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CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q355 Marksparametric_derivatives

CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Question 35 (Continuity And Differentiability)

For \(x=a(\sin t-t\cos t)\) and \(y=b(\cos t+t\sin t)\), find the first and second derivatives of y with respect to x.

✅ Step-by-Step Solution

Final Answer: y′=(b/a)cot t and y″=−b/(a²t sin³t), where a≠0, t≠0 and sin t≠0.
  1. Differentiate in t to get dx/dt=at sin t and dy/dt=bt cos t.
  2. On regular branches divide to obtain dy/dx=(b/a)cot t. Differentiate this in t to obtain −(b/a)csc²t.
  3. Divide again by at sin t to get d²y/dx²=−b/(a²t sin³t).

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Differentiate in t to get dx/dt=at sin t and dy/dt=bt cos t.
  • [3 marks] On regular branches divide to obtain dy/dx=(b/a)cot t. Differentiate this in t to obtain −(b/a)csc²t.
  • [1 mark] Divide again by at sin t to get d²y/dx²=−b/(a²t sin³t).

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