Question

Let $f(x) =\cos^{-1}\left[\frac {1}{\sqrt {13}}(2\cos x-3\sin x)\right]$ Then, f'(0.5) is equal to

• A. 0.5
• B. 1
• C. 0
• D. -1

Answer 1

Answer: (B)

Given, $f(x)=\cos ^{-1}\left\{\frac{1}{\sqrt{13}}(2 \cos x-3 \sin x)\right\}$
$\Rightarrow f(x)=\cos ^{-1}\left\{\frac{2}{\sqrt{13}} \cos x-\frac{3}{\sqrt{13}} \sin x\right\}$

$\Rightarrow f(x)=\cos ^{-1}\{\cos \alpha \cdot \cos x-\sin \alpha \cdot \sin x\}$
$=\cos ^{-1}\{\cos (x+\alpha)\}$
$=x+\alpha$
On differentiating w.r.t. $x$, we get
$ f'(x) =1=$ constant value
$\therefore f'(0.5) =1 $