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If x = (sqrt(3) + 1) / (sqrt(3) - 1) and y = 1/x, then x^2 + y^2 equals
A16
B10
C14
D12
Answer & Solution
Correct answer: C. 14
1. Rationalize x: x = (sqrt(3) + 1)^2 / ((sqrt(3))^2 - 1^2) = (3 + 2sqrt(3) + 1) / 2 = 2 + sqrt(3).
2. y = 1/x = 1/(2 + sqrt(3)) = (2 - sqrt(3))/((2)^2 - 3) = 2 - sqrt(3).
3. x + y = 4, xy = (2 + sqrt(3))(2 - sqrt(3)) = 4 - 3 = 1.
4. x^2 + y^2 = (x + y)^2 - 2xy = 16 - 2 = 14.
_Source: ICAI BoS Foundation Paper 3, Ch 1 Unit III 'Indices', p.14_
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