Q116 / 180 · KCET 2020 · Mathematics

If $f(x)=\left|\begin{array}{ccc}x^{3}-x & a+x & b+x \\ x-a & x^{2}-x & c+x \\ x-b & x-c & 0\end{array}\right|$, then

  • A
    $f(\mathrm{l})=0$
  • B
    $f(2)=0$
  • Correct answer
    C
    $f(0)=0$
  • D
    $f(-1)=0$

Solution

We have, $\begin{aligned} f(x) &=\left|\begin{array}{ccc} x^{3}-x & a+x & b+x \\ x-a & x^{2}-x & c+x \\ x-b & x-c & 0 \end{array}\right| \\ f(0) &=\left|\begin{array}{ccc} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{array}\right| \\ f(0)=0 & \end{aligned}$ $f(0)$ is skew symmetric matrix of order $3 .$

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

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