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CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/1 · Q273 Marksbranch_derivatives

CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/1 · Question 27 (Continuity And Differentiability)

Differentiate \(y=e^{\cos^{-1}\sqrt{1-x^2}}\) and examine the printed claim \(y^\prime=y/\sqrt{1-x^2}\).

✅ Step-by-Step Solution

Final Answer: y′=sign(x)y/√(1−x²) for 0<abs(x)<1; no derivative at 0.
  1. For −1≤x≤1, cos⁻¹√(1−x²)=sin⁻¹(abs(x)).
  2. Thus the exponent derivative is 1/√(1−x²) for 0<x<1 and −1/√(1−x²) for −1<x<0. At zero the one-sided slopes of y are −1 and 1.
  3. The printed positive formula requires 0<x<1. The full real derivative has the stated two branches.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] For −1≤x≤1, cos⁻¹√(1−x²)=sin⁻¹(abs(x)).
  • [2 marks] Thus the exponent derivative is 1/√(1−x²) for 0<x<1 and −1/√(1−x²) for −1<x<0. At zero the one-sided slopes of y are −1 and 1.
  • [0.5 marks] The printed positive formula requires 0<x<1. The full real derivative has the stated two branches.

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All Questions in CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/1

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