The coefficient of $x$ in $f(x) = \left| \begin{matrix} x & 1 + \sin x & \cos x \ 1 & \log{(1 + x)} & 2 \ x^{2} & 1 + x^{2} & 0 \end{matrix} ight|$, $- 1

Question

The coefficient of $x$ in $f(x) = \left| \begin{matrix} x & 1 + \sin x & \cos x \ 1 & \log{(1 + x)} & 2 \ x^{2} & 1 + x^{2} & 0 \end{matrix} ight|$, $- 1 < x \leq 1$, is

Dr. Abubacker · Accepted Answer

(b)
Given , $f(x) = \left| \begin{matrix}
x & 1 + \sin x & \cos x \
1 & \log{(1 + x)} & 2 \
x^{2} & 1 + x^{2} & 0
\end{matrix} ight|$
$= x\left\{ - 2\left( 1 + x^{2} ight) ight\} - (1 + \sin{x)\left( - 2x^{2} ight) + \cos{x\{ 1 + x^{2} - x^{2}\log{(1 + x)\}}}}$
$= - 2x - 2x^{3} + 2x^{2} + 2x^{2}\sin{x + \cos{x\{ 1 + x^{2} - x^{2}\log{(1 + x)\}}}}$
$	herefore Coefficient\ of\ x\ in\ f(x) = - 2.$

Q. The coefficient of $x$ in $f(x) = \left| \begin{matrix} x & 1 + \sin x & \cos x \\ 1 & \log{(1 + x)} & 2 \\ x^{2} & 1 + x^{2} & 0 \end{matrix} \right|$, $- 1 < x \leq 1$, is

Explanation

Reviewed by Dr. Abubacker

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