If D = left| {,begin{array}{*{20}{c}}{frac{1}{z}}&{frac{1}{z}}&{ - frac{{(x + y)}}{{{z^2}}}}

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3 and 4 .Determinants and Matrices

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If $D =$ $\left| {\,\begin{array}{*{20}{c}}{\frac{1}{z}}&{\frac{1}{z}}&{ - \frac{{(x + y)}}{{{z^2}}}}\\{ - \frac{{(y + z)}}{{{x^2}}}}&{\frac{1}{x}}&{\frac{1}{x}}\\{ - \frac{{y(y + z)}}{{{x^2}z}}}&{\frac{{x + 2y + z}}{{xz}}}&{ - \frac{{y(x +y)}}{{x{z^2}}}}\end{array}\,} \right|$ then, the incorrect statement is

A

$D$ is independent of $x$

B

$D$ is independent of $y$

C

$D$ is independent of $z$

D

$D$ is dependent on $x, y, z$

Solution

Multiply $c_1$ by $x; c_2$ by $y$ and $c_3$ by $z$ and divide the determinant by $xyz$.

Use $c_1 \rightarrow c_1 + c_2 + c_3$

$\Rightarrow$ values of determinant is zero.

Standard 12

Mathematics

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(JEE MAIN-2025)