Q91 / 200 · Manipal (MET) 2013 · Mathematics

Coefficient of $x$ in $f(x)=\left|\begin{array}{ccc}x & (1+\sin x)^3 & \cos x \\ 1 & \log (1+x) & 2 \\ x^2 & (1+x)^2 & 0\end{array}\right|$ is

  • A
    $0$
  • B
    $1$
  • Correct answer
    C
    $-2$
  • D
    Cannot be determined

Solution

Coefficient $x$ in $f(x)$ is equal to coefficient of $x$ in $\left|\begin{array}{ccc}x & \left(1+x-\frac{x^3}{3!} \ldots\right)^3 & 1-\frac{x^2}{2!} \\ 1 & x-\frac{x^2}{2} & 2 \\ x^2 & 1+x^2 & 0\end{array}\right|$ $=$ coefficient of $x$ in $\left|\begin{array}{ccc}x & 1 & 1 \\ 1 & x & 2 \\ x^2 & 1 & 0\end{array}\right|$ $\begin{aligned} & =\text { Coefficient of } x \text { in }\left[x(0-2)-1\left(0-2 x^2\right)\right. \\ & \left.+1\left(1-x^3\right)\right]\end{aligned}$ $=-2$

Hence, the correct answer is option: 'C'.

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