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Class 12 Continuity And Differentiability · Long Answer3 Marksparametric_derivatives

Class 12 Maths · Continuity And Differentiability · Long Answer #70

For x=\(a \left(\sin{\left(t \right)} - \cos{\left(t \right)}\right) e^{t}\), y=\(a \left(\sin{\left(t \right)} + \cos{\left(t \right)}\right) e^{t}\), find dy/dx at t=\(\frac{\pi}{4}\). Assume a≠0.

✅ Step-by-Step Solution

Final Answer: \(1\)
  1. Differentiate in t: dx/dt=\(a \left(\sin{\left(t \right)} - \cos{\left(t \right)}\right) e^{t} + a \left(\sin{\left(t \right)} + \cos{\left(t \right)}\right) e^{t}\) and dy/dt=\(a \left(- \sin{\left(t \right)} + \cos{\left(t \right)}\right) e^{t} + a \left(\sin{\left(t \right)} + \cos{\left(t \right)}\right) e^{t}\).
  2. Thus dy/dx=(dy/dt)/(dx/dt)=\(\frac{1}{\tan{\left(t \right)}}\).
  3. Evaluate at the specified t to obtain \(1\).
  4. Therefore dy/dx=\(1\).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Differentiate in t: dx/dt=\(a \left(\sin{\left(t \right)} - \cos{\left(t \right)}\right) e^{t} + a \left(\sin{\left(t \right)} + \cos{\left(t \right)}\right) e^{t}\) and dy/dt=\(a \left(- \sin{\left(t \right)} + \cos{\left(t \right)}\right) e^{t} + a \left(\sin{\left(t \right)} + \cos{\left(t \right)}\right) e^{t}\).
  • [1 mark] Thus dy/dx=(dy/dt)/(dx/dt)=\(\frac{1}{\tan{\left(t \right)}}\).
  • [1 mark] Evaluate at the specified t to obtain \(1\).
  • [0.5 marks] Therefore dy/dx=\(1\).

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