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UPSC CDS 2 2022 Math Paper

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Question : 68 of 100

Marks: +1, -0

If

x = 2 + 2^{\frac{1}{2}}

, then what is the value of

x^4 + 16 x^{-4} ?

152
144
136
132

Solution: 👈: Video Solution

we know

(a-b)^2 = a^2 + b^2 - 2ab

We have,

\Rightarrow x = 2 + 2^{\frac{1}{2}}

\Rightarrow x - 2 = 2^{\frac{1}{2}}

On squaring both the sides, we get

\Rightarrow (x-2)^2 = \left(2^{\frac{1}{2}}\right)^2

\Rightarrow (x-2)^2 = 2

\Rightarrow x^2 - 4x + 4 = 2

\Rightarrow x^2 - 4x + 4 - 2 = 0

\Rightarrow x^2 - 4x + 2 = 0

\Rightarrow x^2 + 2 = 4x

On dividing by '

x

', we get

\Rightarrow x + \; \frac{2}{x} = 4

Again squaring on both the sides, we get

\Rightarrow x^2 + \; \frac{4}{x^2} + 4 = 16

\Rightarrow x^2 + \; \frac{4}{x^2} = 12

Again squaring on both the sides, we get

\Rightarrow x^4 + \; \frac{16}{x^4} + 8 = 144

\Rightarrow x^4 + \; \frac{16}{x^4} = 136

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