Determinants Matrices Question 26
Question: If $ f(x)= \begin{vmatrix} \cos x & x & 1 \\ 2\sin x & x^{2} & 2x \\ \tan x & x & 1 \\ \end{vmatrix}, $ then $ \underset{x\to 0}{\mathop{\lim }}[ \frac{f’(x)}{x} ] $ is
Options:
A) $ 2 $
B) $ -2 $
C) $ 1 $
D) $ -1 $
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] We have, $ f(x)= \begin{vmatrix} \cos & x & 1 \\ 2\sin x & x^{2} & 2x \\ \tan x & x & 1 \\ \end{vmatrix}= \begin{vmatrix} \cos x-\tan x & 0 & 0 \\ 2\sin x & x^{2} & 2x \\ \tan x & x & 1 \\ \end{vmatrix} $ [Applying $ R_1\to R_1-R_3 $ ] $ =(\cos x-\tan x)(x^{2}-2x^{2}) $ [Expanding along $ R_1 $ ] $ =-x^{2}(\cos x-\tan x) $
$ \therefore f’(x)=-2x(\cos x-\tan x)-x^{2}(-sinx-{{\sec }^{2}}x) $
$ \therefore \underset{x\to 0}{\mathop{\lim }}[ \frac{f’(x)}{x} ]=\underset{x\to 0}{\mathop{\lim }}-2(\cos x-\tan x) $
$ +x(\sin x+{{\sec }^{2}}x) $
$ =-2\times 1=-2 $
BETA
