"If x, y, z are in arithmetic progression with common difference d, x $\ne$ 3d, and the determinant of the matrix $$\left[ {\matrix{ 3 & {4\sqrt 2 } & x \cr 4 & {5\sqrt 2 } & y \cr 5 & k & z \cr } } \right]$$ is zero, then the value of k2 is :"

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Accepted Answer

"$$\left| {\matrix{ 3 & {4\sqrt 2 } & x \cr 4 & {5\sqrt 2 } & y \cr 5 & k & z \cr } } \right| = 0$$

${R_1} \to {R_1} + {R_3} - 2{R_2}$

$\Rightarrow$ $$\left| {\matrix{ 0 & {4\sqrt 2 - k - 10\sqrt 2 } & 0 \cr 4 & {5\sqrt 2 } & y \cr 5 & k & z \cr } } \right| = 0$$ { $\because$ 2y = x + z}

$\Rightarrow (k - 6\sqrt 2 )(4z - 5y) = 0$

$\Rightarrow$ k = $6\sqrt 2$ or 4z = 5y (Not possible $\because$ x, y, z in A.P.)

So, k² = 72

$\therefore$ Option (A)"

If x, y, z are in arithmetic progression with common difference d, x $\ne$ 3d, and the determinant of the matrix $$\left[ {\matrix{ 3 & {4\sqrt 2 } & x \cr 4 & {5\sqrt 2 } & y \cr 5 & k & z \cr } } \right]$$ is zero, then the value of k² is :

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If x, y, z are in arithmetic progression with common difference d, x $\ne$ 3d, and the determinant of the matrix $$\left[ {\matrix{ 3 & {4\sqrt 2 } & x \cr 4 & {5\sqrt 2 } & y \cr 5 & k & z \cr } } \right]$$ is zero, then the value of k2 is :

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If x, y, z are in arithmetic progression with common difference d, x $\ne$ 3d, and the determinant of the matrix $$\left[ {\matrix{ 3 & {4\sqrt 2 } & x \cr 4 & {5\sqrt 2 } & y \cr 5 & k & z \cr } } \right]$$ is zero, then the value of k² is :