CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q355 Marksmonotonicity
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Question 35 (Applications Of Derivatives)
Find the intervals of increase and decrease for arctan(sin x−cos x) on (0,π). OR Revolve a rectangle of perimeter 24 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 arctan(sin x−cos x) on (0,π).
Answer: Increasing on (0,3π/4), decreasing on (3π/4,π).
- The derivative is (cos x+sin x)/[1+(sin x−cos x)²].
- Its denominator is positive and its numerator is √2 cos(x−π/4), positive before 3π/4 and negative after it.
- Therefore the function increases before 3π/4 and decreases afterward.
Alternative 2 5 Marks
Revolve a rectangle of perimeter 24 cm around one side to form a cylinder of maximum volume. Find the rectangle dimensions.
Answer: 8 cm by 4 cm, rotated around the 4 cm side.
- Let r be the radius, giving height 12−r and volume πr²(12−r).
- The derivative 3πr(8−r) is positive up to r=8 and negative afterward on 0<r<12.
- The maximum occurs at r=8 cm and height 4 cm.