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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…
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)
AIIn 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 keyThe 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
From ISC 2025 Practice Mathematics, question 95.