CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q384 Marksapplications_of_integrals
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Question 38 (Applications Of Integrals)
Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such roundabout is made such that equation representing its boundary is given by C₁: x² + y² = 64. There is a circular pond with a fountain in the middle of the roundabout whose equation is given by C₂: x² + y² = 4.
Part (1) 1 Marks
Represent the given equations C₁ and C₂ with the help of a diagram.
Answer: Diagram: concentric circles of radii 8 and 2.
- Draw two concentric circles centered at origin with radii 8 and 2 (see figure).
- Diagram: concentric circles of radii 8 and 2.
Part (2) 1 Marks
Express y as a function of x, (y = f(x)), for both C₁ and C₂.
Answer: y = ±√(64−x²), y = ±√(4−x²)
- The upper and lower branches are y=±√(64−x²) for C₁ and y=±√(4−x²) for C₂.
- These are separate upper and lower branches. Their domains are [−8,8] and [−2,2], respectively.
Part (3) 2 Marks
Using integration find the area of region covered by the roundabout.
Answer: 64π sq. units
- Area = 4∫₀^8 √(64−x²) dx = 4[x√(64−x²)/2 + 32sin⁻¹(x/8)]₀^8 = 64π.
- Area = 4∫₀^8 √(64−x²) dx = 4[x√(64−x²)/2 + 32sin⁻¹(x/8)]₀^8 = 64π.
- 64π sq. units
Part (4) 2 Marks
Using integration, find the area of region covered by circular pond.
Answer: 4π sq. units
- Area = 4∫₀^2 √(4−x²) dx = 4[x√(4−x²)/2 + 2sin⁻¹(x/2)]₀^2 = 4π.
- Area = 4∫₀^2 √(4−x²) dx = 4[x√(4−x²)/2 + 2sin⁻¹(x/2)]₀^2 = 4π.
- 4π sq. units