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Solve the following.

There are two curves given in the first quadrant as: . What are the points of intersection of both…

Mathematics20254 marksNumerical
There are two curves given in the first quadrant as: $x^2 + y^2 = \pi^2, y = \sin x$.
(a)[1.0]
What are the points of intersection of both the given curves?
(b)[1.0]
What is the value of K, if $\int_0^\pi \sqrt{\pi^2 - x^2} \, dx = \frac{\pi^3}{K}$?
(c)[1.0]
Sketch the region enclosed by the given curves in the first quadrant and the y–axis.
(d)[1.0]
Find the area of the region enclosed by the given curves in the first quadrant and the y–axis.

Answer

Answer (a)

AI
In the first quadrant the circle $x^2 + y^2 = \pi^2$ and the curve $y = \sin x$ meet where $\sin^2 x = \pi^2 - x^2$. For $0 \le x < \pi$ the circle is higher than the sine curve (e.g. at $x = 0$, $\pi$ against $0$), and they meet only at $x = \pi$, where $y = \sin\pi = 0$ and $\pi^2 + 0 = \pi^2$. So the only point of intersection in the first quadrant is $(\pi, 0)$. (The printed key lists $(0,0)$ and $(0,\pi)$, but neither lies on both curves.)

Final answer: $(\pi, 0)$

Answer (b)

Official answer key
$K = 4$

Final answer: 4

Answer (c)

Official answer key
The region in the first quadrant bounded by the quarter circle $x^2 + y^2 = \pi^2$ (arc from $(0,\pi)$ to $(\pi,0)$), the curve $y = \sin x$ (from $(0,0)$ to $(\pi,0)$) and the y-axis (from $(0,0)$ to $(0,\pi)$), shaded.

Answer (d)

Official answer key
$\frac{\pi^3}{4} - 2$ sq. units

Final answer: $\frac{\pi^3}{4} - 2$ sq. units

Application of Integrals

From ISC 2025 Practice Mathematics, question 95.