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} \right|,-1 < x\le 1, $ is

• A. $1$
• B. $ -2 $
• C. $ -1 $
• D. $0$
• E. $2$

Answer 1

Answer: (B)

$ 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| $

$=x\{-2(1+{{x}^{2}})\}-(1+\sin x)(-2{{x}^{2}}) $ $ +\cos x \{1+{{x}^{2}}-{{x}^{2}}\log (1+x)\} $

$=-2x-2{{x}^{3}}+2{{x}^{2}}+2{{x}^{2}}\sin x $ $ +\cos x\{1+{{x}^{2}}-{{x}^{2}}\log (1+x)\} $

$ \therefore $ Coefficient of $ x $ in $ f(x)=-2 $