Q128 / 200 · Manipal (MET) 2010 · Mathematics
If $f(x)=(a x+b) \sin x+(c x+d) \cos x$, then the values of $a, b, c$ and $d$ such that $f^{\prime}(x)=x \cos x$ for all $x$, are
- Correct answerA$b=c=0, a=d=1$
- B$b=d=0, a=c=1$
- C$c=d=0, a=b=1$
- DNone of the above
Solution
We have, $f(x)=(a x+b) \sin x+(c x+d) \cos x$
$\begin{aligned}
f^{\prime}(x)=a \sin x+(a x+b) \cos x & +c \cos x -(c x+d) \sin x
\end{aligned}$
But $f^{\prime}(x)=x \cos x$ for all $x$ (given)
$\begin{aligned}
& \therefore x \cos x=(a-d) \\
& \quad \sin x+(b+c) \cos x+a x \cos x-c x \sin x
\end{aligned}$
Equating the coefficients of $\sin x, \cos x, x \cos x$ and $x \sin x$, we get
$\begin{aligned}
& \quad a-d=0, b+c=0, a=1, c=0 \\
& \therefore b=c=0 \text { and } a=d=1
\end{aligned}$
Hence, the correct answer is option: 'A'.