Worksheet on Number System for Class 9

Solved Questions on Number System

1. Which of the following compares of the surds A = √7 + √6 and B = √5 + √8?

a) A < B                          
b) A > B
c) A = B                          
d) A² = B² + 2

Answer: b) A > B

Explanation: By squaring both the surds we get,

A² = (√7 + √6)² = 7 +  6 +  2√42   =  13 + 2√42 ………….(1)
B² = (√5 + √8)² = 5 +  8 +  2√40   = 13 + 2√40  ………….(2)
Also, √42 > √40

Comparing equation (1) and (2),                        
⇒ 13 + 2√42 > 13 + 2√40       
⇒ A² > B²

∴ A > B

2. Simplify: (4 + √7) (7 + √10)

a) 28 + 4√10 + 6√7 + 2√70
b) 28 + 4√10 + 7√7 + 2√70
c) 28 + 4√10 + 6√7 + √70
d) 28 + 4√10 + 7√7 + √70

Answer: d) 28 + 4√10 + 7√7 + √70

Explanation: Using the identity,

(√p + √q) (√r + √s) = √pr + √ps + √qr + √qs

(4 + √7) (7 + √10) = 28 + 4√10 + 7√7 + √70

3. Rationalise the numerator of the following expression:

a) 
b) 
c) 
d) 

Answer: c) 

Explanation:

4. Simplify the following by the method of rationalising:

a) 2 + 3√2 + 2√6 + √12
b)2 −3√2 + 2√6 + √12
c)2 + 3√2−2√6 + √12
d) 2 + 3√2 + 2√6 −√12

Answer:  a) 2 + 3√2 + 2√6 + √12

Explanation:

5. What is the value of x in the following equation?

a) 2/k
b) 3/k
c) 4/k
d) 5/k

Answer: a) 2/k

Explanation: By cross-multiplying both sides, we get:

xk = [√3 +√ 12 + √ 4+√ 9] × [√ 3+√ 12 − √ 4−√ 9]

xk = [(√3 +√ 12) + (√ 4+√ 9)] × [(√ 3+√ 12) − (√ 4 +√ 9)]

Using the formula, (a + b) × (a − b) = a² − b², we get:

xk = [(√ 3+√ 12)² − (√ 4 + √ 9)²]

xk = (3 + 12 + 2√36) − (4 + 9 +  2√ 36)

xk = 15 + 2√ 36 − 13 − 2√ 36

xk = 2

∴ x = 2/k