Worksheet – 02 (Continuity and Differentiability)

Class: 12th Subject : Maths Date: 28/05/2025

sin3x  αsinx  βcos3x
1. If the function f  x   , x  R , is continuous at x  0 , then f  0  is equal to
x3
(a) -4 (b) -2 (c) 2 (d) 4
 tan   a  1 x   b tanx
 , x0
 x

2. For a, b  0 , let f  x   3, x0 , be a continuous function at x  0 .

 ax  b x  ax ,
2 2
x0
 b ax x
b
Then is equal to
a
(a) 5 (b) 6 (c) 8 (d) 4

3. If log x y  3sin 1x , then 1  x 2  y  xy at x 

1
is equal to
2
π π π π
(a) 9e 6 (b) 9e 2 (c) 3e 6 (d) 3e 2

 a  bcos2x
 ; x0
x2

4. Let f : R  R be defined as f  x    x 2  cx  2; 0  x  1 If f is continuous everywhere in R and m
 2x  1; x 1


is the number of points where f is NOT differential then m  a  b  c equals
(a) 3 (b) 1 (c) 4 (d) 2
5. Let f  x   2x 2  5 x 3 , x  R . If m and n denote the number of points where f is not continuous

and not differentiable respectively, then m  n is equal to

(a) 5 (b) 3 (c) 2 (d) 0
 1 x2  1
6. Let y  log e  2 
, 1  x  1 . Then at x  , the value of 225  y  y  is equal to
 1 x  2

(a) 746 (b) 736 (c) 742 (d) 732

  x
1  1 a 
 log e   ,x 0
 x  1 x
  b 1 1 4
7. If the function f  x   k , x  0 is continuous at x  0 , then   is equal to
a b k
 cos 2 x  sin 2 x  1
 ,x 0
 x2  1 1




1 Topper’s Choice, SCO 76, 2ND Floor, Sec-40-C, CHD

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(a) 4 (b) 5 (c) -4 (d) -5
  x2  5x  6
 , x2
11 5 x  x  6

2
 
8. Let f : R  R be defined as f  x   tan  x  2 
 x x
c , x2
11, x2


where x is the greatest integer less than or equal to x . If f is continuous at x  2 , then    is

equal to

(a) c  c  1 (b) c  c  2  (c) 1 (d) 2e  1

ae x  be x , 1  x  1
 2
9. If a function f  x  defined by f  x   cx ,1  x  3 be continuous for some a, b, c  R
ax 2  2cx, 3  x  4

and f   0   f   2   e , then the value of a is

1 e c e
(a) (b) (c) (d)
c  3c  13
2
c  3c  13
2
e  3c  13
2
e  3e  13
2

 2cosx  1 
 , x 
   
10. If the function f defined on  ,  by f  x    cotx  1 4 is continuous, then k is
6 3  
k. x

 4
equal to
1 1
(a) 1 (b) (c) 2 (d)
2 2

a   x  1, x  5
11. If the function f  x    is continuous at x  5 , then the value of a  b is
b x    3, x  5

2 2 2 2
(a) (b) (c) (d)
5   5  5  5
 sin    1 x  sinx
 ,x 0
 x
12. If f  x    is continuous at x  0 , then the ordered pair  p, q  is equal to
 xx  x
2

 ,x 0
x3/2

5 1  3 1  3 1  1 3
(a)  ,  (b)   ,  (c)   ,   (d)   , 
2 2  2 2  2 2  2 2

2 Topper’s Choice, SCO 76, 2ND Floor, Sec-40-C, CHD

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  tan8x

  8  tan 7 x π
  , 0x
 7  2

 π
13. Let f :  0, π   R be a function given by f  x   a  8, x
 2
 b
tanx π
(1  cotx ) xπ
a
,
 2

π
where a, b  Z . If f is continuous at x  , then a 2  b2 is equal to_______.
2
14. Let a, b  R, b  0 . Define a function

 
 asin  x  1 , for x  0

f  x  
2
 tan2 x  sin 2 x , for x  0

 bx3
If f is continuous at x  0 , then 10  ab is equal to _____.

1  1  3x 
 logc  , when x  0
15. If the function f defined on f  x    x  1 2x 
 when x  0
 k,

is continuous, then k is equal to_____.

Answer Keys
1. (a) 2. (b) 3. (b) 4. (d) 5. (b)
6. (b) 7. (d) 8. (a) 9. (d) 10. (b)
11. (a) 12. (b) 13. (81) 14. (14) 15. (5)

3 Topper’s Choice, SCO 76, 2ND Floor, Sec-40-C, CHD

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