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UPSC CDS II 2025 Mathematics Paper

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Question : 23 of 100

Marks: +1, -0

If

x = 2 + 2^{1/2} + 2^{3/2}

x^{2} - 4x - 10

equal to?

0
1
4
6

Solution:

Concept:

Simplify

x

by substituting the square roots, then compute

x^2

and evaluate the given expression.

Explanation:

Rewrite the powers:

2^{1/2} = \sqrt{2}

and

2^{3/2} = 2\sqrt{2}

.

So

x = 2 + \sqrt{2} + 2\sqrt{2} = 2 + 3\sqrt{2}

.

Now square

x

:

x^2 = (2 + 3\sqrt{2})^2 = 4 + 12\sqrt{2} + 18 = 22 + 12\sqrt{2}

.

Substitute into

x^2 - 4x - 10

:

(22 + 12\sqrt{2}) - 4(2 + 3\sqrt{2}) - 10

.

Simplify:

22 + 12\sqrt{2} - 8 - 12\sqrt{2} - 10

.

The

12\sqrt{2}

terms cancel. Then

22 - 8 - 10 = 4

.

Answer:

4

, i.e., option C.

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