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Class 12 Continuity And Differentiability · Long Answer3 Markscontinuity_parameters

Class 12 Maths · Continuity And Differentiability · Long Answer #59

At x<π/2 let f(x)=(1−sin³x)/(3cos²x); at x=π/2 let f(x)=p; at x>π/2 let f(x)=q(1−sin x)/(π−2x)². Find p,q for continuity at π/2.

✅ Step-by-Step Solution

Final Answer: p=1/2,q=4
  1. For x below π/2, factor 1−sin³x=(1−sin x)(1+sin x+sin²x) and cos²x=(1−sin x)(1+sin x).
  2. The left limit is (1+1+1)/[3(1+1)]=1/2.
  3. With h=x−π/2, the right expression is q(1−cos h)/(4h²), whose limit is q/8. Continuity requires p=1/2=q/8.
  4. Therefore p=1/2 and q=4.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] For x below π/2, factor 1−sin³x=(1−sin x)(1+sin x+sin²x) and cos²x=(1−sin x)(1+sin x).
  • [1 mark] The left limit is (1+1+1)/[3(1+1)]=1/2.
  • [1 mark] With h=x−π/2, the right expression is q(1−cos h)/(4h²), whose limit is q/8. Continuity requires p=1/2=q/8.
  • [0.5 marks] Therefore p=1/2 and q=4.

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