Exercise VSAQ
Question 1
What can you say about the sum of a rational number and an irrational number?Solution 1
The sum of a rational number and an irrational number is irrational.
Example: 5 + 
Solve 
.Solution 2
Question 3
The number 
will terminate after how many decimal places?Solution 3
Thus, the given number will terminate after 3 decimal places. Question 4
Find the value of (1296)0.17× (1296)0.08. Solution 4
(1296)0.17× (1296)0.08
Question 5
Simplify 
.Solution 5
Question 6
Find an irrational number between 5 and 6.Solution 6
An irrational number between 5 and 6 = 
Question 7
Find the value of 
.Solution 7
Question 8
Rationalise 
Solution 8
Question 9
Solve for x: 
.Solution 9
Question 10
Simplify (32)1/5 + (-7)0 + (64)1/2.Solution 10
Question 11
Evaluate 
.Solution 11
Question 12
Simplify 
.Solution 12
Question 13
If a = 1, b = 2 then find the value of (ab + ba)-1.Solution 13
Given, a = 1 and b = 2
Question 14
Simplify 
.Solution 14
Question 15
Give an example of two irrational numbers whose sum as well as product is rational.Solution 15
Question 16
Is the product of a rational and irrational numbers always irrational? Give an example.Solution 16
Yes, the product of a rational and irrational numbers is always irrational.
For example,
Question 17
Give an example of a number x such that x2 is an irrational number and x3 is a rational number.Solution 17
Question 18
Write the reciprocal of (
).Solution 18
The reciprocal of ()
Question 19
Solution 19
Question 20
Simplify 
Solution 20
Question 21
If 10x = 64, find the value of 
.Solution 21
Question 22
Evaluate 
Solution 22
Question 23
Simplify 
.Solution 23
Exercise MCQ
Question 1
Which of the following is a rational number?
(a) 
(b) π
(c) 
(d) 0Solution 1
Correct option: (d)
0 can be written as 
where p and q are integers and q ≠ 0.Question 2
A rational number between -3 and 3 is
(a) 0
(b) -4.3
(c) -3.4
(d) 1.101100110001….Solution 2
Correct option: (a)
On a number line, 0 is a rational number that lies between -3 and 3. Question 3
Two rational numbers between 
are
(a) 
(b) 
(c) 
(d) 
Solution 3
Correct option: (c)
Two rational numbers between 
Question 4
Every point on number line represents
(a) a rational number
(b) a natural number
(c) an irrational number
(d) a unique numberSolution 4
Correct option: (d)
Every point on number line represents a unique number. Question 5
Which of the following is a rational number?
Solution 5
Question 6
Every rational number is
(a) a natural number
(b) a whole number
(c) an integer
(d)a real numberSolution 6
Question 7
Between any two rational numbers there
(a) is no rational number
(b) is exactly one rational number
(c) are infinitely many rational numbers
(d)is no irrational numberSolution 7
Question 8
The decimal representation of a rational number is
(a) always terminating
(b) either terminating or repeating
(c) either terminating or non-repeating
(d)neither terminating nor repeatingSolution 8
Question 9
The decimal representation of an irrational number is
(a) always terminating
(b) either terminating or repeating
(c) either terminating or non-repeating
(d)neither terminating nor repeatingSolution 9
Question 10
The decimal expansion that a rational number cannot have is
(a) 0.25
(b) 
(c) 
(d) 0.5030030003….Solution 10
Correct option: (d)
The decimal expansion of a rational number is either terminating or non-terminating recurring.
The decimal expansion of 0.5030030003…. is non-terminating non-recurring, which is not a property of a rational number. Question 11
Which of the following is an irrational number?
(a) 3.14
(b) 3.141414….
(c) 3.14444…..
(d) 3.141141114….Solution 11
Correct option: (d)
The decimal expansion of an irrational number is non-terminating non-recurring.
Hence, 3.141141114….. is an irrational number. Question 12
A rational number equivalent to 
is
(a) 
(b) 
(c) 
(d) 
Solution 12
Correct option: (d)
Question 13
Choose the rational number which does not lie between 
(a) 
(b) 
(c) 
(d) 
Solution 13
Correct option: (b)
Given two rational numbers are negative and is a positive rational number.
So, it does not lie between Question 14
Π is
(a) a rational number
(b) an integer
(c) an irrational number
(d) a whole numberSolution 14
Correct option: (c)
Π = 3.14159265359…….., which is non-terminating non-recurring.
Hence, it is an irrational number.Question 15
The decimal expansion of 
is
(a) finite decimal
(b) 1.4121
(c) nonterminating recurring
(d) nonterminating, nonrecurringSolution 15
Correct option: (d)
The decimal expansion of 
, which is non-terminating, non-recurring. Question 16
Which of the following is an irrational number?
(a) 
(b) 
(c) 0.3799
(d) 
Solution 16
Correct option: (a)
The decimal expansion of 
, which is non-terminating, non-recurring.
Hence, it is an irrational number. Question 17
Hoe many digits are there in the repeating block of digits in the decimal expansion of 
(a) 16
(b) 6
(c) 26
(d) 7Solution 17
Correct option: (b)
Question 18
Which of the following numbers is irrational?
(a) 
(b) 
(c) 
(d) 
Solution 18
Correct option: (c)
The decimal expansion of 
, which is non-terminating, non-recurring.
Hence, it is an irrational number. Question 19
The product of two irrational numbers is
(a) always irrational
(b) always rational
(c) always an integer
(d)sometimes rational and sometimes irrationalSolution 19
Question 20
Which of the following is a true statement?
(a) The sum of two irrational numbers is an irrational number
(b) The product of two irrational numbers is an irrational number
(c) Every real number is always rational
(d) Every real number is either rational or irrationalSolution 20
Question 21
Which of the following is a true statement?
(a) 
(b) 
(c) 
(d) 
Solution 21
Question 22
A rational number lying between 
is
(a) 
(b) 
(c) 1.6
(d) 1.9Solution 22
Correct option: (c)
Question 23
Which of the following is a rational number?
(a) 
(b) 0.101001000100001…
(c) π
(d) 0.853853853…Solution 23
Correct option: (d)
The decimal expansion of a rational number is either terminating or non-terminating recurring.
Hence, 0.853853853… is a rational number. Question 24
The product of a nonzero rational number with an irrational number is always a/an
(a) irrational number
(b) rational number
(c) whole number
(d) natural numberSolution 24
Correct option: (a)
The product of a non-zero rational number with an irrational number is always an irrational number. Question 25
The value of 
, where p and q are integers and q ≠ 0, is
(a) 
(b) 
(c) 
(d) 
Solution 25
Correct option: (b)
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
An irrational number between 5 and 6 is
Solution 30
Question 31
Solution 31
Question 32
Solution 32
Question 33
The sum of 
(a) 
(b) 
(c) 
(d) 
Solution 33
Correct option: (b)
Let x = 
i.e. x = 0.3333…. ….(i)
⇒ 10x = 3.3333…. ….(ii)
On subtracting (i) from (ii), we get
9x = 3
Let y = 
i.e. y = 0.4444…. ….(i)
⇒ 10y = 4.4444…. ….(ii)
On subtracting (i) from (ii), we get
9y = 4
Question 34
The value of 
(a) 
(b) 
(c) 
(d) 
Solution 34
Correct option: (c)
Let x = 
i.e. x = 2.4545…. ….(i)
⇒ 100x = 245.4545……. ….(ii)
On subtracting (i) from (ii), we get
99x = 243
Let y = 
i.e. y = 0.3636…. ….(iii)
⇒ 100y = 36.3636…. ….(iv)
On subtracting (iii) from (iv), we get
99y = 36
Question 35
Which of the following is the value of 
?
(a) -4
(b) 4
(c) 
(d) 
Solution 35
Correct option: (b)
Question 36
when simplified is
(a) positive and irrational
(b) positive and rational
(c) negative and irrational
(d) negative and rationalSolution 36
Correct option: (b)
Which is positive and rational number. Question 37
when simplified is
(a) positive and irrational
(b) positive and rational
(c) negative and irrational
(d) negative and rationalSolution 37
Correct option: (b)
Which is positive and rational number. Question 38
When 
is divided by 
, the quotient is
(a) 
(b) 
(c) 
(d) Solution 38
Correct option: (c)
Question 39
The value of 
is
(a) 10
(b) 
(c) 
(d) 
Solution 39
Correct option: (a)
Question 40
The value of 
is
(a) 
(b) 
(c) 
(d) 
Solution 40
Correct option: (b)
Question 41
= ?
(a) 
(b) 
(c) 
(d) None of theseSolution 41
Correct option: (b)
Question 42
=?
(a) 
(b) 2
(c) 4
(d) 8Solution 42
Correct option: (b)
Question 43
(125)-1/3 = ?
(a) 5
(b) -5
(c) 
(d) 
Solution 43
Correct option: (c)
Question 44
The value of 71/2⋅ 81/2 is
(a) (28)1/2
(b) (56)1/2
(c) (14)1/2
(d) (42)1/2Solution 44
Correct option: (b)
Question 45
After simplification, 
is
(a) 132/15
(b) 138/15
(c) 131/3
(d) 13-2/15Solution 45
Correct option: (d)
Question 46
The value of 
is
(a) 
(b) 
(c) 8
(d) 
Solution 46
Correct option: (a)
Question 47
The value of 
is
(a) 0
(b) 2
(c) 
(d) Solution 47
Correct option: (b)
Question 48
The value of (243)1/5 is
(a) 3
(b) -3
(c) 5
(d) 
Solution 48
Correct option: (a)
Question 49
93 + (-3)3 – 63 = ?
(a) 432
(b) 270
(c) 486
(d) 540Solution 49
Correct option: (c)
93 + (-3)3 – 63 = 729 – 27 – 216 = 486 Question 50
Simplified value of 
is
(a) 0
(b) 1
(c) 4
(d) 16Solution 50
Correct option: (b)
Question 51
The value of 
is
(a) 2-1/6
(b) 2-6
(c) 21/6
(d) 26Solution 51
Correct option: (c)
Question 52
Simplified value of (25)1/3× 51/3 is
(a) 25
(b) 3
(c) 1
(d) 5Solution 52
Correct option: (d)
Question 53
The value of 
is
(a) 3
(b) -3
(c) 9
(d) Solution 53
Correct option: (a)
Question 54
There is a number x such that x2 is irrational but x4 is rational. Then, x can be
(a) 
(b)
(c) 
(d) 
Solution 54
Correct option: (d)
Question 55
If 
then value of p is
(a) 
(b) 
(c) 
(d) 
Solution 55
Correct option: (b)
Question 56
The value of 
is
(a) 
(b) 
(c) 
(d) 
Solution 56
Correct option: (b)
Question 57
The value of xp-q⋅ xq – r⋅ xr – p is equal to
(a) 0
(b) 1
(c) x
(d) xpqrSolution 57
Correct option: (b)
xp-q⋅ xq – r⋅ xr – p
= xp – q + q – r + r – p
= x0
= 1 Question 58
The value of 
is
(a) -1
(b) 0
(c) 1
(d) 2Solution 58
Correct option: (c)
Question 59
= ?
(a) 2
(b) 
(c) 
(d) 
Solution 59
Correct option: (a)
Question 60
If 
then x = ?
(a) 1
(b) 2
(c) 3
(d) 4Solution 60
Correct option: (d)
Question 61
If (33)2 = 9x then 5x = ?
(a) 1
(b) 5
(c) 25
(d) 125Solution 61
Correct option: (d)
(33)2 = 9x
⇒ (32)3 = (32)x
⇒ x = 3
Then 5x = 53 = 125 Question 62
On simplification, the expression 
equals
(a) 
(b) 
(c) 
(d) 
Solution 62
Correct option: (b)
Question 63
The simplest rationalisation factor of 
is
(a) 
(b) 
(c) 
(d) 
Solution 63
Correct option: (d)
Thus, the simplest rationalisation factr of 
Question 64
The simplest rationalisation factor of 
is
(a) 
(b) 
(c) 
(d) 
Solution 64
Correct option: (b)
The simplest rationalisation factor of is Question 65
The rationalisation factor of 
is
(a) 
(b) 
(c) 
(d) 
Solution 65
Correct option: (d)
Question 66
Rationalisation of the denominator of 
gives
(a) 
(b) 
(c) 
(d) 
Solution 66
Correct option: (d)
Question 67
(a) 
(b) 2
(c) 4
(d) 
Solution 67
Correct option: (c)
Question 68
(a) 
(b) 
(c) 
(d) None of theseSolution 68
Correct option: (c)
Question 69
(a) 
(b) 14
(c) 49
(d) 48Solution 69
Correct option: (b)
Question 70
(a) 0.075
(b) 0.75
(c) 0.705
(d) 7.05Solution 70
Correct option: (c)
Question 71
(a) 0.375
(b) 0.378
(c) 0.441
(d) None of theseSolution 71
Correct option: (b)
Question 72
The value of 
is
(a) 
(b) 
(c) 
(d) 
Solution 72
Correct option: (d)
Question 73
The value of 
is
(a) 
(b) 
(c)
(d) Solution 73
Correct option: (c)
Question 74
(a) 0.207
(b) 2.414
(c) 0.414
(d) 0.621Solution 74
Correct option: (c)
Question 75
= ?
(a) 34
(b) 56
(c) 28
(d) 63Solution 75
Correct option: (a)
Question 76
Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.
| Assertion (A) | Reason (R) |
 | A rational number between two rational numbers p and q is  . |
The correct answer is: (a)/(b)/(c)/(d).Solution 76
Question 77
Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.
| Assertion (A) | Reason (R) |
 | Square root of a positive integer which is not a perfect square is an irrational number. |
The correct answer is: (a)/(b)/(c)/(d).Solution 77
Question 78
Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.
| Assertion (A) | Reason (R) |
| e is an irrational number. | Π is an irrational number. |
The correct answer is: (a)/(b)/(c)/(d).Solution 78
Question 79
Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.
| Assertion (A) | Reason (R) |
| The sum of a rational number and an irrational number is an irrational number. |
The correct answer is: (a)/(b)/(c)/(d).Solution 79
Question 80
Match the following columns:
| Column I | Column II |
 | (p) 14(q) 6(r) a rational number(s) an irrational number |
The correct answer is:
(a)-…….,
(b)-…….,
(c)-…….,
(d)-…….,Solution 80
Question 81
Match the following columns:
| Column I | Column II |
 |  |
The correct answer is:
(a)-…….,
(b)-…….,
(c)-…….,
(d)-…….,Solution 81
Exercise Ex. 1B
Question 1(i)
Without actual division, find which of the following rationals are terminating decimals.
Solution 1(i)
If the prime factors of the denominator are 2 and/or 5 then the rational number is a terminating decimal.
Since, 80 has prime factors 2 and 5, 
is a terminating decimal.Question 1(ii)
Without actual division, find which of the following rationals are terminating decimals.
Solution 1(ii)
If the prime factors of the denominators of the fraction are other than 2 and 5, then the rational number is not a terminating decimal.
Since, 24 has prime factors 2 and 3 and 3 is different from 2 and 5,
is not a terminating decimal.Question 1(iii)
Without actual division, find which of the following rationals are terminating decimals.
Solution 1(iii)
If the prime factors of the denominators of the fraction are other than 2 and 5, then the rational number is not a terminating decimal.
Since 12 has prime factors 2 and 3 and 3 is different from 2 and 5,
is not a terminating decimal.Question 1(iv)
Without actual division, find which of the following rational numbers are terminating decimals.
Solution 1(iv)
Since the denominator of a given rational number is not of the form 2m × 2n, where m and n are whole numbers, it has non-terminating decimal. Question 2(i)
Write each of the following in decimal form and say what kind of decimal expansion each has.
Solution 2(i)
Hence, it has terminating decimal expansion. Question 2(ii)
Write each of the following in decimal form and say what kind of decimal expansion each has.
Solution 2(ii)
Hence, it has terminating decimal expansion. Question 2(iii)
Write each of the following in decimal form and say what kind of decimal expansion each has.
Solution 2(iii)
Hence, it has non-terminating recurring decimal expansion. Question 2(iv)
Write each of the following in decimal form and say what kind of decimal expansion each has.
Solution 2(iv)
Hence, it has non-terminating recurring decimal expansion. Question 2(v)
Write each of the following in decimal form and say what kind of decimal expansion each has.
Solution 2(v)
Hence, it has non-terminating recurring decimal expansion. Question 2(vi)
Write each of the following in decimal form and say what kind of decimal expansion each has.
Solution 2(vi)
Hence, it has terminating decimal expansion. Question 2(vii)
Write each of the following in decimal form and say what kind of decimal expansion each has.
Solution 2(vii)
Hence, it has terminating decimal expansion. Question 2(viii)
Write each of the following in decimal form and say what kind of decimal expansion each has.
Solution 2(viii)
Hence, it has non-terminating recurring decimal expansion. Question 3(i)
Express each of the following decimals in the form 
, where p, q are integers and q ≠ 0.
Solution 3(i)
Let x = 
i.e. x = 0.2222…. ….(i)
⇒ 10x = 2.2222…. ….(ii)
On subtracting (i) from (ii), we get
9x = 2
Question 3(ii)
Express each of the following decimals in the form , where p, q are integers and q ≠ 0.
Solution 3(ii)
Let x = 
i.e. x = 0.5353…. ….(i)
⇒ 100x = 53.535353…. ….(ii)
On subtracting (i) from (ii), we get
99x = 53
Question 3(iii)
Express each of the following decimals in the form , where p, q are integers and q ≠ 0.
Solution 3(iii)
Let x = 
i.e. x = 2.9393…. ….(i)
⇒ 100x = 293.939……. ….(ii)
On subtracting (i) from (ii), we get
99x = 291
Question 3(iv)
Express each of the following decimals in the form , where p, q are integers and q ≠ 0.
Solution 3(iv)
Let x = 
i.e. x = 18.4848…. ….(i)
⇒ 100x = 1848.4848……. ….(ii)
On subtracting (i) from (ii), we get
99x = 1830
Question 3(v)
Express each of the following decimals in the form , where p, q are integers and q ≠ 0.
Solution 3(v)
Let x = 
i.e. x = 0.235235..… ….(i)
⇒ 1000x = 235.235235……. ….(ii)
On subtracting (i) from (ii), we get
999x = 235
Question 3(vi)
Express each of the following decimals in the form , where p, q are integers and q ≠ 0.
Solution 3(vi)
Let x = 
i.e. x = 0.003232..…
⇒ 100x = 0.323232……. ….(i)
⇒ 10000x = 32.3232…. ….(ii)
On subtracting (i) from (ii), we get
9900x = 32
Question 3(vii)
Express each of the following decimals in the form , where p, q are integers and q ≠ 0.
Solution 3(vii)
Let x = 
i.e. x = 1.3232323..… ….(i)
⇒ 100x = 132.323232……. ….(ii)
On subtracting (i) from (ii), we get
99x = 131
Question 3(viii)
Express each of the following decimals in the form , where p, q are integers and q ≠ 0.
Solution 3(viii)
Let x = 
i.e. x = 0.3178178..…
⇒ 10x = 3.178178…… ….(i)
⇒ 10000x = 3178.178……. ….(ii)
On subtracting (i) from (ii), we get
9990x = 3175
Question 3(ix)
Express each of the following decimals in the form , where p, q are integers and q ≠ 0.
Solution 3(ix)
Let x = 
i.e. x = 32.123535..…
⇒ 100x = 3212.3535…… ….(i)
⇒ 10000x = 321235.3535……. ….(ii)
On subtracting (i) from (ii), we get
9900x = 318023
Question 3(x)
Express each of the following decimals in the form , where p, q are integers and q ≠ 0.
Solution 3(x)
Let x = 
i.e. x = 0.40777..…
⇒ 100x = 40.777…… ….(i)
⇒ 1000x = 407.777……. ….(ii)
On subtracting (i) from (ii), we get
900x = 367
Question 4
Express 
as a fraction in simplest form.Solution 4
Let x = 
i.e. x = 2.3636…. ….(i)
⇒ 100x = 236.3636……. ….(ii)
On subtracting (i) from (ii), we get
99x = 234
Let y = 
i.e. y = 0.2323…. ….(iii)
⇒ 100y = 23.2323…. ….(iv)
On subtracting (iii) from (iv), we get
99y = 23
Question 5
Express in the form of 
Solution 5
Let x = 
i.e. x = 0.3838…. ….(i)
⇒ 100x = 38.3838…. ….(ii)
On subtracting (i) from (ii), we get
99x = 38
Let y = 
i.e. y = 1.2727…. ….(iii)
⇒ 100y = 127.2727……. ….(iv)
On subtracting (iii) from (iv), we get
99y = 126
Question 9(v)
Without actual division, find which of the following rationals are terminating decimals.
Solution 9(v)
If the prime factors of the denominator are 2 and/or 5 then the rational number is a terminating decimal.
Since 125 has prime factor 5 only
is a terminating decimal.
Exercise Ex. 1C
Question 1
What are irrational numbers? How do they differ from rational numbers? Give examples.Solution 1
Irrational number: A number which cannot be expressed either as a terminating decimal or a repeating decimal is known as irrational number. Rather irrational numbers cannot be expressed in the fraction form, 
For example, 0.101001000100001 is neither a terminating nor a repeating decimal and so is an irrational number.
Also, 
etc. are examples of irrational numbers.Question 2(iii)
Classify the following numbers as rational or irrational. Give reasons to support you answer.
Solution 2(iii)
We know that, if n is a not a perfect square, then 
is an irrational number.
Here, 
is a not a perfect square number.
So, is irrational.Question 2(v)
Classify the following numbers as rational or irrational. Give reasons to support you answer.
Solution 2(v)
is the product of a rational number 
and an irrational number 
.
Theorem: The product of a non-zero rational number and an irrational number is an irrational number.
Thus, by the above theorem, 
is an irrational number.
So, 
is an irrational number.Question 2(i)
Classify the following numbers as rational or irrational. Give reasons to support your answer.
Solution 2(i)
Since quotient of a rational and an irrational is irrational, the given number is irrational. Question 2(ii)
Classify the following numbers as rational or irrational. Give reasons to support your answer.
Solution 2(ii)
Question 2(iv)
Classify the following numbers as rational or irrational. Give reasons to support your answer.
Solution 2(iv)
Question 2(vi)
Classify the following numbers as rational or irrational. Give reasons to support your answer.
4.1276Solution 2(vi)
The given number 4.1276 has terminating decimal expansion.
Hence, it is a rational number. Question 2(vii)
Classify the following numbers as rational or irrational. Give reasons to support your answer.
Solution 2(vii)
Since the given number has non-terminating recurring decimal expansion, it is a rational number. Question 2(viii)
Classify the following numbers as rational or irrational. Give reasons to support your answer.
1.232332333….Solution 2(viii)
The given number 1.232332333…. has non-terminating and non-recurring decimal expansion.
Hence, it is an irrational number. Question 2(ix)
Classify the following numbers as rational or irrational. Give reasons to support your answer.
3.040040004…..Solution 2(ix)
The given number 3.040040004….. has non-terminating and non-recurring decimal expansion.
Hence, it is an irrational number. Question 2(x)
Classify the following numbers as rational or irrational. Give reasons to support your answer.
2.356565656…..Solution 2(x)
The given number 2.356565656….. has non-terminating recurring decimal expansion.
Hence, it is a rational number. Question 2(xi)
Classify the following numbers as rational or irrational. Give reasons to support your answer.
6.834834….Solution 2(xi)
The given number 6.834834…. has non-terminating recurring decimal expansion.
Hence, it is a rational number. Question 3
Let x be a rational number and y be an irrational number. Is x + y necessarily an irrational number? Give an example in support of your answer.Solution 3
We know that the sum of a rational and an irrational is irrational.
Hence, if x is rational and y is irrational, then x + y is necessarily an irrational number.
For example,
Question 4
Let a be a rational number and b be an irrational number. Is ab necessarily an irrational number? Justify your answer with an example.Solution 4
We know that the product of a rational and an irrational is irrational.
Hence, if a is rational and b is irrational, then ab is necessarily an irrational number.
For example,
Question 5
Is the product of two irrationals always irrational? Justify your answer.Solution 5
No, the product of two irrationals need not be an irrational.
For example,
Question 6
Give an example of two irrational numbers whose
(i) difference is an irrational number.
(ii) difference is a rational number.
(iii) sum is an irrational number.
(iv) sum is an rational number.
(v) product is an irrational number.
(vi) product is a rational number.
(vii) quotient is an irrational number.
(viii) quotient is a rational number. Solution 6
(i) Difference is an irrational number:
(ii) Difference is a rational number:
(iii) Sum is an irrational number:
(iv) Sum is an rational number:
(v) Product is an irrational number:
(vi) Product is a rational number:
(vii) Quotient is an irrational number:
(viii) Quotient is a rational number:
Question 7
Examine whether the following numbers are rational or irrational.
Solution 7
Question 8
Insert a rational and an irrational number between 2 and 2.5Solution 8
Rational number between 2 and 2.5 = 
Irrational number between 2 and 2.5 = 
Question 9
How many irrational numbers lie between
? Find any three irrational numbers lying between .Solution 9
There are infinite irrational numbers between.
We have
Hence, three irrational numbers lying between 
are as follows:
1.5010010001……., 1.6010010001…… and 1.7010010001……. Question 10
Find two rational and two irrational numbers between 0.5 and 0.55.Solution 10
Since 0.5 < 0.55
Let x = 0.5, y = 0.55 and y = 2
Two irrational numbers between 0.5 and 0.55 are 0.5151151115……. and 0.5353553555…. Question 11
Find three different irrational numbers between the rational numbers 
.Solution 11
Thus, three different irrational numbers between the rational numbers are as follows:
0.727227222….., 0.757557555….. and 0.808008000….. Question 12
Find two rational numbers of the form 
between the numbers 0.2121121112… and 0.2020020002……Solution 12
Let a and b be two rational numbers between the numbers 0.2121121112… and 0.2020020002……
Now, 0.2020020002…… <0.2121121112…
Then, 0.2020020002…… < a < b < 0.2121121112…
Question 13
Find two irrational numbers between 0.16 and 0.17.Solution 13
Two irrational numbers between 0.16 and 0.17 are as follows:
0.1611161111611111611111…… and 0.169669666……. Question 14(i)
State in each case, whether the given statement is true or false.
The sum of two rational numbers is rational.Solution 14(i)
TrueQuestion 14(ii)
State in each case, whether the given statement is true or false.
The sum of two irrational numbers is irrational.Solution 14(ii)
FalseQuestion 14(iii)
State in each case, whether the given statement is true or false.
The product of two rational numbers is rational.Solution 14(iii)
TrueQuestion 14(iv)
State in each case, whether the given statement is true or false.
The product of two irrational numbers is irrational.Solution 14(iv)
FalseQuestion 14(v)
State in each case, whether the given statement is true or false.
The sum of a rational number and an irrational number is irrational.Solution 14(v)
TrueQuestion 14(vi)
State in each case, whether the given statement is true or false.
The product of a nonzero rational number and an irrational number is a rational number.Solution 14(vi)
FalseQuestion 14(vii)
State in each case, whether the given statement is true or false.
Every real number is rational.Solution 14(vii)
FalseQuestion 14(viii)
State in each case, whether the given statement is true or false.
Every real number is either rational or irrational.Solution 14(viii)
TrueQuestion 14(ix)
State in each case, whether the given statement is true or false.
is irrational and is rational.Solution 14(ix)
True
Exercise Ex. 1D
Question 1(i)
Add:
Solution 1(i)
We have:
Question 1(ii)
Add:
Solution 1(ii)
We have:
Question 1(iii)
Add:
Solution 1(iii)
Question 2(i)
Multiply:
Solution 2(i)
Question 2(ii)
Multiply:
Solution 2(ii)
Question 2(iii)
Multiply:
Solution 2(iii)
Question 2(iv)
Multiply:
Solution 2(iv)
Question 2(v)
Multiply:
Solution 2(v)
Question 2(vi)
Multiply:
Solution 2(vi)
Question 3(i)
Divide:
Solution 3(i)
Question 3(ii)
Divide:
Solution 3(ii)
Question 3(iii)
Divide:
Solution 3(iii)
Question 4(iii)
Simplify:
Solution 4(iii)
Question 4(iv)
Simplify:
Solution 4(iv)
Question 4(vi)
Simplify:
Solution 4(vi)
Question 4(i)
Simplify
Solution 4(i)
= 9 – 11
= -2 Question 4(ii)
Simplify
Solution 4(ii)
= 9 – 5
= 4 Question 4(v)
Simplify
Solution 4(v)
Question 5
Simplify
Solution 5
Question 6(i)
Examine whether the following numbers are rational or irrational:
Solution 6(i)
Thus, the given number is rational. Question 6(ii)
Examine whether the following numbers are rational or irrational:
Solution 6(ii)
Clearly, the given number is irrational. Question 6(iii)
Examine whether the following numbers are rational or irrational:
Solution 6(iii)
Thus, the given number is rational. Question 6(iv)
Examine whether the following numbers are rational or irrational:
Solution 6(iv)
Thus, the given number is irrational. Question 7
On her birthday Reema distributed chocolates in an orphanage. The total number of chocolates she distributed is given by 
.
(i) Find the number of chocolates distributed by her.
(ii) Write the moral values depicted here by Reema.Solution 7
(i) Number of chocolates distributed by Reema
(ii) Loving, helping and caring attitude towards poor and needy children.Question 8(i)
Simplify
Solution 8(i)
Question 8(ii)
Simplify
Solution 8(ii)
Question 8(iii)
Simplify
Solution 8(iii)
Exercise Ex. 1G
Question 1(iii)
Simplify:
Solution 1(iii)
Question 1(i)
Simplify
Solution 1(i)
Question 1(ii)
Simplify
Solution 1(ii)
Question 1(iv)
Simplify
Solution 1(iv)
Question 2(i)
Simplify:
Solution 2(i)
Question 2(ii)
Simplify:
Solution 2(ii)
Question 2(iii)
Simplify:
Solution 2(iii)
Question 3(i)
Simplify:
Solution 3(i)
Question 3(ii)
Simplify:
Solution 3(ii)
Question 3(iii)
Simplify:
Solution 3(iii)
Question 4(i)
Simplify:
Solution 4(i)
Question 4(ii)
Simplify:
Solution 4(ii)
Question 4(iii)
Simplify:
Solution 4(iii)
Question 5(i)
Evaluate:
Solution 5(i)
Question 5(ii)
Evaluate:
Solution 5(ii)
Question 5(iii)
Evaluate:
Solution 5(iii)
Question 5(iv)
Evaluate:
Solution 5(iv)
Question 5(v)
Evaluate:
Solution 5(v)
Question 5(vi)
Evaluate:
Solution 5(vi)
Question 6(i)
If a = 2, b = 3, find the value of (ab + ba)-1Solution 6(i)
Given, a = 2 and b = 3
Question 6(ii)
If a = 2, b = 3, find the value of (aa + bb)-1Solution 6(ii)
Given, a = 2 and b = 3
Question 7(i)
Simplify
Solution 7(i)
Question 7(ii)
Simplify
(14641)0.25Solution 7(ii)
(14641)0.25
Question 7(iii)
Simplify
Solution 7(iii)
Question 7(iv)
Simplify
Solution 7(iv)
Question 8(i)
Evaluate
Solution 8(i)
Question 8(ii)
Evaluate
Solution 8(ii)
Question 8(iii)
Evaluate
Solution 8(iii)
Question 8(iv)
Evaluate
Solution 8(iv)
Question 9(i)
Evaluate
Solution 9(i)
Question 9(ii)
Evaluate
Solution 9(ii)
Question 9(iii)
Evaluate
Solution 9(iii)
Question 9(iv)
Evaluate
Solution 9(iv)
Question 10(i)
Prove that
Solution 10(i)
Question 10(ii)
Prove that
Solution 10(ii)
Question 10(iii)
Prove that
Solution 10(iii)
Question 11
Simplify 
and express the result in the exponential form of x.Solution 11
Question 12
Simplify the product 
Solution 12
Question 13(i)
Simplify
Solution 13(i)
Question 13(ii)
Simplify
Solution 13(ii)
Question 13(iii)
Simplify
Solution 13(iii)
Question 14(i)
Find the value of x in each of the following.
Solution 14(i)
Question 14(ii)
Find the value of x in each of the following.
Solution 14(ii)
Question 14(iii)
Find the value of x in each of the following.
Solution 14(iii)
Question 14(iv)
Find the value of x in each of the following.
5x – 3× 32x – 8 = 225Solution 14(iv)
5x – 3 × 32x – 8 = 225
⇒ 5x – 3× 32x – 8 = 52 × 32
⇒ x – 3 = 2 and 2x – 8 = 2
⇒ x = 5 and 2x = 10
⇒ x = 5 Question 14(v)
Find the value of x in each of the following.
Solution 14(v)
Question 15(i)
Prove that
Solution 15(i)
Question 15(ii)
Prove that
Solution 15(ii)
Question 15(iii)
Prove that
Solution 15(iii)
Question 15(iv)
Prove that
Solution 15(iv)
Question 16
If x is a positive real number and exponents are rational numbers, simplify
Solution 16
Question 17
If 
prove that m – n = 1.Solution 17
Question 18
Write the following in ascending order of magnitude.
Solution 18
Exercise Ex. 1A
Question 1
Is zero a rational number? Justify.Solution 1
A number which can be expressed as 
, where ‘a’ and ‘b’ both are integers and b ≠ 0, is called a rational number.
Since, 0 can be expressed as 
, it is a rational number.Question 2(i)
Represent each of the following rational numbers on the number line:
(i) 
Solution 2(i)
(i)
Question 2(ii)
Represent each of the following rational numbers on the number line:
(ii) 
Solution 2(ii)
(ii)
Question 2(iii)
Represent each of the following rational numbers on the number line:
Solution 2(iii)
Question 2(iv)
Represent each of the following rational numbers on the number line:
(iv) 1.3Solution 2(iv)
(iv) 1.3
Question 2(v)
Represent each of the following rational numbers on the number line:
(v) -2.4Solution 2(v)
(v) -2.4
Question 3(i)
Find a rational number lying between
Solution 3(i)
Question 3(ii)
Find a rational number lying between
1.3 and 1.4Solution 3(ii)
Question 3(iii)
Find a rational number lying between
-1 and 
Solution 3(iii)
Question 3(iv)
Find a rational number lying between
Solution 3(iv)
Question 3(v)
Find a rational number between
Solution 3(v)
Question 4
Find three rational numbers lying between 
How many rational numbers can be determined between these two numbers?Solution 4
Infinite rational numbers can be determined between given two rational numbers.Question 5
Find four rational numbers between 
Solution 5
We have
We know that 9 < 10 < 11 < 12 < 13 < 14 < 15
Question 6
Find six rational numbers between 2 and 3.Solution 6
2 and 3 can be represented as
respectively.
Now six rational numbers between 2 and 3 are
. Question 7
Find five rational numbers between 
Solution 7
Question 8
Insert 16 rational numbers between 2.1 and 2.2.Solution 8
Let x = 2.1 and y = 2.2
Then, x < y because 2.1 < 2.2
Or we can say that, 
Or, 
That is, we have, 
We know that,
Therefore, we can have,
Therefore, 16 rational numbers between, 2.1 and 2.2 are:
So, 16 rational numbers between 2.1 and 2.2 are:
2.105, 2.11, 2.115, 2.12, 2.125, 2.13, 2.135, 2.14, 2.145, 2.15, 2.155, 2.16, 2.165, 2.17, 2.175, 2.18
Question 9(i)
State whether the given statement is true or false. Give reasons. for your answer.
Every natural number is a whole number.Solution 9(i)
True. Since the collection of natural number is a sub collection of whole numbers, and every element of natural numbers is an element of whole numbersQuestion 9(ii)
Write, whether the given statement is true or false. Give reasons.
Every whole number is a natural number.Solution 9(ii)
False. Since 0 is whole number but it is not a natural number.Question 9(iii)
State whether the following statements are true or false. Give reasons for your answer.
Every integer is a whole number.Solution 9(iii)
False, integers include negative of natural numbers as well, which are clearly not whole numbers. For example -1 is an integer but not a whole number.Question 9(iv)
Write, whether the given statement is true or false. Give reasons.
Ever integer is a rational number.Solution 9(iv)
True. Every integer can be represented in a fraction form with denominator 1.Question 9(v)
State whether the following statements are true or false. Give reasons for your answer.
Every rational number is an integer.Solution 9(v)
False, integers are counting numbers on both sides of the number line i.e. they are both positive and negative while rational numbers are of the form 
. Hence, Every rational number is not an integer but every integer is a rational number.Question 9(vi)
Write, whether the given statement is true or false. Give reasons.
Every rational number is a whole number.Solution 9(vi)
False. Since division of whole numbers is not closed under division, the value of 
, may not be a whole number.
Exercise Ex. 1E
Question 1
Represent 
on the number line.Solution 1
Draw a number line as shown.
On the number line, take point O corresponding to zero.
Now take point A on number line such that OA = 2 units.
Draw perpendicular AZ at A on the number line and cut-off arc AB = 1 unit.
By Pythagoras Theorem,
OB2 = OA2 + AB2 = 22 + 12 = 4 + 1 = 5
⇒ OB = 
Taking O as centre and OB = as radius draw an arc cutting real line at C.
Clearly, OC = OB =
Hence, C represents on the number line.Question 2
Locate 
on the number line. Solution 2
Draw a number line as shown.
On the number line, take point O corresponding to zero.
Now take point A on number line such that OA = 1 unit.
Draw perpendicular AZ at A on the number line and cut-off arc AB = 1 unit.
By Pythagoras Theorem,
OB2 = OA2 + AB2 = 12 + 12 = 1 + 1 = 2
⇒ OB = 
Taking O as centre and OB = as radius draw an arc cutting real line at C.
Clearly, OC = OB =
Thus, C represents on the number line.
Now, draw perpendicular CY at C on the number line and cut-off arc CE = 1 unit.
By Pythagoras Theorem,
OE2 = OC2 + CE2 = 2 + 12 = 2 + 1 = 3
⇒ OE = 
Taking O as centre and OE = as radius draw an arc cutting real line at D.
Clearly, OD = OE =
Hence, D represents on the number line. Question 3
Locate 
on the number line.Solution 3
Draw a number line as shown.
On the number line, take point O corresponding to zero.
Now take point A on number line such that OA = 3 units.
Draw perpendicular AZ at A on the number line and cut-off arc AB = 1 unit.
By Pythagoras Theorem,
OB2 = OA2 + AB2 = 32 + 12 = 9 + 1 = 10
⇒ OB = 
Taking O as centre and OB = as radius draw an arc cutting real line at C.
Clearly, OC = OB =
Hence, C represents on the number line. Question 4
Locate 
on the number line. Solution 4
Draw a number line as shown.
On the number line, take point O corresponding to zero.
Now take point A on number line such that OA = 2 units.
Draw perpendicular AZ at A on the number line and cut-off arc AB = 2 units.
By Pythagoras Theorem,
OB2 = OA2 + AB2 = 22 + 22 = 4 + 4 = 8
⇒ OB = 
Taking O as centre and OB = as radius draw an arc cutting real line at C.
Clearly, OC = OB =
Hence, C represents on the number line. Question 5
Represent 
geometrically on the number line.Solution 5
Draw a line segment AB = 4.7 units and extend it to C such that BC = 1 unit.
Find the midpoint O of AC.
With O as centre and OA as radius, draw a semicircle.
Now, draw BD ⊥ AC, intersecting the semicircle at D.
Then, BD = units.
With B as centre and BD as radius, draw an arc, meeting AC produced at E.
Then, BE = BD = units. Question 6
Represent 
on the number line.Solution 6
Draw a line segment OB = 10.5 units and extend it to C such that BC = 1 unit.
Find the midpoint D of OC.
With D as centre and DO as radius, draw a semicircle.
Now, draw BE ⊥ AC, intersecting the semicircle at E.
Then, BE = units.
With B as centre and BE as radius, draw an arc, meeting AC produced at F.
Then, BF = BE = units.Question 7
Represent geometrically on the number line.Solution 7
Draw a line segment AB = 7.28 units and extend it to C such that BC = 1 unit.
Find the midpoint O of AC.
With O as centre and OA as radius, draw a semicircle.
Now, draw BD AC, intersecting the semicircle at D.
Then, BD = units.
With D as centre and BD as radius, draw an arc, meeting AC produced at E.
Then, BE = BD = units.Question 8
Represent 
on the number line.Solution 8
Draw a line segment OB = 9.5 units and extend it to C such that BC = 1 unit.
Find the midpoint D of OC.
With D as centre and DO as radius, draw a semicircle.
Now, draw BE ⊥ AC, intersecting the semicircle at E.
Then, BE = 
units.
With B as centre and BE as radius, draw an arc, meeting AC produced at F.
Then, BF = BE = units.
Extend BF to G such that FG = 1 unit.
Then, BG =
Question 9
Visualize the representation of 3.765 on the number line using successive magnification.Solution 9
Question 10
Visualize the representation of 
on the number line up to 4 decimal places.Solution 10
Exercise Ex. 1F
Question 1
Write the rationalising factor of the denominator in 
. Solution 1
The rationalising factor of the denominator in is 
Question 2(i)
Rationalise the denominator of following:
Solution 2(i)
On multiplying the numerator and denominator of the given number by 
, we get
Question 2(ii)
Rationalise the denominator of following:
Solution 2(ii)
On multiplying the numerator and denominator of the given number by 
, we get
Question 2(iii)
Rationalise the denominator of following:
Solution 2(iii)
Question 2(iv)
Rationalise the denominator of following:
Solution 2(iv)
Question 2(v)
Rationalise the denominator of following:
Solution 2(v)
Question 2(vi)
Rationalise the denominator of each of the following.
Solution 2(vi)
Question 2(vii)
Rationalise the denominator of each of the following.
Solution 2(vii)
Question 2(viii)
Rationalise the denominator of each of the following.
Solution 2(viii)
Question 2(ix)
Rationalise the denominator of each of the following.
Solution 2(ix)
Question 3(i)
find the value to three places of decimals, of each of the following.
Solution 3(i)
Question 3(ii)
find the value to three places of decimals, of each of the following.
Solution 3(ii)
Question 3(iii)
find the value to three places of decimals, of each of the following.
Solution 3(iii)
Question 4(i)
Find rational numbers a and b such that
Solution 4(i)
Question 4(ii)
Find rational numbers a and b such that
Solution 4(ii)
Question 4(iii)
Find rational numbers a and b such that
Solution 4(iii)
Question 4(iv)
Find rational numbers a and b such that
Solution 4(iv)
Question 5(i)
find to three places of decimals, the value of each of the following.
Solution 5(i)
Question 5(ii)
find to three places of decimals, the value of each of the following.
Solution 5(ii)
Question 5(iii)
find to three places of decimals, the value of each of the following.
Solution 5(iii)
Question 5(iv)
find to three places of decimals, the value of each of the following.
Solution 5(iv)
Question 5(v)
find to three places of decimals, the value of each of the following.
Solution 5(v)
Question 5(vi)
find to three places of decimals, the value of each of the following.
Solution 5(vi)
Question 6(i)
Simplify by rationalising the denominator.
Solution 6(i)
Question 6(ii)
Simplify by rationalising the denominator.
Solution 6(ii)
Question 7(i)
Simplify: Solution 7(i)
Question 7(ii)
Simplify
Solution 7(ii)
Question 7(iii)
Simplify
Solution 7(iii)
Question 7(iv)
Simplify
Solution 7(iv)
Question 8(i)
Prove that
Solution 8(i)
Question 8(ii)
Prove that
Solution 8(ii)
Question 9
Find the values of a and b if
Solution 9
*Back answer incorrect Question 10
Simplify
Solution 10
Question 11
Solution 11
Thus, the given number is rational. Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
*Question modified Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
.Solution 21
Question 22(i)
Rationalise the denominator of each of the following.
Solution 22(i)
Question 22(ii)
Rationalise the denominator of each of the following.
Solution 22(ii)
Question 22(iii)
Rationalise the denominator of each of the following.
Solution 22(iii)
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
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