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UPSC NDA II 2023 Mathematics Solved Paper

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Question : 3 of 120

Marks: +1, -0

What is the value of

\sqrt{12+5i}+\sqrt{12-5i}

i=\sqrt{-1}

?

24
25
5√2
$5(\sqrt{2} - 1)$

Solution:

Concept:

Square the given expression to eliminate square roots and simplify using the identity

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

.

Explanation:

Let

x = \sqrt{12+5i} + \sqrt{12-5i}

.

Square both sides:

x^2 = 12+5i + 12-5i + 2\sqrt{(12+5i)(12-5i)}

.

Simplify:

x^2 = 24 + 2\sqrt{144 + 25} = 24 + 2\sqrt{169}

.

Since

\sqrt{169} = 13

, we get

x^2 = 24 + 26 = 50

.

Thus,

x = \sqrt{50} = 5\sqrt{2}

(positive root, as the sum is positive).

Answer:

5\sqrt{2}

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