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UPSC CDS 2 2022 Math Paper

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Question : 38 of 100

Marks: +1, -0

If

x = 2 + 2^{1/2}

, then what is the value of

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

?

152
144
136
132

Solution:

Concept:

We use algebraic identities and repeated squaring to find the value of the expression without directly computing

x^4

.

Explanation:

Given

x = 2 + \sqrt{2}

.

Subtract 2:

x - 2 = \sqrt{2}

.

Square both sides:

(x - 2)^2 = 2

.

Expand:

x^2 - 4x + 4 = 2

.

Simplify:

x^2 - 4x + 2 = 0

.

Rewrite:

x^2 + 2 = 4x

.

Divide by

x

(since

x \neq 0

):

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

.

Square again:

\left(x + \frac{2}{x}\right)^2 = 16

.

Expand:

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

.

Thus

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

.

Square one more time:

\left(x^2 + \frac{4}{x^2}\right)^2 = 144

.

Expand:

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

.

Therefore

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

.

Answer:

136

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