CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q212 Marksabsolute_extrema
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Question 21 (Applications Of Derivatives)
Find the absolute maximum of cos x+sin²x on [0,π]. OR The volume of a solid hemisphere increases at a uniform rate. Prove that the rate of change of its surface area is inversely proportional to its radius.
Alternative 1 2 Marks
Find the absolute maximum of cos x+sin²x on [0,π].
Answer: 5/4 at x=π/3.
- Put u=cos x, so −1≤u≤1 and f=1+u−u².
- Complete the square to get 5/4−(u−1/2)². The upper bound is attained at u=1/2.
- The absolute maximum is 5/4 at x=π/3.
Alternative 2 2 Marks
The volume of a solid hemisphere increases at a uniform rate. Prove that the rate of change of its surface area is inversely proportional to its radius.
Answer: dS/dt=3k/r for total surface area, where k=dV/dt.
- For a solid hemisphere, V=2πr³/3 and total surface area S=3πr².
- Differentiate to get k=2πr² dr/dt and dS/dt=6πr dr/dt.
- Substitution gives dS/dt=3k/r, which is inversely proportional to r.