Class 12 RD Sharma Solutions – Chapter 11 Differentiation – Exercise 11.7 | Set 3

Question 21. If  and  , find 

Solution:

Here,

Differentiate it with respect to t using chain rule,

And,

Differentiate it with respect to t using quotient rule,

Question 22. Find , if y = 12(1 – cos t), x = 10(t – sin t), 

Solution:

It is given that, 

y = 12(1 – cos t),

x = 10(t – sin t)

Therefore,

Therefore,

Question 23. If x = a(θ – sin θ) and y = a(1 – cos θ), find , at θ =

Solution:

Here,

x = a(θ – sin θ)

and

y = a(1 – cos θ)

Then,

Therefore,

Question 24. If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t), show that at t = 

Solution:

Consider the given functions,

x = a sin 2t (1 + cos 2t)

and 

y = b cos 2t (1 – cos 2t)

Write again the functions,

x = a sin 2t + sin 4t

Differentiate the above function with respect to t,

y = b cos 2t (1 – cos 2t)

y = b cos 2t – b cos2 2t

From equation (1) and (2)

Question 25. If x = cos t (3 – 2cos²t) and y = sin t (3 – 2 sin²t), find the value of  at t = 

Solution:

Here, the given function:

x = cos t (3 – 2cos2t)

x = cos t – 2cos3t

y = sin t (3 – 2 sin2t)

y = 3cos t – 2sin3t

Question 26. If ,  find 

Solution:

Here,

 and

Question 27. If x = 3sin t – sin3t, y = 3cos t – cos3t, find 

Solution:

x = 3sin t – sin3t

and,

y = 3cos t – cos3t

When, 

Question 28. If ,  find 

Solution:

and,

and