Question

If $x , y , z$ are in arithmetic progression with common difference $d , x \neq 3 d$, and the determinant of the matrix $\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]=0$ is zero then the value of $k ^{2}$ is

• A. 72
• B. 12
• C. 36
• D. 6

Answer 1

Answer: (A)

$\begin{vmatrix}3&4\sqrt{2}&x\\ 4&5\sqrt{2}&y\\ 5&k&z\end{vmatrix}=o$
$R _{2} \rightarrow R _{1}+ R _{3}-2 R _{2} $
$\Rightarrow \begin{vmatrix}3 & 4 \sqrt{2} & x \\ 0 & k -6 \sqrt{2} & 0 \\ 5 & k & z \end{vmatrix}=0$
$ \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)
$\Rightarrow k =6 \sqrt{2} \Rightarrow k ^{2}=72$