#### Chapter 1 Real Numbers R.D. Sharma Solutions for Class 10th Math Exercise 1.5

**Exercise 1.5**

**Level 1**

1. Show that the following numbers are irrational.

(i) 1/âˆš2

(ii) 7âˆš5

(iii) 6+âˆš2

(iv) 3-âˆš5

2. Prove that following numbers are irrationals.

(i) 2/âˆš7

(ii) 3/2âˆš5

(iii) 4+âˆš2

(iv) 5âˆš2

3. Show that 2-âˆš3Â is an irrational number.

4. Show that 3+âˆš2Â is an irrational number.

5. Prove that 4-âˆš5 is an irrational number.

6. Show that 5-2âˆš3 is an irrational number.

7. Prove that 2âˆš3-1 is an irrational number.

8. Prove that 2-3âˆš5 is an irrational number.

9. Prove that âˆš5+âˆš3Â is irrational.

10. Prove that âˆš2+âˆš3 is an irrational number.

11. Prove that for any prime positive integer p, âˆšpÂ is an irrational number.

12. If p, q are prime positive integers, prove that âˆšpÂ + âˆšq is a irrational number.

(i) 1/âˆš2

(ii) 7âˆš5

(iii) 6+âˆš2

(iv) 3-âˆš5

**Answer**(i) 2/âˆš7

(ii) 3/2âˆš5

(iii) 4+âˆš2

(iv) 5âˆš2

**Answer**3. Show that 2-âˆš3Â is an irrational number.

**Answer**4. Show that 3+âˆš2Â is an irrational number.

**Answer**5. Prove that 4-âˆš5 is an irrational number.

**Answer**6. Show that 5-2âˆš3 is an irrational number.

**Answer**7. Prove that 2âˆš3-1 is an irrational number.

**Answer**8. Prove that 2-3âˆš5 is an irrational number.

**Answer**9. Prove that âˆš5+âˆš3Â is irrational.

**Answer**10. Prove that âˆš2+âˆš3 is an irrational number.

**Answer****Level 2**11. Prove that for any prime positive integer p, âˆšpÂ is an irrational number.

**Answer**12. If p, q are prime positive integers, prove that âˆšpÂ + âˆšq is a irrational number.

**Answer**