CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q384 Marksellipse_area
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Question 38 (Applications Of Integrals)
A 3 m wide track surrounds an elliptical ground given by 9x²+16y²=144. The outer edge meets the positive x- and y-axes at P and Q respectively. O is the origin.
Part (1) 1 Marks
Express y as a function of x for the ellipse.
Answer: Upper branch y=(3/4)√(16−x²); lower branch is its negative, for −4≤x≤4.
- Rearrange to y²=(9/16)(16−x²).
- Take the positive and negative square-root branches for −4≤x≤4. Use the upper branch for area.
Part (2) 1 Marks
Integrate the upper-branch function with respect to x.
Answer: (3x/8)√(16−x²)+6 arcsin(x/4)+C.
- Factor 3/4 out of the integral of the upper branch.
- Apply the standard primitive of √(a²−x²) with a=4. For the lower branch negate this primitive up to the constant.
Part (3) 2 Marks
Use integration to find the area of the elliptical ground excluding the track.
Answer: 12π m².
- By symmetry integrate the upper branch from 0 to 4 and multiply by four.
- The primitive has value 3π at x=4 and zero at x=0 when the integration constant is omitted.
- Therefore the area is 4·3π=12π m².
Part (4) 2 Marks
Find P and Q, then use integration to find the area of triangle POQ.
Answer: P=(7,0), Q=(0,6), area 21 m².
- The inner intercepts are 4 and 3. Add 3 m in the outward axis directions to get P=(7,0) and Q=(0,6).
- The line PQ is y=6−6x/7 on [0,7].
- The area is ∫₀⁷(6−6x/7)dx=21 m².