Differentiation Test- 5 - Continuity & Differentiability - Class 12

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CBSE TEST PAPER-05 
CLASS - XII MATHEMATICS (Calculus: Continuity and Differentiability) 
Topic: - Differentiation 
1. For what value of K is the following function continuous at x = 2? [4] 
2x+1     ;     x < 2
( ) K          ;     x = 2
3 1    ;     x > 2
f x
x
?
?
=
?
?
- ?
2. Differentiate the following w.r.t. to x 
1
1 1
tan
1 1
x x
x x
- ? ?
+ - - ? ?
+ + - ? ?
[4] 
3. If
2
1
5 12 1
sin
13
x x
y
- ? ?
+ - = ? ?
? ?
? ?
find 
dy
dx
[4] 
4. Discuss the continuity of the following function at x = 0 [4] 
4 3 2
1
2
 ,    x  0
( )
tan
0  ,    x = 0
x x x
f x
x
- ? + +
? ?
=
?
?
?
5. Verify L.M.V theorem for the following function f(x) = x
2
 + 2x + 3, for [4, 6] [4] 
6. If 
sec 1
( ) ,
sec 1
x
f x
x
- =
+
find ( ) ' f x also find '
2
f
p
? ?
? ?
? ?
 [4] 
7. If ( )
p q p q
x y x y
+
= + prove that 
dy y
dx x
= [4] 
8. If sin , cos x a pt y b pt = = find the value of
2
2
d y
dx
at t = 0 [4] 
9. If 
( )
2 2
1 log 1 y x x x + = + - prove that 
( )
2
1 1 0
dy
x xy
dx
+ + + = [4] 
10. If sin sin sin ........... y x x x = + + + + 8 prove that
cos
2 1
dy x
dx y
=
- [4] 
OR 
If sin sin( ) y x a y = + prove that
2
sin ( )
sin
dy a y
dx a
+
=