Question

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

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

Answer 1

Answer: (B)

$f(x)=\left|\begin{array}{ccc}x& 1+\sin x& \cos x\\ 1& \log (1+x)& 2\\ x^2& 1+x^2& 0\end{array}\right|$

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

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

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