If \( \Large x = \frac{\sqrt{3}+1}{\sqrt{3}-1} \) and \( \Large y = \frac{\sqrt{3}-1}{\sqrt{3}+1} \) then the value of \( \Large \frac{x^{2}}{y} + \frac{y^{2}}{x} \)

A) 52

B) 76

C) 4

D) 64

Correct answer:
A) 52

Description for Correct answer:

\( \Large x = \frac{\sqrt{3}+1}{\sqrt{3}-1} \)

\( \Large y = \frac{\sqrt{3}-1}{\sqrt{3}+1} \)

\( \Large x + y = \frac{ \left(\sqrt{3}+1\right)^{2} + \left(\sqrt{3}-1\right)^{2} }{ \left(\sqrt{3}-1\right) \left(\sqrt{3}-1\right) } \)

= \( \Large \frac{3+1+2\sqrt{3}+3+1-2\sqrt{3}}{3 - 1} = \frac{8}{2} \)

= x + y = 4

and xy = \( \Large \frac{\sqrt{3}+1}{\sqrt{3}-1} \times \frac{\sqrt{3}-1}{\sqrt{3}+1} = 1 \)

Therefore, \( \Large \frac{x^{2}}{y}+\frac{y^{2}}{x} = \frac{x^{3} + y^{3}}{xy} \)

= \( \Large \frac{ \left(x+y\right)^{3} - 3xy \left(x+y\right) }{xy} \)

= \( \Large \frac{ \left(4\right)^{3} - 3 \times 1 \times 4 }{1} \)

= \( \Large \frac{64 - 12}{1} = 52 \)



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