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Class 12 Continuity And Differentiability · Long Answer3 Marksparametric_derivatives

Class 12 Maths · Continuity And Differentiability · Long Answer #87

For x=\(a \left(t - \sin{\left(t \right)}\right)\), y=\(a \left(1 - \cos{\left(t \right)}\right)\), find d²y/dx² at t=\(\frac{\pi}{3}\). Assume a>0.

✅ Step-by-Step Solution

Final Answer: \(- \frac{4}{a}\)
  1. Differentiate in t: dx/dt=\(a \left(1 - \cos{\left(t \right)}\right)\) and dy/dt=\(a \sin{\left(t \right)}\).
  2. Thus dy/dx=(dy/dt)/(dx/dt)=\(\frac{\sin{\left(t \right)}}{1 - \cos{\left(t \right)}}\).
  3. Use d²y/dx²=[d(dy/dx)/dt]/(dx/dt)=\(- \frac{1}{a \left(\cos{\left(t \right)} - 1\right)^{2}}\).
  4. Therefore d²y/dx²=\(- \frac{4}{a}\).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Differentiate in t: dx/dt=\(a \left(1 - \cos{\left(t \right)}\right)\) and dy/dt=\(a \sin{\left(t \right)}\).
  • [1 mark] Thus dy/dx=(dy/dt)/(dx/dt)=\(\frac{\sin{\left(t \right)}}{1 - \cos{\left(t \right)}}\).
  • [1 mark] Use d²y/dx²=[d(dy/dx)/dt]/(dx/dt)=\(- \frac{1}{a \left(\cos{\left(t \right)} - 1\right)^{2}}\).
  • [0.5 marks] Therefore d²y/dx²=\(- \frac{4}{a}\).

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