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 

Summary

Exercise 11.7 Set 3 in RD Sharma's Class 12 Mathematics textbook further expands on the application of derivatives to solve complex rate of change problems. This set introduces more advanced scenarios, often involving multiple interrelated variables or sophisticated real-world applications. Students are challenged to apply higher-order differentiation techniques, interpret complex physical situations, and analyze rates of change in multivariable systems.