RD Sharma Class 10 Solutions Chapter 1 Exercise 1.5 (Updated for 2021-22)

RD Sharma Class 10 Solutions Chapter 1 Exercise 1.5: Numbers that cannot be stated in the usual form of p/q are referred to as irrational numbers. This exercise explains how to prove the irrationality of numbers. Kopykitab’s subject experts have prepared the RD Sharma Solutions Class 10 to help students understand how to do workout problems correctly. If you need help with any of the questions in this exercise, you can use the RD Sharma Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.5, which are available in PDF format below.

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RD Sharma Class 10 Solutions Chapter 1 Exercise 1.5 PDF

Access answers to RD Sharma Solutions Class 10 Maths Chapter 1 Exercise 1.5- Important Question with Answers

Question 1.
Show that the following numbers are irrational
(i) 1√2
(ii) 7 √5
(iii) 6 + √2
(iv) 3 – √5
Solution:



But it contradicts that because √5 is irrational
3 – √5 is irrational

Question 2.
Prove that the following numbers are irrationals :
(i) 2√7
(ii) 32√5
(iii) 4 + √2
(iv) 5 √2
Solution:




5 √2 is an irrational number

Question 3.
Show that 2 – √3 is an irrational number. [C.B.S.E. 2008]
Solution:
Let 2 – √3 is not an irrational number

√3 is a rational number
But it contradicts because √3 is an irrational number
2 – √3 is an irrational number
Hence proved.

Question 4.
Show that 3 + √2 is an irrational number.
Solution:
Let 3 + √2 is a rational number

and √2 is irrational
But our supposition is wrong
3 + √2 is an irrational number

Question 5.
Prove that 4 – 5√2 is an irrational number. [CBSE 2010]
Solution:
Let 4 – 5 √2 is not are irrational number
and let 4 – 5 √2 is a rational number
and 4 – 5 √2 = ab where a and b are positive prime integers

√2 is a rational number
But √2 is an irrational number
Our supposition is wrong
4 – 5 √2 is an irrational number

Question 6.
Show that 5 – 2 √3 is an irrational number.
Solution:
Let 5 – 2 √3 is a rational number
Let 5 – 2 √3 = ab where a and b are positive integers

and √3 is a rational number
Our supposition is wrong
5 – 2 √3 is a rational number.

Question 7.
Prove that 2 √3 – 1 is an irrational number. [CBSE 2010]
Solution:
Let 2 √3 – 1 is not an irrational number
and let 2 √3 – 1 a ration number
and then 2 √3 – 1 = ab where a, b positive prime integers

√3 is a rational number
But √3 is an irrational number
Our supposition is wrong
2 √3 – 1 is an irrational number

Question 8.
Prove that 2 – 3 √5 is an irrational number. [CBSE 2010]
Solution:
Let 2 – 3 √5 is not an irrational number and let 2 – 3 √5 is a rational number
Let 2 – 3 √5 = ab where a and b are positive prime integers

⟹2b−a3b=√5
√5 is a rational
But √5 is an irrational number
Our supposition is wrong
2 – 3 √5 is an irrational

Question 9.
Prove that √5 + √3 is irrational.
Solution:
Let √5 + √3 is a rational number
and let √5 + √3 = ab where a and b are co-primes

√3 is a rational number
But it contradicts as √3 is an irrational number
√5 + √3 is irrational.

Question 10.
Prove that √2 + √3 is an irrational number.
Solution:
Let us suppose that √2 + √3 is rational.
Let √2 + √3 = a, where a is rational.
Therefore, √2 = a – √3
Squaring on both sides, we get

which is a contradiction as the right-hand side is a rational number while √3 is irrational.
Hence, √2 + √3 is irrational.

Question 11.
Prove that for any prime positive integer p, √p is an irrational number.
Solution:
Suppose √p is not a rational number
Let √p be a rational number
and let √p = ab
Where a and b are co-prime number


But it contradicts that a and b are co-primes
Hence our supposition is wrong
√p is an irrational

Question 12.
If p, q are prime positive integers, prove that √p + √q is an irrational number
Solution:


Hence proved.

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