Question

If $f\left(x\right)=(ax+b)\cos x+(cx+d)\sin x$ and $f^{{}^{\prime }}\left(x\right)=x\cos x$, for all values of $x\in R$, then $a,b,c,d$ are given by

• A. $a=b=c=d$
• B. $0,1,-1,0$
• C. $1,0,-1,0$
• D. $0,1,1,0$

Answer 1

The correct option is D $0,1,1,0$

Given, $f\left(x\right)=(ax+b)\cos x+(cx+d)\sin x$
Thus $f^{\prime }\left(x\right)=(ax+b)(-\sin x)+a\cos x+c\sin x+(cx+d)\cos x$
$=(-ax-b+c)\sin x+(a+cx+d)\cos x$
Also given, $f^{\prime }\left(x\right)=x\cos x$

Comparing the coefficients, we get
$a=0,b=1,c=1,d=0$