Q46 / 160 · TS EAMCET 2023 (14 May Shift 2) · Mathematics

$f(x)=\left|\begin{array}{ccc}1 & \text { x } & \text x+1 \\ 2 x & x(x-1) & (x+1) x \\ 3 x(x-1) & x(x-1)(x-2) & (x+1) x(x-1)\end{array}\right|$ then $f(100)$ is equal to

  • Correct answer
    A
    0
  • B
    1
  • C
    100
  • D
    -100

Solution

Given $f(x)=\left\lvert\, \begin{array}{ccc}1 & x & x+1 \\ 2 x & x(x-1) & (x+1) x \\ 3 x(x-1) & x(x-1)(x-2) & (x+1) x(x-1)\end{array}\right.$ On applying $C_3 \rightarrow C_3-C_2$, we get $\Rightarrow f(x)=\left|\begin{array}{ccc}1 & x & 1 \\ 2 x & x(x-1) & 2 x \\ 3 x(x-1) & x(x-1)(x-2) & 3 x(x-1)\end{array}\right|=0$ $\therefore f(x)=0$ $\Rightarrow f(100)=0$

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

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