If p, q, r, s are in A.P. and f (x) = $left| {,begin{array}{*{20}{c}} {p,, + ,,sin ,x}&{q,, + ,,

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3 and 4 .Determinants and Matrices

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If $p, q, r, s$ are in $A.P.$ and $f (x) =$ $\left| {\,\begin{array}{*{20}{c}} {p\,\, + \,\,\sin \,x}&{q\,\, + \,\,\sin \,x}&{p\,\, - \,\,r\,\, + \,\,\sin \,x}\\ {q\,\, + \,\,\sin \,x}&{r\,\, + \,\,\sin \,x}&{ - \,1\,\, + \,\,\sin \,x}\\ {r\,\, + \,\,\sin \,x}&{s\,\, + \,\,\sin \,x}&{s\,\, - \,\,q\,\, + \,\,\sin \,x} \end{array}\,} \right|$ such that $f (x)dx = - 4$ then the common difference of the $A.P.$ can be :

A$-1$

B$\frac{1}{2}$

C$1$

DBoth $(A)$ and $(C)$

Solution

$p = a$ ;$ q = a + d ; r = a + 2 d ; s = a + 3 d$
==> $f (x) = – 2 d^2$
Also use $R_1 \rightarrow R_1 -R_2$ and $R_2 \rightarrow R_2 -R_3$

Standard 12

Mathematics

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