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UPSC NDA I 2023 Mathematics Solved Paper

Question Numbers: 76-78

Consider the following for the next three (03) items that follow :

Let f(x) =

\begin{vmatrix} cosx & x & 1 \\ 2sinx & x^{2} & 2x \\ tanx & x & 1 \end{vmatrix}

Question : 76 of 120

Marks: +1, -0

What is f(0) equal to ?

Solution:

Concept:

A determinant with a constant factor in one row or column can be simplified by taking the factor outside.

Explanation:

Given

f(x) = \begin{vmatrix} \cos x & x & 1 \\ 2\sin x & x^{2} & 2x \\ \tan x & x & 1 \end{vmatrix}

Simplify the determinant by factoring common terms.

Observe that the third column has

1

2x

1

. No direct factoring.

However, using row operations or direct expansion yields

f(x) = -x^{2}(\cos x - \tan x)

Now evaluate at

x = 0

f(0) = -(0)^{2}(\cos 0 - \tan 0) = 0 \times (1 - 0) = 0

Thus

f(0) = 0

Answer:

0

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