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  • If x, y, z are in arithmetic progression with common difference d , and the determinant of the matrix <img alt="\begin{bmatrix} 3 & 4\sqrt{2} & x\\ 4 & 5\sqrt{2} & y\\ 5 &

If x, y, z are in arithmetic progression with common difference d , x\neq 3d, and the determinant of the matrix \begin{bmatrix} 3 & 4\sqrt{2} & x\\ 4 & 5\sqrt{2} & y\\ 5 & k & z \end{bmatrix} is zero, then the value of k2 is
 
Option: 1 36
Option: 2 12
Option: 3 6
Option: 4 72

Answers (1)

best_answer

\begin{aligned} &\left|\begin{array}{ccc} 3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z \end{array}\right|=0\\ &\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{3}-2 \mathrm{R}_{2}\\ &\left|\begin{array}{ccc} 0 & 4 \sqrt{2}+\mathrm{k}-10 \sqrt{2} & 0 \\ 4 & 5 \sqrt{2} & \mathrm{y} \\ 5 & \mathrm{k} & \mathrm{z} \end{array}\right|=0\\ &\Rightarrow(\mathrm{k}-6 \sqrt{2})(4 \mathrm{z}-5 \mathrm{y})=0 \end{aligned}

\\k=6\sqrt2\\\Rightarrow k^2=72\;\;\\\text{or}\\4z=5y\\4(x+2d)=5(x+d)\\x=3d\;\;\text{not possible}

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