CBSE 2026 · Region 1 · Set 1 · Q38 · 4 marks
Roundabouts are often made on busy roads to ease the traffic and avoid red lights.
One such round-about is made such that equation representing its boundary is given by $\displaystyle \mathrm{C}_{1} ; x^{2}+\mathrm{y}^{2}=64$. There is a circular pond with a fountain in the middle of the roundabout whose equation is given by $\displaystyle \mathrm{C}_{2}: x^{2}+\mathrm{y}^{2}=4$. Based on the given information, answer the following questions :(i)Represent the given equations $\displaystyle \mathrm{C}_{1}$ and $\displaystyle \mathrm{C}_{2}$ with the help of a diagram. $\displaystyle 1$(ii)Express y as a function of $\displaystyle x,(\mathrm{y}=\mathrm{f}(x))$, for both $\displaystyle \mathrm{C}_{1}$ an $\displaystyle \mathrm{C}_{2} . \quad \mathbf{1}$(iii)Using integration find the area of region covered by the roundabout.Using integration, find the area of region covered by circular pond.
Roundabouts are often made on busy roads to ease the traffic and avoid red lights.
One such round-about is made such that equation representing its boundary is given by $\displaystyle \mathrm{C}_{1} ; x^{2}+\mathrm{y}^{2}=64$.
There is a circular pond with a fountain in the middle of the roundabout whose equation is given by $\displaystyle \mathrm{C}_{2}: x^{2}+\mathrm{y}^{2}=4$.
Based on the given information, answer the following questions :
(i)
Represent the given equations $\displaystyle \mathrm{C}_{1}$ and $\displaystyle \mathrm{C}_{2}$ with the help of a diagram. $\displaystyle 1$
(ii)
Express y as a function of $\displaystyle x,(\mathrm{y}=\mathrm{f}(x))$, for both $\displaystyle \mathrm{C}_{1}$ an $\displaystyle \mathrm{C}_{2} . \quad \mathbf{1}$
(iii)
Using integration find the area of region covered by the roundabout.
Using integration, find the area of region covered by circular pond.
Official answer
From CBSE’s own marking scheme for this paper.
(i)
Diagram showing two concentric circles centered at origin with radii $\displaystyle 8$ and $\displaystyle 2$; (ii) For C₁: y = ±√($\displaystyle 64$-x²), for C₂: y = ±√($\displaystyle 4$-x²); (iii)(a) Area of roundabout = $\displaystyle 60$π square units OR (iii)(b) Area of pond = $\displaystyle 4$π square units
Marking-scheme solution
(i)
(ii)
Circle $\displaystyle C_{1}: y=\sqrt{64-x^{2}}$ or $\displaystyle y=-\sqrt{64-x^{2}}$
Circle $\displaystyle C_{2}: y=\sqrt{4-x^{2}}$ or $\displaystyle y=-\sqrt{4-x^{2}}$
(iii)
Required area $\displaystyle =4 \int_{0}^{8} \sqrt{64-x^{2}}\, d x$
$\displaystyle =2\left[x \sqrt{64-x^{2}}+64 \sin^{-1} \dfrac{x}{8}\right]_{0}^{8}=64 \pi$
Required area $\displaystyle =4 \int_{0}^{2} \sqrt{4-x^{2}}\, d x$
$\displaystyle =2\left[x \sqrt{4-x^{2}}+4 \sin^{-1} \dfrac{x}{2}\right]_{0}^{2}=4 \pi$
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CBSE Class 12 Mathematics past-paper question from the 2026board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.