Q50 / 110 · VITEEE 2023 · Mathematics

If \(\left|\begin{array}{ccc}3^2+k & 4^2 & 3^2+3+k \\ 4^2+k & 5^2 & 4^2+4+k \\ 5^2+k & 6^2 & 5^2+5+k\end{array}\right|=0\), then the value of \(k\) is

  • A
    2
  • Correct answer
    B
    1
  • C
    -1
  • D
    0

Solution

Breaking the given determinant into two determinants, we get \(\begin{aligned} & \left|\begin{array}{lll} 3^2+\mathrm{k} & 4^2 & 3^2+\mathrm{k} \\ 4^2+\mathrm{k} & 5^2 & 4^2+\mathrm{k} \\ 5^2+\mathrm{k} & 6^2 & 5^2+\mathrm{k} \end{array}\right|+\left|\begin{array}{lll} 3^2+\mathrm{k} & 4^2 & 3 \\ 4^2+\mathrm{k} & 5^2 & 4 \\ 5^2+\mathrm{k} & 6^2 & 5 \end{array}\right|=0 \\ & \Rightarrow 0+\left|\begin{array}{ccc} 9+\mathrm{k} & 16 & 3 \\ 7 & 9 & 1 \\ 9 & 11 & 1 \end{array}\right|=0 \end{aligned}\) [Applying \(R_3-R_2\) and \(R_2-R_1\) in second det.] \(\begin{aligned} & \Rightarrow\left|\begin{array}{ccc} 9+\mathrm{k} & 16 & 3 \\ 7 & 9 & 1 \\ 2 & 2 & 0 \end{array}\right|=0\left[\text { Applying } \mathrm{R}_3-\mathrm{R}_2\right] \\ & \left.\Rightarrow\left|\begin{array}{ccc} 9+\mathrm{k} & 7-\mathrm{k} & 3 \\ 7 & 2 & 1 \\ 2 & 0 & 0 \end{array}\right|=0 \text { [Applying } \mathrm{C}_2-\mathrm{C}_1\right] \\ & \Rightarrow 2(7-\mathrm{k}-6)=0 \Rightarrow \mathrm{k}=1 \end{aligned}\)

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

Relevant Questions