CBSE Class 12 Mathematics 2026 Main Paper 65(B) · Q374 Marksparametric_derivatives
CBSE Class 12 Mathematics 2026 Main Paper 65(B) · Question 37 (Continuity And Differentiability)
A class is studying dy/dx=(dy/dt)/(dx/dt) for parametrically defined curves on regular branches.
Part (1) 1 Marks
For x=at³, y=at⁵, find dy/dx.
Answer: 5t²/3 for a≠0,t≠0.
- Differentiate to get dx/dt=3at² and dy/dt=5at⁴.
- Take the quotient on regular branches.
Part (2) 1 Marks
For x=t+1/t, y=t−1/t, find dy/dx.
Answer: (t²+1)/(t²−1), t≠0,±1.
- The derivatives are 1−1/t² and 1+1/t².
- Take their quotient where the first is nonzero.
Part (3) 2 Marks
For x=2sin t+sin2t and y=2cos t+cos2t, find dy/dx at t=π/6.
Answer: −1.
- The t-derivatives are 2cos t+2cos2t and −2sin t−2sin2t.
- At π/6 these are √3+1 and −1−√3.
- Their quotient is −1.
Part (4) 2 Marks
For x=a(cos t+ln tan(t/2)) and y=a sin t, find dy/dx on regular real branches.
Answer: tan t.
- The derivative of ln tan(t/2) is csc t on its real domain.
- Thus dx/dt=a(−sin t+csc t)=a cos²t/sin t and dy/dt=a cos t.
- Their quotient is tan t where a≠0, sin t≠0 and cos t≠0.