## RD Sharma Class 10 Solutions Chapter 1 Real Numbers Ex 1.5

These Solutions are part of RD Sharma Class 10 Solutions. Here we have given RD Sharma Class 10 Solutions Chapter 1 Real Numbers Ex 1.5

Other Exercises

- RD Sharma Class 10 Solutions Chapter 1 Real Numbers Ex 1.1
- RD Sharma Class 10 Solutions Chapter 1 Real Numbers Ex 1.2
- RD Sharma Class 10 Solutions Chapter 1 Real Numbers Ex 1.3
- RD Sharma Class 10 Solutions Chapter 1 Real Numbers Ex 1.4
- RD Sharma Class 10 Solutions Chapter 1 Real Numbers Ex 1.5
- RD Sharma Class 10 Solutions Chapter 1 Real Numbers Ex 1.6
- RD Sharma Class 10 Solutions Chapter 1 Real Numbers VSAQS
- RD Sharma Class 10 Solutions Chapter 1 Real Numbers MCQS

Question 1.

Show that the following numbers are irrational

(i) \(\frac { 1 }{ \surd 2 }\)

(ii) 7 √5

(iii) 6 + √2

(iv) 3 – √5

Solution:

But it contradics that because √5 is irrational

3 – √5 is irrational

Question 2.

Prove that following numbers are irrationals :

(i) \(\frac { 2 }{ \surd 7 }\)

(ii) \(\frac { 3 }{ 2\surd 5 }\)

(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 suppositon 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 = \(\frac { a }{ b }\) 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 = \(\frac { a }{ b }\) 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 = \(\frac { a }{ b }\) 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 = \(\frac { a }{ b }\) where a and b are positive prime integers

\(\Longrightarrow \frac { 2b-a }{ 3b } =\surd 5\)

√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 = \(\frac { a }{ b }\) where a and b are co-primes

√3 is a rational number

But it contradics as √3 is 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 = \(\frac { a }{ b }\)

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.

Hope given RD Sharma Class 10 Solutions Chapter 1 Real Numbers Ex 1.5 are helpful to complete your math homework.

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