NIMCET 2025 — Mathematics PYQ

Solution

Step 1: Find the Point of Intersection

To find where the curves meet between $0$ and $\frac{\pi }{2}$, we set $y=\sin x$ equal to $y=\cos x$:

So, the curves intersect at $x=\frac{\pi }{4}$.

Step 2: Determine the Upper and Lower Curves

The interval $[0,\frac{\pi }{2}]$ is split into two parts:

1. From $0$ to $\frac{\pi }{4}$: $\cos x\ge \sin x$

2. From $\frac{\pi }{4}$ to $\frac{\pi }{2}$: $\sin x\ge \cos x$

Step 3: Set Up the Integral

The total area $A$ is the sum of the areas in these two regions:

Step 4: Evaluate the Integrals

First Part:

Second Part:

Step 5: Total Area

Final Answer:

The area is $2(\sqrt{2}-1)$. (Option D)