Class 12 RD Sharma Solutions – Chapter 11 Differentiation – Exercise 11.7 | Set 2

Question 11. 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 12. Find, when and

Solution:

Here,

Differentiating it with respect to t using chain rule,

Now,

Differentiating it with respect to t using chain rule,

Dividing equation (2) by (1)

Question 13. Find, when and

Solution:

Here,

Differentiating it with respect to t using quotient rule,

and,

Differentiating it with respect to t using quotient rule,

Question 14. If x = 2cosθ – cos2θ and y = 2sinθ – sin2θ, prove that

Solution:

Here,

x = 2cosθ – cos2θ

Differentiating it with respect to θ using chain rule,

and,

y = 2sinθ – sin2θ

Differentiating it with respect to θ using chain rule,

Dividing equation (2) by equation (1),

Question 15. If x = e^cos2t and y = e^sin2t prove that,

Solution:

Here,

x = e^cos2t

Differentiating it with respect to t using chain rule,

and,

y = e^sin2t

Differentiating it with respect to t using chain rule,

Dividing equation (2) by (1)

Question 16. If x = cos t and y = sin t, prove that

Solution:

Here,

x = cos t

Differentiating it with respect to t,

and,

y = sin t

Differentiating it with respect to t,

Dividing equation (2) by (1),

Question 17. Ifand , Prove that

Solution:

Here,

Differentiating it with respect to t,

and,

Differentiating it with respect to t,

Dividing equation (2) by (1)

Question 18. Ifand, -1 < 1 < 1, prove that

Solution:

Here,

Put t = tan θ

Differentiating it with respect to t,

Further,

Put t = tan θ

Differentiating it with respect to t,

Dividing equation (2) by (1),

Question 19. If x and y are connected parametrically by the equation, without eliminating the parameter, find, when: ,

Solution:

Here, the given equations areand

Thus,

Therefore,

Question 20. Ifand, find

Solution:

Here,

Differentiating it with respect to t using chain rule,

And,

Differentiating it with respect to t using chain rule,

Dividing equation (2) by (1)