If x, y, z are in arithmetic progression with common difference d… - Vidyadip

MCQ

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

✓
$72$
B
$12$
C
$36$
D
$6$

✓

Answer

Correct option: A.

$72$

a
$\left|\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right|=0$

$R _{2} \rightarrow R _{1}+ R _{3}-2 R _{2}$

$\Rightarrow\left|\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 0 & k-6 \sqrt{2} & 0 \\ 5 & k & z\end{array}\right|=0$

$\Rightarrow(k-6 \sqrt{2})(3 z-5 x)=0$

if $3 z-5 x=0 \Rightarrow 3(x+2 d)-5 x=0$

$\Rightarrow x=3 d$ (Not possible)

$\Rightarrow k =6 \sqrt{2} \quad \Rightarrow k ^{2}=72$

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