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


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




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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. 







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