CBSE Class 12 Maths Chapter-5 Important Questions – Free PDF Download
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CBSE Class 12 Mathematics Important Questions Chapter 5 – Continuity and Differentiability
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. If
show that 
Ans. 
Square both side
Differentiation
Dividing (2) and (1)
4. If y = (tan-1x)2 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, x3 + x2y + xy2 + y3 = 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.
Squaring and adding
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
