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UPSC NDA 2 2021 Mathematics Solved Paper

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Question : 9 of 120

Marks: +1, -0

If

\Delta=\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}

\begin{vmatrix} 3d+5g & 4a+7g & 6g \\ 3e+5h & 4b+7h & 6h \\ 3f+5i & 4c+7i & 6i \end{vmatrix}

$\Delta$
$7 \Delta$
$72 \Delta$
$-72 \Delta$

Solution: 👈: Video Solution

Given,

\Delta=\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}

=\begin{vmatrix} 3d+5g & 4a+7g & 6g \\ 3e+5h & 4b+7h & 6h \\ 3f+5i & 4c+7i & 6i \end{vmatrix}

=6\begin{vmatrix} 3d+5g & 4a+7g & g \\ 3e+5h & 4b+7h & h \\ 3f+5i & 4c+7i & i \end{vmatrix}

\text{By}_{1} \rightarrow C_{1}-5C_{3}, C_{2} \rightarrow C_{2}-7C_{3}

=6\begin{vmatrix} 3d & 4a & g \\ 3e & 4b & h \\ 3f & 4c & i \end{vmatrix}

=6 \times 3 \times 4\begin{vmatrix} d & a & g \\ e & b & h \\ f & c & i \end{vmatrix}

By

_{2} \leftrightarrow C_{1}

=-72\begin{vmatrix} a & d & g \\ b & e & h \\ c & f & i \end{vmatrix} \;\;

By

R \leftrightarrow C

=-72\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}=-72 \Delta

Hence, option (d) is correct.

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