Question

Assertion :Let m & n be positive integers. a = $\cos \left\{{\int }_{-\pi }^{\pi }(\sin mx.\sin nx)dx\right\},$ if $m\ne n$ & $b=\cos \left\{{\int }_{-\pi }^{\pi }(\sin mx.\sin nx)\right\}$ if M = n then a + b = 2. Reason: ${\int }_{-\pi }^{\pi }(\sin mx.\sin nx)dx$ =$\{\begin{array}{cc}0,& m\ne n\\ \pi & m=n,\end{array}$ where m & n are positive integers.

• 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 incorrect but Reason is correct
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is incorrect but Reason is correct

Reason is true
Assertion
${\int }_{-x}^{\pi }(\sin mx\cdot \sin nx)dx=0$ if $m\ne n$ and ${\int }_{-\pi }^x(\sin mx\cdot \sin nx)dx=\pi$ if m=n
$\therefore a=\cos 0=1$ and

$\therefore b=\cos \pi =-1$
$a+b=0$