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
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