QJEEM21Ma85e3: JEE Main 2021 Mathematics - Class 12 JEEM21Ma85e3 Solution: If x, y, z are in arithmetic progression with common difference …

Question

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

\neq

3d, and the determinant of the matrix

[\begin{matrix} 3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z \end{matrix}]

is zero, then the value of k² is :

Accepted Answer

| \begin{matrix} 3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z \end{matrix} | = 0

R_{1} \to R_{1} + R_{3} - 2 R_{2}

\Rightarrow

| \begin{matrix} 0 & 4 \sqrt{2} - k - 10 \sqrt{2} & 0 \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z \end{matrix} | = 0

{

∵

2y = x + z}

\Rightarrow (k - 6 \sqrt{2}) (4 z - 5 y) = 0

\Rightarrow

k =

6 \sqrt{2}

or 4z = 5y (Not possible

∵

x, y, z in A.P.)

So, k² = 72

∴

Option (A)

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