If x, y, z are in arithmetic progression with common difference d, x≠ 3d, and the determinant of the matrix 342x452y5kz is zero, then the value of k2 is

R2→R1+R3-2R23 42 x0k-6205kz=0(k-62) (3z-5x)=03z-5x=0 ⇒ 3(x+2d)-5x=0 ∴ n≠3d∴ k=62k2=72

If x, y, z are in arithmetic progression with common difference d, x≠ 3d, and the determinant of the matrix 342x452y5kz is zero, then the value of k2 is

R2→R1+R3-2R23 42 x0k-6205kz=0(k-62) (3z-5x)=03z-5x=0 ⇒ 3(x+2d)-5x=0 ∴ n≠3d∴ k=62k2=72

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If x, y, z are in arithmetic progression with common difference d, $x \neq 3 d$ , and the determinant of the matrix $[\begin{matrix} 3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z \end{matrix}]$ is zero, then the value of $k^{2}$ is

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answer is D.

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Detailed Solution

$\begin{array}{rcl} R_{2} & \to & R_{1} + R_{3} - 2 R_{2} \\ |\begin{matrix} 3 & 4 \sqrt{2} & x \\ 0 & k - 6 \sqrt{2} & 0 \\ 5 & k & z \end{matrix}| & = & 0 \\ (k - 6 \sqrt{2}) (3 z - 5 x) & = & 0 \\ 3 z - 5 x & = & 0 \Rightarrow 3 (x + 2 d) - 5 x = 0 \end{array}$