CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q335 Marksmonotonicity
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Question 33 (Applications Of Derivatives)
For f(x)=arccot(sin x+cos x), x∈(0,π), find its increasing and decreasing intervals. OR A rectangle of perimeter 36 cm is revolved around one side. Find the dimensions that maximize the cylinder’s volume.
Alternative 1 5 Marks
For f(x)=arccot(sin x+cos x), x∈(0,π), find its increasing and decreasing intervals.
Answer: Decreasing on (0,π/4) and increasing on (π/4,π).
- Differentiate to get f′=(sin x−cos x)/[1+(sin x+cos x)²].
- The denominator is positive. The numerator is √2 sin(x−π/4), which is negative before π/4 and positive after it within the domain.
- Thus f decreases before π/4 and increases after it.
Alternative 2 5 Marks
A rectangle of perimeter 36 cm is revolved around one side. Find the dimensions that maximize the cylinder’s volume.
Answer: 12 cm by 6 cm, rotated around the 6 cm side.
- Let the radius be r, so the axis side is 18−r and the volume is πr²(18−r).
- For 0<r<18 its derivative is 3πr(12−r), positive before 12 and negative after 12.
- The maximum occurs at radius 12 cm and height 6 cm.