✓ Board-verified
Mathematics · 2026 · 4 marks
CBSE 2026 · Region 1 · Set 1 · Q38
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…
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)
(a)
Using integration find the area of region covered by the roundabout.
OR
(b)
Using integration, find the area of region covered by circular pond.
Official answer
(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)
(a)
Required area $\displaystyle =4 \int_{0}^{8} \sqrt{64-x^{2}}\, d x$
$\displaystyle =2\left[x \sqrt{64-x^{2}}+64 \sin^{-1} \dfrac{x}{8}\right]_{0}^{8}=64 \pi$
OR
(b)
Required area $\displaystyle =4 \int_{0}^{2} \sqrt{4-x^{2}}\, d x$
$\displaystyle =2\left[x \sqrt{4-x^{2}}+4 \sin^{-1} \dfrac{x}{2}\right]_{0}^{2}=4 \pi$
All Application of Integrals questions2026 question paperNCERT solutions: Application of IntegralsPractice Application of Integrals →
More from Application of Integrals
- The area of the region enclosed between the curve y=x|x|, x -axis, x=-2 and x=2 is:2025
- Find the area of the region bounded by the lines y=4 x+5, x+y=5 and 4 y=x+5, using integration.2023
- Using integration, find the area of the smaller region enclosed by the curve 4 x^2+4 y^2=9 and the line 2 x+2 y=3. OR If the area of the region…2022
- Sketch the curve described by {(x, y): 9 x^2+16 y^2=144} and find the area of the region enclosed by it, using integration.2026
- Find the area of the region bounded by the lines x-2 y=4, x=-1, x=6 and x -axis, using integration.2024
- Using integration, find the area of the region bounded by y=m x( m>0), x=1, x=2 and the x -axis.2023
- Find the area of the region bounded by curve 4 x^2=y and the line y=8 x+12, using integration.2022
- Find the area of the region bounded by the curve 4 x^2+y^2=36 using integration.2024
CBSE Class 12 Mathematics past-paper question from the 2026 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.