JEE Mathematics 2025 Question with Solution

Given: The curves are $|y| = 1 - x^2$ and $x^2 + y^2 = 1$.

Find: The value of $|\beta - \gamma|$ when $9\alpha = \beta\pi + \gamma$, where $\alpha$ is the enclosed area.

Using symmetry, the enclosed area is taken as four times the area in the first quadrant.

The area of the quarter circle is

Now evaluate the integral:

Therefore,

Now calculate $9\alpha$:

Comparing with $9\alpha = \beta\pi + \gamma$, we get $\beta = 9$ and $\gamma = 24$.

Hence,

Therefore, the correct option is D.

Given: The region is enclosed between the parabola-like curve $|y| = 1 - x^2$ and the circle $x^2 + y^2 = 1$.

Find: The required integer value $|\beta - \gamma|$.

Because the figure is symmetric about both axes, compute the area in the first quadrant and multiply by $4$.

In the first quadrant, the circle contributes area $\frac{\pi}{4}$, while the curve $y = 1 - x^2$ contributes the area under it from $x=0$ to $x=1$.

So the enclosed area is

Then

Thus, from $9\alpha = \beta\pi + \gamma$, the extracted integers are $\beta = 9$ and $\gamma = 24$, giving the required value as $33$.