Q88 / 180 · KCET 2022 · Mathematics

If there are two values of ' $a$ ' which makes determinant $$ \Delta=\left|\begin{array}{ccc} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a \end{array}\right|=86 $$ Then, the sum of these number is

  • Correct answer
    A
    -4
  • B
    9
  • C
    4
  • D
    5

Solution

Given, $\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a\end{array}\right|=86 \Rightarrow$ $$ \begin{aligned} & \left|\begin{array}{ccc} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a \end{array}\right|=86 \\ & \Rightarrow 1\left[2 a^2+4\right]+2[4 a+0]+5[8-0]=86 \\ & \Rightarrow \quad 2 a^2+4+8 a+40=86 \\ & \Rightarrow \quad \quad 2 a^2+8 a-42=0 \Rightarrow a^2+4 a-21=0 \\ & \Rightarrow \quad a^2+7 a-3 a-21=0 \Rightarrow(a+7)(a-3)=0 \\ & \therefore \text { Required sum }=-7+3=-4 \end{aligned} $$

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

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