Jee 2014 Booklet5 Hwt Differential Calculus 1

Vidyamandir Classes DATE :

TIME : 30 Minutes

MARKS : [ ___ /10]

TEST CODE : DC-1 [1]

START TIME :

END TIME :

TIME TAKEN:

PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

4  x2 , where, [x] is the greatest integer function is x    ,  2    1, 2   x   2

Statement : 1

The domain of f  x  

Statement : 2

The domain of f  x    x   4  x   3 , where, [x] represents the greatest integer function x    , 2  2

 2 n  2 2 n  1   The domain of f  x   sin log 2 log3 x x  32 ,3  nI   Choose the correct choice : (A) FTF (B) TFT (C) TTT (D)

Statement : 3

2.

If f  x  2    x  3  2 x , then f (x) = 2

x2  2

(A)

3.

4.

If  tan x    tan y  , then y

x2  5

x

log  tan y  log  tan x 

dy  dx

(B)

log  tan y   1 2 log  tan x 

(C)

x2  4 x  9

(D)

 x  5 2  2  x  2 

(C)

1

(D)

2

(C)

log (tan y ) 

(C)

24

(D)

43

(C)

19

(D)

1 + log 2

3a tan

(D)

a sin  cos 

2y (D) sin 2 x

log  tan y   2 y cos ec 2 x log  tan x   2 x cos ec 2 y

If log 4 log3 log 2 x   1 , then x is : (A)

6.

(B)

dy  x 1 1  x  1  If y  sec 1    sin  x  1  , x  1 , then dx is :  x 1    (A) 0 (B) 1

(A)

5.

FFF

4

23

(B)

9

 3| x | 5 ,   If f : R  R is defined by f  x    logx ,   x2  4 ,  

x

2

1 2

1 x2 2 x2

 1 1 2  3 The value of f     f    f    f    f  3 is :  3 3 3 2

(A)

7.

0

(B)

1

  dy 2  1       dx   3 3 If x  a cos  , y  a sin  , then  d2y

(A)

sec

VMC/Differential Calculus - 1

(B)

3/ 2

is equal to :

dx 2 3a sin  cos 

(C)

46

HWT/Mathematics

Vidyamandir Classes 8.

If f  x   (A)

9.

f  x   sin  x   sin x  , 0  x 

(A)

(C)

10.

6x  3 , then f 1  x  is : 2x  4 2x  4 (B) 6x  3

lim

, 0  x 1  0  f  x     sin 1 , 1  x  2  3x  | x |

(C)

4x  3 6  2x

(D)

Does not exist

 , where [ ] represents the greatest integer function can also be represented as : 2

, 0  x 1  0  f  x    1  sin 1 , 1  x  2 

x   7 x  5| x |

(A)

6x  4 2x  3

 4   , x 4 2

(B)

 1   2 f  x   1  1  2

(D)

  , 0 x  0 4    f  x   1 ,  x 1 4     sin 1 , 1  x  2 

,

0 x



3/2

VMC/Differential Calculus - 1

(B)

3/7

(C)

47

3/5

(D)

2

HWT/Mathematics

Vidyamandir Classes DATE :

TIME : 30 Minutes

MARKS : [ ___ /10]

TEST CODE : DC-1 [2]

START TIME :

END TIME :

TIME TAKEN:

PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

 x1 / 3  a1 / 3  dy If y  tan 1  is :  , x  0 , then  1  x1 / 3a1 / 3  dx   1 2 (A) (B) x2 / 3 1  x2 / 3 x1 / 3 1  x 2 / 3



2.

4.

(B)

2(x2 + 7)

a x  log 1  x   sin x  cos 2 x

1 2

(B)

0





1

3x1 / 3 1  x 2 / 3

(C)

2(x2 + 11)

(D)

2(x2 + 10)

(C)

x2  2

(D)

1  x x  

1 2

(D)

2

(C)

1   x  2 A function f is defined on 1, 1 as f  x     x 1  2



, 1  x  0 ,

f (x)

(B)

2f (x)

. Then value of f  x   f | x | is :

0  x 1 1   x 2  1   x  2  1   x 2 

, 1  x 

1 2

,

1 x0 2

,

0  x 1

(D)

 1 , 1  x  10   4 2 x  1 , 0  x  1 

(C)

  ,  2    2, 3

(D)

R  3,  2 , 2 , 3

Domain of the function f  x   5| x |  x 2  6 is : (A)

  , 2    3,  

(B)

3,  2  2 , 3

If [ ] denotes the greatest integer function,

lim x

(A) 8.



(D)

where, a > 0, is :

ex  1

x 0

(C)

7.

2x2 + 11

The value of lim

(A)

6.



1 1  If f  x    x 2  ; then f (x) is : x  x2 1  (A) (B) x2  2 x x  

(A)

5.



1

3x 2 / 3 1  x 2 / 3

If f  x  2   x 2  5 x  11, f  x   f   x  is : (A)

3.



(C)

0

(B)

 n

If 0  x  y, lim y n  x n (A)



e

VMC/Differential Calculus - 1

1/ n

 2

5 sin cos x  cos x   2

is :

1

(C)



(D)

Does not exist

x

(C)

y

(D)

nx n  1

is :

(B)

48

HWT/Mathematics

Vidyamandir Classes 9.

If  0.5 (A)

10.

4  log3 log1 / 3  x 2   5 

 1 then | x | belongs to :

1,  

(B)

 2  , 1   5 

(C)

 2  ,    5 

(D)

 17 3 ,  5  15

(D)

2 3

  

  x 1    lim x  sin 1     is : x   2x  1  6  (A)

1 2 3

VMC/Differential Calculus - 1

(B)

2 3

(C)

49

3 2

HWT/Mathematics

Vidyamandir Classes DATE :

TIME : 30 Minutes

MARKS : [ ___ /10]

TEST CODE : DC-1 [3]

START TIME :

END TIME :

TIME TAKEN:

PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

2.

3.

A single formula that gives f (x) for all x  0 where,  3 x , 0  x  3 is : f  x   x3 3 x  3 , (A)

f  x   |2 x  1|  4 x

(B)

f  x   | x  3|  2 x

(C) (D)

6.

4 cos3 x sin x

f  x   |3 x  9 |  x

(C)

4 sin3 x cos x

(D)

u

f  x   | x  3|  3x

x x   log 1    log 1   2 2   If f  x   for x  0 x and f  0   a and f (x) is continuous at x = 0, then a is :





(A)

64 y

(B)

(C)

64y

(D)

 ddx y  x dydx equals : 2

7.

2

y 64 y  64

(A) (C)

8.

Let f  x    x  x  1 where [ ] denotes the greatest   2

1 5 2 x=1 x

3

lim

x

(A) (C) 5.

The

dy is : dx

(B)

integer function. Then, in (0, 2), f (x) is discontinuous at the point :

4.

If y  u 4 where u  cos x,

[0, 1]

4u3



(C)

(D)

(A)

If y  sin 8 sin 1 x then 1  x 2

(A)

5   2 ,   

(C)

(D)

1 5 2 Both (A) and (C)

(B) (D)

2 2

(B)

x

2 x3  1

9.

(B) (D)

1 1/2



 27 log  x  2  2

x 9

x 3

(A)

12

(B)

8

(C)

9

(D)

e 1

x



   e x  1  x 1 e x   lim   x  0  x 

is :

4 1 domain

3



2

4x  8x  5x  2

x lim

0 1

of

the

10.

function

1/ 4

2    x 2  2 x 1 is x  f  x   9 x  27 3  219  3        (A) (B) 3, 3 3,  

VMC/Differential Calculus - 1

(A)

12

(B)

8

(C)

9

(D)

e 1

If lim

x0

(A) (C)

50

x  a  cos x   b sin x x a=b a=b–1

 1 then : (B) (D)

a=b+1 None of these

HWT/Mathematics

Vidyamandir Classes DATE :

TIME : 30 Minutes

MARKS : [ ___ /10]

TEST CODE : DC-1 [4]

START TIME :

END TIME :

TIME TAKEN:

PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

If f (x) is a thrice differentiable function such that lim

4.

5.

6.

x  R, x  2

9

(D)

e 1

(D)

x    ,  1

(C)

21 8

(D)

21 8

(C)

2y 1 2y 1

(D)

cos x 2y 1

(C)

If y  sin 1  cos x   cos 1  sin x  , 0  x 

 dy , then is : 2 dx (C)

3

2

Range of the function f  x  

  ,  

(B)

(B)

2

5 sin  log 5  12

15

(D)

15

1

(D)

0

(C)

1 1 4 , 2  

(D)

 5 , 5   

(C)

1 1 4 , 2  

(D)

 5 , 5   

(C)

cos

(D)

 cos

1 is : 3  sin 5 x

(B)

 11  1, 3   

Range of the function f  x    x  1 x  3 x  5  is : (A)

10.

(C)

3

(A) 9.

8

dy  2x  3  If f  x   sin  log x  and y  f   , then dx at x  1 is equal to :  3  2x  12 (A) (B) (C) (D) cos  log 5  cos  log 5  sin  log 5  5  x3   k  4  x  2.k  , x  3 is continuous at x = 3, then k is : If the function f  x    x3  8 , x3 

(A)

8.

(B)

7 dy 1 If x 2  xy  y 2  , then at x = 1 and y  is : 4 dx 2 3 5 (A) (B) 4 4 dy Let y  sin x  sin x  sin x  . . . . . , then is : dx 2y cos x (A) (B) cos x 2y 1

(A) 7.

12

 12 , then f   0  is equal to :

   Let f  x    x  cos   where, [ ] denotes the greatest integer function. Then, the domain of f is :   x  2    (A) (B) x    ,  2    1,   x  R, x not an integer

(C) 3.

x3

x0

(A) 2.

f  4 x   3 f  3x   3 f  2 x   f  x 

  ,  

(B)

 11  1, 3   





If cos y  x cos   y  then 1  x 2  2 x cos  . (A)

sin

VMC/Differential Calculus - 1

(B)

 sin

dy is : dx

37

HWT/Mathematics

Vidyamandir Classes DATE :

TIME : 30 Minutes

MARKS : [ ___ /10]

TEST CODE : DC-1 [5]

START TIME :

END TIME :

TIME TAKEN:

PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than one correct option. 1.

   1  tan x   log    2 1    is : lim     4x      x  sin  x    4  4    

(A) *2.

3.

(B)

1/4

(C)

0

(D)

does not exist

(B)

discontinuous when x  n , n  Z

The function f  x   1 |tan x | is : (A)

continuous everywhere

(C)

not differentiable when x   2n  1

(D)

discontinuous at x   2n  1

If x  sin and y  cos3  then 2 y (A) (C)

4.

4

 n , n  Z and not differentiable at x  ,nZ 2 2

d2y

  2 2 2 3 cos   cos   13 sin  

 dy   4   dx  dx 2

2

is :

6 cos 2  7 sin 2   cos 2 

x  x 4  x9  . . .  x x 1 x 1

n2

lim

(A)

 ,nZ 2

n  n  1 n  2 

n

(D)



cos 2  13 sin 2   cos 2  2



2

2



3 cos  17 sin   cos 



is :

n  n  1 2n  1

(B)

6

(B)

6

(C)

n  n  1 2n  1 6

(D)

n  n  1 2n  1 6

1/ x

5.

 sin x  lim   x 0 x  (A)

is :

1



6.

 sin  x   x  If f  x     x   x  1  (A) (C)

7.

1

(B)



, x0

0

(D)

e

, where [ ] denotes the greatest integer function, then :

, x0

lim f  x   sin 1

(B)

lim f  x  does not exist

(D)

x0

x0

If f  p   2, f   p   6 and lim

g  x f  p  g  p f  x x p

x p

(A)

(C)

3:1

VMC/Differential Calculus - 1

(B)

1:3

 0 then

(C)

38

lim f  x   1

x0

lim f  x  exists but f (x) is not continuous at x = 0

x0

g p g  p

1 : 12

is : (D)

12 : 1

HWT/Mathematics

Vidyamandir Classes 8.

 x3 , x  xn Let f  x    . If the two roots of px 2  qx  r  0 are reciprocal to one another, then, the value of p, q, r 2  px  qx  r , x  xn for which f (x) is continuous and differentiable at x = x0 are respectively : (A)

(C)

9.

pr

pr

2 x03 x02

1

2 x03 x02

1

,q

,q



x02 3  x02 1  x02



x02 3  x02 x02

 

, 0 x ,

,

pr

(D)

1

 p  1 | cos x ||cos x|    If f  x    q          cot   x  2  cot m  x  2   e 

pr

(B)

x

2 x03 1  x02 2 x03 1

x02

,q

,q



x02 3  x02 1  x02



x03 3  x02 1





x02

 2

 2

  x 2

is a continuous function on  0,   then the value of p and q are respectively : (A) 10.

m m/ ,e 

(B)

em /  ,

m 

(C)

The set of all points of discontinuity of the inverse of f  x   (A)



VMC/Differential Calculus - 1

(B)

  ,  1

m , e m

(D)

e m /  , m

(D)

R   1, 1

e x  e x

is : e x  e x (C) 1,  

39

HWT/Mathematics

Vidyamandir Classes DATE :

TIME : 30 Minutes

MARKS : [ ___ /10]

TEST CODE : DC-1 [6]

START TIME :

END TIME :

TIME TAKEN:

PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than one correct option. 1.

*2.

In x  (0, 1), f  x   3 x 2  1 , where [x] stands for the   greatest integer not exceeding x is : (A) continuous (B) continuous except at one point (C) continuous except at two points (D) continuous except at three points

lim 2  25 x 2  x  5 x  is equal to :  

x0

(A)

(B)

(D) 3.

6.

lim

2 x  log e 1  x 

lim

e x  1  x

x2 x sin 5 lim x0 x x0



2 1  cos x 2 (C)

lim

x0



(A) (C)

0 22



(D)

f (x) is not differentiable at x 

 e x   x  4 x Let f  x     x   x  3 

5x4

7.

dx

*8.

12 22

(B) (D)

(A)

6x 6

(B)

36x11

(C)

6x5

(D)

3x 6

*5.

1

 k  2 

 n  1 n  2  tan

(A)

tan

(B)

tan 1  n  1  tan 1 2

(C)

tan 1 2

1

(D)

9.

f (x) is continuous in 2  x  2 f (x) is not differentiable at x = 1

(C)

 3  35 f  1  f    2 8

(D)

27  3 f   1  f       2 2

(A) (C)

Let f  x   |2 x  9|  |2 x |  |2 x  9|

10.

Which of the following are true ? (A)

9 f (x) is not differentiable at x  2

(B)

9 f (x) is not differentiable at x  2

VMC/Differential Calculus - 1

3x  (A) (C)

40





3 x  log 1  3 x  2 x 2  0 for x 

2

n  tan 1 2





Let f  x   max x, x 2 , x3 in 2  x  2 . Then :

k 1

2

None of these

du If u  3x12 and v  x 6 , then is : dv

(A) (B)

 tk is equal to where

cot 1 1  k  k  1  tk = 2  k  1

, x0

x0

n

4.

9 9 , 0, 2 2

, x0

lim f  x  exists (D)

(C)

2 3d y If x 2  xy  3 y 2  1 then x  6 y is : 2



f (x) is not differentiable at x = 0

Where [ ] denotes the greatest integer function. Then, (A) f (x) is discontinuous at x = 0 (B) f (x) is continuous at x = 0

2

5x2

x0

(C)

 1  0, 2    (0, 2)



(B)

(0, 1)

(D)

(0, 3)



5 2 x  log 1  3 x  2 x 2  3 x in : 2  1  0, 2    (0, 2)

(B)

(0, 1)

(D)

(0, 3)

HWT/Mathematics

Vidyamandir Classes DATE :

TIME : 30 Minutes

MARKS : [ ___ /10]

TEST CODE : DC-1 [7]

START TIME :

END TIME :

TIME TAKEN:

PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

If f  x   cos1

domain is : (A) 2 , 6

(B)

 6, 2   2, 3

6 , 2

(D)

2, 2    2 , 3

(C) 2.

(A) (C) (D)

4.

1 6  3 x

4 2 10   3 , 3   

 cos ec

1 x  1

(B)

2

, x  R , is :

7.

 10  3, 3   

2   10     , 3    3 ,       None of these



(B)

(C)

 4

(D)

0

x5  243

x  3 x x  aa

 1 , then a is equal to :

(B) (D)

is continuous (B) is differentiable is non-differentiable

0 None

VMC/Differential Calculus - 1

is discontinuous

None

2



et

2



2t 2t 2  2t  1



 et

dy  dx



(B)

 et

2



2t 2t 2  2t  1

2



t 2t 2  2t  1

  4  x2  If f  x   sin log e   1 x  

(D)

None of these

   , x  R , then domain and  

range of f (x) are given by : (A)  2, 1 , 1, 1 (C)

  ,  1  1,   , 0, 1  0, 1 , 1, 1

(D)

None of these

(B)

x  1  1 :  f  x    x : 1  x  1 . Then f (x) at x = 0 : 1 : x 1  (A) (C) (D)

(D)

2 3

If x  e t , y  tan 1  2t  1 , then

(C)

9.

1 e

 

(A)

(A)

removable discontinuity infinite discontinuity no discontinuity essential discontinuity

1 If y  log e x3  3 sin 1 x  kx 2 and y     2 3 , then 2 k= (A) 6 (B) 6

(C) 8.

 2

If lim

Let f  x   |2 sgn  2 x |  2 . Then f (x) has : (A) (B) (C) (D)

1  cos   x   2  is equal to : lim x 1 1  x

(A) (C) 5.

6.

Domain of the function f  x   cos

3.

1 1  2 | x |  loge  3  x  , then its 4

10.

 x   x The function f  x   sin    cos   is :  n!    n  1 ! 

(A) (B) (C) (D)

37

non periodic periodic with period 2 n ! periodic with period 2 (n + 1) ! periodic with period of 2(n + 1) 

HWT/Mathematics

Vidyamandir Classes DATE :

TIME : 30 Minutes

MARKS : [ ___ /10]

TEST CODE : DC-1 [8]

START TIME :

END TIME :

TIME TAKEN:

PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than one correct option. 1.

1 2 1  cos 2 x   is : x x0

(A) (C)

2.

(A)

lim

1 Does not exist

   b  2 lim  1  a x sin  2 x  1 a x  







(B) (D)



(C)

1 None of these

    , a, b,  R, a  0 ,  

6.

3.

b

(B)

(C)

a2/b

(D)

a  R, b  0

*4.

is

7.

a 2b None

8.

If

dx

9.

10.

at

   is : 2

VMC/Differential Calculus - 1

 2,  1

Let f  x  



sin  x 

(D)



2

x  2x  4

None of these

, [.] G.I.F., then which one is

(B) (D)

f is even f is onto

1 Does not exist

(B) (D)

0 None of these

x a sinb x

, a, b, c  R  0 , exists and is nonsinc x zero, then : (A) (B) abc  0 abc  0

If lim

x0

(C)

37

1 2

 tan 1 x  lim   , [.] = G.I.F. is : x  x  0 

(A) (C)

2

(B) (D)

not true : (A) f is periodic (C) f is many-one

f (x) is continuous f (x) is differentiable Discontinuous None of these y  a 1  cos   , x  a   sin   , then

1 2

If log0.3  x  1  log0.09  x  1 , then x lies in the interval : (A) (B) (1, 2)  2,   (C)

None of these

d2y

(D)

t

If x  e sin t, y  e cos t

(A) (C)

1  x  1 If f  x    x     ; ([.] = G.I.F.), then at x  2 :  2 

(A) (B) (C) (D) 5.

(D)

Does not exist t

1 a 2 None of these

2

ax 2  b : x  0 If f  x    possesses derivative at x : x0  x 2 = 0, then : (A) (B) a  0, b  0 a  0, b  0 (C)

(B)

and y   x  y   k  xy   y  , then k =

equal to : (A)

4

abc  0

(D)

None of these

HWT/Mathematics

Vidyamandir Classes DATE :

TIME : 30 Minutes

MARKS : [ ___ /10]

TEST CODE : DC-1 [9]

START TIME :

END TIME :

TIME TAKEN:

PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than one correct option. 1.

Let

f   x  be continuous at x = 0 and f   0   k .

The value of lim

2 f  x  3 f  2x  f  4x x2 (B) (D)

x0

(A) (C) 2.

If

(C)

,

2

then

(B)

7 24 , 3 11

(C)

y  y  x log y  x  y log x  x 



y  y  x log y  x  x  y log x 

(B) (D)

 2  f   x2

2,

(A)

(C)

2

(t  1)

(B)

(t  1) (t 2  1)

(D)

f is odd f is periodic

(B) (D)

 1  tan x  lim   x  0  1  sin x 

7 24 , 3 11

y  x  y log x 

*9.

x  y log x  y 

lim

ax 1 ax 1 a x  a x a x  a x

n a sin 2  n! n 1

(A) (C)

1 0

f  x  x

(D)

f  x   sin x

ax 1

cos ec x



1/e 1

n

ax 1

(B)

(B) (D)

e None

,  0  a  1 

(B) (D)

 None

For a real number x, let [x] denote the greatest integer less than or equal to x. Then f  x  

None of these





tan   x    2

1   x 

is :

(A)

Continuous at some x

(B)

Continuous at all x but f   x  does not exist for some x

(t 2  1)

(C)

2

(t  1)

f   x  exists for all x but f  x  does not exist

for some x (D)

2t

f   x  exists for all x

(1  t 2 )

10.

f is even None of these

Which of the following functions is an even function ?

VMC/Differential Calculus - 1

f  x 

(A) (C)

If f  x   min| x  1|,| x |,| x  1| , then : (A) (C)

6.

is

None of these

2

5.

7.

1 1 If x  t  , y  t  , then dy/dx is equal to : t t

2t

(C) 2k None of these

8. (D)

f  x 

is :

If x y y x  1 , then dy/dx is : (A)

4.

1

f  x 

x 2  17 x  66 discontinuous at x = (A)

3.

k 3k

(A)

37

 e1 / x  1 : x0  has at x = 0 : f  x    e1 / x  1  : x0  0 (A) Removable discontinuity (B) Non-removable discontinuity (C) No discontinuity (D) None of these

HWT/Mathematics

Vidyamandir Classes DATE :

TIME : 30 Minutes

MARKS : [ ___ /10]

TEST CODE : DC-1 [10]

START TIME :

END TIME :

TIME TAKEN:

PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

If x 

y

0

1

du , then

1  9u 2 1

(A)

1  9 y 

d2y dx 2

2.

3.

9y

is equal to :





(B)

 1  9 y2

(D)

None of these

2

(C)

6.

7.

If g is the inverse of f and f   x   sin x , then g   x  



(A)

 cos ec g  x 

(C)

cos ec g  x 









sin g  x 

(D)

None of these

5.

f  x    sin x  cos x  , where denotes the greatest integer function, is : (A) {0} (B) {0, 1} (C) {1} (D) None Range

of

If p, q  0 and

lim

x

8.

9.

(B)

2 , 1

(C)

2 ,  1

(D)

2,  1

Let f  x   max 2 sin x, 1  cos x , x   0,   . Then set

(A)



(B)

    2

(C)

 1 3    cos  5 

(D)

 1 3  cos  5 

(C)

(B) 2

q /p

2

VMC/Differential Calculus - 1

(D)

Let f  x  

10.

p 2 / q2

x 1  a cos x   b sin x

, x  0, f  0   1 . If f x3 (x) is continuous at x = 0, a and b are given by : (A)

5 3 , 2 2

(B)

5 ,  3

(C)

5 3  , 2 2

(D)

None of these

If xe xy  y  sin 2 x , then dy/dx at x = 0 is : (A) (C)

px  q px  q  a, lim b, qx  p x  0 qx  p

1

 b , then a and b, respectively,

2, 1

[.]

then a/b = (A)

x3

are : (A)

Period of the function f ( x )  sin 3  x   tan   x , where [.] & {.} denote the integral part and fractional parts respectively, is given by : (A) 1 (B) 2 (C) 3 (D) 

4.

sin 2 x  a sin x

x0

of points of non-differentiability is :



(B)

If lim

0

(B) (D)

1

Let f  x   log

u  x v  x

1 None

, u   2   4, v   2   2, u  2   2,

v  2   1 , f   2  is equal to :

None

(A) (C)

37

0

1

(B) (D)

1 None

HWT/Mathematics