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CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q283 Marksintegrals

CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Question 28 (Integrals)

If I₁ = ∫₋π/4^π/4 dx/(1+cos2x) and I₂ = ∫₋1/2^1/2 |x| dx, then show that I₁ − 4I₂ = 0.

✅ Step-by-Step Solution

Final Answer: I₁ − 4I₂ = 0
  1. I₁ = 2∫₀^{π/4} dx/(2cos²x) = ∫₀^{π/4} sec²x dx = [tan x]₀^{π/4} = 1.
  2. I₂ = 2∫₀^{1/2} x dx = 2 [x²/2]₀^{1/2} = (1/2)² = 1/4. I₁ − 4I₂ = 1 − 1 = 0.
  3. I₁ − 4I₂ = 0

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] I₁ = 2∫₀^{π/4} dx/(2cos²x) = ∫₀^{π/4} sec²x dx = [tan x]₀^{π/4} = 1.
  • [2 marks] I₂ = 2∫₀^{1/2} x dx = 2 [x²/2]₀^{1/2} = (1/2)² = 1/4. I₁ − 4I₂ = 1 − 1 = 0.
  • [0.5 marks] I₁ − 4I₂ = 0

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All Questions in CBSE Class 12 Mathematics 2026 Main Paper 65/1/1

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