NIMCET 2012 — Mathematics PYQ

Step 1: Understanding Continuity

For a function $f(x)$ to be continuous at a point $x=c$, the Left-Hand Limit (LHL), the Right-Hand Limit (RHL), and the value of the function at that point must all be equal.

In this problem, the "break point" where the function might be discontinuous is $x=\frac{\pi }{2}$.

Step 2: Find the Left-Hand Limit (LHL) and $f(\frac{\pi }{2})$

For $x\le \frac{\pi }{2}$, $f(x)=\sin x$.

Also, $f\left(\frac{\pi }{2}\right)=\sin \left(\frac{\pi }{2}\right)=1$.

Step 3: Find the Right-Hand Limit (RHL)

For x > \frac{\pi}{2}, $f(x)=ax$.

Step 4: Equate LHL and RHL

For the function to be continuous at $x=\frac{\pi }{2}$:

Step 5: Solve for $a$

Conclusion:

The value of $a$ that makes the function continuous is $\frac{2}{\pi }$.

Correct Option: (c)