Test Index

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 : 78 of 120

Marks: +1, -0

What is

\underset{x→0}{lim}\frac{f(x)}{x^{2}}

equal to ?

Solution:

Concept:

Use determinant properties—factoring a scalar from a column and column operations—to simplify

f(x)

and then evaluate the limit.

Explanation:

Given

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

Factor

x

from the second column:

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

Apply

C_3 \to C_3 - C_2

f(x) = x \begin{vmatrix} \cos x & 1 & 0 \\ 2\sin x & x & x \\ \tan x & 1 & 0 \end{vmatrix}

Expand along the third column (two zeros):

f(x) = x \cdot \left[ -x \cdot \begin{vmatrix} \cos x & 1 \\ \tan x & 1 \end{vmatrix} \right] = -x^{2}(\cos x - \tan x)

Thus

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

Take limit

x \to 0

\displaystyle \lim_{x\to 0} \frac{f(x)}{x^{2}} = -\bigl(\cos 0 - \tan 0\bigr) = -(1 - 0) = -1

Answer:

\displaystyle \lim_{x\to 0} \frac{f(x)}{x^{2}} = -1

, which corresponds to option A.

Go to Question:

Prev Question

Next Question