If x, y, z are in arithmetic progression with common difference d, x [] 3d, and the determ…

Question

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

Accepted Answer

$$\left| {\begin{pmatrix} 3 & {4\sqrt 2 } & x  \
   4 & {5\sqrt 2 } & y  \
   5 & k & z  \

\end{pmatrix} } \right| = 0$$

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

$$ \Rightarrow $$ $$\left| {\begin{pmatrix} 0 & {4\sqrt 2  - k - 10\sqrt 2 } & 0  \
   4 & {5\sqrt 2 } & y  \
   5 & k & z  \

\end{pmatrix} } \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)

Solutions · 1