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3 and 4 .Determinants and Matrices
hard
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 ..... .
A
$72$
B
$12$
C
$36$
D
$6$
(JEE MAIN-2021)
Solution
$\left|\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right|=0$
$R _{2} \rightarrow R _{1}+ R _{3}-2 R _{2}$
$\Rightarrow\left|\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 0 & k-6 \sqrt{2} & 0 \\ 5 & k & z\end{array}\right|=0$
$\Rightarrow(k-6 \sqrt{2})(3 z-5 x)=0$
if $3 z-5 x=0 \Rightarrow 3(x+2 d)-5 x=0$
$\Rightarrow x=3 d$ (Not possible)
$\Rightarrow k =6 \sqrt{2} \quad \Rightarrow k ^{2}=72$
Standard 12
Mathematics
