Jee 2014 Booklet3 Hwt Functions

August 28, 2017 | Author: varunkohliin | Category: Function (Mathematics), Sine, Trigonometric Functions, Logarithm, Integer
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Jee 2014 Booklet3 Hwt Functions...

Description

Vidyamandir Classes

DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : FNC [1]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

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, question marked with ‘*’ may have More than One correct option. 1.

Let f  x   6 x  7  cos  x  6 x where [ ] is greatest integer function, then : (A)

Period of the function is 1 1 Period of the function is 2

(C) 2.



 1,   f  a  b  e

6.

9.

0,  

(D)

 1,  

(B)

f  a  b  e2

(C)

f a  b

(D)

None of these

x2 , x  2 is : x2

1

(C)

1, 1

(D)

1, 0, 1

 2 x  2 x   is : sin   2 x  2 x    (A) Even function

(B)

Has domain R

(C)

Odd function

(D)

None of these

If f  x  2 y, x  2 y   xy then f (x, y) is : x2  y2 8

(B)

x2  y2 4

(C)

x2  y2 4

(D)

x2  y2 2

If  x   5  x   6  0 (where [ ] denotes greatest integer function) then x belongs to : 2

[2, 4)

(B)

[2, 4) – {3}

(C)

{3}

(D)

{2}

(C)

[0, 1)

(D)

(0, 1]

  , 1

(C)

1,  

(D)

1,  

odd

(C)

neither even nor odd

 x2  e   , then range of f (x) is : If f  x   ln   x2  1    (A) (0, 1) (B) [0, 1]

Range of y  (A)

10.

(C)

(B)

(A) 8.

 0,  

{1}

(A) 7.

Function is non periodic



(B)

Range of the function f  x   (A)

*5.

(D)

If f  x   e px  2 then f  a  f  b  is : (A)

4.

Period of the function is 2

Domain of f  x   log10 log10 1  x 3 is : (A)

3.

(B)

ex ,x 0 :  x  1

 0,  

(B)

If f  x   x 2 sin x , then f (x) is : (A)

even

VMC/Functions

(B)

61

(D)

None of these

HWT/Mathematics

Vidyamandir Classes

DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : FNC [2]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

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, question marked with ‘*’ may have More than One correct option. 1.

2.

  x , x  0 [{ } denotes fractional part] f  x    x , x  0 Then f (x) is : (A) even (B) both even and odd (C)

*8.

2

(D)

3

even

(C)

both

(D)

None of these

periodic

(C)

odd

(D)

None of these



(B)

even

(B)

even natural number irrational number

(B) (D)

odd natural number zero

If f  x   cos  2  x  cos   2  x , where [ ] stands for greatest integer function, then :    

  f    1 2

(B)

f    1

(C)

f     0

(B)

f  x  y  f  x  f  y

(D)

  f   1 4

Let f  x   x 1 then :

  

(A)

f x2  f  x 

(C)

f



2

 x   f  x

(D)

None of these

(B) (D)

periodic with period 2 odd function

 0,    1

(C)

 e,  

(D)

1,  

R  0 , 1,  1

(C)

R  0

(D)

None of these

ln sin x   tan3 x  cos ec  3x  5  is : Function f  x   e 

periodic with period  non periodic

Domain of f  x   log e log e x is : (A)

10.

(C)

1

If f (x) is odd and even simultaneously then f  3  f  2  is :

(A) (C) 9.



odd

(A) 7.

(B)

Function f  x   x  tan x is :

(A) (C) *6.

0



(A) 5.

None of these

If h  x   f  x   f   x  g  x   g   x  , then h (x) is : (A)

4.

(D)

Number solution of : x 2  4   x   0 (A)

3.

odd

1,  

(B)

Domain of f  x   log e x 2 is : (A)

R

VMC/Functions

(B)

62

HWT/Mathematics

Vidyamandir Classes

DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : FNC [3]

ROLL NO.

START TIME :

STUDENT’S SIGNATURE :

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 x  x   x  is satisfied by k values of x. Then k is :

(A) 2.

f  x  (A)

3.

9.

10.

2

(D)

3

(B)

Odd

(C)

Period

(D)

Positive always

(B) (D)

Periodic but with the fundamental period None of these

(C)

 x  2 2

(D)

x2  2

(C)

R  0

(D)

R   1

(D)

None of these

x  1 is : 2

Periodic with period Non periodic

 x  2 2 3

R+

x2

 x  2 2

(B)

2

4

x then domain of f (x) is : (B)

R  1

(C)

2

1

(B)

0

(B) (D)

Periodic with different period Periodic with same or different period

If log x  x ; then number of such value of x are : (A) 0 (B) 1

(C)

2

(D)

3

The number of solutions of 2 x  4 x  5 x  0 are : (A) 0 (B) 1

(C)

2

(D)

3

Total number solutions of 2 x  4 x  5 x  0 are : (A) 0 (B) 1

(C)

2

(D)

3

If f (x) is periodic then f  x  is : (A) (C)

8.

(C)

If f  x   e x and g  x   e  x , then f  x   g  x  has how many solutions : (A)

7.

e 1 Even

If f  x   log (A)

6.



1

f  x   2 f 1  x   x 2  2,  x  R , then f (x) is :

(A)

5.

x x

(B)

f  x   sin x   x is :

(A) (C) 4.

0

Periodic with same period Non periodic

VMC/Functions

63

HWT/Mathematics

Vidyamandir Classes

DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : FNC [4]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

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, question marked with ‘*’ may have More than One correct option. 1.

Period of sin x  cos x is :

 2

(A) 2.



4.

1 sq. unit

  ,  2   4,  

*10.

(B)

2 sq. unit

1

 4

(C)

No point

(D)

None of these

(C)

f (1) = 2

(D)

f (x) = 0 has 1 real solution

(C)

3 sq. unit

(D)

4 sq. unit

(C)

  ,  2    4,  

(D)

None of these

(C)

2

(D)

3

is :

2

 x    x   6

  ,  2  4,  

Number of integers in the range of f  x   1

(B)

x 1  x2

0

are :

Domain of the function f  x   x 2   x  is : 2

R

(B)

0,  

(C)

  , 0 

(D)

None of these

(B)

 4

(C)

2

(D)

 2

7

(D)

4

Period of sin 4 x  cos 4 x is : 

(A) 9.

2 points

(B)

Domain of the function f  x  

(A) 8.

(D)

Area bounded by the curve x  y  1 is :

(A) 7.

(B)

1 1 If f  x  f    f  x   f   and f  3  28 then : x    x (A) f (x) is even (B) f (x) is odd

(A) 6.

2



1 point

(A) 5.

(C)

f  x   sgn e x  1 is discontinuous at : (A)

*3.



(B)

Number of break points in the curve f  x   2 sin x  in 0 , 2  : (A) 6 (B) 5 (C) f  x

e e If e Choose correct statements : (A) Only 1 natural number is in domain of the function (B) Only 1 natural number is in range of the function (C) Only 1 whole number is in the domain of the function (D) Only 1 whole number is in the range of the function

VMC/Functions

x

64

HWT/Mathematics

Vidyamandir Classes

DATE :

IITJEE :

NAME :

  MARKS :    10 

TIME : 25 MINUTES

TEST CODE : FNC [5]

ROLL NO.

START TIME :

STUDENT’S SIGNATURE :

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, question marked with ‘*’ may have More than One correct option. 1.

(A)

If f (x) is defined for x  0 , 1 , then function f  2 x  3

(C)

is defined for :

*2.

3.

(A)

x  0 , 1

(B)

 3  x    ,  1  2 

(C)

xR

(D)

 3  x    , 1  2 

Which of the following is(are) odd function : (A) f (x) = constant (B) f (x) = sin x + cos x (C)

  f  x   sin  log  x  x 2  1     

(D)

f  x   1  x  2 x3

If f  x   (A)

f  x  3

Domain set of definition of f  x  

6.

3 f  x 1 f  x  3 f  x  3

1 x 2 x

(C) (D)

None of these

Domain of f  x   log10 (A)

[4, 6]

(C)

(2, 3)



x4  6 x (B)

(D) 1  sin x sin x Function f  x     2  cos x cos x

9.

3 f  x 1

  ,    2, 2 (B)   ,    1, 1 1, 1    ,  2    2 ,  

(A)

5.

(D)

f  x 1

(D)

2  4

 2   x 2  lies in Value of the function f  x   3 sin   16    the interval :  3     (A) (B) 0,   4 , 4  2    (C)

8.

(B)

 3, 3

Range of the function f  x  

(D)

None of these

1 is : 2  cos 3 x

 1  (B) R   3 , 0   1  (C) (D) None of these  3 , 1   A polynomial function satisfies the condition 1 1 f  x  f    f  x   f   . If f 10   1001 , then x  x f (20) is : (A) 2002 (B) 8008 (C) 8001 (D) None of these (A)

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

(C)

4.

7.

  2

10.

:

  , 6  (4, 6)   is periodic with  

 1 x  The function f  x   log   satisfies the equation :  1 x  (A) f  x  2   2 f  x  1  f  x   0





(B)

f  x  2   f  x   f x  x  1

(C)

f  x1  f  x2   f  x1  x2 

(D)

 x x  f  x1   f  x2   f  1 2   1  x1 x2 

period :

VMC/Functions

65

HWT/Mathematics

Vidyamandir Classes

DATE :

IITJEE :

NAME :

  MARKS :    10 

TIME : 25 MINUTES

TEST CODE : FNC [6]

ROLL NO.

START TIME :

STUDENT’S SIGNATURE :

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, question marked with ‘*’ may have More than One correct option. 1.

2.

If a function f (x) satisfies the condition 1 1  f  x    x 2  2 , x  0 , then f (x) equals : x  x (A)

x 2  2 for  x  0

(B)

x 2  2 for  x  2

(C)

x 2  2 for  x  2

(D)

None of these

4.

5.

6.

7.

1 The function f  x     is : 2 (A) Periodic (B) An odd function (C) Expressible as the sum of an even function and an odd function (D) None of these





(B)

 5,  

(C)

(D)

None of these

   Domain definition of the function f  x   tan     x  2    is :

VMC/Functions

[8, 10] [1, 10]

None of these



(B) (D)

(A) (C) 8.

12

One Zero



64 1 7 x is equals to : 3 (B) Two (D) None of these

Minimum value of f  x   x  1  x  2  x  3 is equal to : (A) 1 (C) 3

9.

(B) (D)

 

 1 If 2 f x 2  3 f  2 x

1  x 

(A) (C)

1  x  2

(B)

5x

2

1  x  4

(D)

None of these

Domain of the function a x  a y  a  a  1 is : (A) (C)

66

5x2

5x

2 zero

 2 2   x  1 then f (x ) is : 

4

10.

(8, 10) None of these

Total number of real values of x such that

x

is defined for all x

belonging to : (A) 5,  

  ,  5

(D)



12  x 1 7  12  x 1 7

f (x) is periodic with period 4k f (x) is periodic with period 2k f (x) is periodic with period k f (x) is not periodic but an odd function

f  x   log 2 x  5 x 2  3x  10

 2,  1

Domain definition of the function

(A) (C)

sin x

*3.

(C)

(B)

f  x   log 4 log5 log3 18 x  x 2  77 is :

If f  x  k   f  x   0 for  x  0 then : (A) (B) (C) (D)

2 , 1 R  2 ,  1

(A)

0  x 1   x  1

(B) (D)

0  x 1   x  0

HWT/Mathematics

Vidyamandir Classes

DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : FNC [7]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

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.

If f  x  3 y, x  3 y   12 xy , then f  x, y  is : (B)

2 x2  y 2

(C)

x2  y 2

(D)

None of these

7.

(B)

f (x) is periodic with period

(C) (D)

f (x) is periodic with period 42 f (x) is non periodic

8.

  , 1 1,  

(B) (D)

Domain of the function : f  x  

(C)

(B) (D)

Domain of the function f  x  



x x

9.

  ,     , 0   log0.3  x  1

(B)

 2, 4 

(C)

(2, 4)

(D)

None of these

f  x   x   x 

(B)

1 if x is a rational number f  x   0 if x is a irrational number

 



(A)

f  x   sin 2 x and g  x   x

(B)

f  x   sin x, g  x   x

(C)

f  x   x 2 , g  x   sin x



2

then :

f and g cannot be determined



If f  x   a  x n

(A) (C) 10.



1n

, where a > 0 and n  N , then

a xn

(B) (D)

x an

Let f  x   x and g  x   x for all x  R . Then the function   x  satisfying

  x   f  x      x   g  x    0 is : 2

Which of the following is non-periodic : (A)



fof  x  is equal to :

 x2  2 x  8

(1, 4)

 2 –1

(B) (D)

2

1

If g f  x   sin x and f g  x   sin x

(D)

1

(A)

VMC/Functions

x , x  1 . Then, for what value of  is x 1



(A) (C)



ex ,x 0 : Range of y   x  1

  ,    0  0,  

Let f  x  



f (x) is periodic with period 2

 0,   1,  

None of these

f f  x  x ?

(A)

(A)

6.

(D)

f  x   sin x , then :

(C)

5.



2xy

(A)

4.



(A)

8 8  1  cos x 1  cos x

f  x 

(C)

67

2

(A)

  x   x, x  0 ,  

(B)

  x   x, x  R

(C)

  x    x, x    , 0

(D)

  x  x  x , x  R

HWT/Mathematics

Vidyamandir Classes

DATE :

IITJEE :

NAME :

  MARKS :    10 

TIME : 25 MINUTES

TEST CODE : FNC [8]

ROLL NO.

START TIME :

STUDENT’S SIGNATURE :

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.

1

f  x 

x x

5.

1 x x

then :

(A)

dom f   and dom g  

(B)

dom f   and dom g  

(C) (D)

f and g have the same domain dom f   and dom g  

Let f  x  

6.

7.

d  a

(C)

a  b  c  d  1 (D)

(B)

a  b 1 8.

9.

(A) (C) (D)

(B)

(C)

x

(D)

The domain of definition of



10.

(B)

2,  

{2, 3} [2, 3]



(B) (D)

(2, 3) None of these

(A)

0  x 1

(B)

0  x 1

(C)

  x  1

(D)

  x  0

Let f : R  R be a function defined by



x

3

 x

 then graph of f (x) lies is :

1  x2 Only III quadrant Only IV quadrant I, II quadrant III and IV quadrant

Period of the function f  x   sin 4  3x   cos 4  3x  is : (A)

 2

(B)

 3

(C)

 6

(D)

None of these

f  x     x   f  x     x  are :

  , 1



1 x x

Domain of the function a x  a y  a  a  1 is :

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

Let f  x   x  1 and   x   x  2 . Then the value of x satisfying

1 x

f  x 

f   x  is an even function

x x x  2 cos  tan The function f  x   sin is 2 3 4 periodic with period : (A) 6 (B) 3 (C) 4 (D) 12

(A)

(A) (C)

d=a

0, x0     , x  1 then f (x) is : Let f  x    x 2 sin 2 x  x 1  x x , (A) Even function (B) Odd function (C) Neither even nor odd function

 

1 If 3 f  x   f    log x 4 then f e  x equals : x

f  x   log 1  log x 2  5 x  16

ax  b . Then fof  x   x provided that : cx  d

(A)

(D) 4.

, g  x 

  ,  2   ,  1  2 ,  

VMC/Functions

68

HWT/Mathematics

Vidyamandir Classes

DATE :

IITJEE :

NAME :

  MARKS :    10 

TIME : 25 MINUTES

TEST CODE : FNC [9]

ROLL NO.

START TIME :

STUDENT’S SIGNATURE :

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.

The domain of definition of   3x  1   f  x   log0.5   log2    is :  3x  2    (B)

 1    3 ,   

(A)

(C)

1   3 ,   

(D)

1  3 ,   

(C)

If f  x   sin  log x  , then the value of

7.

x f  xy   f    2 f  x  cos  log y  is :  y 1 (B) 0 1 (D) None of these

If f is even function defined on the interval  5, 5 then 9.

1 

If f  x   x 2  x then if

(B) (D)

f

x

1

VMC/Functions

(B)

 

1 , x  0 , then f x 2 equals : x

f  x  f x

f  x

f x

(B)

f  x f x

(D)

None of these

Number of solutions of log x 3  2 x  3 is : (A) 3 (B) 1 (C) 2 (D) 0 Domain of the function f  x   2  x  1  x (A)

[2, 6]

(B)

 2 , 6 

(C)

[8, 12]

(D)

None of these

If f  x  and g  x  be periodic and non-periodic



2 None of these 10.

 k then for what



(A)

always periodic

(B)

never periodic

(C)

can’t say

(D)

periodic it g(x) is linear function of x

Number of solutions of x 2  2  2  x   0 (where [.] denotes greatest integer function) (A) 1 (B) (C) 0 (D)

value of k equation has exactly 4 roots : (A)

3 4

(D)

functions respectively, then f g  x  is :

Period of f  x   sin 2 x  cos 2 x is : (A) (C)

5.

8.

1 4

If f  x   x 

1    ,  3   

number of real values of x satisfying the equation  x 1  f  x  f   are :  x2 (A) 1 (B) 2 (C) 3 (D) 4 4.

6.

(A)

(A) (C) 3.

(C)

1 2

69

2 infinite

HWT/Mathematics

Vidyamandir Classes

DATE :

  MARKS :    10 

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : FNC [10]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

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, question marked with ‘*’ may have More than One correct option. 1.

Let f  x  

4x x

4 2

(C)

. Then the value of

 1   2   3   1996  is : f  f  f . . . . f   1997   1997   1997   1997 

(A) (C) 2.

*3.

4.

6.

(B) (D)

999 None of these

*7.

Range of the function f  x   3 sin x  2 cos x is : (A)

 2 , 13   

(B)

3, 13   

(C)

 13 ,  13   

(D)

 2 , 3

If

(A) (C) 8.

10  x f x e   , x   10, 10  10  x

 200 x  1 f  x   kf  , then is : 2 k  100  x  a prime (B) a composite even (D) odd

If h  x   2 max  f ( x )  g ( x ) , 0 , then h(x) is equal to :

If f  x   27 tan 3 x  cot 3 x and  ,  are two values

(A)

f  x  g  x  g  x  f  x

of x for which 3 tan x  cot x  2 , then :

(B)

f  x  g  x  g  x  f  x

(C)

f  x  g  x  g  x  f  x

(D)

f  x  g  x  g  x  f  x

(A)

f    f   

(C)

f    2

Domain of f  x   (A) (C)

5.

998 997

(D)

 ,nI 2 None of the above x : x   2n  1

R+ R

(B)

f    26

(D)

f    f

1 x12  x 9  x 4  x  1 (B) (D)

2

 

is :

9.

1  x 1  f  x   log0.4  is :  2 x  5   x  36

R can’t be easily specified

(A)

Range of f  x    sin x  cos x  is : (A)

{0, 1}

(B)

1, 0, 1

(C)

{1, 2}

(D)

{1}

Domain of the function

(C) *10.

 , 0  6 1,    6

f  x   x3

and g  x   sec 2 x  tan 2 x . The two functions are

(B)

f  x   e x  e x

equal over the set :  (A)

(C)

f  x   sin x

(D)

e x  e x

VMC/Functions

x x  cos 2 2 2

(B)

R

70

(D)

 0,    1, 6 1,    6

Function(s) whose graph is symmetrical about origin is(are) : (A)

Let f  x   sin 2

(B)

HWT/Mathematics

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