Exercise 13.1

1. Evaluate the Given limit:

Solution: –

Given

Substituting x = 3, we get

= 3 + 3

= 6

2. Evaluate the Given limit:

Solution: –

Given limit:

Substituting x = π, we get

= (π – 22 / 7)

3. Evaluate the Given limit:

Solution: –

Given limit:

Substituting r = 1, we get

π(1)2

= π

4. Evaluate the Given limit:

Solution: –

Given limit:

Substituting x = 4, we get

= [4(4) + 3] / (4 – 2)

= (16 + 3) / 2

= 19 / 2

5. Evaluate the Given limit:

Solution: –

Given limit:

Substituting x = -1, we get

= [(-1)10 + (-1)5 + 1] / (-1 – 1)

= (1 – 1 + 1) / – 2

= – 1 / 2

6. Evaluate the Given limit:

Solution: –

Given limit:

= [(0 + 1)5 – 1] / 0

=0

Since, this limit is undefined

Substitute x + 1 = y, then x = y – 1

7. Evaluate the Given limit:

Solution: –

8. Evaluate the Given limit:

Solution: –

9. Evaluate the Given limit:

Solution: –

= [a (0) + b] / c (0) + 1

= b / 1

= b

10. Evaluate the Given limit: 

Solution: –

11. Evaluate the Given limit:

Solution: –

Given limit:

Substituting x = 1

= [a (1)2 + b (1) + c] / [c (1)2 + b (1) + a]

= (a + b + c) / (a + b + c)

Given

= 1

12. Evaluate the Given limit:

Solution: –

By substituting x = – 2, we get

13. Evaluate the Given limit:

Solution: –

Given 

14. Evaluate the given limit: 

Solution: –

15. Evaluate the given limit: 

Solution: –

16. Evaluate the given limit: 

Solution: –

17. Evaluate the given limit: 
NCERT Solutions Mathematics Class 11 Chapter 13 - 41

Solution: –

18. Evaluate the given limit: 

Solution: –

19. Evaluate the given limit: 

Solution: –

20. Evaluate the given limit: 

Solution: –

21. Evaluate the given limit: 

Solution: –

22. Evaluate the given limit: 

Solution: –

23. 

Solution: –

24. Find 

, where

Solution: –

25. Evaluate 

, where f(x) = 

Solution: –

26. Find 

, where f (x) = 

Solution: –

27. Find 

, where 

Solution: –

28. Suppose 

and if 

what are possible values of a and b

Solution: –

29. Let a1, a2,………an be fixed real numbers and define a function

f (x) = (x – a1) (x – a2) ……. (x – an).

What is 

For some a ≠ a1, a2, ……. an, compute 

Solution: –

30. If 

 For what value (s) of a does

 exists?

Solution: –

31. If the function f(x) satisfies

 , evaluate

Solution: –

32. If 

 For what integers m and n does both 

and

 exist?

Solution: –


Exercise 13.2

1. Find the derivative of x2– 2 at x = 10

Solution: –

Let f (x) = x2 – 2

2. Find the derivative of x at x = 1.

Solution: –

Let f (x) = x

Then,

3. Find the derivative of 99x at x = l00.

Solution: –

Let f (x) = 99x,

From first principle

= 99

4. Find the derivative of the following functions from first principle

(i) x3 – 27

(ii) (x – 1) (x – 2)

(iii) 1 / x2

(iv) x + 1 / x – 1

Solution: –

(i) Let f (x) = x3 – 27

From first principle

(ii) Let f (x) = (x – 1) (x – 2)

From first principle

(iii) Let f (x) = 1 / x2

From first principle, we get

(iv) Let f (x) = x + 1 / x – 1

From first principle, we get

5. For the function 

 .Prove that f’ (1) =100 f’ (0).

Solution: –

6. Find the derivative of 

 for some fixed real number a.

Solution: –

7. For some constants a and b, find the derivative of
(i) (x − a) (x − b)

(ii) (ax2 + b)2

(iii) x – a / x – b

Solution: –

(i) (x – a) (x – b)

(ii) (ax2 + b)2

= 4ax (ax2 + b)

(iii) x – a / x – b

8. Find the derivative of 

 for some constant a.

Solution: –

9. Find the derivative of

(i) 2x – 3 / 4

(ii) (5x3 + 3x – 1) (x – 1)

(iii) x-3 (5 + 3x)

(iv) x5 (3 – 6x-9)

(v) x-4 (3 – 4x-5)

(vi) (2 / x + 1) – x2 / 3x – 1

Solution: –

(i)

(ii)

(iii)

(iv)

(v)

(vi)

10. Find the derivative of cos x from first principle

Solution: –

Let f (x) = cos (x + h)

Accordingly, f (x + h) = cos (x +h)

11. Find the derivative of the following functions:

(i) sin x cos x

(ii) sec x

(iii) 5 sec x + 4 cos x

(iv) cosec x

(v) 3 cot x + 5 cosec x

(vi) 5 sin x – 6 cos x + 7

(vii) 2 tan x – 7 sec x

Solution: –

(i) sin x cos x

(ii) sec x

(iii) 5 sec x + 4 cos x

(iv) cosec x

(v) 3 cot x + 5 cosec x

(vi)5 sin x – 6 cos x + 7

Let f (x) = 5 sin x – 6 cos x + 7

(vii) 2 tan x – 7 sec x