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CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q325 Marksparametric_derivatives

CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Question 32 (Continuity And Differentiability)

For \(x=3\sin t-\sin 3t\) and \(y=3\cos t-\cos 3t\), find \(dy/dx\) and prove \(d^2y/dx^2=-\csc^3(2t)\csc t/3\).

✅ Step-by-Step Solution

Final Answer: dy/dx=cot(2t), and d²y/dx²=−csc³(2t)csc t/3 on regular branches.
  1. Differentiate to get dx/dt=3(cos t−cos 3t) and dy/dt=3(sin 3t−sin t).
  2. Sum-to-product gives dx/dt=6 sin 2t sin t and dy/dt=6 cos 2t sin t. Their quotient is cot 2t where dx/dt≠0.
  3. Differentiate cot 2t in t and divide by dx/dt, giving −2 csc²(2t)/(6 sin 2t sin t)=−csc³(2t)csc t/3.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Differentiate to get dx/dt=3(cos t−cos 3t) and dy/dt=3(sin 3t−sin t).
  • [3 marks] Sum-to-product gives dx/dt=6 sin 2t sin t and dy/dt=6 cos 2t sin t. Their quotient is cot 2t where dx/dt≠0.
  • [1 mark] Differentiate cot 2t in t and divide by dx/dt, giving −2 csc²(2t)/(6 sin 2t sin t)=−csc³(2t)csc t/3.

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