# Important Questions for CBSE Class 12 Maths Chapter 5 - Continuity and Differentiability

## CBSE Class 12 Maths Chapter-5 Important Questions – Free PDF Download

Free PDF download of Important Questions for CBSE Class 12 Maths Chapter 5 – Continuity and Differentiability prepared by expert Maths teachers from latest edition of CBSE(NCERT) books, On CoolGyan.Org to score more marks in CBSE board examination.

## 4 Marks Questions

1. Find the values of K so that the function f is continues at the given value of x.

Ans.

K = 6

2. Differentiate the function
Ans. Let y = u + v
When u = x sinx, v = (sinx)cosx

Taking log both side
log u = log xsinx
log u = sinx . logx
diff. both side w.r. to x

Taking log both side
log v = log (sinx)cosx

Differentiation both side w.r. to x

Hence

3. Ifshow that
Ans.
Square both side

Differentiation

Dividing (2) and (1)

4. If y = (tan-1x)show that (x2 + 1)2 y2 + 2x (x2 + 1)y1 = 2
Ans. y = (tan-1 x)2 (given)
Differentiation both side w.r. to x

Again differentiation both side w.r. to

5. Verify Rolle’s Theorem for the function y = x2 +2 , [ -2 , 2]
Ans. y = x2 + 2 is continuous in [-2, 2] and differentiable in (-2, 2). Also f (-2) = f(2) = 6
Hence all the condition of Rolle’s Theorem are verified hence their exist value c such that
(c) = 0
0 = 2c.
C = 0
Hence prove.

6. Differentiate
Ans.

7. Differentiate sin2x w.r. to ecosx
Ans.

8. If prove that
Ans.

Square both side

9. If cosy = x cos (a + y) prove that
Ans.

10. If x = a (cos t + t sin t)
y = a (sin t – t cos t )
find
Ans.

11. Find all points of discontinuity if

Ans. At x = -3
f(-3) = |-3| + 3 = 3 + 3 = 6

Hence continuous at x = -3
At x = 3

Hence it is continuous

12. Differentiate
Ans.

13. Find if
Ans. Differentiate both side w.r.t. to x, x+ x2y + xy+ y= 81

14. Differentiate xy = e(x-y)
Ans.
Taking log both side

Diff. both side w.r.t. to x

15. Find if
Ans.

16. If y = 3 cos (log x) + 4 sin (log x). Show that x2y2 + xy1 + y = 0
Ans.
Diff. both side w.r.t. to x

Again diff.

17. Verify Rolle’s Theorem for the function f(x) = x2 + 2x – 8, x[-4, 2]
Ans. The function
Continuous in [-4, 2] and differentiable in (-4, 2)
Also
Hence all the condition of all Rolle ’s Theorem, is verified
Their exist a value C
Such that (c) = 0
(c) = 2c +2
0 = 2C+2
C = -1

18. Find
Ans.

19. If x = a (cos t + t sin t) and y = a (sin t – t cos t), find
Ans.

20. If Prove that
Ans.

21. Find the value of K so that function is continuous at the given value.

Ans.

22. Differentiate
Ans.

23. Find
Ans.

24. Find
Ans. Let

Therefore — (1)

Taking log both side

Differentiate both side w.r.t. to x

— (2)

Taking log both side

— (3)

Taking log both side

— (4)
(by putting 2,3 and 4 in 1)

25. Find when
Ans.

26. If Prove that
Ans.

=

LHS

27. If Show that
Ans.

28. If
Prove is a constant independent of a & b.

Ans.
Diff. both side w.r.t. to x

Again diff. both side

Put (y-b) in equation (1)

Put the value of (x-a) and (y-b) in equation (1)

Hence prove

29. Find if
Ans.
Differentiate both side w.r.t. x

30. Find
Ans.
Taking log both side

Differentiate both side w.r.t. x

31. Discuss the continuity of the function

Ans. At x = -1
f(-1) = -2

Hence continuous at x = -1

Continuous

32. Find if
Ans.

33. Find if
Ans.

Diff.

34. Find , if y=
Ans. Let
Where

Taking log both side

Differentiate

Taking log both side

Differentiate

35. find
Ans.

36. If show that
Ans.
Differentiate

37. Find
Ans.

38.
Ans.

39. If Prove that
Ans. Let

Squaring both side

Differentiate

40. Show that
Ans.

,
hence

41. For what value of K is the following function continuous at x = 2?

Ans.

A T

42. Differentiate the following w.r.t. to x
Ans.

43. If find
Ans.

44. Discuss the continuity of the following function at x = 0

Ans.

Hence continuous

45. Verify L.M.V theorem for the following function f(x) = x2 + 2x + 3, for [4, 6]
Ans. Since f(x) is polynomial hence continuous in the interval [4, 6] thus f(x) is differentiable in (4, 6) both condition of L.M.V theorem are satisfied.

46. If find also find
Ans.

47. If prove that
Ans.
Taking log both side

Differentiate both side w.r.t. to x

48. If find the value of at t = 0
Ans.

49. If prove that
Ans.

50. If
prove that OR
If prove that
Ans. Let

Squaring both side

Differentiate both side w.r.t. to x

OR

Differentiate both side w.r.t. to x