If \(\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 4\) and \(\begin{vmatrix} p & b \\ q & d \end{vmatrix} = -1\), then \(\begin{vmatrix} a + 2p & b \\ c + 2q & d \end{vmatrix}\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(2\)
When every element of one column is a sum of two terms, the determinant splits into the sum of two determinants (the other column stays the same):
\( \displaystyle \begin{vmatrix} a + 2p & b \\ c + 2q & d \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} + \begin{vmatrix} 2p & b \\ 2q & d \end{vmatrix} \)
The common factor 2 of the first column of the second determinant comes outside:
\( \displaystyle = \begin{vmatrix} a & b \\ c & d \end{vmatrix} + 2\begin{vmatrix} p & b \\ q & d \end{vmatrix} \)
\(= 4 + 2 \times (-1) = 2\)
Check by expansion: \((a + 2p)d - b(c + 2q) = (ad - bc) + 2(pd - bq) = 4 - 2 = 2\).