Class 12 RD Sharma Solutions - Chapter 11 Differentiation - Exercise 11.7 | Set 1

Question 1. Find , when: x = at² and y = 2at

Solution:

Given that x = at², y = 2at

So,

Therefore,

Question 2. Find , when: x = a(θ + sinθ) and y = a(1 - cosθ)

Solution:

Here,

x = a(θ + sinθ)

Differentiating it with respect to θ,

and,

y = a(1 - cosθ)

Differentiate it with respect to θ,

Using equation (1) and (2),

Question 3. Find , when: x = acosθ and y = bsinθ

Solution:

Then x = acosθ and y = bsinθ

Then,

Therefore,

Question 4. Find , when: x = ae^Θ (sinθ -cosθ), y = ae^Θ (sinθ +cosθ)

Solution:

Here,

x = ae^Θ (sinθ - cosθ)

Differentiating it with respect to θ,

And,

y = ae^Θ(sinθ+cosθ)

Differentiating it with respect to θ

Dividing equation (2) by equation (1)

Question 5. Find , when: x = bsin²θ and y = acos²θ

Solution:

Here,

x = bsin²θ and y = acos²θ

Then,

Question 6. Find , when: x = a(1 - cosθ) and y = a(θ +sinθ) at θ =

Solution:

Here,

x = a(1 - cosθ) and y = a(θ + sinθ)

Then,

Therefore,

Question 7. Find , when: and

Solution:

Here,

Differentiate it with respect to t,

and,

Differentiating it with respect to t,

Dividing equation (2) and (1)

Question 8. Find , when: and

Solution:

Here,

Differentiating it with respect to t using quotient rule,

and,

Differentiating it with respect to t using quotient rule,

Dividing equation (2) by (1)

Question 9. If x and y are connected parametrically by the equation, without eliminating the parameter, find when: x = a(cosθ +θsinθ), y = a(sinθ -θcosθ)

Solution:

The given equations are

x = a(cosθ +θ sinθ) and y = a(sinθ -θcosθ)

Then,

= a[-sinθ + θcosθ + sinθ] = aθcosθ

= a[cosθ +θsinθ -cosθ]

= aθsinθ

Therefore,

Question 10. Find , when: and

Solution:

Here,

Differentiating it with respect to θ using product rule,

and,

Differentiating it with respect to θ using product rule and chain rule

Summary

Exercise 11.7 Set 1 in RD Sharma's Class 12 Mathematics textbook focuses on the application of derivatives to solve problems related to rates of change. This set covers problems involving instantaneous rates, average rates, and related rates in various contexts. Students are required to apply differentiation techniques to analyze how quantities change with respect to time or other variables, often in real-world scenarios involving motion, growth, or other dynamic processes.