If a, b, c are in A.P. then the value of \( \begin{vmatrix}2y+4 & 5y+7 & 8y+a\\3y+5 & 6y+8 & 9y+b\\4y+6 & 7y+9 & 10y+c\end{vmatrix} \) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0
a, b, c in A.P. means 2b = a + c, that is \( a-2b+c=0 \). In each column the first and third entries add up to twice the middle entry, so the operation \( R_1\to R_1-2R_2+R_3 \) gives:
\( \displaystyle (2y+4)-2(3y+5)+(4y+6)=0 \)
\( \displaystyle (5y+7)-2(6y+8)+(7y+9)=0 \)
\( \displaystyle (8y+a)-2(9y+b)+(10y+c)=a-2b+c=0 \)
The new first row is entirely zero, so
\( \displaystyle \begin{vmatrix}0 & 0 & 0\\3y+5 & 6y+8 & 9y+b\\4y+6 & 7y+9 & 10y+c\end{vmatrix}=0 \)