CBSE Class 9 – Page 38

RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1D

January 16, 2023

RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1D

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 1 Real Numbers Ex 1D.

Other Exercises

Question 1.
Solution:
RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1D 1

Question 2.
Solution:
RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1D 4

Question 3.
Solution:
RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1D 6

Question 4.
Solution:
RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1D 7

Question 5.
Solution:
(i) Draw a line segment AB = 3.2 units (cm) and extend it to C such that BC = 1 unit.
RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1D 10
(ii) Find the mid-point O of AC.
(iii) With centre O and OA as radius draw a semicircle on AC
(iv) Draw BD ⊥ AC meeting the semicircle at D.
(v) Join BD which is √3.2 units.
(vi) With centre B and radius BD, draw an arc meeting AC when produced at E.
Then BE = BD = √3.2 units. Ans.

Question 6.
Solution:
(i) Draw a line segment AB = 7.28 units and produce is to C such that BC = 1 unit (cm)
RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1D 11
(ii) Find the mid-point O of AC.
(iii) With centre O and radius OA, draw a semicircle on AC.
(iv) Draw a perpendicular BD at AC meeting the semicircle at D
Then BD = √7.28 units.
(v) With centre B and radius BD, draw an arc which meet AC produced at E.
Then BE = BD = √7.28 units.

Question 7.
Solution:
(A) For Addition
(i) Closure property: The sum of two real numbers is always a real number.
(ii) Associative Law : (a + b) + c = a + (b + c), for all values of a, b and c.
(iii) Commutative Law : a + b = b + a for all real values of a and b.
(iv) Existance of Additive Identity : 0 is the real number such that: 0 + a = a + 0 = afor every real value of a.
(v) Existance of addtive inverse : For each real value of a, there exists a real value (-a) such that a + (-a) = (-a) + a = 0, Then (a) and (-a) are called the additive inverse of each other.
(v) Existence of Multiplicative Inverse. For each non zero real number a, there exists a real number \(\frac { 1 }{ a }\) such that a . \(\frac { 1 }{ a }\) = \(\frac { 1 }{ a }\) . a = 1
a and \(\frac { 1 }{ a }\) are called multiplicative inverse or reciprocal of each other.
(B) Multiplication
(i) Closure property: The product of two real numbers is always a real number.
(ii) Associative law : ab(c) = a(bc) for all real values of a, b and c
(iii) Commutative law : ab=ba for all real numbers a and b
(iv) Existance of Multiplicative Identity: clearly is a real number such that 1.a = a.1 = a for every value of a.

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RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS

January 16, 2023

RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS

Question 1.
Write (625)^–^1/4 in decimal form.
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS Q1.1

Question 2.
State the product law of exponents:
Solution:
x^m x xⁿ = x^{m +n}

Question 3.
State the quotient law of exponents.
Solution:
x^m ÷ xⁿ = x^{m -n}

Question 4.
State the power law of exponents.
Solution:
(x^m)ⁿ =x^{m x n} = x^mn

Question 5.
If 2⁴ x 4² – 16^x, then find the value of x.
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS Q5.1

Question 6.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS Q6.1
Solution:

Question 7.
Write the value of \(\sqrt [ 3 ]{ 7 }\) x \(\sqrt [ 3 ]{ 49 }\) .
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS Q7.1

Question 8.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS Q8.1
Solution:

Question 9.
Write the value of \(\sqrt [ 3 ]{ 125×27 }\)
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS Q9.1

Question 10.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS Q10.1
Solution:

Question 11.

Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS Q11.2

Question 12.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS Q12.1
Solution:

Question 13.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS Q13.1
Solution:

Question 14.
If (x – 1)³ = 8, what is the value of (x + 1)²?
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers VSAQS Q14.1

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RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1C

January 16, 2023

RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1C

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 1 Real Numbers Ex 1C.

Other Exercises

Question 1.
Solution:
Irrational numbers : Numbers which are not rational numbers, are called irrational numbers. Rational numbers can be expressed in terminating decimals or repeating decimals while irrational number can’t.
\(\frac { 1 }{ 2 } \) , \(\frac { 2 }{ 3 } \) , \(\frac { 7 }{ 5 } \) etc.are rational numbers and π, √2, √3, √5, √6….etc are irrational numbers

Question 2.
Solution:
(i) √4 = ±2, it is a rational number
(ii) √196 = ±14 it is a rational number
(iii) √21 It is irrational number.
(iv) √43 It is irrational number.
(v) 3 + √3 It is irrational number because sum of a rational and an irrational number is irrational
(vi) √7 – 2 It is irrational number because difference of a rational and irrational number is irrational
(vii) \(\frac { 2 }{ 3 } \)√6 . It is irrational number because product of a rational and an irrational number is an irrational number.
(viii) 0.\(\overline { 6 } \) = 0.6666…. It is rational number because it is a repeating decimal.
(ix) 1.232332333…. It is irrational number because it not repeating decimal
(x) 3.040040004…. It is irrational number because it is not repeating decimal.
(xi) 3.2576 It is rational number because it is a terminating decimal.
(xii) 2.3565656…. = 2.3 \(\overline { 56 } \) It is rational number because it is a repeating decimal.
(xiii) π It is an irrational number
(xiv) \(\frac { 22 }{ 7 } \). It is a rational number which is in form of \(\frac { p }{ q } \) Ans.

Question 3.
Solution:
(i) Let X’OX be a horizontal line, taken as the x-axis and let O be the origin. Let O represent 0.
Taken OA = 1 unit and draw AB ⊥ OA such that AB = 1 unit. Join OB, Then,
RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1C 1

Question 4.
Solution:
Firstly we represent √5 on the real line X’OX. Then we will find √6 and √7 on that real line.
Now, draw a horizontal line X’OX, taken as x-axis
RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1C 4

Question 5.
Solution:
(i) 4 + √5 : It is irrational number because in it, 4 is a rational number and √5 is irrational and sum of a rational and an irrational is also an irrational.
(ii) (-3 + √6) It is irrational number because in it, -3 is a rational and √6 is irrational and sum or difference of a rational and irrational is an irrational.
(iii) 5√7 : It is irrational because 5 is rational and √7 is irrational and product of a rational and an irrational is an irrational.
(iv) -3√8 : It is irrational because -3 is a rational and √8 is an irrational and product of a rational and an irrational is also an irrational.
(v) \(\frac { 2 }{ \sqrt { 5 } } \) It is irrational because 2 is a rational and √5 is an irrational and quotient of a rational and an irrational is also an irrational.
(vi) \(\frac { 4 }{ \sqrt { 3 } } \) It is irrational because 4 is a rational and √3 is an irrational number and quotient of a rational and irrational is also an irrational.

Question 6.
Solution:
(i) True.
(ii) False, as the sum of two irrational number is irrational is not always true.
(iii) True.
(iv) False, as the product of two irrational numbers is irrational is not always true.
(v) True.
(vi) True.
(vii) False as a real number can be either rational or irrational.

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RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2

January 12, 2023

RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2

Question 1.
Assuming that x, y, z are positive real numbers, simplify each of the following:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q1.1
Solution:

Question 2.
Simplify:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q2.1
Solution:

Question 3.
Prove that:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q3.1
Solution:

Question 4.
Show that:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q4.1

Solution:

Question 5.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q5.1
Solution:

Question 6.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q6.1
Solution:

Question 7.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q7.1
Solution:

Question 8.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q8.1
Solution:

Question 9.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q9.1
Solution:

Question 10.
Find the values of x in each of the following:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q10.1
Solution:

Question 11.
If x = 2¹^/3 + 2^2/3, show that x³ – 6x = 6.
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q11.1

Question 12.
Determine (8x)^x, if 9^x^{+ 2} = 240 + 9^x.
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q12.1

Question 13.
If 3^x⁺¹ = 9^x-², find the value of 2^{1 +x}.
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q13.1

Question 14.
If 3^4x = (81)^-1 and 10^1/y = 0.0001, find the value of 2^-x+4ySolution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q14.1

Question 15.
If 5^3x = 125 and 10^y = 0.001 find x and y.
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q15.1

Question 16.
Solve the following equations:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q16.1
Solution:

Question 17.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q17.1
Solution:

Question 18.
If a and b are different positive primes such that
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q18.1
Solution:

Question 19.
If 2^x x 3^y x 5^z = 2160, find x, y and z. Hence, compute the value of 3^x x 2^-y x 5^-z.
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q19.1

Question 20.
If 1176 = 2^a x 3^b x 7^c, find the values of a, b and c. Hence, compute the value of 2^a x 3^b x 7^-c as a fraction.
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q20.1

Question 21.
Simplify:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q21.1
Solution:

Question 22.
Show that:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q22.1
Solution:

Question 23.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.2 Q23.1
Solution:

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RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1B

January 12, 2023

RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1B

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Solutions Class 9 Chapter 1 Real Numbers Ex 1B.

Other Exercises

Question 1.
Solution:
We know that a fraction \(\frac { p }{ q } \) is terminating if prime factors of q are 2 and 5 only.
Hence.
(i) \(\frac { 13 }{ 80 } \) and \(\frac { 16 }{ 125 } \) are the terminating decimals.

Question 2.
Solution:
RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1B 1

Question 3.
Solution:
(i) Let,x = 0.\(\overline { 3 } \) = 0.3333…(i)
Then, 10x = 3.3333….
RS Aggarwal Class 9 Solutions Chapter 1 Real Numbers Ex 1B 6

Question 4.
Solution:
(i) True, because set of natural numbers is a subset of whole number.
(ii) False, because the number 0 does not belong to the set of natural numbers.
(iii) True, because a set of integers is a subset of a rational numbers.
(iv) False, because the set of rational numbers is not a subset of whole numbers.
(v) True, because rational number can be expressed as terminating or repeating decimals.
(vi) True, because every rational number can be express as repeating decimals.
(vii) True, because 0 = \(\frac { 0 }{ 1 } \), which is a rational number Ans.

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RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1

January 12, 2023

RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1

Question 1.
Simplify the following:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q1.1
Solution:

Question 2.
If a = 3 and b =-2, find the values of:
(i) a^a+ b^b
(ii) a^b + b^a(iii) (a+b)^abSolution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q2.1

Question 3.
Prove that:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q3.1
Solution:

Question 4.
Prove that
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q4.1
Solution:

Question 5.
Prove that
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q5.1
Solution:

Question 6.
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q6.1
Solution:

Question 7.
Simplify the following:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q7.1
Solution:

Question 8.
Solve the following equations for x:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q8.1
Solution:

Question 9.
Solve the following equations for x:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q9.1
Solution:

Question 10.
If 49392 = a⁴b²V³, find the values.of a, b and c, where a, b and c are different positive primes.
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q10.1

Question 11.
If 1176 = 2^a x 3^b x T^c, find a, 6 and c.
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q11.1

Question 12.
Given 4725 = 3^a5^b7^c, find:
(i) the integral values of a, b and c
(ii) the value of 2^-a 3^b 7^c
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q12.1

Question 13.
If a = xy^p-1, b = xy ^q^-1 and c = xy^r-1, prove that a^q-rb^r-pc^p-q = 1
Solution:
RD Sharma Class 9 Solutions Chapter 2 Exponents of Real Numbers Ex 2.1 Q13.1

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RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS

January 12, 2023

RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS

Other Exercises

the correct alternative in each of the following:
Question 1.
Which one of the following is a correct statement?
(a) Decimal expansion of a rational number is terminating
(b) Decimal expansion of a rational number is non-terminating
(c) Decimal expansion of an irrational number is terminating
(d) Decimal expansion of an irrational number is non-terminating and non-repeating
Solution:
Decimal expansion of an irrational number is non-terminating and non-repeating . (d)

Question 2.
Which one of the following statements is true?
(a) The sum of two irrational numbers is always an irrational-number
(b) The sum of two irrational numbers is always a rational number
(c) The sum of two irrational numbers may be a rational number or an irrational number
(d) The sum of two irrational numbers is always an integer
Solution:
The sum of two irrational numbers may be a rational number or an irrational number (c)

Question 3.
Which of the following is a correct statement?
(a) Sum of two irrational numbers is always irrational
(b) Sum of a rational and irrational number is always an irrational number
(c) Square of an irrational number is always a rational number
(d) Sum of two rational numbers can never be an integer
Solution:
Sum of a rational and irrational number is always an irrational number (b)

Question 4.
Which of the following statements is true?
(a) Product of two irrational numbers is always irrational
(b) Product of a rational and an irrational number is always irrational
(c) Sum of two irrational numbers can never be irrational
(d) Sum of an integer and a rational number can never be an integer
Solution:
Product of a rational and an irrational number is always irrational (b)

Question 5.
Which of the following is irrational?
RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS Q5.1
Solution:

Question 6.
Which of the following is irrational?
(a) 14
(b) 0.14\(\overline { 16 }\)
(c) 0.\(\overline { 1416 }\)
(d) 0.1014001400014
Solution:
0.1014001400014…….. is irrational as it is non-terminating nor repeating decimal, (d)

Question 7.
Which of the following is rational?
RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS Q7.1
Solution:

Question 8.
The number 0.318564318564318564… is:
(a) a natural number
(b) an integer
(c) a rational number
(d) an irrational number
Solution:
The number = 0.318564318564318564…………
= 0.\(\overline { 318564 }\)
∵ The decimal is non-terminating and recurring
∴ It is rational number. (c)

Question 9.
If n is a natural number, then \(\sqrt { n } \)is
(a) always a natural number
(b) always a rational number
(c) always an irrational number
(d) sometimes a natural number and sometimes an irrational number
Solution:
If n is a natural number then \(\sqrt { n } \) may sometimes a natural number and sometime an irrational number e.g.
If n = 2 then \(\sqrt { n } \) =\(\sqrt { 2 } \) which is are irrational and if n = 4, then \(\sqrt { n } \)= \(\sqrt { 4 } \) = 2 which is a rational number. (d)

Question 10.
Which of the following numbers can be represented as non-terminating, repeating decimals?

Solution:
RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS Q10.2

Question 11.
Every point on a number line represents
(a) a unique real number
(b) a natural number
(c) a rational number
(d) an irrational number
Solution:
Every point on a number line represents a unique real number. (a)

Question 12.
Which of the following is irrational?
(a) 0.15
(b) 0.01516
(c) 0.\(\overline { 1516 }\)
(d) 0.5015001500015..
Solution:
As it is non-terminating non-repeating decimals while others are terminating or non-terminating repeating decimals. (d)

Question 13.
The number 1.\(\overline { 27 }\) in the form \(\frac { p }{ q }\) , where p and q are integers and q ≠ 0, is
RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS Q13.1
Solution:

Question 14.
RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS Q14.1
Solution:

Question 15.
RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS Q15.1
Solution:

Question 16.
RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS Q16.1
Solution:

Question 17.
RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS Q17.1
Solution:

Question 18.
RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS Q18.1
Solution:

Question 19.
RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS Q19.1
Solution:
An irrational number between 2 and 2.5 is \(\sqrt { 5 } \) as it has approximate value 2.236… (b)

Question 20.
The number of consecutive zeros in 2³ x 3⁴x 5⁴ x 7, is
(a) 3
(b) 2
(c) 4
(d) 5
Solution:
In 2³ x 3⁴ x 5⁴ x 7, number of consecutive zero will be 3 as 2³ x 5⁴ = 2 x 2 x 2 x 5x 5 x 5 x 5 = 5000 (a)

Question 21.
The smallest rational number by which \(\frac { 1 }{ 3 }\) should be multiplied so that its decimal expansion terminates after one place of decimal, is
RD Sharma Class 9 Solutions Chapter 1 Number Systems MCQS Q21.1
Solution:

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RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.6

January 12, 2023

RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.6

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.6

Question 1.
Visualise 2.665 on the number line, using successive magnification.
Solution:
2.665
∵ It lies between 2 and 3
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.6 Q1.1

Question 2.
Visualise the representation of 5.3\(\overline { 7 }\) on the number line upto 5 decimal places, that is upto 5.37777. [NCERT]
Solution:
5.37777.
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.6 Q2.1

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RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.5

January 12, 2023

RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.5

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.5

Question 1.
Complete the following sentences:
(i) Every point on the number line corresponds to a … number which many be either … or
(ii) The decimal form of an irrational number is neither … nor …
(iii) The decimal representation of a rational number is either … or …
(iv) Every real number is either … number or … number.
Solution:
(i) Every point on the number line corresponds to a real number which many be either rational or irrational.
(ii) The decimal form of an irrational number is neither terminating nor repeating.
(iii) The decimal representation of a rational number is either terminating or nonterminating, recurring.
(iv) Every real number is either rational number or an irrational number.

Question 2.
Find whether the following statements are true or false:
(i) Every real number is either rational or irrational.
(ii) π is an irrational number.
(iii) Irrational numbers cannot be represented by points on the number line.
Solution:
(i) True. (Value of π = 3.14)
(ii) False : we can represent irrational number also.

Question 3.
Represent \(\sqrt { 6 } \), \(\sqrt { 7 } \), \(\sqrt { 8 } \) on the number line.
Solution:
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.5 Q3.1

Question 4.
Represent \(\sqrt { 3.5 } \) , \(\sqrt { 9.4 } \)and \(\sqrt { 10.5 } \) on the real number line.
Solution:
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.5 Q4.1

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RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.4

January 12, 2023

RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.4

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.4

Question 1.
Define an irrational number.
Solution:
A number which cannot be expressed in the form of \(\frac { p }{ q }\) where p and q are integers and q ≠ 0 is called an irrational number.

Question 2.
Explain, how irrational numbers differ from rational numbers?
Solution:
A rational number can be expressed in either terminating decimal or non-terminating recurring decimals but an irrational number is expressed in non-terminating non-recurring decimals.

Question 3.
Examine, whether the following numbers are rational or irrational:
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.4 Q3.1
Solution:

Question 4.
Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.4 Q4.1
Solution:

Question 5.
In the following equation, find which variables x, y, z etc. represent rational or irrational numbers:
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.4 Q5.1
Solution:

Question 6.
Given two rational numbers lying between 0.232332333233332… and 0.212112111211112.
Solution:
Two rational numbers lying between 0.232332333233332… and 0.212112111211112… will be 0.232 and 0.212

Question 7.
Give two rational numbers lying between 0.515115111511115… and 0.5353353335…
Solution:
Two rational numbers lying between 0.515115111511115… and 0.535335333533335… will be 0.515, 0.535

Question 8.
Find one irrational numbers between 0.2101 and 0.2222… = 0.\(\overline { 2 }\).
Solution:
One irrational number lying between 0.2101 and 0.2222… = 0.\(\overline { 2 }\) will be 2201.0010001…

Question 9.
Find a rational number and also an irrational number lying between the numbers, 0.3030030003… and 0.3010010001…
Solution:
Between two numbers 0.3030030003… and 0.3010010001…, a rational will be 0.301 and irrational number will be 0.3020020002…

Question 10.
Find three different irrational numbers between the rational numbers \(\frac { 5 }{ 7 }\) and \(\frac { 9 }{ 11 }\). [NCERT]
Solution:
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.4 Q10.1

Question 11.
Give an example of each, of two irrational numbers whose:
(i) difference is a rational number.
(ii) difference is an irrational number.
(iii) sum is a rational number.
(iv) sum is an irrational number.
(v) product is a rational number.
(vi) product is an irrational number.
(vii) quotient is a rational number.
(viii) quotient is an irrational number.
Solution:
(i) Two numbers whose difference is also a rational number, e.g. \(\sqrt { 2 } \)
, \(\sqrt { 2 } \) which are irrational numbers.
∴ Difference = \(\sqrt { 2 } \) – \(\sqrt { 2 } \) = 0 which is also a rational number.
(ii) Two numbers whose difference is an irrational number.
e.g. \(\sqrt { 3 } \) and \(\sqrt { 2 } \) which are irrational numbers.
Now difference = \(\sqrt { 3 } \) –\(\sqrt { 2 } \) which is also an irrational number.
(iii) Let two irrational numbers be \(\sqrt { 3 } \) and –\(\sqrt { 2 } \) which are irrational numbers.
Now sum = \(\sqrt { 3 } \) + (-\(\sqrt { 3 } \)) = \(\sqrt { 3 } \)
– \(\sqrt { 3 } \) = 0 Which is a rational number.
(iv) Let two numbers be \(\sqrt { 5 } \) , \(\sqrt { 3 } \) which are irrational numbers.
Now sum = \(\sqrt { 5 } \) + \(\sqrt { 3 } \) which is an irrational number.
(v) Let numbers be \(\sqrt { 3 } \) +\(\sqrt { 2 } \)and \(\sqrt { 3 } \) –\(\sqrt { 2 } \)which are irrational numbers.
Now product = (\(\sqrt { 3} \) +\(\sqrt { 2 } \) ) (\(\sqrt { 3 } \) –\(\sqrt { 2 } \))
= 3-2 = 1 which is a rational number.
(vi) Let numbers be \(\sqrt { 3 } \) and \(\sqrt { 5 } \) , which are irrational number.
Now product = \(\sqrt { 3 } \) x \(\sqrt { 5 } \) = \(\sqrt { 3×5 } \)
⁼ \(\sqrt { 15 } \)
which is an irrational number.
(vii) Let numbers be 6 \(\sqrt { 2 } \) and 2 \(\sqrt { 2 } \) which are irrational numbers.
Quotient =\(\frac { 6\sqrt { 2 } }{ 2\sqrt { 2 } }\) = 3 which is a rational number.
(viii) Let numbers be \(\sqrt { 3 } \)and \(\sqrt { 5 } \) which are irrational numbers.
Now quotient =\(\frac { \sqrt { 3 } }{ \sqrt { 5 } }\) = \(\sqrt { \frac { 3 }{ 5 } }\) which is an irrational number.

Question 12.
Find two irrational numbers between 0.5 and 0.55.
Solution:
Two irrational numbers between 0.5 and 0.55 will be 0.51010010001… and 52020020002…

Question 13.
Find two irrational numbers lying betwee 0.1 and 0.12.
Solution:
Two irrational numbers lying between 0.1 and 0.12 will be 0.1010010001… and 0.1020020002…

Question 14.
Prove that \(\sqrt { 3 } \)+\(\sqrt { 5 } \) is an irrational number.
Solution:
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.4 Q14.1

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RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.3

January 12, 2023

RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.3

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.3

Question 1.
Express each of the following decimals in the form \(\frac { p }{ q }\):
(i) 0.39
(ii) 0.750
(iii) 2.15
(iv) 7.010
(v) 9.90
(vi) 1.0001
Solution:
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.3 Q1.1

Question 2.
Express each of the following decimals in the form \(\frac { p }{ q }\):
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.3 Q2.1
Solution: