# RD Sharma Solutions Class 10 Real Numbers Exercise 1.5

### RD Sharma Class 10 Solutions Chapter 1 Ex 1.5 PDF Free Download

#### Exercise 1.5

1. Show that the following numbers are irrational.

(i) $\frac{1}{\sqrt{2}}$

(ii) $7\sqrt{5}$

(iii) $6+\sqrt{2}$

(iv) $3-\sqrt{5}$

Solution:

(i) $\frac{1}{\sqrt{2}}$

(ii) $7\sqrt{5}$

(iii) $6+\sqrt{2}$

(iv) $3-\sqrt{5}$

2. Prove that the following numbers are irrationals.

(i) $\frac{2}{\sqrt{7}}$

(ii) $\frac{3}{2\sqrt{5}}$

(iii) $4+\sqrt{2}$

(iv) $5\sqrt{2}$

Solution:

(i) $\frac{2}{\sqrt{7}}$

(ii) $\frac{3}{2\sqrt{5}}$

(iii) $4+\sqrt{2}$

(iv) $5\sqrt{2}$

3. Show that $2-\sqrt{3}$ is an irrational number.

Solution:

4. Show that $3+\sqrt{2}$ is an irrational number.

Solution:

5. Prove that $4-5\sqrt{2}$ is an irrational number.

Solution:

6. Show that $5-2\sqrt{3}$ is an irrational number.

Solution:

7. Prove that $2\sqrt{3}-1$ is an irrational number.

Solution:

8. Prove that $2-3\sqrt{5}$ is an irrational number.

Solution:

9. Prove that $\sqrt{5}+\sqrt{3}$ is irrational.

Solution:

10. Prove that $\sqrt{3}+\sqrt{4}$ is irrational.

Solution:

11. Prove that for any prime positive integer p, $\sqrt{p}$ is an irrational number.

Solution:

12. If p, q are prime positive integers, prove that $\sqrt{p}+\sqrt{q}$ is an irrational number.

Solution: