\frac { x }{ a } =\frac { y }{ b } =\frac { z }{ c } \)
Solution:
\(\frac { by+cz }{ b^{ 2 }+{ c }^{ 2 } } =\frac { cz+ax }{ { c }^{ 2 }+{ a }^{ 2 } } =\frac { ax+by }{ { a }^{ 2 }+{ b }^{ 2 } } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q23.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test Q23.2

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test are helpful to complete your math homework.

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CA Foundation Business Laws Study Material Chapter 11 Conditions and Warranties

CA Foundation Business Laws Study Material Chapter 11 Conditions and Warranties

Sec. 12 of the Sale of Goods Act states that a stipulation (or term) in a contract of sale with reference to goods may be a condition or a warranty.
CA Foundation Business Laws Study Material Chapter 11 Conditions and Warranties 1

CONDITION

A condition is a stipulation essential to the main purpose of the contract, the breach of which gives rise to a right to treat the contract as repudiated. [Sec. 12(2)]
For example, A wants to buy a car which can give a mileage of 20 kms/litre. B, the car dealer, points out at a particular car and says “this car will suit you”. A buys the car. But later on he finds that the car is giving a mileage of only 10 kms/litre. THERE IS A BREACH OF CONDITION, because the stipulation made by B forms the very basis of the contract.

WARRANTY

A warranty is stipulation collateral to the main purpose of the contract, the breach of which gives rise to claim for damages but not a right to reject the goods and treat the contract as repudiated- I [Sec. 12(3)]
For example : A goes to B, a car dealer, and says, “I want a good car” The car dealer shows him a car and says, “it can give you a mileage of 20 kms/litre”. A buys the car. Later on, A finds that the car is giving a mileage of 10 kms/litre only. THERE IS A BREACH OF WARRANTY, because the stipulation made by the seller was only collateral one.
Whether a stipulation in a contract of sale is a condition or a warranty depends in each case on the construction of the contract. A stipulation may the a condition, though called a warranty in the contract – [Sec. 12(4)]
Conditions and warranties may be expressly stated or may be implied by law. Implied conditions and warranties are enumerated in sections 14 to 17. They are deemed to be incorporated in every contract of sale unless the terms of the contract show a contrary intention.

When a condition can be treated as a warranty:
Voluntary waiver of a condition [Sec. 13(1)]
1. Where a contract of sale is subject to a condition to be fulfilled by the seller, the buyer may—

  1. waive the condition, for example a buyer may accept defective goods or accept goods
    beyond stipulated time.
  2. elect to treat a breach of condition as a breach of warranty, i.e. instead of repudiating the contract he may accept performance and sue for damages, if he has suffered any.

Once a buyer decides to waive, he cannot afterwards insist on its fulfilment.

Compulsory waiver of a condition [Sec. 13(2)]
Where a contract of sale is not severable (Le. indivisible) and the buyer has accepted the goods or a part thereof, he cannot repudiate the contract but can only sue for damages. In such a case, the breach of condition can only be treated as a breach of war ranty, unless there is a contract to the contrary. [Sec. 13(2)]
E.g: W bought laptops from M and resold it to C without examining the laptops. The laptops were defective. It was held that W must be deemed to have accepted the goods and therefore he could not repudiate the contract but could claim only damages.

  • However, there may be an agreement between the parties which may be contrary to section 13(2). In that case the parties may agree between themselves that the provision of section 13(2) will not apply in their case and the buyer shall have a right to reject the goods even though he has accepted the indivisible goods.
  • If the contract of sale is divisible and the buyer has accepted a part of the goods, he can still exercise the right to reject the remaining goods.
  • Impossibility [Sec.13(3)] : The above provisions of Section 13(1) and 13(2) do not affect the cases where the fulfilment of any condition or warranty is excused by law by reason of impossibility or otherwise. This means that under section 13(3) the seller has the right to rely upon impossibility as an excused in appropriate cases, if sued by the buyer.

    CONDITION

    WARRANTY

    Condition is a term, which is essential to the main purpose of the contract. Warranty is only a collateral term. It is subsidiary to the main purpose of the contract.
    Breach of a condition gives the aggrieved party a right to repudiate the contract and also to claim damages. Breach of warranty entitles the aggrieved party to claim damages only. He cannot repudiate the contract.
    A breach of condition may under certain circumstances, be treated as breach of warranty But a warranty cannot become a condition.

IMPLIED CONDITIONS AND WARRANTIES
A stipulation (or term) in a contract of sale of goods may be express or implied. Express terms are those which have been expressly agreed upon by the parties. Implied terms are those which have been enacted in the Sale of Goods Act. Sections 14 to 17 of the Act contain a list of conditions and warranties which are implied in a contract for the sale of goods, unless the circumstances of the contract are such as to show a different intention. The implied conditions and warrants are stated below:
(a) Implied conditions
1. Implied condition as to title. – [Sec. 14]
There is an implied condition on the part of the seller that, in the case of a sale he has the right to sell the goods, and in the case of an agreement to sell, he will have the right to sell the goods at the time when the property is to pass. If the seller’s title turns out to be defective, the buyer is entitled to reject the goods and claim refund of the price plus damages.

  • A bought a motor car from B. He used it for 3 months and thereafter the car was detected to have been stolen. A was compelled to return it to the true owner. Could A recover the sale price from B?
    (Ans: Yes)
  • A sells to B tins of condensed milk labelled “Nissly Brand” and this is proved to be an infringement of Nestle Company’s trade mark. Is it a breach of implied condition as to title?
    (Ans: When a person sell the goods by infringing a copyrights or trademark of the others, he is considered as not having right to sell such goods.)

2. Implied condition in a sale by description. – [Sec. 14]
Where there is a contract for the sale of goods by description, there is an implied condition that the goods shall correspond with the description. “Correspond with the description” means that the buyer must get the goods that he has asked for. The description may be given—

  1. by mentioning qualities or characteristics of the goods e.g. Basmati rice.
  2. by mentioning the trademark or brand name e.g. Videocon TV.
  3. by the type of packing e.g. 1 kg. packing of tea in plastic jar.

If the buyer does not get the goods he has described he can reject the goods. The rule is “If you contract to sell peas, you cannot oblige a party to take beans. If the description of the article tendered is different in any respect, it is not the article bargained for, and the other party is not bound to take it”. E.g. A car is sold as a “new maruti car”. The buyer finds it to be a used one. The buyer may reject the car or retain the car and claim damages.

3. Implied condition in a sale by sample as well as by description. – [Sec. 15]
When goods are sold by sample as well as by description, the goods shall correspond both with the sample and with the description.

4. Implied condition as to fitness or quality. – [Sec. 16(1)]
The general rule is, there is no implied condition as to quality or fitness for the purpose of the buyer. This is based on the doctrine of “caveat emptor” that is, let the buyer beware. It means that while buying the goods, it is the responsibility of the buyer to check that the goods he is buying would suit his purpose or not. However, in the following situation, the responsibility as to fitness of goods falls upon the seller:
a. where the buyer, expressly or by implication, makes known to the seller the particular purpose for which the goods are required,
b. so as to show that the buyer relies on the seller’s skill, or judgment, and
c. the goods are of a description which it is in the course of the seller’s business to supply (whether he is the manufacturer or not), there is an implied condition that the goods shall be reasonably fit for such purpose.

  • A contracts to make and deliver a set of false teeth to B. The false teeth do not fit in the mouth of B. B is entitled to reject the goods.
  • X places order for lorries to be used for ‘heavy traffic in a hilly country’. The lorries were unfit for this purpose and broke down. It was held that there was breach of condition as to fitness.

Sale under patent or trade name. Proviso to section 16(1) lays down that in the case of a contract for the sale of a specified article under its patent or other trade name, there is no implied condition as to its fitness for any particular purpose. It is so because in such a case the buyer is not relying on the skill and judgment of the seller but relies on the patent name. For example, a hotelier orders ‘Sujeet’ juicer and mixer (patent product) for his business. The juicer and mixer supplied was found to be unsuitable for commercial use. The buyer has no cause of action against the seller, since he purchased the juicer by its patent name.

5. Implied condition as to merchantability. – [Sec. 16(2)]
Where goods are brought by description from a seller who deals in goods of that description, there is an implied condition that goods shall be of merchantable quality. Merchantable means that the goods are commercially saleable and that they are hit for the purpose for which they are generally used.
Where the buyer examines the goods prior to sale, there is no implied condition as to merchantability as regards defects which such examination ought to have revealed. However, inspite of exil amination, if the goods have certain latent defects which no examination could reveal, the implied condition remains.

  • X bought a colour TV from M/s Concord Electronics. The TV was defective right from the beginning and it did not work inspite of repairs by expert technicians. There is a breach of implied condition as to merchantability and the dealer will have to take back the defective TV and refund the amount.
  • X orders motor horns from a manufacturer. The horns supplied are defective. X is entitled to reject them as unmerchantable.

6. Implied condition in a sale by sample. – [Sec. 17]
When goods are to be supplied according to a sample agreed upon, the following conditions are implied:

  1. The bulk shall correspond with the sample in quality.
  2. The buyer shall have a reasonable opportunity of comparing the goods with the sample.
  3. The goods shall be free from any latent defect ( hidden defect) rendering them unmerchantable. Latent defects are the defects which would not be apparent on reasonable examination of the sample and they can be discovered only when the goods are put to use. If the defect is easily discoverable on inspection and the buyer takes delivery after inspection, he has no remedy.

A sale is by sample where there is a term in contract, express or implied to that effect. The effect of the section is that where goods are sold by sample, there should not be any latent defect therein which renders them unmerchantable.

7. Implied condition as to wholesomeness
In case of food stuff and eatables, in addition to the implied condition as to merchantability, there is another implied condition that the goods shall be wholesome that is fit for human consumption.
X bought milk from Y, a dairy owner. The milk was contaminated with germs of typhoid fever. X’s
wrfe, on taking the milk, became infected and died of it. Y was held liable in damages.

(b) Implied warranties
In the absence of an agreement to the contrary, the following warranties are implied in every con-tract of sale:
1. The buyer must get quiet possession [Sec. 14(b)]
The buyer shall have and enjoy quiet possession of the goods. For e.g.: X has given his car on hire for a period of one month to Y. Thereafter, X sold it to Z without disclosing to him that Y was en-titled to use the car on account of the hire agreement. Z claims the car from Y. Y’s possession is disturbed. He can claim damages from X.

2. The goods must be free from encumbrance [Sec. 14(c)]
There is an implied warranty that the goods shall be free from any charge or encumbrance in favour of a third party not declared or known to the buyer before or at the time when the contract iis made. The effect of this clause is that if the buyer pays off the charge of encumbrance, he will be entitled to recover the money from the seller.

3. Warranty for quality or use by usage of trade [Sec. 16(3)]
A warranty as to fitness for a particular purpose may be annexed to a contract of sale by a custom usage of trade.

4. Disclosure of dangerous nature of goods
Where the goods are dangerous in nature and the buyer is ignorant of the danger, the seller must warn the buyer of the probable danger. If there is a breach of this warranty, the seller may be liable i in damages.
Note:

  1. Express terms – [Sec. 16(4)]: An express warranty or condition does not negate a warranty or condition implied by the Act. (Unless the express terms are inconsistent with the implied conditions). This means that implied warranty or condition may co-exist with express warranty or condition. Thus, for example, where sleepers supplied to a railway company were required to be approved by its experts, it was held that it did not exclude the implied condition of merchantableness.
  2. Exclusion of implied terms – [Sec. 62]: These implied conditions and warranties may be ex- eluded or modified by the parties to the contract by express contract, by course of dealing i and by usage of trade.

THE DOCTRINE OF CAVEAT EMPTOR

Caveat Emptor is a Latin expression, which means, “Buyers Beware”. The doctrine of caveat emptor means that, ordinarily, a buyer must buy goods after satisfying himself of their quality and fitness. If he makes a bad choice he cannot blame the seller or recover damages from him. This doctrine is stated in the opening words of section 16: Subject to the provisions of this Act and of any other law for the time being in force, there is no implied warranty or condition as to the quality or fitness for any particular purpose of goods supplied under a contract of sale.

  • It is buyer’s duty to examine goods thoroughly.
  • The buyer should ensure at the time of purchase that the goods conform to his requirements.
  • If the goods turn out to be defective, buyer cannot hold the seller responsible.

EXCEPTIONS:
The doctrine of caveat emptor does not apply in the following situations:
1. Fitness as to quality or use. [Sec. 16(1)]

  1. Where the buyer, expressly or by implication, makes known to the seller the particular purpose for which the goods are required,
  2. soas to show that the buyer relies on the seller’s skill, or judgment, and
  3. the goods are of a description which it is in the course of the seller’s business to supply (whether he is the manufacturer or not, there is an implied condition that the goods shall be reasonably fit for such purpose.)

In Priest Vs. Last, P purchased a hot water bottle from a chemist. The chemist informed him that the bottle was specially meant for holding hot water. At the time of use, the bottle burst as soon as hot water was poured into it and injured P’s wife. Chemist was held liable to pay damages to P’.
However, this rule does not apply when the goods are sold under a patent or a brand name.

2. Sale of goods by description. [Sec. 16(2)]
Where there is a sale of goods by description, there is an implied condition that the goods are merchantable that is, fit for particular purpose.

3. Trade usage. [Sec. 16(3)]
An implied condition of fitness may be annexed to a contract of sale by usage of trade.
E.g. In readymade garment business, there is an implied condition by usage of trade that the garments shall be reasonably fit on the buyer.

4. Where the seller is guilty of fraud.
Where the seller makes a false representation and buyer relies on that representation, the doctrine of caveat emptor will not apply. In such a case the buyer will be entitled to the goods according to that representation.

5. Where seller actively conceals a defect
Where the seller actively conceals a defect in the goods so that the same could not be discovered on a reasonable examination, the doctrine of caveat emptor will not apply. Such a contract will be voidable.

6. Sale by sample
When goods are purchased by sample, the bulk must correspond with the sample and the buyer must have reasonable opportunity of inspecting the goods.

7. Sale by sample as well as description
The doctrine of Caveat Emptor is not applicable if the goods do not correspond to both, sample as well as description.

MULTIPLE CHOICE QUESTIONS:

1. Conditions are stipulations
(a) essential to the main purpose of the contract
(b) collateral to the main purpose of the contract
(c) either ‘a’ or ‘b’
(d) neither ‘a’ nor ‘b’

2. A warranty is stipulation
(a) essential to the main purpose of the contract
(b) collateral to the main purpose of the contract
(c) very important to the seller
(d) very important to the buyer

3. Breach of a condition gives rise to
(a) claim for damages
(b) a right to treat the contract as repudiated
(c) both ‘a’ and ‘b’
(d) either ‘a’ or ‘b’

4. Breach of a warranty gives rise to
(a) claim for damages
( b) a right to treat the contract as repudiated
(c) both ‘a’ and ‘b’
(d) either ‘a’ or ‘b’

5. What are implied stipulations of a contract?
(a) agreed by the parties.
(b) incorporated by law unless specifically agreed otherwise.
(c) implied by the circumstances
(d) implied by trade customs

6. If the condition as to the title of goods is not fulfilled, the buyer
(a) may reject the goods
(b) has no alternative but to buy the goods
(c) may reject the goods and claim damages
(d) all the above

7. In a sale by sample and description, there is an implied condition
(a) that bulk of the goods correspond with the sample
(b) that bulk of goods must correspond to the j description as well as the sample thereof
(c) the bulk of goods must correspond either to the description or to the sample
(d) the bulk of goods must correspond to the description only

8. “If you contract to sell peas, you cannot oblige a party to have beans”: this statement applies to
(a) a implied condition as to be description of goods
(b) the implied condition as to fitness of goods
for a particular purpose
(c) implied condition as to sample
(d) implied condition as to title

9. Under section 11 of the Sale of Goods Act, 1930, the time of payment can be of the essence of the contract —
(a) by agreement between the parties
(b) by operation of law
(c) both (a) and (b)
(d) either (a) or (b)

10. Whether or not any stipulation other than time of payment is of the essence of the contract depends upon —
(a) application of section 11
(b) operation of law
(c) terms of the contract
(d) all of the above

11. A stipulation in a contract of sale which is collateral to the main purpose of contract is called as
(a) guarantee
(b) warranty
(c) condition
(d) term

12. Merchantable quality of goods means
(a) that the goods are commercially saleable
(b) they are fit for the purpose for which they are generally used
(c) both ‘a’ and ‘b’
(d) the quality should be of high standard

Answers:
CA Foundation Business Laws Study Material Chapter 11 Conditions and Warranties 2

STATE WHETHER THE FOLLOWING ARE TRUE OR FALSE:

1. Where the buyer is deprived to goods by their true owner then the buyer may recover the price for breach of the condition as to title.
2. A stipulation essential to the main purpose of the contract is called as guarantee.
3. There is an implied condition that the goods shall be free from all encumbrances.
4. If a contract of sale contains a stipulation which has become impossible to perform the law excuses such a stipulation by reason of impossibility.
5. Breach of implied warranties leads to the repudiation of a contract.
6. Once the buyer decides to waive the condition he cannot insist on its fulfilment later on.
7. Implied condition as to description can be given by the type of packing.

Answers:
CA Foundation Business Laws Study Material Chapter 11 Conditions and Warranties 3

CA Foundation Business Laws Study Material Chapter 1 Nature of Contract

CA Foundation Business Laws Study Material Chapter 1 Nature of Contract

WHAT IS LAW?

Law is a mechanism for regulating the human conduct in a society. It consists of rules and principles enforced by an authority to regulate people’s behaviour with a view to secure justice, peaceful living and social security.

WHAT IS MERCANTILE LAW?

  • There are various branches of law such as civil law, criminal law, tax law, labour law, business law etc.
  • Mercantile Law, Commercial Law or Business Law is that branch of law, which regulates business and commercial transactions. It includes the laws relating to Contract, Sale of Goods, Partnership, Companies, Negotiable Instruments, Insurance, Carriage of goods etc.

LAW OF CONTRACTS

  • The law of contract forms the basis of the commercial/business law. It is concerned with enforceability of promises.
  • For example, if a supplier ‘S’ has promised to supply goods to a manufacturer ‘M’ on a spe¬cific date, there is a binding contract. Based on this promise, the manufacturer M will plan his production schedule and accept orders from his customers. Now if the supplier fails to supply the goods in time. (i.e. commits a breach of promise) M can claim damages for the loss he has suffered. Thus the purpose of the Law of Contract is to ensure that the expectations created by promises of the parties are fulfilled and obligations created by agreements are enforced.
  • In the absence of the Law of Contract it will be impossible to carry on trade and commerce. The businessman who has made a promise should fulfil it or else he will be liable to pay damages to the other party. The object of law of contract is to introduce certainty and defi¬niteness in business transactions. To quote Anson, “The law of contract is intended to ensure that what a man has been led to expect shall dome to pass; and that what has been promised to him shall be performed”.

(a) Applicability to business community as well as others
The law of contract is applicable not only to the business community, but also to others. Every one of us enters into contracts day after day. When you buy a book, or keep your vehicle at the cycle/ scooter stand or travel in a bus, or take a DVD for home viewing, in all these transactions of daily life, you are entering into a contract.

(b) Sources of Law of Contract

  • The law of contract in India is contained in the Indian Contract Act, 1872. The Act came into force on the first day of September, 1872 and it applies to the whole of India except the State of Jammu & Kashmir.
  • It mentions elements necessary for a valid contract; it says which persons are capable of entering into enforceable agreements; it mentions the cases in which agreements are avoid-able; it declares certain kinds of agreements void; it deals with performance of contract and it prescribes remedies for breach of contracts.
  • Apart from Indian Contract Act, 1872, the other sources of law of contract are: Judicial de-cisions or precedents; and customs and usages of trade. The decisions of the Supreme Court are binding on the lower courts. The judicial decisions constitute an important source of the law of contract, especially when the Act is silent on a point or there is ambiguity.
  • Customs /usages refer to a generally accepted practice or behaviour among members of a business community. A custom or usage to be legally binding must not be inconsistent with statutory law and must be widely known, certain and reasonable.
  • The Contract Act will prevail over any usage or custom of trade. However, any usage, custom or trade will be valid and binding as long as it is not inconsistent with the provisions of the Contract Act.

(c) The Act is not exhaustive
The Contract Act is not exhaustive. It does not deal with all the branches of the law of contract. There are separate Acts which deal with contracts relating to negotiable instruments, transfer of 8 property, sale of goods, partnership, insurance, etc. For example, if you are buying a house the law applicable will be the Transfer of Property Act while if you are buying a car, the governing law is g the Sale of Goods Act. The Partnership Act regulates the partnership agreements. The Contract Act thus, contains the general principles of contract and does not deal with contractual relationships H which are dealt under special statutes.

(d) What is the Scheme of the Act?
The scheme of the Act may be divided into two groups:
(a) General Principles of the law of contract (Secs. 1-75).
(b) Specific kinds of contracts, viz.;

  1. Contracts of Indemnity and Guarantee (Secs. 124-147).
  2. Contracts of Bailment and Pledge (Secs. 148-181)
  3. Contracts of Agency (Secs. 182-238).

Sections 76-123 relating to Contracts of Sale of Goods were repealed in 1930 and a separate Act called the Sale of Goods Act was enacted. Similarly, sections 239-266 relating to partnership were repealed in 1932 when the Indian Partnership Act was passed.

(e) The subject matter of contract can be discussed under the following heads

  • The Nature of contract.
  • Formation of contract i.e., how a contract is made, what things are necessary for the formation of a contract.
  • Operation of Contract, Le. whom the contract affects, and how the contract is performed.
  • Discharge of contract, Le. when the rights and obligations arising out of a contract are extin-guished.
  • Remedies for a breach of contract.

WHAT IS A CONTRACT?

According to Section 2(h) of the Indian Contract Act: “An Agreement enforceable by law is a con-tract”.
Thus a contract consists of two elements:

(a) An agreement
(b) Legal obligation Le. a duty enforceable by law.
CA Foundation Business Laws Study Material Chapter 1 Nature of Contract 1

(a) Agreement
An agreement is defined in section 2(e)
Every promise and every set of promises, forming the con-sideration for each other is an agreement”.
Now, what is promise?
Promise is defined as an accepted proposal, for section 2(b) says. “A proposal, when, accepted becomes promise ”. Thus an agreement is an accepted proposal  OR

      AGREEMENT = OFFER + ACCEPTANCE

The process of definition comes down to this:
An agreement comes into existence when one party makes a proposal or offer to the other party i=md that other party gives his acceptance thereto. Thus there should be exchange of promises. There must be two or more persons to make an agreement because one person cannot enter into an agreement with himself. There should also be consensus-ad-idem Le. both the parties must agree on the same thing in the same sense.
(b) Legal Obligation
For an agreement to become a contract, it must give rise to a legal obligation Le., a legal duty which is enforceable by law. The parties must have the intention to impose a duty on the promisor to fulfil the promise and bestow a right on the promisee to claim its fulfilment. This obligation must not be merely moral alone; it must be legal.

For example, A invites B to join his marriage party and B promises to do so. But B eventually fails to keep up his promise. In this case, there is a full-fledged agreement between A and B. But behind this agreement there is no intention on the part of the parties to impose a duty on the promisor (Le., A) and bestow a right on the promisee (Le., B) to claim the fulfilment of the contract. Therefore, the agreement is not enforceable by law.

ALL CONTRACTS ARE AGREEMENTS BUT ALL AGREEMENTS ARE NOT CONTRACTS

Agreement is the genus of which contract is the species. An agreement is a wider term than a contract. It may be a legal agreement (Le. enforceable by law) or a social agreement (Le. not enforceable by law). Agreements relating to social matters like an agreement to go to movie together or a visit to a hotel do not create legal obligations between the‘parties and hence are not contracts. Only those agreements grow into contracts, which create legal obligations.

DISTINCTION BETWEEN AGREEMENT AND CONTRACT

Sr. No.

AGREEMENT

CONTRACT

1.

Agreement is a promise. Offer and acceptance together constitute an agreement. Contract is an agreement enforceable by law.

2.

Agreement is a wider term. It is a genus. It includes legal as well as social agreement. Contract is a specie of an agreement. It is a narrower term.

3.

Agreement may not create any legal obligation. A contract necessarily creates a legal obligation.

4.

All agreements are not contracts. All contracts are agreements.

WHAT TYPE OF LEGAL OBLIGATIONS ARE DEALT WITH BY THE LAW OF CONTRACTS?
Obligations may arise from different sources. The law of contract deals only with such legal obli-gations which arise from agreements. Obligations which are not contractual in nature are outside the purview of the law contract. For example, obligation to observe traffic rules does not fall within the scope of the Contract Act.
The other sources of obligations are: obligations under the trust law or the law of tort or the fun-damental duties under the Constitution etc. They are outside the purview of the Contract law since they are not voluntarily created through an agreement. Salmond has rightly observed:

“The law of contracts is not the whole law of agreements, nor is the whole law of obligation.
It is the law of those agreements which create obligations and those obligations, which have their source in agreements.”

Contract creates Right in Personam
“The law of contract creates ‘right in personam’as against ‘right in rem. ”Right in personam means a right available against a particular person. For example, A buys TV from B for Rs. 20,000. B has a right to recover this amount. This right can be exercised only by B and only against A. This right of B is right in personam.

Right in rem
Right in rem means a right available against the whole world. If A is the owner of a house property he has the right of peaceful possession and enjoyment of the property against the whole world.

WHAT ARE THE ESSENTIAL ELEMENTS OF A VALID CONTRACT?

Section 10 provides “all agreements are contracts if they are made by the free consent of parties competent to contract for a lawful consideration and with a lawful object, and are not hereby expressly declared to be void”.
The essential elements or essentials of a valid contract (or enforceable agreement) are:

  • An offer or proposal by one party and an acceptance of that offer by another party resulting in an agreement.
  • An intention to create legal relations or an intent to have legal consequence.
  • Free consent between the parties.
  • The parties to contract are legally capable of contracting.
  • The object of the contract is legal and is not opposed to public policy.
  • The agreement is supported by consideration.
  • The agreement must not have been expressly declared to be void under the Act.
  • The terms of the contract are certain.
  • The agreement is capable of being performed, Le. it is not impossible to perform the contract.
  • Where agreement is required to be in writing under any law it must be in writing; and where both writing and registration are required by some Act or Law, the agreement must be in writing and registered.

Offer and acceptance
There must be a “lawful offer” and a “lawful acceptance” of the offer, thus resulting in an agree-ment. The adjective lawful implies that the offer and acceptance must satisfy the requirements of the Contract Act in relation thereto.

Intention to create legal relations
There must be an intention among the parties that the agreements should be attended by legal consequences and create legal obligations. Agreements of a social or domestic nature do not con-template a contract. An agreement to dine at a friend’s house is not an agreement intended to create legal relations and therefore is not a contract.

Balfour Vs Balfour, 1919, 2 KB 571.
Mr. & Mrs. Balfour who were living in Ceylon went to England. Mrs. Balfour fell ill. Mr. Balfour had to come back to Ceylon to join his duties. However he promised to pay 30 pounds per month to his wife. On his failure to pay, Mrs. Balfour sued him for the recovery of the amount. It was held that it was a domestic agreement and the husband never intended to create any legal rela-tions out of it.
In commercial agreements an intention to create legal relations is presumed. Thus, an agreement to buy and sell goods intends to create legal relationship, and hence is a contract, provided other requisites of a valid contract are present.

Lawful consideration
Consideration means ‘something in return.’ An agreement is enforceable when each of the parties to it gives something and gets something in return. If A agrees to sell his house to B for Rs. 5 lac, the consideration for A’s promise is Rs. 5 Lac and B’s promise is a house. Thus consideration is the price paid by one party for the promise of the other. The payment of money is a common form of consideration. But it may also consist of an act, forbearance, and a promise to do or not to do something. Consideration must be real, valuable and lawful.

Capacity of parties
The parties to an agreement must be competent to contract; otherwise it cannot be enforced by a court of law. Every person is competent to contract who is
(a) of the age of majority,
(b) of sound mind and
(c) is not disqualified from contracting by any law. (Sec. 11)

Free consent
The consent of the parties must be free i.e. the parties should enter into contract voluntarily and free will. Section 14 lays down that consent is not free if it is caused by
(a) coercion,
(b) undue influence,
(c) fraud,
(d) misrepresentation or
(e) mistake.

Lawful object
The object of the agreement should be lawful. It should be authorised or sanctioned by law. The object of an agreement is unlawful if it is forbidden by law or is fraudulent or is immoral or opposed to public policy. For example a “supart” contract for unlawful recovery of money or a smuggling agreement is unlawful hence unenforceable.

Agreement not expressly declared void
The Indian Contract Act, 1872, has expressly declared certain agreements to be not enforceable at law, e.g. agreements in restraint of marriage, agreements in restraint of trade, wagering agreements etc. The parties to the agreement should ensure that their agreement do not fall in the category of these void agreements, otherwise the agreement will not be enforceable even if all the other essentials of valid contract are present.

Certainty
The terms of the contract should be certain and definite and not vague. Section 29 says “Agree-ments, the meaning of which is not certain or capable of being made certain are void.” For example, A agrees to sell B “a hundred tons of oil”. There is nothing whatever to show what kind of oil was intended. The agreement is not enforceable because it is vague and uncertain.

Possibility of performance
Yet another essential feature of a valid contract is that it must be capable of performance. Section 56 lays down that “An agreement to do an act impossible in itself is void.” If the act is impossible in itself, physically or legally, the agreement cannot be enforced at law. For example, A agrees with B to discover treasure by magic. The agr eement is void due to impossibility.

Writing and registration
According to the Indian Contract Act, a contract may be oral or in writing. An oral contract is as much enforceable as a written contract. However, if there is a provision in any law prescribing that contracts should be in writing/registered then, this formality of writing and registration should be followed.

For example, in certain special cases the Contract Act prescribes that the contract should be in writing or/and registered. Section 25 of the Contract Act requires that an agreement to pay a time barred debt must be in writing and an agreement to make a gift for natural love and affection must be in writing and registered.

Similarly, certain other Acts also require writing or/and registration to make the agreement enforce-able by law which must be complied with.
Thus

  1. an arbitration agreement must be in writing as per the Arbitration Act, 1996,
  2. an agreement for a sale of immovable property must be in writing and registered under the Transfer of Property Act, 1882 before they can be legally enforced,
  3. for example, contract with the Government should be in writing. Article 299, Constitution of India.

KINDS OF CONTRACTS

On the basis of enforceability or validity a contract can be classified under following heads:
(a) Valid Contracts
(b) Void Agreement
(c) Voidable Contract
(d) Void Contract
(e) Unenforceable Contract
(f) Illegal or Unlawful Agreement

On the basis of Formation a contract can be classified as:
(g) Express Contract
(h) Implied Contract
(i) Quasi-Contract
(k) E.com. Contract

On the basis of performance it can be classified as:
(k) Executed Contract
(l) Executory Contract

Executory contract can further be classified as:
(m) Unilateral Contract
(n) Bilateral Contracts

(a) Valid Contract
A valid contract is one which contains all the essential elements of a valid contract. It is an agree¬ment which is binding and enforceable by law.

(b) Void agreement
“An agreement not enforceable by law is said to be void. [Sec. 2(g)]

Features
(a) A void agreement does not give rise to any legal consequences. It is void ab-initio, Le., from the very beginning. If any of the essentials of a valid contract, other than free consent, is missing, the agreement is void, Le., it cannot be enforced at courts of law. For example, an agreement with a minor or an agreement without consideration.
(b) Certain agreements have been expressly declared as void by the Indian Contracts Act, in sections 11, 20, 23-30 and section 56.
(c) There cannot be restitution of benefit under a void agreement and if something has been paid it cannot be recovered. However, when an agreement is discovered to be void or when a contract becomes void, any person who has received any advantage under such agreement or contract is bound to restore it, or to make compensation for it, to the person from whom he received it. (Sec. 65).
For example, A pays B Rs. 50,000 in consideration of B’s promising to sell his car to him. The car is destroyed in an accident at the time of the promise though neither party was aware of the fact. In this case the agreement is discovered to be void and B must repay A Rs. 50,000. It should be noted that when the agreement is known to be void, no restitution is allowed. Thus if A pays Rs. 10,000 to B to assault. C, the money cannot be recovered.

(c) Voidable contract
An agreement which is enforceable by law at the option of one or more of the parties thereto, but not at the option of the other or others, is a voidable contract.” [Sec. 2(i)].

Features

  • A voidable contract is enforceable at the option of one party. For ex. if X is forced to sign a contract the contract is voidable at the option of X. X may either rescind (avoid or repudiate) the contract or elect to be bound by it.
  • A voidable contract continues to be good until it is avoided by the party entitled to do so.
    The aggrieved party must exercise his option of rejecting the contract

    • within a reasonable time and
    • before the rights of third parties intervene, otherwise the contract cannot be repudiated.
  • The party rescinding a voidable contract shall if he has received any benefit thereunder from another party to such contract, restore such benefit, so far as may be, to the person from whom it was received (Sec. 64)

The various circumstances in which a contract is voidable are depicted by the following chart:

CA Foundation Business Laws Study Material Chapter 1 Nature of Contract 2

VOIDABLE IN INCEPTION

VOIDABLE BY SUBSEQUENT DEFAULT

Consent caused by coercion, undue influence, fraud, mis­representation (Secs. 14, 19 A) ♦ Where offer of performance is not accepted (Sec. 38)
♦ When one party prevents performance of reciprocal promise. (Sec. 53)
♦ When a party fails to perform at the time fixed, if time is essence of the contract (Sec. 55)

(d) Void contract
“A contract which ceases to be enforceable by law becomes void when it ceases to be enforceable”. [Section 2(j)].

Features
a. The term void contract appears to be contradictory, but it is a nice way of describing a situation where a contract is valid in the beginning but becomes void subsequently. Note that a CONTRACT BECOMES VOID. IT IS NEVER VOID AB INITIO.
b. A void contract is one, which was valid when it was made but becomes void later on. For example, A agrees to supply liquor to B but before he gives delivery, the Government declares total prohibition. The contract becomes void. A void contract is not void from its inception and its valid and binding on the parties when originally entered but subsequent to its formation it becomes invalid.
c. Restitution of benefit allowed when contract becomes void: According to Section 65 when a contract becomes void, the party who received any advantage under such agreement, should restore or make compensation for it to the party from whom he received it. For example, A takes an advance of Rs. 1000 for singing at a concert for B. A is too ill to sing. A must refund to B the 1000 rupees paid in advance.

The reasons which transform a valid contract into a void contract as given in the Contract Act are as follows:

  • Supervening impossibility (Section 56): A Contract becomes void if it becomes impossible to perform, after it is made. A and B contracted to marry each other. Before the time fixed for the marriage A goes mad. In this case the contract becomes void due to subsequent impossibility.
  • Subsequent illegality (Section 56): A contract becomes void if it becomes illegal after it is made. A agrees to sell B 100 bags of wheat at Rs. 550 per bag. Before delivery, the government bans private trading in wheat. The contract becomes void due to subsequent illegality.
  • Repudiation of a voidable contract: When a voidable contract is rescinded, the contract be¬comes void.
  • Subsequent impossibility of contingent event (Sec. 32): A contingent contract to do or not to do something on the happening of an uncertain future event, becomes void, when the event becomes impossible.

The following are the points of differences between various types of contracts discussed above:

Void agreement

Void contract

It is void ab-initio It is not void ab-initio. Initially a valid contract comes into existence but it becomes void and unenforceable later on due to reasons like impossibility of performance, illegality etc.

Void agreement

Void contract

No restitution of benefit is allowed When a contract becomes void, restitution of benefit is allowed under section 65.

The legal effect of void agreements and void contract is the same. Both cannot be enforced in a Court of Law. Note that a contract cannot be void ab-initio and only an agreement can be void ab-initio.

Void agreement

Voidable contract

It is void ab-initio It is not void ab-initio. It becomes void and unenforceable only when the aggrieved party chooses to void it.
No contract comes into existence Contract comes into existence and remains valid unless it is avoided.
No restitution of benefit is allowed The party rescinding the contract shall restore the benefit, if he has received any, to the other party under section 64.
No question of compensation since a void agreement has no legal effect. If a party rightfully avoids the contract it can claim compensation from other party for loss suffered by him on account of non­performance of contract.
A third party cannot acquire any title to the goods under a void agreement. A third party acquires a valid title to the goods obtained under a voidable contract if it has been obtained in good faith for a value and before the contract is avoided.

Void contract

Voidable contract

A void contract is one which is valid when it is made but becomes void later on A voidable contract is one, which is enforceable by law at the option of one of the parties.
A void contract cannot be enforced A voidable contract can be enforced if the aggrieved party elects to carry out the contract.
A contract becomes void due to certain reasons like impossibility of perfor­mance, subsequent illegality etc. A contract becomes voidable, if consent is caused by coercion, un­due influence, fraud and misrepresentation or failure to perform at the time fixed if time is essence of the contract.
Compensation is not payable except only when party knows beforehand about the impossibility of the performance In a voidable contract the aggrieved party can claim damages.

(e) Unenforceable contract
An unenforceable contract is one, which suffers from some technical defect. It is valid in itself, but Is not capable of being enforced in a court of law because of non-observance of some technical formalities such as insufficiency of stamp, want of registration, attestation etc. In some cases such contracts can be enforced if their technical defects are removed, for example, the defect of under stamping can be removed by affixing the right value of stamps.

(f) Illegal or unlawful agreement
An illegal agreement is one, which is contrary to law. According to section 23 an agreement is illegal and void if its object or consideration.

  1. is forbidden by law, or
  2. is of such a nature that, if permitted, it would defeat the provisions at any law, or
  3. is fraudulent, or
  4. involves or implies injury to the person or property of another, or
  5. the court regards it as immoral or opposed to public policy (Sec. 23)

An illegal agreement may attract punishment and prosecution under criminal law. An agreement which is collateral to an illegal agreement also becomes illegal. It is like an contagious disease and is fatal not only to the main contract but to collateral transactions as well.

Difference between Void & Illegal agreements 
a. Scope : An illegal agreement is narrower in scope than a void agreement. All illegal agreements are void but all void agreements are not necessarily illegal. E.g. an agreement with a minor is void, but not illegal.
b. Collateral Transactions : When an agreement is illegal, other agreements which are incidental or collateral to it are also tainted with illegality, hence void.
Example: India and Pakistan are playing test match in Nagpur. X of Nagpur, agrees to pay Rs. 1 lac to Y, if India wins. The match is won by India and in order to pay Y, X borrows 11 lac from Z, who is aware of the purpose.
The agreement between X and Y is void being wagering (betting) agreement and it is also illegal in Maharashtra. The agreement between X and Z being collateral agreement is also void because the main agreement is between X and Y is illegal.
c. Restitution: In the case of illegal agreement, no right/remedy is available to either party. Hence money paid under an illegal agreement cannot be recovered. Under sec. 65 if an agreement is discovered to be void any person who has received advantage/benefit must restore it or make compensation for it.
d. Punishment: In case of an illegal agreement the parties may be punished under the criminal law, in case of a void agreement (which is not illegal) there is no such punishment.

(g) Express contract
An express contract is created by the words of the parties, whether oral or written. Section 9 of the Act provides that if a proposal or acceptance of any promise is made in words the promise is said to be express. For example: A tells B that he offers to sell his house for Rs. 20 lakhs and B replies that he accepts the offer.

(h) Implied contract
An implied contract is created by implication of law or by the conduct of the parties. For example; A coolie in uniform picks up the luggage of Mr. S to be carried out of the railway station without being asked by S and S allows him to do so. Here, S is compelled to pay to the coolie for his services.
Tacit Contracts: Tacit means Silent. These are the contracts that are inferred through action of the conduct of the parties without any words spoken or written. For example; Mr. V steps into a bus to go to a certain location. V is bound to pay the fare, although he has not in words promised to do so. Other examples of Tacit contracts are obtaining cash from an ATM, sale by fall of hammer at an auction sale etc.
Tacit contracts are not separate forms of contracts but they fall within the scope of implied contracts.

(i) Quasi-Contract
Quasi contract is a contract in which there is no intention on the part of either party to make a contract but law imposes a contract upon parties. These are not actual contracts but they resemble a contract which is created by law under certain circumstances. Here, law creates legal rights and obligations when there is no real contract. For example; obligation of finder of lost goods to return them or liability of person whom money is paid by,mistake to repay it back.

(j) E-Com Contract
These are also known as e-commerce contracts, EDI contracts, Cyber contracts, mouse click con¬tracts or e-contracts. These contracts are created by parties using electronic means such as email. Different parties create networks which are linked to other networks through Electronic Data Interchange (EDI). When you buy a mobile phone from an online shopping website or through a mobile application, you enter into an e-contract.

(k) Executed contracts
An executed contract is one that has been performed by all parties. A buys a TV set from B for Rs. 20,000. A pays the price and B delivers the TV. It is an executed contract. Both the parties have performed their respective obligations.

(l) Executory contracts
An executory contract is one where both the parties have still to perform their respective contractual obligations. A contract may be partly executed and partly executory. For example: A contracts to sell and deliver a TV to B for Rs. 20,000 to be paid in 3 weeks. A delivers the TV. The contract is executed as to A, executory as to B, as B has not yet paid the agreed price.

(m) Unilateral contracts
In case of a unilateral contract, only one partly has to perform his obligation and the other party has performed his obligation at the time of formation of contract or before. If A buys a railway ticket for his journey from Nagpur to Bombay. A has performed his duty under the contract by paying the fare but the railways are yet to perform their promise ie. of carrying him from Nagpur to Bombay. A unilateral contract is partly executed and partly executory. Such contracts are also called as contracts with executed consideration or one-sided contracts.

(n) Bilateral contracts
A bilateral contract is one in which both the parties are yet to perform their respective obligations at the time of formation of contract. They are similar to executory contracts and are called as con¬tracts with executory consideration.

(o) Formal and simple contracts
This classification is made in the English Law.

FORMAL CONTRACT
Formal Contract is expressed in a particular form. Its validity depends on form alone. It is in writing. The signature is usually attested Le. witnessed. No consideration is necessary. The Indian Contract Act does not recognize these contracts since consideration is a necessary element in a contract subject to certain exceptions mentioned in Sec. 25.
Formal contracts can be sub-divided into:

(a) Contract of Record
(b) Contracts under seal

(a) Contract of Record: A contract of record consists of either a judgment of a court or recognizance. They derive their binding force from the authority of the Court.
A COURT JUDGMENT on being recorded is called a contract of record. It is an obligation imposed upon the parties by the court as a judicial authority. It is not a contract in the real sense since it is not based on any agreement.
RECOGNISANCE is conditional judgment arising in criminal proceedings binding a person to be of good behaviour or to appear as a witness, subject to a money penalty if the obligation is broken. It sort of a written acknowledgement to the State by an accused that on his default to be of good conduct etc. he is bound to pay to the State a certain some of money.

(b) Contract under seal: They are also called as specialty contracts or deeds. All the terms of such contracts are reduced to writing and then the contract is signed, sealed and delivered. Consideration is not essential to support a deed or a contract under a seal.

SIMPLE CONTRACTS
These contracts are also called as parol contracts. This class includes all contracts not under seal and for their enforcement they require the fulfilment of the essential elements of the contract Le. consideration, free consent etc. Simple contracts may be made orally or in writing.

Multiple Choice Questions:

Question 1.
Law of contract
(a) Is the whole law of obligations
(b) Is the whole law of agreements
(c) Deals with only such legal obligation which arise from agreement
(d) Deals with social agreements
Answer:
(c).

Question 2.
Social agreements are
(a) Enforceable in the courts
(b) Not enforceable in the courts
(c) Subject to legal obligations
(d) Made by social workers.
Answer:
(b).

Question 3.
All contracts
(a) Are agreements
(b) Are not agreements
(c) Do not have legal obligations
(d) Should be in writing
Answer:
(a).

Question 4.
Obligation between parties that form contract
(a) Are all kinds of obligations
(b) Are legal obligation which spring from agreements
(c) Are not voluntary in nature
(d) None of the above
Answer:
(b).

Question 5.
A contract means an agreement
(a) Which is enforceable by law
(b) Which is not enforceable by law
(c) Which creates social obligation.
(d) Which is in writing.
Answer:
(a).

Question 6.
Voidable contract
(a) are enforceable by law if they are not avoided
(b) are not enforceable by law
(c) can be enforced if the court directs
(d) can be enforced with prior permission of Court/ Government
Answer:
(a).

Question 7.
The terms of agreement
(a) must be certain
(b) must be capable of made certain
(c) unambiguous and clear
(d) all the above.
Answer:
(d).

Question 8.
All illegal agreements
(a) are not void
(b) are not void ab-initio
(c) are void
(d) none of the above
Answer:
(c).

Question 9.
A void agreement
(a) is illegal
(b) is not void ab-inition
(c) may or may not be illegal
(d) none of the above
Answer:
(c).

Question 10.
All kinds of obligations between the parties form part of the contract. This statement is
(a) True
(b) False
(c) Partially true
(d) None of the above
Answer:
(b).

Question 11.
A contract is made where:
(a) A buys a book from a shop
(b) X bids at a public auction.
(c) X agrees with Y to discover a treasure by magic.
(d) Z agrees to attend the birthday party of his friend.
Answer:
(a).

Question 12.
Right in rem implies:
(a) a right available against the whole world.
(b) a right available against a particular individual.
(c) a right available against the Government.
(d) none of the above
Answer:
(a).

Question 13.
A void contract
(a) is void from the very beginning.
(b) is valid in the beginning but becomes void later on.
(c) is enforceable at the option of one of the contracting parties only.
(d) none of the above.
Answer:
(b).

Question 14.
A void agreement is one
(a) which is forbidden by law
(b) enforceable at the option of one of the parties
(c) which is not enforceable by law
(d) enforceable by law
Answer:
(c).

Question 15.
An agreement created by words spoken or written is called
(a) express agreement
(b) execute agreement
(c) implied agreement
(d) voidable agreement
Answer:
(a).

Question 16.
Which of the following statements is false
(a) Law of contract is the whole law of obliga¬tions
(b) Certain contracts must be in writing
(c) All contracts are agreements
(d) All illegal agreements are void
Answer:
(a).

Question 17.
Parol contracts are also known as
(a) Simple Contract
(b) Format Contract
(c) Void Contract
(d) Conditional Contract
Answer:
(a).

Question 18.
A agree to sell to B a hundred tons of oil. There is nothing whatever to show what kind of oil was intended. The agreement is
(a) Valid
(b) Void for uncertainty
(c) Voidable
(d) Illegal
Answer:
(b).

Question 19.
A agrees to sell to B my white horse for Rs. 500 or Rs. 1,000. There is nothing to show which of the two prices was to be given. The agreement is
(a) Valid
(b) Void
(c) Voidable
(d) Unenforceable
Answer:
(b).

State Whether The Following Are True Or False:

Question 1.
An agreement and a contract are one and the same thing.
Answer:
False.

Question 2.
Law of Contract is the whole law of obligations.
Answer:
False.

Question 3.
Social agreements are enforceable in the Courts.
Answer:
False.

Question 4.
All contracts should be in writing.
Answer:
False.

Question 5.
A foreigner is not competent to enter into a contract.
Answer:
False.

Question 6.
All contracts are not agreements.
Answer:
False.

Question 7.
The Indian Contract Act, 1872 is a complete code on contracts.
Answer:
False.

Question 8.
Precedents (Judicial decisions) form an important source of mercantile law.
Answer:
True.

Question 9.
Voidable contracts are enforceable by law, if they are not avoided.
Answer:
False.

Question 10.
Mercantile law is applicable to businessman only.
Answer:
False.

Question 11.
An agreement the meaning of which is not capable of being made certain is void.
Answer:
True.

Question 12.
An unenforceable contract can be enforced if the technical defect is removed.
Answer:
True.

Question 13.
A contract is a contract from the time its performance is due and not from the time it is made.
Answer:
False.

Question 14.
A contract consists in actionable promise or promises.
Answer:
True.

Question 15.
An agreement with intention to create legal liability is not enforceable in law.
Answer:
False.

Question 16.
In an executed contract both parties have yet to fulfil their obligations.
Answer:
False.

Question 17.
All illegal agreements are void but all void agreements are not necessarily illegal.
Answer:
True.

Question 18.
All void agreements are illegal.
Answer:
False.

Question 19.
There cannot be a contract to make a contract.
Answer:
True.

Question 20.
All kinds of obligations between the parties form part of the contract.
Answer:
False.

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3

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 5 Quadratic Equations in One Variable Ex 5.3

More Exercises

Solve the following (1 to 8) equations by using formula :

Question 1.
(i) 2x² – 7x + 6 = 0
(ii) 2x² – 6x + 3 = 0
Solution:
(i) 2x² – 7x + 6 = 0
Here a = 2, b = -7, c = 6
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q1.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q1.2

Question 2.
(i) x² + 7x – 7 = 0
(ii) (2x + 3)(3x – 2) + 2 = 0
Solution:
(i) x² + 7x – 7 = 0
Here a = 1, b = 7, c = -7
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q2.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q2.2

Question 3.
(i)256x² – 32x + 1 = 0
(ii) 25x² + 30x + 7 = 0
Solution:
(i) 256x² – 32x + 1 = 0
Here a = 256, b = -32, c = 1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q3.1

Question 4.
(i) 2x² + √5x – 5 = 0
(ii) √3x² + 10x – 8√3 = 0
Solution:
(i) 2x² + √5x – 5 = 0
Here a = 2, b = √5, c = -5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q4.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q4.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q4.3

Question 5.
(i) \(\frac { x-2 }{ x+2 } +\frac { x+2 }{ x-2 } =4\)
(ii) \(\frac { x+1 }{ x+3 } =\frac { 3x+2 }{ 2x+3 } \)
Solution:
(i) \(\frac { x-2 }{ x+2 } +\frac { x+2 }{ x-2 } =4\)
⇒ \(\frac { { (x-2) }^{ 2 }+{ (x+2) }^{ 2 } }{ (x+2)(x-2) } =4\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q5.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q5.2

Question 6.
(i) a (x² + 1) = (a² + 1) x , a ≠ 0
(ii) 4x² – 4ax + (a² – b²) = 0
Solution:
(i) a (x² + 1) = (a² + 1) x
ax² – (a² + 1)x + a = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q6.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q6.2

Question 7.
(i)\(x-\frac { 1 }{ x } =3,x\neq 0\)
(ii)\(\frac { 1 }{ x } +\frac { 1 }{ x-2 } =3,x\neq 0,2\)
Solution:
(i)\(x-\frac { 1 }{ x } =3\)
x² – 1 = 3x
⇒ x² – 3x – 1 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q7.2

Question 8.
\(\frac { 1 }{ x-2 } +\frac { 1 }{ x-3 } +\frac { 1 }{ x-4 } =0\)
Solution:
\(\frac { 1 }{ x-2 } +\frac { 1 }{ x-3 } +\frac { 1 }{ x-4 } =0\)
⇒ \(\frac { 1 }{ x-2 } +\frac { 1 }{ x-3 } =-\frac { 1 }{ x-4 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q8.2

Question 9.
Solve for \(x:2\left( \frac { 2x-1 }{ x+3 } \right) -3\left( \frac { x+3 }{ 2x-1 } \right) =5,x\neq -3,\frac { 1 }{ 2 } \)
Solution:
\(x:2\left( \frac { 2x-1 }{ x+3 } \right) -3\left( \frac { x+3 }{ 2x-1 } \right) =5 \)
Let \(\frac { 2x-1 }{ x+3 } =y\) then \(\frac { x+3 }{ 2x-1 } =\frac { 1 }{ y } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q9.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q9.2

Question 10.
Solve the following equation by using quadratic equations for x and give your
(i) x² – 5x – 10 = 0
(ii) 5x(x + 2) = 3
Solution:
(i) x² – 5x – 10 = 0
On comparing with, ax² + bx + c = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q10.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q10.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q10.3

Question 11.
Solve the following equations by using quadratic formula and give your answer correct to 2 decimal places :
(i) 4x² – 5x – 3 = 0
(ii) 2x – \(\\ \frac { 1 }{ x } \) = 1
Solution:
(i) Given equation 4x² – 5x – 3 = 0
Comparing with ax² + bx + c = 0, we have
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q11.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q11.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q11.3

Question 12.
Solve the following equation: \(x-\frac { 18 }{ x } =6\). Give your answer correct to two x significant figures. (2011)
Solution:
\(x-\frac { 18 }{ x } =6\)
⇒ x² – 6x – 18 = 0
a = 1, b = -6, c = -18
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q12.1

Question 13.
Solve the equation 5x² – 3x – 4 = 0 and give your answer correct to 3 significant figures:
Solution:
We have 5x² – 3x – 4 = 0
Here a = 5, b = – 3, c = – 4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 Q13.1

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.3 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 7 Ratio and Proportion MCQS

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion 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 7 Ratio and Proportion MCQS

More Exercises

Choose the correct answer from the given options (1 to 10):

Question 1.
The ratio of 45 minutes to \(5 \frac { 3 }{ 4 } \) hours is
(a) 180:23
(b) 3:23
(c) 23:3
(d) 6:23
Solution:
ratio of 45 minutes to \(5 \frac { 3 }{ 4 } \) hours is
45 minutes to : \(5 \frac { 3 }{ 4 } \) hours
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion MCQS Q1.1

Question 2.
The ratio of 4 litres to 900 mL is
(a) 4 : 9
(b) 40 : 9
(c) 9 : 40
(d) 20 : 9
Solution:
4l : 900 ml
= 4000 ml : 900 ml
= 4000 : 900
= 40 : 9 (b)

Question 3.
When the number 210 is increased in the ratio 5 : 7, the the new number is
(a) 150
(b) 180
(c) 294
(d) 420
Solution:
210 is increased in the ratio 5 : 7, then
New increased number will be
= 210 × \(\\ \frac { 7 }{ 5 } \)
= 294 (c)

Question 4.
Two numbers are in the ratio 7 : 9. If the sum of the numbers is 288, then the smaller number is
(a) 126
(b) 162
(c) 112
(d) 144
Solution:
Ratio in two number = 7 : 9
Sum of numbers = 288
Sum of ratios = 7 + 9
= 16
Smaller number = \(\\ \frac { 288\times 7 }{ 16 } \)
= 126 (a)

Question 5.
A ratio equivalent to the ratio \(\\ \frac { 2 }{ 3 } \) : \(\\ \frac { 5 }{ 7 } \) is
(a) 4:6
(b) 5:7
(c) 15:14
(d) 14:15
Solution:
\(\\ \frac { 2 }{ 3 } \) : \(\\ \frac { 5 }{ 7 } \)
Multiply and divide \(\\ \frac { 2 }{ 3 } \) by 7 and
Multiply and divide \(\\ \frac { 5 }{ 7 } \) by 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion MCQS Q5.1

Question 6.
The ratio of number of edges of a cube to the number of its faces is
(a) 2 : 1
(b) 1 : 2
(c) 3 : 8
(d) 8 : 3
Solution:
No. of edges of the cube = 12
No. of faces = 6
Ratio in edges a cube to the number of faces = 12 : 6
= 2 : 1 (a)

Question 7.
If x, 12, 8 and 32 are in proportion, then the value of x is
(a) 6
(b) 4
(c) 3
(d) 2
Solution:
x, 12, 8, 32 are in proportion, then
x × 32 = 12 × 8 (∵ ad = bc)
⇒ x = \(\\ \frac { 12\times 8 }{ 32 } \) = 3
x = 3 (c)

Question 8.
The fourth proportional to 3, 4, 5 is
(a) 6
(b) \(\\ \frac { 20 }{ 3 } \)
(c) \(\\ \frac { 15 }{ 4 } \)
(d) \(\\ \frac { 12 }{ 5 } \)
Solution:
The fourth proportion to 3, 4, 5 will be
= \(\\ \frac { 4\times 5 }{ 3 } \)
= \(\\ \frac { 20 }{ 3 } \) (b)

Question 9.
The third proportional to \(6 \frac { 1 }{ 4 } \) and 5 is
(a) 4
(b) \(8 \frac { 1 }{ 2 } \)
(c) 3
(d) none of these
Solution:
The third proportional to \(6 \frac { 1 }{ 4 } \) and 5 is
⇒ \(6 \frac { 1 }{ 4 } \) : 5 :: 5 : x
⇒ \(\\ \frac { 25 }{ 4 } \) : 5 :: 5 : x
⇒ x = \(\\ \frac { 5\times 5 }{ 25 } \) × 4
⇒ 4 (a)

Question 10.
The mean proportional between \(\\ \frac { 1 }{ 2 } \) and 128 is
(a) 64
(b) 32
(c) 16
(d) 8
Solution:
The mean proportional between \(\\ \frac { 1 }{ 2 } \) and 128 is
= \(\sqrt { \frac { 1 }{ 2 } \times 128 } \)
= √64
= 8 (d)

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion 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.

CA Foundation Business Laws Study Material Chapter 10 Formation of Contract of Sale

CA Foundation Business Laws Study Material Chapter 10 Formation of Contract of Sale

INTRODUCTION

The Sale of Goods Act, 1930, governs transfer of property in goods. It does not include transfer of immovable property which is governed by the Transfer of Property Act, 1882.

  • Contract of Sale of Goods is a special contract. Originally, it was part of Indian Contract Act itself in chapter VII (sections 76 to 123). Later these sections in Contract Act were deleted, and separate Sale of Goods Act was passed in 1930.
  • The Sale of Goods Act, 1930, contains 66 sections in VII Chapters. It came into force on the 1st of July 1930 as, ‘The Indian Sale of Goods Act, 1930’. Later in 1963, the word “Indian” was omitted and it became “The Sale of Goods Act, 1930”.
  • The Sale of Goods Act, extends to the whole of India except the State of Jammu and Kashmir.
  • As per section 3 of the Sale of Goods Act, the principles of the Contract Act relating to formation of contract, performance of contract, law of damages etc. are also applicable to contract of the sale of goods insofar as they are not inconsistent with the express provisions of the Sale of Goods Act.

A. WHAT IS A CONTRACT OF SALE?
Sec. 4(1) of the Sale of Goods Act defines a contract of sale of goods as -“a contract whereby the seller transfers or agrees to transfer the property in goods to the buyer for a price”.
 contract of sale of goods, like any other contract, results by an offer by one party and its acceptance by the other. The parties are free to decide the terms and conditions of performance of their contract. Wherever the contract is silent, rules provided by the Sale of Goods Act apply to the relevant issue.

Buyer means a person who buys or agrees to buy goods. [Sec. 2(1)]
Seller means a person who sells or agrees to sell goods. [Sec. 2(13)]
Property means the general property in goods, and not merely a special property. Sec. 2(11). Property means ownership. If A who owns goods pledges them for raising money to B, A has the general property in the goods, while B (pledgee, person with whom goods are pledged) has a special property or interest in them, e.g. pledgee has a right to retain the pledged goods until he is paid by A (pledgor) the entire amount of his loan with interest.
CA Foundation Business Laws Study Material Chapter 10 Formation of Contract of Sale 1

Essential characteristics of a contract of sale

  1. Two parties – there must be two parties a buyer and a seller.
  2. Transfer of property – a transfer of property i.e. ownership, in goods from the seller to the buyer must take place (in the case of sale) or ownership should be agreed to be transferred (in the case of agreement to sell)
  3. Goods – the subject matter of sale must be goods.
  4. Price – transfer of property must take place for some money consideration called price.
  5. It includes both a ‘sale’and ‘an agreement to sell’.
  6. A contract of sale may be absolute or conditional [Sec. 4(2)].
  7. It may be in writing/oral or implied
  8. Essential elements of a valid contract must be present.

B. SALE & AGREEMENT TO SELL
A contract for the sale of goods may be either a sale or an agreement to sell.
Sale
Where under a contract of sale the property in the goods (Le. the ownership) is transferred from the seller to the buyer the contract is called a sale. Sec. 4(3). The transaction is a sale even though the price is payable at a later date or delivery is to be given in the future, provided the ownership of the goods is transferred from the seller to the buyer.
Example: S makes a contract with P for sale of his Nano Car for Rs. 80,000. P makes the payment and takes the delivery of car. This is the transaction of sale where the ownership has passed from S to P for a price.

Agreement to sell
When the transfer of ownership is to take place at a future time or subject to some condition to be fulfilled later, the contract is called an agreement to sell. [Sec. 4(3)]
Example: S agrees to sell his Car to P for Rs. 2,00,000 after one month. P agrees to buy the car and make payment after one month. This an agreement to sell and it will become a sale after one month when P make the payment and gets the ownership of car.
The conditional contract of sale like goods sent op “sale or return” basis are in the nature of an agreement to sell.

When an agreement to sell becomes a sale?
An agreement to sell becomes a sale when the prescribed time elapses or the conditions, subject to which the property in the goods is to be transferred, are fulfilled. [Sec. 4(4)]
Thus, if goods are delivered to the buyer on approval Le. “on sale or return”, the transaction is an agreement to sell, but it becomes a sale and the property in the goods passes to the buyer where the buyer gives his approval or acceptance to the seller.

SALE                  

AGREEMENT TO SELL

1. Transfer of property

The title to the goods passes to the buyer immediately. The title to the goods passes to the buyer on future date or on fulfilment of some condition.

2. Nature of Contract

It is an executed contract. It is an executory contract.

3. Burden of risk

Risk of loss is that of buyer since risk follows ownership. Risk of loss is that of seller.

4. Nature of rights

It creates jus in rem that is the buyer as a owner gets the right to enjoy the goods against the whole world. If the seller refuses to deliver the goods the buyer may sue for recovery of goods by specific performance. It creates jus in personam that is the buyer has only a personal remedy against the seller. He can sue only for damages for breach and not for recovery of goods.

5. Remedies for breach

If the buyer fails to pay for the goods, the seller may sue for the price (suit for price sec. 55) and also has other remedies available to an unpaid seller. If the buyer fails to accept and pay for the goods, the seller can only sue for damages and not for price. (Damages for non­acceptance sec. 56)

6. Insolvency of Buyer

If the buyer becomes insolvent before paying the price, the seller shall have to deliver the goods to the Official Receiver on his demand because the ownership of the goods has passed to the buyer. Since the seller continues to be the owner, he can refuse to deliver the goods to the Official Receiver unless he is paid the price because the seller continuous to be the owner of the goods.

7. Insolvency of Seller

If the seller becomes insolvent while the goods are still in his possession, the buyer shall have a right to claim the goods from the Official Receiver because the ownership of goods has passed to the buyer. If the seller becomes insolvent, the buyer cannot claim the goods. If the buyer has paid the price he can claim ratable dividend from the estate of the insolvent seller.

Sale & Hire-Purchase
Hire purchase agreement is a contract for the hire of an asset, which contains a provision giving the hirer an option to purchase. A hire purchase agreement has two elements:

  1. Element of bailment, since the possession of goods is given to the buyer
  2. Element of sale, since it contemplates an eventual sale.

The hirer is given an option either to become the owner after the payment of the stipulated hire charges/instalments or to return the goods and put an end to the hiring. The agreement must give the hirer an option to terminate the agreement and to refuse payment for further instalments, if he so desires. If the hirer defaults in paying the instalments, the seller can terminate the agreement and resume the possession of the goods.
If there is an immediate transfer of ownership of goods, it is a sale, even though, the price is paid by instalments.

SALE

HIRE-PURCHASE

(1)

In a contract of sale, the seller transfers or agrees to transfer the property in goods to the buyer for a price. In hire purchase there is an agreement for the hire of an asset conferring an option to purchase.

(2)

The ownership in goods passes on making the contract even if price is paid in instalments. The ownership passes when the option to purchase is finally exercised by the intending purchaser after complying with the terms of agreement.

(3)

The purchaser becomes owner of goods In a hire-purchase the hirer is not the owner but only a bailee of goods.

(4)

After a sale takes place the buyer cannot terminate the contract and refuse to pay the price of the goods. In a hire-purchase the hire purchaser can terminate the contract at any time and he is not bound to pay any further instalments.

(5)

On default by the buyer the seller cannot claim back the goods. On default of any payment by the hirer, the owner of the article has the right to terminate the agreement and to regain the possession of the article.

Sale and contract for work and labour
A contract of sale involves transfer of property in goods for a price. A contract for work and labour involves exercise of skill or labour. The main object is providing service by using skills, though goods are also delivered under the contract. For example, where a goldsmith is given gold for making ornaments or an artist is given paint and canvas to paint a picture, These are contracts of work and labour.

  • Nagpur Computer Services Ltd. has taken a comprehensive maintenance contract of computers which covers not only the maintenance of computers but also the supply of spares. This is a contract of work and labour.
  • A lady gave a plain saree to Jariwala Brothers for embroidering with Jari, to be purchased by Jariwala Brothers. It was held by the court that it was contract for work and labour and not a sale.

Sale and bailment
In case of bailment possession of goods is transferred from the bailor to bailee for some purpose, e.g., safe custody, repair, etc. The goods are to be returned on the fulfilment of purpose. In case of sale there is transfer of ownership, and the question of return of goods does not arise. The following are the points of distinction:

SALE

BAILMENT

(1)

In a contract of sale, the seller transfers or agrees to transfer the property in goods to the buyer for a price. In case of bailment possession of goods is transferred from the bailor to bailee for some purpose, e.g., safe custody, repair, etc.

(2)

The buyer can deal with the goods the way he likes. The bailee can use the goods only for the intended purpose of bailment

(3)

The buyer gets ownership of the goods. The bailee only acquires possession.

(4)

Generally, the goods are not returnable in a contract of sale. The goods are returnable after a specified period or when the purpose for which they were delivered is achieved.

(5)

The consideration for a sale is the price in terms of money. The consideration for bailment may be gratuitous or non-gratuitous.

C. FORMALITIES OF CONTRACT OF SALE [SEC. 5]
A contract of sale is formed by offer and acceptance. There is an offer to sell or buy goods for a price and the acceptance of such an offer.
– The contract shall provide for delivery of goods. Delivery may be immediate, simultaneous, by instalments or in future.
– The contract shall provide for payment of price. Payment of price may be immediate, simultaneous, by instalments or in future.
Contract of Sale. – How it is Made?

  1. May be in writing
  2. May be by word of mouth
  3. May be partly in writing and partly oral
  4. May be implied from the conduct of parties or by course of their business.

D. GOODS: SUBJECT MATTER OF CONTRACT OF SALE
Goods means—
every kind of movable property other than actionable claims and money.
and includes – stock and shares, growing crops, grass, and things attached to or forming part of the land which are agreed to be severed before sale or under the contract of sale. Section 2(7).
Actionable claim means a right to a debt or to any beneficial interest in movable property not in the possession of the claimant, which can be recovered by a suit or legal action. Money means the legal tender or currency of the country and It does not include old coins and foreign currency.
A. Classification of Goods

  1. Existing Goods
    1. Specific
    2. Ascertained
    3. Unascertained
  2. Future Goods
  3. Contingent Goods

1. Existing goods
A. Specific goods
The goods which are identified and agreed upon at the time when the contract of sale is §; made, are called ‘specific goods’ (Section 2(14). For example, a Videocon washing machine, a specified and finally decided car or scooter etc.

B. Ascertained goods
The term ‘ascertained goods’ is not defined in the Sale of Goods Act but has been judicially interpreted. Ascertained goods are those goods which are identified in accordance with the agreement after the contract of sale is made. When out of a large number or large quantity * of unascertained goods, the number or quantity contracted for is identified and set aside for such contract, such number or quantity is said to be ‘ascertained goods’. E.g. A whole seller of wheat has 100 bags in his godown. He agrees to sell 10 bags of wheat and these bags are identified and set aside. On selection the goods become ascertained.
Both, specific or ascertained goods in the ultimate analysis mean identified goods. The dif-ference is in the point of time when identified. In case of specific goods, they are identified at the time of making of the contract, while in case of ascertained goods, they are identified after the making but before the performance of the contract, the process being conducted in conformity with the agreement. ,

C. Unascertained goods
The goods which are not specifically identified and agreed upon at the time when the contract of sale is made, are called ‘un-ascertained goods’. For example, X is a wholesaler dealing in wheat. He agrees to sell 50 bags of wheat to Y. This contract is for the sale of un-ascertained goods because the bags of wheat have not been identified at the time of the contract of sale.
If I have 3 cars of the same kind and I offer to sell one particular car, the goods are un-ascertained till one particular car is appropriated towards the contract. On appropriation the goods become ascertained. If the identity of contract goods is not established by appropriating them towards the contract, the contract remains in respect of un-ascertained goods.

2. Future goods
Those goods which are yet to be manufactured or produced or acquiredby the seller after the making of the contract of sale, are called ‘future goods’. Sec. 2(6). For e.g. A gives an advance of t 2 lakhs for booking a Maruti car which is to be delivered after three months. This is the contract for the sale of future goods. A contract for the sale of future goods is always an agreement to sell It is never actual sale because a man cannot transfer what is not in existence.

3. Contingent goods
As per section 6(2) of the Act, contingent goods are those goods the acquisition of which by the seller depends upon a contingency (uncertain event) which may or may not happen. It may be noted that although the contingent goods are a type of future goods but they are different from future goods in the sense that the procurement of contingent goods is dependent upon an uncertain event or uncertainty of occurrence, whereas the obtaining of future goods does not depend upon any uncertainty of occurrence.
Example: A car dealer agrees to sell a yellow colour car to a customer provided it is available with the manufacturer. This agreement is for a sale of contingent goods and it will become void if the yellow colour car is not available with the manufacturer.
Quality of Goods includes their stai^ or condition. [Sec. 2(12)]

B. Effect of Destruction or Perishing of Goods
The destruction or perishing of goods may take at any of the following stages:
a. Goods perishing before making the contract [Section 7]

  • Where specific goods had perished or become damaged
  • before the contract was made
  • without the knowledge of the seller, the contract is void.

Thus, the contract of sale shall be void on the perishing of goods, if the following conditions are satisfied:

  1. It must be a contract for sale of specific goods;
  2. The goods must have perished before making the contract; and
  3. The seller must not be aware of the perishing or damaging.

Example: A agrees to sell B a certain horse. It turns out, that the horse was dead at the time of agreement, though neither party was aware of the fact. The agreement is void.

b. Goods perishing before sale but after agreement to sell [Section 8]

  • Where specific goods had perished or became damaged
  • without the fault of seller or buyer
  • after the agreement to sell is made and before the risk passes to the buyer
  • the contract becomes void.

Thus, the agreement to sell become void in the following circumstances:

  1. The contract of sale must be an agreement to sale and an actual sale
  2. The agreement to sale must be for specific goods
  3. The goods must perish or become damaged after agreement to sale but before sale
  4. The goods get perished or damaged without any wrongful act or default on the part of the seller or the buyer.

For example, an agreement to sell a car after a certain period becomes void, if the car is de-stroyed or damaged in the intervening period.
Note:

  1. Perishing of goods means not only physical destruction of the goods but it also covers loss by theft or the loss in the commercial value of the goods (e.g. where cement is spoiled by water and becomes stone)
  2. It should be noted that both the Sections 7 and 8 as mentioned above, apply only to ‘specific goods’. It is only perishing of specific and ascertained goods that affects a contract of sale. Where, un-ascertained goods are perished the contract will remain valid and the seller is bound to supply the goods. For example if X agrees to sell to Y 10 bags of wheat out of 100 bags lying in his godown and the bags in the godown are totally destroyed by fire, the contract does not become void. X must supply 10 bags of wheat or pay damages for the breach.

E. PRICE
Price is an essential condition of a contract of sale of goods. According to Section 2(10), price is the
money consideration for a sale of goods. Money means legal tender money in circulation. Old and
rare coins are not included in the definition of money.
How is the price of the goods ascertained?
Section 9 provides 4 modes of ascertainment of price. The price in a contract of sale may be—

  1. fixed by the contract
  2. may be left to be fixed in an agreed manner (such as market price or fixation of price by a third party).
  3. may be determined by the course of dealings between parties, (such as manufacturing cost, market price).
  4. a reasonable price (if price cannot be fixed in accordance with the above provisions).

What is a reasonable price is a question of fact dependent on the circumstances of each particular case. [Sec. 9(2)]
Consequence of Non-Fixation of Price by Third Party [Section 10]

  1. The parties may agree to sell and buy goods on the terms that the price is to be fixed by the valuation of a third party. If such third party fails to make the valuation the contract becomes void.
  2. However, if the buyer has received and appropriated the goods or any part thereof, he becomes bound to pay reasonable price.
  3. If the third party is prevented from making the valuation by the fault of the seller or the buyer, the innocent party may maintain suit for damages against the party in fault.

Stipulations regarding payment of price [Sec. 11]
In a contract of sale, stipulations as to time may be of two kinds:

– Stipulations relating to time of payment, and
– Stipulations not relating to time of payment, for e.g. relating to time of delivery of goods

  • Stipulations as to time for payment of price are not regarded as essence of contract, unless a different intention appears from the terms of the contract. Thus if the payment is not made in time, the seller cannot avoid the contract but can claim damages. For example A sells a laptop computer to B with a stipulation that payment should be made within 3 days. B makes the payment after 7 days of the contract. Here A cannot avoid the contract on the ground of breach of stipulation as to time of payment.
    However, time of payment can be made essence of the contract, if there is an express provision in the contract of sale. If there is no express provision in the contract of sale, with regard to time of payment, then time of payment is not deemed to be the essence of contract.
  • Whether any other stipulation as to time (c.g. of delivery of goods) is of the essence of contract, will depend upon the terms agreed upon. It means that time of delivery of goods etc., can also be made essence of the contract of sale if an express provision to this effect is made in it. If no such provision is made, then time of delivery of goods will not be the essence of contract. (Sec. 11) Suppose if time of delivery of goods is made the essence of the contract of sale by providing express terms in this regard – what will be the remedy for the buyer, if the seller does not make the delivery within the stipulated time? (The buyer can avoid the contract)
  • It may be noted that in ordinary commercial contracts for sale of goods, time is prima facie of the essence with respect to delivery.

MULTIPLE CHOICE QUESTIONS:

1. The code governing sale of goods was earlier contained in
(a) the Indian Contract Act
(b) the Transfer of Property Act
(c) the Hire Purchase Act
(d) None of the above

2. The Sale of Goods Act, 1930 governs the transfer of property in
(a) movable property
(b) immovable property
(c) both movable and immovable property
(d) all type of properties

3. “Goods” means
(a) every kind of movable property other than actionable claims and money
(b) some kinds of immovable property only
(c) every kind of movable property including actionable claims and money
(d) Both ‘a’ and ‘b’

4. Where under a contract of sale the property in goods is transferred from the seller to the buyer, the contract is called.
(a) an agreement to sell
(b) a sale
(c) both ‘a’ and ‘b’
(d) either ‘a’ or ‘b’

5. A valid sale must have two parties who
(a) must be competent to contract
(b) may not be competent to contract
(c) must be Indian citizens
(d) must be residents of the same state

6. An agreement to sell is
(a) an executory contract
(b) an executed contract
(c) neither ‘a’ or ‘b’
(d) sometime ‘a’ or ‘b’

7. Specific goods are such goods which are
(a) existing and identified at the time of making the contract
(b) identified after the making of contract but before the performance of contract
(c) both ‘a’ and ‘b’
(d) neither ‘a’ nor ‘b’

8. ‘Future goods’
(a) can be the subject matter of sale
(b) cannot be subject matter of sale
(c) sometimes may be the subject matter of sale
(d) depends on circumstances

9. When there is a contract for un-ascertained goods, and goods perish without the fault of the seller or buyer before the risk passes to the buyer, the contract
(a) can be avoided
(b) cannot be avoided
(c) becomes void
(d) becomes unenforceable

10. To constitute a Contract of Sale, the transfer of property in goods
(a) must be for monetary consideration
(b) may be for non-monetary consideration
(c) must be for both monetary and non-monetary consideration
(d) may be either monetary or non-monetary consideration

Answers:
CA Foundation Business Laws Study Material Chapter 10 Formation of Contract of Sale 2

STATE WHETHER THE FOLLOWING ARE TRUE OR FALSE:

1. The term “goods” under Sale of Goods Act, 1930 includes actionable Claims.
2. The Sale of Goods Act, 1930 deals with movable goods only.
3. The Sale of Goods Act, 1930 covers mortgage and pledge of goods.
4. The provisions of Sale of Goods were originally contained in the Indian Contract Act, 1872.
5. In case of hire purchase the hirer can pass title to a bona fide purchaser.
6. In a contract of sale, subject matter of the contract must always be money.
7. In a contract of sale. The agreement may be expressed or implied from the conduct of the parties.
8. The property in goods means possession of goods.
9. The goods are at the risk of the party who has the ownership of the goods.
10. A lady gave a plain saree to Jariwala Brothers for embroidering with lari, to be purchased by Jariwala Brothers. It was held by the court that it was contract of sale.

Answers:
CA Foundation Business Laws Study Material Chapter 10 Formation of Contract of Sale 3

CA Foundation Business Laws Study Material

CA Foundation Business Laws Study Material

The aim of this study material is to provide an introduction to the core concepts of Business Laws in a student friendly style. The study material is primarily written for the students of Foundation Course of Chartered Accountancy under the new syllabus introduced by the Institute of Chartered Accountants of India. It is also useful for the students of other professional exams.

We have attempted to present the subject in a lucid and simple style keeping in mind the students of under graduate level The study material has been presented in a tabular form so that it becomes easier for the students to revise as well during the time of examination. To test the understanding of the student and also to enable them to have sufficient knowledge, a large number of exercise has been provided at the end of the chapters. The exercise covers a lot of Multiple Choice Questions, True and False type questions and also Theoretical Questions as well as questions in the pattern of Case Studies.

The study material is divided into five units depending upon their respective Acts. The new syllabus and answers and hints for writing the answers further adds up to the value of the study material.

Unit 1 Indian Contract Act, 1872

Unit 2 Sale of Goods Act, 1930

Unit 3 Partnership Act, 1932

Unit 4 The Limited Liability Partnership Act, 2008

Unit 5 The Companies Act, 2013

In preparation of the study material we have received encouragement and support from various quarters. We would be failing in our duty if we do not acknowledge the deep sense of gratitude that we owe to various people for providing us an insight and inspiration in writing the study material.

Any comment and criticism relating to the present work will be most welcome.

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6

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 6 Factorization Ex 6

More Exercises

Question 1.
Find the remainder (without divisions) on dividing f(x) by x – 2, where
(i) f(x) = 5x2 – 1x + 4
(ii) f (x) = 2x3 – 7x2 + 3
Solution:
Let x – 2 = 0, then x = 2
(i) Substituting value of x in f(x)
f(x) = 5x2 – 7x + 4
⇒ f(2) = 5(2)2 – 7(2) + 4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q1.1

Question 2.
Using remainder theorem, find the remainder on dividing f(x) by (x + 3) where
(i) f(x) = 2x2 – 5x + 1
(ii) f(x) = 3x3 + 7x2 – 5x + 1
Solution:
Let x + 3 = 0
⇒ x = -3
Substituting the value of x in f(x),
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q2.1

Question 3.
Find the remainder (without division) on dividing f(x) by (2x + 1) where
(i) f(x) = 4x2 + 5x + 3
(ii) f(x) = 3x3 – 7x2 + 4x + 11
Solution:
Let 2x + 1 = 0, then x = \(– \frac { 1 }{ 2 } \)
Substituting the value of x in f(x):
(i) f(x) = 4x2 + 5x + 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q3.1

Question 4.
(i) Find the remainder (without division) when 2x3 – 3x2 + 7x – 8 is divided by x – 1 (2000)
(ii) Find the remainder (without division) on dividing 3x2 + 5x – 9 by (3x + 2)
Solution:
(i) Let x – 1 = 0, then x = 1
Substituting value of x in f(x)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q4.1

Question 5.
Using remainder theorem, find the value of k if on dividing 2x3 + 3x2 – kx + 5 by x – 2, leaves a remainder 7. (2016)
Solution:
f(x) = 2x2 + 3x2 – kx + 5
g(x) = x – 2, if x – 2 = 0, then x = 2
Dividing f(x) by g(x) the remainder will be
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q5.1

Question 6.
Using remainder theorem, find the value of a if the division of x3 + 5x2 – ax + 6 by (x – 1) leaves the remainder 2a.
Solution:
Let x – 1 = 0, then x = 1
Substituting the value of x in f(x)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q6.1

Question 7.
(i) What number must be subtracted from 2x2 – 5x so that the resulting polynomial leaves the remainder 2, when divided by 2x + 1 ?
(ii) What number must be added to 2x3 – 7x2 + 2x so that the resulting polynomial leaves the remainder – 2 when divided by 2x – 3?
Solution:
(i) Let a be subtracted from 2x2 – 5x,
Dividing 2x2 – 5x by 2x + 1,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q7.2

Question 8.
(i) When divided by x – 3 the polynomials x2 – px2 + x + 6 and 2x3 – x2 – (p + 3) x – 6 leave the same remainder. Find the value of ‘p’
(ii) Find ‘a’ if the two polynomials ax3 + 3x2 – 9 and 2x3 + 4x + a, leaves the same remainder when divided by x + 3.
Solution:
By dividing
x3 – px2 + x + 6
and 2x3 – x2 – (p + 3) x – 6
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q8.2

Question 9.
By factor theorem, show that (x + 3) and (2x – 1) are factors of 2x2 + 5x – 3.
Solution:
Let x + 3 = 0 then x = – 3
Substituting the value of x in f(x)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q9.1

Question 10.
Show that (x – 2) is a factor of 3x2 – x – 10 Hence factorise 3x2 – x – 10.
Solution:
Let x – 2 = 0, then x = 2
Substituting the value of x in f(x),
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q10.1

Question 11.
Show that (x – 1) is a factor of x3 – 5x2 – x + 5 Hence factorise x3 – 5x2 – x + 5.
Solution:
Let x – 1 = 0, then x = 1
Substituting the value of x in f(x),
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q11.1

Question 12.
Show that (x – 3) is a factor of x3 – 7x2 + 15x – 9. Hence factorise x3 – 7x2 + 15 x – 9
Solution:
Let x – 3 = 0, then x = 3,
Substituting the value of x in f(x),
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q12.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q12.2

Question 13.
Show that (2x + 1) is a factor of 4x3 + 12x2 + 11 x + 3 .Hence factorise 4x3 + 12x2 + 11x + 3.
Solution:
Let 2x + 1 = 0,
then x = \(– \frac { 1 }{ 2 } \)
Substituting the value of x in f(x),
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q13.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q13.2

Question 14.
Show that 2x + 7 is a factor of 2x3 + 5x2 – 11x – 14. Hence factorise the given expression completely, using the factor theorem. (2006)
Solution:
Let 2x + 7 = 0, then 2x = -7
x = \(\\ \frac { -7 }{ 2 } \)
substituting the value of x in f(x),
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q14.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q14.2

Question 15.
Use factor theorem to factorise the following polynominals completely.
(i) x3 + 2x2 – 5x – 6
(ii) x3 – 13x – 12.
Solution:
(i) Let f(x) = x3 + 2x2 – 5x – 6
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q15.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q15.3

Question 16.
(i) Use the Remainder Theorem to factorise the following expression : 2x3 + x2 – 13x + 6. (2010)
(ii) Using the Remainder Theorem, factorise completely the following polynomial: 3x2 + 2x2 – 19x + 6 (2012)
Solution:
(i) Let f(x) = 2x3 + x2 – 13x + 6
Factors of 6 are ±1, ±2, ±3, ±6
Let x = 2, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q16.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q16.2

Question 17.
Using the Remainder and Factor Theorem, factorise the following polynomial: x3 + 10x2 – 37x + 26.
Solution:
f(x) = x3 + 10x2 – 37x + 26
f(1) = (1)3 + 10(1)2 – 37(1) + 26
= 1 + 10 – 37 + 26 = 0
x = 1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q17.1

Question 18.
If (2 x + 1) is a factor of 6x3 + 5x2 + ax – 2 find the value of a
Solution:
Let 2x + 1 = 0, then x = \(– \frac { 1 }{ 2 } \)
Substituting the value of x in f(x),
f(x) = 6x3 + 5x2 + ax – 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q18.1

Question 19.
If (3x – 2) is a factor of 3x3 – kx2 + 21x – 10, find the value of k.
Solution:
Let 3x – 2 = 0, then 3x = 2
⇒ x = \(\\ \frac { 2 }{ 3 } \)
Substituting the value of x in f(x),
f(x) = 3x3 – kx2 + 21x – 10
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q19.1

Question 20.
If (x – 2) is a factor of 2x3 – x2 + px – 2, then
(i) find the value of p.
(ii) with this value of p, factorise the above expression completely
Solution:
(i) Let x – 2 = 0, then x = 2
Now f(x) = 2x3 – x2 + px – 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q20.1

Question 21.
Find the value of ‘K’ for which x = 3 is a solution of the quadratic equation, (K + 2)x2 – Kx + 6 = 0.
Also, find the other root of the equation.
Solution:
(K + 2)x2 – Kx + 6 = 0 …(1)
Substitute x = 3 in equation (1)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q21.1

Question 22.
What number should be subtracted from 2x3 – 5x2 + 5x so that the resulting polynomial has 2x – 3 as a factor?
Solution:
Let the number to be subtracted be k and the resulting polynomial be f(x), then
f(x) = 2x3 – 5x2 + 5x – k
Since, 2x – 3 is a factor of f(x),
Now, converting 2x – 3 to factor theorem
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q22.1

Question 23.
Find the value of the constants a and b, if (x – 2) and (x + 3) are both factors of the expression x3 + ax2 + bx – 12.
Solution:
Let x – 2 = 0, then x = 0
Substituting value of x in f(x)
f(x) = x3 + ax2 + bx – 12
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q23.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q23.2

Question 24.
If (x + 2) and (x – 3) are factors of x3 + ax + b, find the values of a and b. With these values of a and b, factorise the given expression.
Solution:
Let x + 2 = 0, then x = -2
Substituting the value of x in f(x),
f(x) = x3 + ax + b
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q24.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q24.2

Question 25.
(x – 2) is a factor of the expression x3 + ax2 + bx + 6. When this expression is divided by (x – 3), it leaves the remainder 3. Find the values of a and b. (2005)
Solution:
As x – 2 is a factor of
f(x) = x3 + ax2 + bx + 6
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q25.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q25.2

Question 26.
If (x – 2) is a factor of the expression 2x3 + ax2 + bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b.
Solution:
f(x) = 2x3 + ax2 + bx – 14
∴ (x – 2) is factor of f(x)
f(2) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q26.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q26.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q26.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q26.4

Question 27.
If ax3 + 3x2 + bx – 3 has a factor (2x + 3) and leaves remainder – 3 when divided by (x + 2), find the values of a and 6. With these values of a and 6, factorise the given expression.
Solution:
Let 2x + 3 = 0 then 2x = -3
⇒ x = \(\\ \frac { -3 }{ 2 } \)
Substituting the value of x in f(x),
f(x) = ax3 + 3x2 + 6x – 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q27.1

Question 28.
Given f(x) = ax2 + bx + 2 and g(x) = bx2 + ax + 1. If x – 2 is a factor of f(x) but leaves the remainder – 15 when it divides g(x), find the values of a and b. With these values of a and b, factorise the expression. f(x) + g(x) + 4x2 + 7x.
Solution:
f(x) = ax2 + bx + 2
g(x) = bx2 + ax + 1
x – 2 is a factor of f(x)
Let x – 2 = 0
⇒ x = 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q28.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 Q28.2

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6 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 9 Discharge of Contract

CA Foundation Business Laws Study Material Chapter 9 Discharge of Contract

CA Foundation Business Laws Study Material Chapter 9 Discharge of Contract 1
METHODS OF TERMINATION OF CONTRACT

When the obligations created by a contract come to an end, the contract is said to be discharged or terminated. A contract may be discharged or terminated in any of the following ways:

I. DISCHARGE BY PERFORMANCE
The obligations of a party to a contract come to an end where he performs his promise. Performance by all the parties, of the respective obligations, put’s an end to the contract completely. This is the normal and natural mode of discharging a contract.

II. DISCHARGE BY ATTEMPTED PERFORMANCE
The offer of performance or tender has the same effect as performance. If a party to a contract offers to perform his promise but the offer is not accepted by the other party, the obligations of the first party are terminated.

III. DISCHARGE BY MUTUAL AGREEMENT
By agreement of all parties, a contract may be cancelled or its terms altered or a new agreement substituted for it. Whenever any of these things happen, the old contract is terminated. “If the parties to a contract agree to substitute a new contract for it, or to rescind or alter it, the original contract need not be performed. ” Sec. 62.

Termination by mutual agreement may occur in any one of the following ways
A. NOVATION
Novation occurs when a new contract is substituted for an existing contract either between the same parties or between different parties. The consideration for the new contract is the discharge of the old contract.

  • To effect a novation, there must be a valid enforceable new substituted contract.
  • Consent of all parties is necessary for novation.
  • Novation should take place before the breach or expiry of old contract.

Example: A is indebted to B and B to C. By mutual agreement B’s debt to C and A’s debt to B is cancelled and C accepts A as his debtor. There is novation. (See Scarf v. Jardine in leading case laws at the end of the chapter)

B. ALTERATION
Alteration of a contract means change in one or more of the terms of a contract. Alteration is valid if it is done with the consent of all the parties to the contract.
In alteration there is change in the terms of the contract but no change of the parties to it. In novation there may be change of parties.

NOVATION

ALTERATION

Novation is substitution of old contract by a new contract by mutual agreement between the parties. Alteration means change in the terms of the existing contract by mutual agreement between the parties.
The parties may either remain the same or a third party may be introduced. Parties remain the same. No third party is involved.
Novation rescinds the original contract as a result the original contract need not be performed. Alteration does not rescind the original contract. As the same original contract in a modified manner is performed.

C. REMISSION
Remission means acceptance of lesser amount, or lesser degree of performance than what was contracted for in full discharge of the contract.
According to Sec. 63 a party may:

  • Dispense with or remit performance wholly or in part; or
  • Extend the time for performance or
  • Accept any other satisfaction instead of performance

For such a release or promise there no need for consideration or new agreement.
Example: A owes B Rs. 5,000. A pays to B and B accepts in full satisfaction for the whole debt Rs. 2,000. The old debt is discharged.
A promise by the promise to give concession to the promisor in one or the other form is binding even if without consideration. In Gopala v. Venkata, it was stated that after the remission has been communicated to the promisor and accepted by him, the promise cannot claim the remitted (sacrificed) amount.

D ACCORD AND SATISFACTION
Under the English law, these terms are used as counter part of the term remission. Under the English Law, “accord” means the promise to accept less than what is due under the original contract. ‘Satisfaction’ means the actual payment or the fulfilment of the smaller obligation. In the English Law a promisee cannot remit a part of the amount unless a fresh promise is supported by consideration. However, this doctrine of accord and satisfaction as applied in England, has no place in India. Sec.63 clearly states that if the promisee agrees to accept a lesser amount in full satisfaction of the whole claim, this promise is valid and therefore enforceable.

E. RESCISSION
Rescission occurs when the parties to a contract agree to dissolve the contract. In the case of rescission only the old contract is cancelled and no new contract comes to exist in its place. The parties come out of the contract by mutual agreement.

F. WAIVER
Waiver means the abandonment of a right. A party to a contract may relinquish (waive) his rights under the contract. Thereupon the other party is released from his obligations. For example, waiver of former’s loan by bank.

G. MERGER
When a superior right and an inferior right coincide and meet in one and the same person, the inferior right vanishes into the superior right. This is known as merger.
Illustration:

  1. A man holding property under a lease buys the property. His rights as a lessee ^ vanish. They are merged into the rights of ownership which he has now acquired.
  2. A may agree to work as a part-time employee of B. Later, they may decide that A will work as full-time employee.

IV. DISCHARGE BY BREACH OF CONTRACT
When a contract is broken by one party the other party or parties are freed from the obligation of performing the contract. They can also take the remedial measures to which they are entitled. Breach of contract may arise in two ways:

  1. By actual breach or present breach.
  2. By anticipatory breach.

A. ACTUAL BREACH OF CONTRACT
Actual breach of contract occurs when during the performance of the contract or at the time when the performance of the contract is due, one party either fails or refuses to perform his obligations under the contract. The refusal of performance may be express (i.e. by word or by writing) or implied (le. by conduct of the party or by non-action) or abstaining from doing something. D agrees to deliver to B, 5 tons of sugar on 1st June. He fails to do so. There is breach of contract by D.

B. ANTICIPATORY BREACH OF CONTRACT (Sec. 39)
Anticipatory breach of contract occurs :

  • when a party before the time for performance is due announces that he is not going to perform the contract or,
  • when a party by his own act disables himself from performing the contract.
    • C enters into a contract to supply B with certain articles on the 1st June. Before 1st June he informs B that he will not be able to supply the goods.
    • X agrees to marry Y. Before the agreed date of marriage, he marries Z.

CONSEQUENCES OF ANTICIPATORY BREACH
When anticipatory breach occurs, the aggrieved party can take the following steps:
(A) May treat the contract as discharged-

  1. He can treat the contract as discharged, so that he is no longer bound by any obligations under the contract; &
  2. He can immediately adopt the legal remedies available to him for breach of contract, viz., hie a suit for damages or specific performance or injunction.

(B) May not treat the contract as discharged-
Anticipatory breach, by itself, does not discharge the contract. The contract is discharged, when the aggrieved party chooses to treat it as discharged. The aggrieved party may decide not to rescind the contract but to treat the contract as alive and operative and wait for the time of performance. In such a case the consequences are as follows:

  1. The contract will be operative for the benefit of both the parties. The contract will continue to exist and may even be performed by the other party.
  2. If the contract is not rescinded and subsequently an event happens which discharges the contract legally (e.g. a supervening impossibility) the aggrieved party loses his right to sue for damages.

For example, A agrees to supply one ton of sugar to B by 20th August. On 10th August, A informs B that he cannot supply sugar B did not accept the refusal and preferred to wait till 20th August. On 15th August, the Minister declares nationalisation of sugar industry. Now the contract is discharged and B has no remedy against A.

V. DISCHARGE BY OPERATION OF LAW
A contract terminates by operation of law in case of death insolvency, and merger.

A. Death
In contracts involving personal skill or ability, death terminates the contract. In other cases, the rights and liabilities pass on to the legal representatives of the dead man.

B. Insolvency
When a person is adjudged insolvent, he is discharged from all liabilities incurred prior to his adjudication. Upon insolvency, the rights and liabilities of the insolvent are, with certain * exceptions, transferred to an officer of the court, known as the Official Assignee/Receiver.

C. Merger
Means coinciding and meeting of inferior and superior right in one and the same person. In such a case, inferior right available to a party under the contract will automatically vanish.

D. Lapse of time
Contracts may be terminated by lapse of time. In civil suits the obligations and liabilities in contracts are barred by limitation. The provisions of law are stated in the Limitation Act.

E. Unauthorised material alteration 
If the terms of a contract is materially altered by a party to the contract without the consent of the other parties, the contract is discharged and cannot be enforced any more.

VI. SUBSEQUENT OR SUPERVENING IMPOSSIBILITY
Pre-contractual Impossibility

A contract which at the time was entered into was impossible to perform, is void ab-initio and creates no rights and obligations. Sec. 56(1) states that “An agreement to do an act impossible in itself is void. “Such fact of impossibility may be-

  1. Known to the parties:
    In such a case the agreement is void ab-initio and creates no rights and obligations. For example a promise to ride a horse to the Sun or A agrees with B to discover treasure by magic. The agreement is void.
  2. Unknown to the parties:
    When both the parties are ignorant of the impossibility at the time of making the contract, the contract, is void on the ground of mutual mistake. For example: A agrees to sell his horse to B but unknown to both the parties the horse had already died at the time of making the contract. The contract is void.
  3. Known only to the promisor:
    On the contrary, if the promisor alone knew about the impos-sibility of performance at the time of making the contract, he shall have to compensate the promisee for any loss which such promisee sustains through the non-performance of the promise. [Sec. 56(3)]

Post-contractual Impossibility
A contract, which at the time was entered into, was capable of being performed may subsequently become impossible to perform or unlawful. In such cases the contract becomes void. This is known as the doctrine of Supervening Impossibility. It is also known as the Doctrine of Frustration. Frustration occurs where it is established that due to subsequent change in circumstances, the contract has become impossible to perform or it has been deprived of its commercial purpose.

“A contract to do an act which, after the contract is made, becomes impossible, or, by reason of some event which the promisor could not prevent, unlawful, becomes void when the act becomes impossible or unlawful”. [Sec.56]Para 2.

Grounds of Frustration
Supervening impossibility may occur in many ways, some of which are explained below:
(i) Destruction of the subject matter of contract
On the destruction of the subject matter, a contract is discharged and no party is liable to perform. (Refer Taylor v. Caldwell in leading case laws at the end of the chapter.)

(ii) Change of Law
The performance of a contract may become unlawful by a subsequent change of law. In such cases, the original contract becomes void. (Refer Shiam Sunder v. Durga’m leading case laws at the end of the chapter.)

(iii) Failure of pre-conditions
When a contract is entered into on the basis of the continued existence of a certain state of things, the contract is discharged if the state of things change.

  • Illustration: A & B contract to marry each other. Before the time fixed for the marriage, A goes mad. The contract becomes void.
  • Illustration: H hired a room from K for two days with the object (as both parties know) of using the room to view the coronation procession of Edward VII although the contract continued no reference to the procession. Owing to the king’s illness the procession was abandoned. Held, that the contract was discharged and H was excused from paying rent of the room as the existence of the procession was the basis of the agreement.

(iv) Death or incapacity for personal services
Where the personal qualification of a party is the basis of the contract the contract is discharged in cases of death or personal incapacity.
G contracts to act of a theatre for six months in consideration of a sum paid in advance by H. On several occasions G is too ill to act. The contract to act on these occasions becomes void.

(v) Outbreak of war
A contract entered into during war with an alien enemy is void ab-initio. A contract entered into before the war commenced between citizens of countries subsequently at war, remains suspended during the pendency of the war. After the termination of the war, the contract revives and may be enforced.
Exceptions
Impossibility of performance, is, as a rule, not an excuse from performance. Only physical or legal impossibility will excuse the parties. The performance of the contract should have become impossible due to any of the circumstances mentioned above. The doctrine of frustration or supervening impossibility does not apply in the following cases. Le. in these cases the contract is not discharged,

1. Difficulty of performance
Difficulty does not excuse performance of contract.
The contract will not be affected if performance has become difficult because of disruption of traffic routes. In the case of Tsakiroglou & Co v. Noblee Thori (1962) AC 93, the closure of Suez Canal in 1956 because of outbreak of war there had caused problems in completion of many contracts involving transportation via the Suez Canal. But, the judicial view was that unless it could be proved that transport via Suez Canal was an express or implied term of the contract, its closure could not be said to make the contract impossible.

2. Commercial Impossibility
A wholesale dealer’s contract to deliver goods is not discharged because a manufacturer has not produced the goods concerned. Similarly increase of wages of prices of raw materials, unseasonable weather or lack of adequate profits do not excuse performance. The reason is that if the parties did not stipulate to the contrary, they must have intended to take the risk of occurrences like these.

3. Strikes, lock-outs, civil disturbances and riots
These events do not terminate contracts unless there is a clause in the contract providing that in such cases the contract is not be performed or that the time of performance is to be extended.

4. Failure of one of the objects
When there are several purposes for which a contract is entered into, failure of one of the objects does not terminate the contract.

THE DOCTRINE OF FRUSTRATION

When the common object of a contract can no longer be carried out, the court may declare the contract to be at an end. This is known as the Doctrine of frustration. Anson says: “Most legal systems make provisions for the discharge of a contract where, subsequent to its formation, a change of circumstances renders the contract legally or physically impossible of performance.’’
In Satyabharata Ghoshv. MugniramBengur A\R(\954) S.C. 44 : The Supreme Court of India discussed the English cases relating to frustration and came to the following conclusions:

The doctrine of frustration of contract comes into play when a contract becomes impossible of performance, after it is made, on account of circumstances beyond the control of the parties. It ; comes within the purview of Sec. 56 of the Indian Contract Act. The word ‘impossible’ in this section has not been used in the sense of physical or literal impossibility. The performance of an act may not | be literally impossible but it may be impracticable and useless from the point of view of the object [ and purpose which the parties had in view; and if an untoward event or change of circumstances totally upsets the very foundation upon which the parties rested their bargain, it can be said that the promisor finds it impossible to do the act which he promised to do.

“A contract to do an act which, after the contract is made, becomes impossible, or, by reason : of some event which the promisor could not prevent, unlawful, becomes void when the act becomes impossible or unlawful”- [Sec. 56] para-2.
Grounds of frustration of contract and supervening impossibility are similar.
The effect of frustration – Frustration automatically discharges a contract from the date of the frustrating event.

REMEDIES FOR BREACH OF CONTRACT

I. Rescission of the contract
When there is a breach of contract by one party, the other party may rescind the contract and need not perform his part of obligations under the contract. This is called the right of rescission which means a right to cancel or to set aside (i.e., reject) the contract.
Further, section 75 provides that a person who rightfully rescinds a contract is entitled to compen- j sation for any damages which he has sustained through the non fulfilment of the contract.
A contracts to supply 100 Kg. of tea leaves to B on 25 April. If A does not supply the tea leaves on the j appointed day, B need not pay the price. B may also file a ‘suit for rescission’ and claim damages.

II. Suit for damages
Damages are the monetary compensation allowed by a court of law to the aggrieved party for the loss or injury suffered by him. The loss or injury suffered is known as damage. This is the difference j between “Damage” and “Damages”.
The foundation of modern law of damages in India is based on the judgment in a case of Hadley v. Baxendale (1854) 9E. 341, and is incorporated in sec.73 of the Indian Contract Act.

H’s mill was stopped by a breakage of the crankshaft. He delivered the shaft to B, a common carrier, to take it to the manufacturers at Greenwich as a pattern for a new one. By some neglect on ; the part of B the delivery of the shaft was delayed beyond a reasonable time. H claimed from B s compensation for the wages of workers and depreciation charges during the period the factory was idle for the delayed delivery and for loss of profits which might have been made if the factory was working. The first two items, were allowed because they were natural consequences of the breach. The last item, loss of profits was disallowed because it was a remote consequence. (Hadley v. Baxendale).

COMPENSATION FOR LOSS OR DAMAGE CAUSED BY BREACH OF CONTRACT: (Sec. 73)
“When a contract has been broken, the party who suffers by such breach is entitled to receive, from the party who has broken the contract, compensation for any loss or damage caused to him thereby, which naturally arose in usual course of things from such breach, or which the parties knew, when they made the contract to be likely to result from the breach of it. ”
Such compensation is not to be given for any remote and indirect loss or damage sustained by reason of the breach.

  • Nature
    Damages under the Contract Act are not granted by way of punishment. They are compensatory in nature. The object is to compensate the injured party and bring him on the same position in which he would have been, if there was no breach of contract.
  • If there is no damage, damages cannot be claimed.
    Damages are claimed to compensate any loss or damage caused by breach of contract. If there is no actual loss than damages cannot be claimed. For example, A contracts to buy B’s car for X 60,000/- but breaks his promise. B then sells to C for the same price. B cannot claim any damages Under the law of contract damages are not granted by way of punishment.

RULES
1. Ordinary damages are recoverable
Ordinary damages are those which naturally arise in the usual course of things from such breach. They are the natural and probable consequence of the breach, which the aggrieved
party suffering can recover from the defaulting party.

2. Special damages are recoverable only if the parties knew about them.
Apart from ordinary damages, special damages can also be claimed. Special damages are for loss which arises on account of the unusual circumstances affecting the plaintiff. They are not recoverable unless the special circumstances were brought to the knowledge of the defendant so that the possibility of the special loss was in the contemplation of the parties.
Thus, special damages are those which result from a breach of contract under some special or unusual circumstances which are in the knowledge of the parties at the time of making the contract.

  • The extent of liability depends upon the knowledge of the parties at the time of the contract about the probable result of the breach.
    A, having contracted with B to supply B with 1,000 tons of iron at 100 rupees per ton to be delivered at a stated time, contracts with C for the purchase of 1,000 tons of iron at 80 rupees a ton, telling C that he does so for the purpose of performing his contract with B. C fails to perform his contract with A, who cannot procure other iron and B, in consequence, rescinds the contract. C must pay to A 20,000 rupees being the profit which A would have made by the performance of his contract with B.
    Compensation is recoverable for:
  • Damages arising naturally in the usual course of things.
  • Damages arising out of special circumstances in contemplation of parties.

3. Remote or indirect damages are not recoverable
Damages cannot be claimed for any remote or indirect loss or damage sustained by reason of the breach. Remote damages are those which are not reasonably foreseeable. The party breaking the contract shall not make compensation in respect of loss or damage indirectly
or remotely caused.
For example, A contracts to pay a sum of money to B on a day specified. A does not pay the
money on that day. B in consequence of not receiving the money on that day, is unable to pay his debts and is totally ruined. A is not liable to make good B anything except the principal sum he contracted to pay together with interest up to the day of payment. This is so because B’s total ruination is an indirect loss.

Nominal damages for no loss suffered: Where the injured party has not in fact suffered any loss by reason of the breach of a contract, the damages recoverable by him are nominal,
Le., very small. For example a rupee or cent. These damages merely acknowledge that the plaintiff has proved his case and won. In Charter v. Sullivan, the defendant refused to buy a car from the plaintiff which was sold to another customer and no loss was occasioned.
He was, thus, held entitled to only nominal damages and not for any loss of profits.

Vindictive or exemplary damages: Damages for breach of contract are given by way of compensation for loss suffered, and not by way of punishment for wrong inflicted. Hence, ‘vindictive’ or ‘exemplary’ damages have no place in the law of contract because they are punitive by nature. But in case of

  1. breach of a promise to marry, and
  2. dishonour of a cheque by a banker wrongfully when he possesses sufficient funds to the credit of the customer, the Court may award exemplary damages.

Other Heads for Damages:
Damages for inconvenience caused by breach.
Damages for mental pain and suffering. In%a Scottish case, a photographer who agreed to take photographs at a wedding, failed in breach of his contract to appear there. As a result the bride had no photographs of her wedding. She was allowed damages for resulting injury to her feelings.
Damages for pre-contract and wasted expenditure.
Damages for delay in delivery.

4. Defaulting party liable to compensate as per market price,
5. Such damages are also payable in case of breach of a quasi-contract too.
6. It is the duty of the injured party to minimise the damage suffered.
7. The injured party is entitled to get the costs of getting the decree for damages from the defaulter party.
8. Liquidated Damages and Penalty
Liquidated Damages: Where the party fixes a genuine pre-estimate of the probable damage, it is called liquidated damages. Liquidated damages are predetermined damages agreed at the time of contract, which are considered reasonable by both the parties. It is a genuine estimate of the actual loss or damage likely to be suffered by the aggrieved party.
Penalty: Where the sum fixed before-hand for the breach of contract does not bear the relationship to the actual damage which the aggrieved party is likely to suffer in the event j of actual breach of contract, it is called penalty. Such an amount acts as a deterrent from committing a breach of contract.

A contract sometimes contains a clause in which a sum of money is named as the amount payable f in case of breach of contract. According to English law, the amount of money payable is interpreted either as liquidated damages or as a penalty. It is considered to be liquidated damages when the amount is fixed by the parties on the basis of a reasonable estimate of the probable actual loss , which a party will suffer in case of breach. On the other hand, the amount fixed is considered to be a penalty if it is not based upon a reasonable calculation of actual loss but is fixed by way of punishment and as a threat. A penalty will not be enforced by the Court.

In India, the distinction between liquidated damages and penalty is not recognised. Sec. 74 of the Contract Act which deals with predetermined damages, lays down that if the parties have fixed what the damages will be, the courts will never allow more. But the court may allow less. A decree is to be passed only for reasonable compensation, not exceeding the sum named by the parties.
Thus, section 74 makes no distinction between a liquidated damages and penalty and the aggrieved party is entitled to reasonable compensation not exceeding the amount so named, regardless whether it is penalty or not.

Under section 73, the actual loss or damage has to be proved but under section 74, the proof of actual loss or damage is not essential.
The difference between the liquidated damages and penalty depends on the facts and circumstances of each case and the intention of the parties which is to be gathered from the whole contract.

  • If the intention is to secure performance of The contract by imposition of a fine or penalty, the sum specified is penalty; but if on the other hand, the intention is to assess the damages for breach of contract, it is liquidated damages.
  • Liquidated damages are the amount assessed on the basis of actual or probable loss by both the parties. Penalty is not based on actual or probable loss. Penalty is payable in the event of breach with a view to prevent a party from committing breach.
  • Liquidated damages are imposed by way of compensation. Penalty is imposed by way of punishment. The amount of penalty is exorbitant, extravagant and unconscionable.
  • Courts in England usually allow liquidated damages without any regard to the actual loss sustained and treat penalty clause as invalid. But under the Indian law, section 74 of the Contract

Act does not recognize any difference between liquidated damages and penalty. The courts are required to allow reasonable compensation so as to cover the actual loss sustained, not exceeding the amount so mentioned in the contract.

A stipulation for higher rate from the date of default may be taken as a penalty if the enhanced rate is exorbitant. If it is reasonable, then, it shall be allowed. The leading case on the matter is Mackintosh v. Crow (1883) 9 Cal 689. If the normal rate of interest is fixed at 12% and the rate of interest from the date of default is to be 36%, this may be taken by court as very unreasonable and may be reduced.
Earnest Money and Security Deposit : Sometimes, when a contract is formed, one party gives to the other a sum as a deposit. This deposit may take two forms: earnest money and security deposit. Earnest money is a kind of advance payment of price by one party to the other out of a larger amount payable. B agrees to buy goods from C and pays him Rs. 5,000 in advance as earnest money. This amount shall be adjusted against the total price payable by B to C. The earnest money paid may or may not be liable to be forfeited under the contract if buyer breaks the contract.
Security deposit is a payment made as a guarantee that the contract shall be fulfilled by the person who has paid the deposit. If the contract is not fulfilled, then, the deposit shall be forfeited under the contract.
Where a buyer has paid earnest money which is liable to be forfeited if buyer breaks the contract, then, on the contract being broken by him, the seller may forfeit the amount if it is a reasonable amount (Shree Hanuman Cotton Mills v. Tata Aircraft Ltd. [1969] 3 SCC 522).

III. SUIT FOR SPECIFIC PERFORMANCE OF THE CONTRACT
There are cases where the damage or loss suffered cannot be measured in terms of money. The court, may, in such cases where the ordinary remedy by a claim for damages is not adequate compensation, direct the defaulting party to perform the contract specifically. (Under Sec. 12. of the Specific Relief Act, 1963). Specific performance is an order of the Court directing the defendant to fulfil his obligations under the contract. Specific performance is a discretionary remedy and is only available where damages are not an adequate remedy.
Some of the cases where specific performance is ordered by the court are:

  1. Where the act itself is such that monetary relief for its non-performance is not adequate.
  2. Where no standard is available to ascertain the value of the actual damage caused by non-performance.
  3. Where it is not probable that the compensation money will be available.

Examples: The specific performance is granted in contracts connected with land, buildings, rare articles and unique goods having special value etc. because injured party will not be able to get an exact substitute in the market.
Specific performance is not allowed in the following cases:

  • Where monetary compensation is an adequate relief.
  • Where the contract is of personal nature, e.g. a contract to marry or a contract to paint a picture or,
  • Where it is not possible for the court to supervise the performance of the contract e.g. a building contract.
  • Where one of the parties to the contract is not competent to contract like a minor.

IV. SUIT FOR AN INJUNCTION
‘Injunction’ is an order of a court restraining a person from doing a particular act. It is a mode of securing the specific performance of the negative terms of the contract. To put it differently, where a party is in breach of a negative term of the contract (ie., where he is doing something which he promised not to do); the court may, by issuing an injunction, restrain him from doing what he promised not do so. Thus ‘injunction’ is a preventive relief. It is particularly appropriate in case ‘anticipatory breach of contract’ where damages would not be an adequate relief.
N, a him actress agreed to act exclusively for Warner Bros, for one year. During the year she contracted to act for X. Held, she could be restrained by an injunction from acting for X. Warns Bros. v. Nelson. It is to be noted that in this case an order directing N to act for Warner Bros. (Specific performance of the contract) was not passed because the contract was of a personal nature and performance could not have been supervised by the courts.

V. SUIT FOR QUANTUM MERUIT
Quantum Meruit means ‘as much as merits ‘or ‘as much as deserves or earns’. In legal sense, it means ‘payment in proportion to the work done’. In other words, quantum meruit means that a person can recover compensation in proportion to the work done or service rendered by him.
Normally, a person cannot claim performance from the other party, unless he has performed his obligation in full. But under the claim of quantum meruit, a person who has performed some work can claim payment in proportion to the work done. The right to claim quantum meruit is not conferred by the contract but it is conferred by the law. It is quasi-contractual in nature. This right is not similar to that of the damages, which arises out of the contract. Two things should be noted.

  1. The claim for quantum meruit can be made only when the original contract has been discharged. If the original contract exists, the party who has done something for the other party cannot have quantum meruit remedy but he is to rely on the remedy in damages, and
  2. Generally, the party who is not in default should bring the claim for quantum meruit.

The claim on quantum meruit arises in the following cases:-
1. Where there is breach of the contract
Where a party performs a part of the contract, but the other party breaks it in between, then the injured party can claim compensation for the work done or the service rendered.

  1. A agreed to write an article for B. The article was to be published in instalments. After 3 instalments were published, the magazine was closed. Can A claim on the basis of quantum meruit?
    Ans: Yes, Example (Planche v. Colburn).
  2. A boy was engaged for a complete journey against a lump sum payment of Rs. 500. The boy died before the journey was completed. Can his legal representatives claim the amount? Can a claim on the basis of quantum meruit arise in this case?
    Ans: No, as the contract is not divisible. (Cutter v. Powell).

2. When an agreement is discovered to be void
Where some work has been done and accepted under a contract which is subsequently dis-covered to be void, then the person who has performed the part of the contract is entitled to recover the amount for the work done. (Sec. 65)
Example: C was appointed as ‘managing director’ of a company at certain remuneration, by the board of directors. Subsequently it was discovered that the board was not qualified to make this appointment and hence it was void C, in the meantime, rendered services to the company. He sued the company for remuneration for the period he provided services. The court held that C could recover on ‘Quantum Meruit’  [Craven Ellis v. Canons Ltd. [1936] 2 K.B. 403]

3. When something has been done non-gratuitously
When something has been done non-gratuitously: Le., has been done with the intention of getting payment. (Sec. 70)

4. Where work has been done by the person guilty of breaking the contract
In such a case defaulting party would be liable for consequences of breach, but for the work done by him he may be entitled to get payment in the following circumstances:
(a) Where the work to be done was divisible: A contract is divisible and a party performs a part of it and refuses to perform the remaining part, the defaulting party can claim reasonable compensation for the part performed, on the basis of quantum meruit. Thus two conditions should exist :

  1. If the contract is divisible and,
  2. If the party not at fault has enjoyed the benefit of part performance.

Example: A. agreed to supply 1000 bales of cotton to B @ Rs. 11000 per bale. The bales were to be supplied in two instalments of 500 each. A supplied the first instalments but failed to supply the second. B must pay for 500 bales.

(b) On the other hand, if the contract is not divisible, i.e., it requires complete performance as a condition for payment, the party in default cannot claim payments for work done, on the basis of quantum meruit.

5. When the indivisible contract is performed substantially/fully
If a lump sum is to be paid for the completion of the entire work and the work has been completed in full, though badly, the person who has performed the contract can claim the lump sum; but the other party can also claim a deduction for bad work.
Example: A agreed to do decorate the flat of B for Rs. 1,00,000. The work was done but B complained of faulty workmanship. It was estimated that it could be rectified by spending Rs. 30,000 more. It was held that B could adjust this amount from the total amount due (of Rs. 1,00,000).

MULTIPLE CHOICE QUESTIONS:

1. Ordinary damages will be awarded in cases where
(a) The loss naturally flows from the breach of contract
(b) The loss is remotely connected with the breach of contract
(c) The loss is unusual and arises out of special circumstances peculiar to the contract
(d) None of these

2. Where the parties to a contract have agreed that a certain sum of money would be paid in case of breach of contract, the Court will ensure that
(a) The exact amount mentioned in the contract is paid to the injured party
(b) An amount not exceeding the stipulated amount is awarded
(c) Reasonable compensation not exceeding the amount stipulated is awarded
(d) A sum exceeding the amount stipulated is awarded

3. The word ‘impossible’ in section 56 connotes
(a) Physical impossibility
(b) Literal impossibility
(c) Commercial impossibility
(d) Impracticability of performance

4. In the case of wrongful dishonour of a cheque by a banker the damages awarded will be
(a) Nominal
(b) Special
(c) Exemplary
(d) Ordinary

5. If loss or damage arose naturally and directly in the usual course of things from a breach of contract, the aggrieved party would be eligible for
(a) Special damages
(b) Nominal damages
(c) Ordinary damages
(d) Exemplary damages

6. Anticipatory breach of contract takes place when there is
(a) Breach of contract when performance is actually due
(b) Breach of contract in the course of perfor-mance of the contract
(c) Breach of contract prior to the date of performance
(d) None of the above

7. Impossibility of performance occurs due to:
(a) Strike
(b) Lock-out
(c) Partial failure of object
(d) Destruction of subject-matter.

8. Object of granting damages is:
(a) to penalize the party,
(b) to monetarily compensate the party,
(c) to set an example before the society,
(d) none of the above.

9. Specific performance is ordered where:
(a) the contract is of personal nature,
(b) monetary compensation is an adequate remedy,
( c) monetary compensation is not an adequate remedy,
(d) performance is illegal.

10. An injunction order is granted by the Court in case:
(a) specific performance of the contract is possible.
(b) specific performance of the contract is impossible.
(c) the Court wants to restrain a party from committing a breach of contract.
(d) the contract is against public interest.

11. In the Indian Contract Act, Novation means
(a) Substitution of an existing contract with a new one
(b) No frustration of executed contracts
(c) Frustration due to change of circumstances
(d) Impossibility does not mean mere commercial difficulty

12. Hadley v. Baxendale case is a leading case on
(a) Breach of implied term
(b) Anticipatory breach
(c) Law of damages
(d) None of these

13. The remedies open to a person, suffering from breach of contract are
(a) Damages
(b) Injunction
(c) Quantum Meruit
(d) All of the above

14. Where the parties to a contract agree to substitute a new contract for it, it is known as
(a) Injunction
(b) Novation
(c) Rescission
(d) Alteration

15. A party to a contract committing breach, is liable to pay compensation in respect of
(a) The direct consequences flowing from the breach
(b) Loss or damage caused indirectly
( c) Losses caused whether directly or indirectly
(d) Losses caused remotely

16. A borrows Rs. 10,000 from B with interest at 12 per cent per annum, with a stipulation that in case of default A shall be liable to pay interest at 75 per cent from the date of default. A commits the default. B is entitled to recover from A
(a) 12% interest
(b) 75% interest
(c) 8796 interest
(d) Such compensation as the Court considers reasonable

Answers:
CA Foundation Business Laws Study Material Chapter 9 Discharge of Contract 2

STATE WHETHER THE FOLLOWING ARE TRUE OR FALSE:

1. A claim for Quantum Meruit cannot succeed when an indivisible contract for a lump sum is partly performed.
2. Nominal damages are never granted by way of compensation for loss.
3. Penalty can be recovered under the Indian Contract Act.
4. A person who himself is guilty of breach of contract cannot get compensation under the doctrine of quantum meruit.
5. The order for injunction and specific performances are simultaneously issued by the court.
6. The aggrieved party is always entitled to compensation no matter whether he has suffered some loss or not.
7. Special damages are recoverable only when the parties knew about them.
8. The aggrieved party is not responsible to mitigate the loss caused by the breach.
9. The measure of ordinary damages is the difference between the contract price and the market price.
10. The claim for quantum meruit can be made only when the original contract has been discharged.
11. Commercial impossibility is not a valid excuse for the non performance of a contract.
12. For the default in the repayment of loan on the agreed date, interest can be increased retrospectively from the date of lending.
13. Commercial impossibility does not make the contract void.
14. Cancellation of a contract by mutual consent of the parties is called waiver.

Answers:
CA Foundation Business Laws Study Material Chapter 9 Discharge of Contract 3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2

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 5 Quadratic Equations in One Variable Ex 5.2

More Exercises

Solve the following equations (1 to 24) by factorization

Question 1.
(i) 4x² = 3x
(ii) \(\frac { { x }^{ 2 }-5x }{ 2 } =0\)
Solution:
(i) 4x² = 3x
x(4x – 3) = 0
Either x = 0,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q1.1

Question 2.
(i) (x – 3) (2x + 5) = 0
(ii) x (2x + 1) = 6
Solution:
(i) (x – 3) (2x + 5) = 0
Either x – 3 = 0,
Then x = 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q2.1

Question 3.
(i) x² – 3x – 10 = 0
(ii) x(2x + 5) = 3
Solution:
(i) x² – 3x – 10 = 0
⇒ x² – 5x + 2x – 10 = 0
⇒ x(x – 5) + 2(x – 5) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q3.1

Question 4.
(i) 3x² – 5x – 12 = 0
(ii) 21x² – 8x – 4 = 0
Solution:
(i) 3x² – 5x – 12 = 0
⇒ 3x² – 9x + 4x – 12 = 0
⇒ 3x (x – 3) + 4(x – 3) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q4.1

Question 5.
(i) 3x² = x + 4
(ii) x(6x – 1) = 35
Solution:
(i) 3x² = x + 4
⇒ 3x² – x – 4 = 0
⇒ 3x² – 4x + 3x – 4 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q5.1

Question 6.
(i) 6p² + 11p – 10 = 0
(ii) \(\frac { 2 }{ 3 } { x }^{ 2 }-\frac { 1 }{ 3 } x=1 \)
Solution:
(i) 6p² + 11p – 10 = 0
⇒ 6p² + 15p – 4p – 10 = 0
⇒ 3p(2p + 5) – 2(2p + 5) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q6.1

Question 7.
(i) (x – 4)² + 5² = 13²
(ii) 3(x – 2)² = 147
Solution:
(i) (x – 4)² + 5² = 13²
x² – 8x + 16 + 25 = 169
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q7.2

Question 8.
(i) \(\\ \frac { 1 }{ 7 } \)(3x – 5)² = 28
(ii) 3(y² – 6) = y(y + 7) – 3
Solution:
(i) \(\\ \frac { 1 }{ 7 } \)(3x – 5)² = 28
(3x – 5)² = 28 × 7
⇒ 9x² – 30x + 25 = 196
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q8.2

Question 9.
x² – 4x – 12 = 0,when x∈N
Solution:
x² – 4x – 12 = 0
⇒ x² – 6x + 2x – 12 = 0
⇒ x (x – 6) + 2 (x – 6) = 0
⇒ (x – 6) (x + 2) = 0
Either x – 6 = 0, then x = 6
or x + 2 = 0, then x = -2
But -2 is not a natural number
∴ x = 6

Question 10.
2x² – 8x – 24 = 0 when x∈I
Solution:
2x² – 8x – 24 = 0
⇒ x² – 4x – 12 = 0 (Dividing by 2)
⇒ x² – 6x + 2x – 12 = 0
⇒ x (x – 6) + 2 (x – 6) = 0
⇒ (x – 6) (x + 2) = 0
Either x – 6 = 0, then, x = 6
or x + 2 = 0, then x = – 2
Hence x = 6, – 2

Question 11.
5x² – 8x – 4 = 0 when x∈Q
Solution:
5x² – 8x – 4 = 0
∵ 5 × ( – 4) = – 20
-20 = – 10 + 2
-8 = – 10 + 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q11.1

Question 12.
2x² – 9x + 10 = 0,when
(i)x∈N
(ii)x∈Q
Solution:
2x² – 9x + 10 = 0
⇒ 2x² – 4x – 5x + 10 = 0
⇒ 2x(x – 2) – 5(x – 2) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q12.1

Question 13.
(i) a²x² + 2ax + 1 = 0, a≠0
(ii) x² – (p + q)x + pq = 0
Solution:
(i) a²x² + 2ax + 1 = 0
⇒ a²x² + ax + ax + 1 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q13.1

Question 14.
a²x² + (a² + b²)x + b² = 0, a≠0
Solution:
a²x² + (a² + b²)x + b² = 0
⇒ a²x(x + 1) + b²(x + 1) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q14.1

Question 15.
(i) √3x² + 10x + 7√3 = 0
(ii) 4√3x² + 5x – 2√3 = 0
Solution:
(i) √3x² + 10x + 7√3 = 0
[ ∵ √3 x 7√3 = 7 x 3 = 21]
⇒ √3x(x + √3) + 7(x + √3) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q15.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q15.3

Question 16.
(i) x² – (1 + √2)x + √2 = 0
(ii) \(x+ \frac { 1 }{ x } \) = \(2 \frac { 1 }{ 20 } \)
Solution:
(i) x² – (1 + √2)x + √2 = 0
⇒ x² – x – √2x + √2 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q16.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q16.2

Question 17.
(i) \(\frac { 2 }{ { x }^{ 2 } } -\frac { 5 }{ x } +2=0,x\neq 0 \)
(ii)\(\frac { { x }^{ 2 } }{ 15 } -\frac { x }{ 3 } -10=0 \)
Solution:
(i) \(\frac { 2 }{ { x }^{ 2 } } -\frac { 5 }{ x } +2=0,x\neq 0 \)
⇒ 2 – 5x + 2x² = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q17.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q17.2

Question 18.
(i) \(3x-\frac { 8 }{ x } =2 \)
(ii) \(\frac { x+2 }{ x+3 } =\frac { 2x-3 }{ 3x-7 } \)
Solution:
(i) \(3x-\frac { 8 }{ x } =2 \)
\(\frac { { 3x }^{ 2 }-8 }{ x } =2 \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q18.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q18.2

Question 19.
(i) \(\frac { 8 }{ x+3 } -\frac { 3 }{ 2-x } =2 \)
(ii) \(\frac { x }{ x-1 } +\frac { x-1 }{ x } =2\frac { 1 }{ 2 } \)
Solution:
(i) \(\frac { 8 }{ x+3 } -\frac { 3 }{ 2-x } =2 \)
\(\frac { 16-8x-3x-9 }{ (x+3)(2-x) } =2 \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q19.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q19.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q19.3

Question 20.
(i) \(\frac { x }{ x+1 } +\frac { x+1 }{ x } =\frac { 34 }{ 15 } \)
(ii) \(\frac { x+1 }{ x-1 } +\frac { x-2 }{ x+2 } =3 \)
Solution:
(i) \(\frac { x }{ x+1 } +\frac { x+1 }{ x } =\frac { 34 }{ 15 } \)
\(\frac { { x }^{ 2 }+{ x }^{ 2 }+2x+1 }{ x(x+1) } =\frac { 34 }{ 15 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q20.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q20.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q20.3

Question 21.
(i) \(\frac { 1 }{ x-3 } -\frac { 1 }{ x+5 } =\frac { 1 }{ 6 } \)
(ii) \(\frac { x-3 }{ x+3 } +\frac { x+3 }{ x-3 } =2\frac { 1 }{ 2 } \)
Solution:
(i) \(\frac { 1 }{ x-3 } -\frac { 1 }{ x+5 } =\frac { 1 }{ 6 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q21.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q21.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q21.3

Question 22.
(i) \(\frac { a }{ ax-1 } +\frac { b }{ bx-1 } =a+b,a+b\neq 0,ab\neq 0\)
(ii) \(\frac { 1 }{ 2a+b+2x } =\frac { 1 }{ 2a } +\frac { 1 }{ b } +\frac { 1 }{ 2x } \)
Solution:
(i) \(\frac { a }{ ax-1 } +\frac { b }{ bx-1 } =a+b\)
⇒ \(\left( \frac { a }{ ax-1 } -b \right) +\left( \frac { b }{ bx-1 } -a \right) =0\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q22.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q22.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q22.3

Question 23.
\(\frac { 1 }{ x+6 } +\frac { 1 }{ x-10 } =\frac { 3 }{ x-4 } \)
Solution:
\(\frac { 1 }{ x+6 } +\frac { 1 }{ x-10 } =\frac { 3 }{ x-4 } \)
⇒ \(\frac { x-10+x+6 }{ (x+6)(x-10) } =\frac { 3 }{ x-4 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q23.1

Question 24.
(i) \(\sqrt { 3x+4 } =x\)
(ii) \(\sqrt { x(x-7) } =3\sqrt { 2 } \)
Solution:
(i) \(\sqrt { 3x+4 } =x\)
Squaring on both sides
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q24.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q24.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q24.3

Question 25.
Use the substitution y = 3x + 1 to solve for x : 5(3x + 1 )² + 6(3x + 1) – 8 = 0
Solution:
y = 3x + 1
Now, 5(3x + 1)² + 6(3x + 1) – 8 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q25.1

Question 26.
Find the values of x if p + 1 =0 and x² + px – 6 = 0
Solution:
p + 1 = 0, then p = – 1
Substituting the value of p in the given quadratic equation
x² + ( – 1)x – 6 = 0
⇒ x² – x – 6 = 0
⇒ x² – 3x + 2x – 6 = 0
⇒ x (x – 3) + 2 (x – 3) = 0
⇒ (x – 3) (x + 2) = 0
Either x – 3 = 0, then x = 3
or x + 2 = 0, then x = – 2
Hence x = 3, -2

Question 27.
Find the values of x if p + 7 = 0, q – 12 = 0 and x² + px + q = 0,
Solution:
p + 7 = 0, then p = – 7
and q – 12 = 0, then q = 12
Substituting the values of p and q in the given quadratic equation,
x² – 7x + 12 = 0
⇒ x² – 3x – 4x + 12 = 0
⇒ x (x – 3) – 4 (x – 3) = 0
⇒ (x – 3) (x – 4) = 0
Either x – 3 = 0, then x = 3
or x – 4 = 0, then x = 4
Hence x = 3, 4

Question 28.
If x = p is a solution of the equation x(2x + 5) = 3, then find the value of p.
Solution:
Given, x = p and x(2x + 5) = 3
Substituting the value of p, we get
p(2p + 5) = 3
⇒ 2p² + 5p – 3 = 0
⇒ 2p² + 6p – p – 3 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q28.1

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 are helpful to complete your math homework.

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ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3

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 7 Ratio and Proportion Ex 7.3

More Exercises

Question 1.
If a : b : : c : d, prove that
(i) \(\frac { 2a+5b }{ 2a-5b } =\frac { 2c+5d }{ 2c-5d } \)
(ii) \(\frac { 5a+11b }{ 5c+11d } =\frac { 5a-11b }{ 5c-11d } \)
(iii) (2a + 3b)(2c – 3d) = (2a – 3b)(2c + 3d)
(iv) (la + mb) : (lc + mb) :: (la – mb) : (lc – mb)
Solution:
(i) a : b : : c : d
then \(\frac { a }{ b } =\frac { c }{ d } \)
⇒ \(\frac { 2a }{ 5b } =\frac { 2c }{ 5d } \) (multiply by \(\\ \frac { 2 }{ 5 } \) )
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q1.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q1.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q1.3

Question 2.
(i) If \(\frac { 5x+7y }{ 5u+7v } =\frac { 5x-7y }{ 5u-7v } \) , Show that \(\frac { x }{ y } =\frac { u }{ v } \)
(ii) \(\frac { 8a-5b }{ 8c-5d } =\frac { 8a+5b }{ 8c+5d } \) , prove that \(\frac { a }{ b } =\frac { c }{ d } \)
Solution:
(i) \(\frac { 5x+7y }{ 5u+7v } =\frac { 5x-7y }{ 5u-7v } \)
Applying alternendo \(\frac { 5x+7y }{ 5u+7v } =\frac { 5x-7y }{ 5u-7v } \)
Applying componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q2.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q2.2

Question 3.
If (4a + 5b) (4c – 5d) = (4a – 5d) (4c + 5d), prove that a, b, c, d are in proporton.
Solution:
(4a + 5b) (4c – 5d) = (4a – 5d) (4c + 5d)
⇒ \(\frac { 4a+5b }{ 4a-5b } =\frac { 4c+5d }{ 4c-5d } \)
Applying componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q3.1

Question 4.
If (pa + qb) : (pc + qd) :: (pa – qb) : (pc – qd) prove that a : b : : c : d
Solution:
(pa + qb) : (pc + qd) :: (pa – qb) : (pc – qd)
⇒ \(\frac { pa+qb }{ pc+qd } =\frac { pq-qb }{ pc-qd } \)
⇒ \(\frac { pa+qb }{ pc-qd } =\frac { pq+qb }{ pc-qd } \)
Applying componendo and dividendo
⇒ \(\frac { pa+qb+pa-qb }{ pa+qb-pa+qb } =\frac { pc+qd+pc-qd }{ pc-qd-pc+qd } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q4.1

Question 5.
If (ma + nb): b :: (mc + nd) : d, prove that a, b, c, d are in proportion.
Solution:
(ma + nb): b :: (mc + nd) : d
⇒ \(\frac { ma+nb }{ b } =\frac { mc+nd }{ d } \)
⇒ mad + nbd = mbc + nbd
⇒ mad = mbc
⇒ ad = bc
⇒ \(\frac { a }{ b } =\frac { c }{ d } \)
Hence a : b :: c : d.

Question 6.
If (11a² + 13b²) (11c² – 13d²) = (11a² – 13b²)(11c² + 13d²), prove that a : b :: c : d.
Solution:
(11a² + 13b²) (11c² – 13d²) = (11a² – 13b²)(11c² + 13d²)
⇒ \(\frac { 11a+{ 13b }^{ 2 } }{ { 11a }^{ 2 }-{ 13b }^{ 2 } } =\frac { { 11c }^{ 2 }+{ 13d }^{ 2 } }{ { 11c }^{ 2 }-{ 13d }^{ 2 } } \)
Applying componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q6.1

Question 7.
If (a + 3b + 2c + 6d) (a – 3b – 2c + 6d) = (a + 3b – 2c – 6d) (a – 3b + 2c – 6d), prove that a : b :: c : d.
Solution:
\(\frac { a + 3b + 2c + 6d }{ a – 3b + 2c – 6d } =\frac { a + 3b – 2c – 6d }{ a – 3b – 2c + 6d } \)
⇒ \(\frac { a + 3b + 2c + 6d }{ a + 3b – 2c – 6d } =\frac { a – 3b + 2c – 6d }{ a – 3b – 2c + 6d } \) (by altenendo)
Applying componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q7.2

Question 8.
If \(x=\frac { 2ab }{ a+b } \) find the value of \(\frac { x+a }{ x-a } +\frac { x+b }{ x-b } \)
Solution:
\(x=\frac { 2ab }{ a+b } \)
⇒ \(\frac { x }{ a } =\frac { 2b }{ a+b } \)
Applying componendo and dividendo,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q8.2

Question 9.
If \(x=\frac { 8ab }{ a+b } \) find the value of \(\frac { x+4a }{ x-4a } +\frac { x+4b }{ x-4b } \)
Solution:
\(x=\frac { 8ab }{ a+b } \)
⇒ \(\frac { x }{ 4a } =\frac { 2b }{ a+b } \)
Applying componendo and dividendo,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q9.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q9.2

Question 10.
If \(x=\frac { 4\sqrt { 6 } }{ \sqrt { 2 } +\sqrt { 3 } } \) find the value of \(\frac { x+2\sqrt { 2 } }{ x-2\sqrt { 2 } } +\frac { x+2\sqrt { 3 } }{ x-2\sqrt { 3 } } \)
Solution:
\(x=\frac { 4\sqrt { 6 } }{ \sqrt { 2 } +\sqrt { 3 } } \)
⇒ \(\frac { 4\sqrt { 2 } \times \sqrt { 3 } }{ \sqrt { 2 } +\sqrt { 3 } } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q10.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q10.2

Question 11.
Solve \(x:\frac { \sqrt { 36x+1 } +6\sqrt { x } }{ \sqrt { 36x+1 } -6\sqrt { x } } =9 \)
Solution:
\(\frac { \sqrt { 36x+1 } +6\sqrt { x } }{ \sqrt { 36x+1 } -6\sqrt { x } } =\frac { 9 }{ 1 } \)
Applying componendo and dividendo,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q11.1

Question 12.
Find x from the following equations :
(i) \(\frac { \sqrt { 2-x } +\sqrt { 2+x } }{ \sqrt { 2-x } -\sqrt { 2+x } } =3 \)
(ii) \(\frac { \sqrt { x+4 } +\sqrt { x-10 } }{ \sqrt { x+4 } -\sqrt { x-10 } } =\frac { 5 }{ 2 } \)
(iii) \(\frac { \sqrt { 1+x } +\sqrt { 1-x } }{ \sqrt { 1+x } -\sqrt { 1-x } } =\frac { a }{ b } \)
(iv) \(\frac { \sqrt { 12x+1 } +\sqrt { 2x-3 } }{ \sqrt { 12x+1 } -\sqrt { 2x-3 } } =\frac { 3 }{ 2 } \)
(v) \(\frac { 3x+\sqrt { { 9x }^{ 2 }-5 } }{ 3x-\sqrt { { 9x }^{ 2 }-5 } } =5 \)
(vi) \(\frac { \sqrt { a+x } +\sqrt { a-x } }{ \sqrt { a+x } -\sqrt { a-x } } =\frac { c }{ d } \)
Solution:
(i) \(\frac { \sqrt { 2-x } +\sqrt { 2+x } }{ \sqrt { 2-x } -\sqrt { 2+x } } =3 \)
Applying componendo and dividendo,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q12.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q12.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q12.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q12.4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q12.5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q12.6
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q12.7
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q12.8
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q12.9

Question 13.
Solve \(\frac { 1+x+{ x }^{ 2 } }{ 1-x+{ x }^{ 2 } } =\frac { 62\left( 1+x \right) }{ 63\left( 1-x \right) } \)
Solution:
\(\frac { 1+x+{ x }^{ 2 } }{ 1-x+{ x }^{ 2 } } =\frac { 62\left( 1+x \right) }{ 63\left( 1-x \right) } \)
⇒ \(\frac { \left( 1-x \right) \left( 1+x+{ x }^{ 2 } \right) }{ \left( 1+x \right) \left( 1-x+{ x }^{ 2 } \right) } =\frac { 62 }{ 63 } \)
⇒ \(\frac { \left( 1+x \right) \left( 1-x+{ x }^{ 2 } \right) }{ \left( 1-x \right) \left( 1+x+{ x }^{ 2 } \right) } =\frac { 63 }{ 62 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q13.1

Question 14.
Solve for \(x:16{ \left( \frac { a-x }{ a+x } \right) }^{ 3 }=\frac { a+x }{ a-x } \)
Solution:
\(x:16{ \left( \frac { a-x }{ a+x } \right) }^{ 3 }=\frac { a+x }{ a-x } \)
⇒ \(\left( \frac { a+x }{ a-x } \right) \times { \left( \frac { a+x }{ a-x } \right) }^{ 3 }=16 \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q14.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q14.2

Question 15.
If \(x=\frac { \sqrt { a+x } +\sqrt { a-1 } }{ \sqrt { a+1 } -\sqrt { a-1 } } \) , using properties of proportion , show that x² – 2ax + 1 = 0
Solution:
We have \(x=\frac { \sqrt { a+x } +\sqrt { a-1 } }{ \sqrt { a+1 } -\sqrt { a-1 } } \)
⇒ \(\frac { x+1 }{ x-1 } =\frac { 2\sqrt { a+1 } }{ 2\sqrt { a-1 } } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q15.1

Question 16.
Given \(x=\frac { \sqrt { { a }^{ 2 }+{ b }^{ 2 } } +\sqrt { { a }^{ 2 }-{ b }^{ 2 } } }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } -\sqrt { { a }^{ 2 }-{ b }^{ 2 } } } \) Use componendo and dividendo to prove that \({ b }^{ 2 }=\frac { { 2a }^{ 2 }x }{ { x }^{ 2 }+1 } \)
Solution:
If \(\frac { x }{ 1 } =\frac { \sqrt { { a }^{ 2 }+{ b }^{ 2 } } +\sqrt { { a }^{ 2 }-{ b }^{ 2 } } }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } -\sqrt { { a }^{ 2 }-{ b }^{ 2 } } } \)
Applying componendo and dividendo both sides
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q16.1

Question 17.
Given that \(\frac { { a }^{ 3 }+3{ ab }^{ 2 } }{ { b }^{ 3 }+{ 3a }^{ 2 }b } =\frac { 63 }{ 62 } \). Using componendo and dividendo find a: b. (2009)
Solution:
Given that \(\frac { { a }^{ 3 }+3{ ab }^{ 2 } }{ { b }^{ 3 }+{ 3a }^{ 2 }b } =\frac { 63 }{ 62 } \)
By componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q17.1
a : b = 3 : 2

Question 18.
Give \(\frac { { x }^{ 3 }+12x }{ { 6x }^{ 2 }+8 } =\frac { { y }^{ 3 }+27y }{ { 9y }^{ 2 }+27 } \) Using componendo and dividendo find x : y.
Solution:
Give \(\frac { { x }^{ 3 }+12x }{ { 6x }^{ 2 }+8 } =\frac { { y }^{ 3 }+27y }{ { 9y }^{ 2 }+27 } \)
Using componendo-dividendo, we have
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q18.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q18.2

Question 19.
Using the properties of proportion, solve the following equation for x; given
\(\frac { x^{ 3 }+3x }{ { 3x }^{ 2 }+1 } =\frac { 341 }{ 91 } \)
Solution:
\(\frac { x^{ 3 }+3x }{ { 3x }^{ 2 }+1 } =\frac { 341 }{ 91 } \)
Applying componendo and dividendo
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q19.1

Question 20.
If \(\frac { x+y }{ ax+by } =\frac { y+z }{ ay+bz } =\frac { z+x }{ az+bx } \) , prove that each of these ratio is equal to \(\\ \frac { 2 }{ a+b } \) unless x + y + z = 0
Solution:
\(\frac { x+y }{ ax+by } =\frac { y+z }{ ay+bz } =\frac { z+x }{ az+bx } \)
= \(\frac { x+y+y+z+z+x }{ ax+by+ay+bz+az+bx } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Ex 7.3 Q20.1

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