QJEEM18Ma9a3f: JEE Main 2018 Mathematics - Class 12 JEEM18Ma9a3f Solution: If f ( x ) =

Question

Verified Solution • 2025–26 Session moderate

If

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

then

\mathop {\lim }\limits_{x \to 0} {{f'\left( x \right)} \over x}

Accepted Answer

Given,

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

= cosx(x² - 2x²) - x(2 sinx - 2x tanx) + (2x sinx - x² tanx)
= x² (tanx - cosx)

\therefore

{f^{'}}(x)

= 2x (tanx - cosx) + x²(sec²x + sinx)

\therefore

\mathop {\lim }\limits_{x \to 0} {{f'\left( x \right)} \over x}

=

\mathop {\lim }\limits_{x \to o} {{2x(\tan x - \cos x) + {x^2}({{\sec }^2}x + \sin x)} \over x}

=

\mathop {\lim }\limits_{x \to o} \,\,2(\tan x - \cos x) + x({\sec ^2}x + \sin x)

= 2 (0-1) + 0

= -2

Step-by-Step Verified Solution