Question

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

• A. 72
• B. 12
• C. 36
• D. 6

Answer 1

Answer: (A)

$â\begin{array}{ccc}3& 4\sqrt{2}& x\\ 4& 5\sqrt{2}& y\\ 5& k& z\end{array}â=o$
$R_2→R_1+R_3−2R_2$
$⇒â\begin{array}{ccc}3& 4\sqrt{2}& x\\ 0& k−6\sqrt{2}& 0\\ 5& k& z\end{array}â=0$
$⇒(k−6\sqrt{2})(3z−5x)=0$
if $3z−5x=0⇒3(x+2d)−5x=0$
$⇒x=3d$ (Not possible)
$⇒k=6\sqrt{2}⇒k^2=72$