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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Determinants
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Q)

Suppose $D=\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}$ and $D'=\begin{vmatrix}a_1+pb_1&b_1+qc_1&c_1+ra_1\\a_2+pb_2&b_2+qc_2&c_2+ra_2\\a_3+pb_3&b_3+qc_3&c_3+ra_3\end{vmatrix}$ then

$\begin{array}{1 1}(A)\;D'=D&(B)\;D'=D(1-pqr)\\(C)\;D'=D(1+p+q+r)&(D)\;D'=D(1+pqr)\end{array}$

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A)
Operate $c_1+pc_2,c_2+qc_3$ on D we get
$D=\begin{vmatrix}a_1+pb_1&b_1+qc_1&c_1\\a_2+pb_2&b_2+qc_2&c_2\\a_3+pb_3&b_3+qc_3&c_3\end{vmatrix}$
$D'=\begin{vmatrix}a_1+pb_1&b_1+qc_1&c_1\\a_2+pb_2&b_2+qc_2&c_2\\a_3+pb_3&b_3+qc_3&c_3\end{vmatrix}+r\begin{vmatrix}a_1+pb_1&b_1+qc_1&a_1\\a_2+pb_2&b_2+qc_2&a_2\\a_3+pb_3&b_3+qc_3&a_3\end{vmatrix}$
$\Rightarrow D+r\begin{vmatrix}a_1+pb_1&b_1+qc_1&a_1\\a_2+pb_2&b_2+qc_2&a_2\\a_3+pb_3&b_3+qc_3&a_3\end{vmatrix}$
$\Rightarrow D+pr\begin{vmatrix}b_1&b_1+qc_1&a_1\\b_2&b_2+qc_2&a_2\\b_3&b_3+qc_3&a_3\end{vmatrix}$
$\Rightarrow D+pqr\begin{vmatrix}b_1&c_1&a_1\\b_2&c_2&a_2\\b_3&c_3&a_3\end{vmatrix}$
$\Rightarrow (1+pqr)D$
Hence (D) is the correct answer.
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