Question

Assertion :The least period of $f\left(x\right)=\cos \left(\sin x\right)+\cos \left(\cos x\right)+\sin 4x$ is equal to $\pi$ Reason: If $f(x+\pi )=f\left(x\right)$ then $\pi$ is the period of $f\left(x\right)$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct.

Answer 1

The correct option is D Assertion is incorrect but Reason is correct.

If $f\left(x\right)$ is periodic with $\lambda$ then

$f(x+\lambda )=f\left(x\right)$
$\Rightarrow \cos \left(\sin (x+\lambda )\right)+\cos \left(\cos (x+\lambda )\right)+\sin (4\times (x+\lambda ))$

$=\cos \left(\sin x\right)+\cos \left(\cos x\right)+\sin 4x$
Substitute $x=0$
$\Rightarrow \cos \left(\sin \lambda \right)+\cos \left(\cos \lambda \right)+\sin 4\lambda =\cos \left(0\right)+\cos \left(1\right)+\sin 0$

$\Rightarrow \cos \left(\sin \lambda \right)+\cos \left(\cos \lambda \right)+\sin \left(4\lambda \right)$ $=\cos \left(\cos \frac{\pi }{2}\right)+\cos \sin \frac{\pi }{2}+\sin 2\pi$

$\therefore 4\lambda =2\pi \Rightarrow \lambda =\frac{\pi }{2}$
$\Rightarrow$ Assertion is false, Reason (R) is obviously true.