question_answer

If  \[p,q,r,s\] are in arithmetic progression and \[f\left( x \right)=\left| \begin{align}   & p+\sin x\,\,\,\,\,\,\,q+\sin x\,\,\,\,\,\,\,p-r+\sin x \\  & q+\sin x\,\,\,\,\,\,\,\,\,r+\sin x\,\,\,\,\,\,\,-1+\sin x \\  & r+\sin x\,\,\,\,\,\,\,\,\,\,s+\sin x\,\,\,\,\,\,\,s-q+\sin x \\ \end{align} \right|\]. Such that \[\int\limits_{0}^{2}{f\left( x \right)dx=-4,}\] then the common difference of the progession is

A) \[\pm 1\]           

B) \[\frac{1}{2}\]

C)             \[\pm 2\]           

D) none of these

Correct Answer: A

Solution :

let \[q=p+d,r=p+2d,s=p+3d\]

\[\therefore \]

Applying \[R\to {{R}_{1}}+{{R}_{3}}-2{{R}_{2}},\]we get

\[=2[\left( p+d+\sin x \right)\left( p+3d+\sin x \right)\]\[-{{\left( p+2d+\sin x \right)}^{2}}=-2{{d}^{2}}\]\[Given\,\int\limits_{0}^{2}{f\left( x \right)dx=-4\Rightarrow \int\limits_{0}^{2}{\left( -2{{d}^{2}} \right)dx=-4}}\]\[\Rightarrow {{d}^{2}}=1\Rightarrow d=\pm 1\]