1. Real Numbers

Real Numbers

Class X

Chapter 1

NCERT Text Book Solution

Exercise 1.3

Q.No. 1. Prove that √5 is irrational.

Solution:

Let √5 is a rational number.

Therefore,

√5 = a/b

Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.

So,

b√5 = a

Now squaring both side

5b² = a²

a² divisible by 5, therefore, a is divisible by 5.

Let a = 5c

Therefore,

5b² = 25c²

b² = 5c²

Hence, b² is divisible by 5, therefore, b is divisible by 5.

Therefore, a and b have at least 5 as a common factor.

But this contradicts the fact that a and b are coprime.

This contradiction has arisen because of our incorrect assumption that √5 is a rational.

So, we conclude that √5 is irrational. Proved.

Q.No. 2. Prove that 3 + 2√5 is irrational.

Solution:

From the solution of question no. 1. It is proved that √5 is irrational.

We know that product of rational and irrational number is irrational.

So, 2√5 is irrational.

We also know that sum of rational and irrational number is irrational.

So, 3 + 2√5 is irrational. Proved.

Q.No. 3. Prove that the following are irrationals:

(i) 1/√2

(ii) 7√5

(iii) 6 + √2

Solution:

(i) Let √2 be rational.

Therefore,

√2 = a/b 

Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.

So, b√2 = a

Now squaring both side

2b² = a²

a² is divisible by 2, therefore a is divisible by 2.

Let a = 2c

Therefore,

2b² = 4c²

b² = 2c²

b² is divisible by 2, therefore, b is divisible by 2.

Therefore, a and b have at least 2 as a common factor.
But this contradicts the fact that a and b are coprime.
This contradiction has arisen because of our incorrect assumption that √2 is a rational.
So, we conclude that √2 is irrational.
We know that quotient of rational and irrational is irrational.
So, 1/√2 is irrational. Proved.

Solution:
(ii) 7√5
From the solution of question no. 1 it is proved that √5 is irrational.
We know that product of rational and irrational is irrational.
So, 7√5 is irrational. Proved.

Solution:
(iii) 6 + √2

From the solution of question no. 3 (i) it is proved that √2 is irrational.
We know that sum of rational and irrational is irrational.
So, 6 + √2 is irrational. Proved.

1. Real Numbers

Exercise 1.2

Class X

Q.No. 1. Express each number as a product of its prime factors:

(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Solution:

(i)                 140

Prime factors of 140 = 2x2x5x7 Ans.

(ii)               156

Prime factors of 156 = 2x2x3x13 Ans.

(iii)             3825

Prime factors of 3825 = 3x3x5x5x17 Ans.

(iv)             5005

Prime factors of 5005 = 5x7x11x13 Ans.

(v)               7429

Prime factors of 7429 = 17x19x23 Ans.

Q.No. 2. Find the LCM and HCF of the following pairs of integers and verify that LCMxHCF = Product of the two numbers.

(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54

Solution:

(i)                 26 and 91

Prime factors of 26 = 2x13

Prime factors of 91 = 13x7

LCM = 2x7x13 = 182

HCF = 13

LCM x HCF = 182x13=2366

Product of number = 26x91 = 2366

LCMxHCF = Product of number is verified.

(ii)               510 and 92

Prime factors of 510 = 2x3x5x17

Prime factors of 92 = 2x2x23

LCM = 2x2x3x5x17x23 = 23460

HCF = 2

LCMxHCF = 23460x2 = 46920

Product of number = 46920

Hence, LCMxHCF = Product of number is verified.

(iii)             336 and 54

Prime factors of 336 = 2x2x2x2x3x7

Prime factors of 54 = 2x3x3x3

LCM = 2x2x2x2x3x3x3x7 = 3024

HCF = 2x3 = 6

LCMxHCF = 18144

Product of number = 336x54 = 18144

Hence, LCMxHCF = Product of number is verified.

Q.No. 3. Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15and 21 (ii) 17, 23 and 29 (iii) 8,9 and 25

Solution:

(i)                 12, 15 and 21

Prime factorisation of 12 = 2x2x3

Prime factorisation of 15 = 3x5

Prime factorisation of 21 = 3x7

LCM = 2x2x3x5x7 = 420

HCF = 3

(ii)               17, 23 and 29

Prime factorisation of 17 = 17

Prime factorisation of 23 = 23

Prime factorisation of 29 = 29

LCM = 17x23x29 = 11339

HCF = 1

(iii)             8, 9 and 25

Prime factorisation of 8 = 2x2x2 = 8

Prime factorisation of 9 = 3x3

Prime factorisation of 25 = 5x5

LCM = 2x2x2x3x3x5x5 = 1800

HCF = 1

Q.No. 4.Given that HCF (306, 657) = 9 , find LCM (306, 657)

Solution:

LCMxHCF = Product of numbers

or, LCM = Product of numbers/HCF

or, LCM = 306x657/9

or, LCM = 22338 Ans.

Q.No. 5. Check whether  can end with the digit 0 for any natural number n.

Solution:

Prime factorisation of 6 = 2x3

So, the prime factorisation of 6 is not divisible by 5.

Hence, cannot end with 0 for any natural number n according to Fundamental Theorem of Arithmetic.

Q.No. 6. Explain why 7x11x13 + 13 and 7x6x5x4x3x2x1 + 5 are composite numbers.

Solution:

These numbers can be expressed as the form of the factorisation of prime numbers.

So, these numbers are composite numbers.

Q.No. 7. There is circular path around a sports field. Sonia takes 18 minutes to drive one round of the field. while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go the same direction. After how many minutes will they meet again at the starting point.

Solution:

Prime factorisation of 18 = 2x3x3

Prime factorisation of 18 = 2x2x3x3

LCM of 12 and 18 = 2x2x3x3 = 36

So, they will meet again after 36 minutes. Ans.