CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q325 Marksmonotonicity
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Question 32 (Applications Of Derivatives)
Find the intervals of increase and decrease for ln(sin x+cos x) on (0,π/2). OR Revolve a rectangle of perimeter 30 cm around one side to form a cylinder of maximum volume. Find the rectangle dimensions.
Alternative 1 5 Marks
Find the intervals of increase and decrease for ln(sin x+cos x) on (0,π/2).
Answer: Increasing on (0,π/4) and decreasing on (π/4,π/2).
- Differentiate to get (cos x−sin x)/(sin x+cos x).
- The denominator is positive. The numerator is positive before π/4 and negative after π/4 on the domain.
- Thus the function increases before π/4 and decreases afterward.
Alternative 2 5 Marks
Revolve a rectangle of perimeter 30 cm around one side to form a cylinder of maximum volume. Find the rectangle dimensions.
Answer: 10 cm by 5 cm, rotated around the 5 cm side.
- Let the cylinder radius be r. Its height is 15−r and its volume is πr²(15−r).
- The derivative is 3πr(10−r), positive before r=10 and negative afterward on 0<r<15.
- Therefore the maximum occurs at radius 10 cm and height 5 cm.