Function ASSIGNMENT FOR IIT-JEE

13
1. The period of the function, f(x) = [sin 3x] + |cos 6x| is : ( [.] denotes the greatest integer less than or equal to x) (a) π (b) 3 2π (c) 2 π (d) 2 π 2. The function f(x) = [] sec - - 1 x x x , where [x] denotes the greatest integer less than or equal to x is defined for all x belonging to : (a) R (b) R - {(-1, 1) {n : n I)} (c) R- (0, 1) (d) R- {n : n N} 3. The function f(x) = ) x ( log 2 x is defined for x belonging to : (a) (- , 0) (b) (1, ) (c) (0, ) (d) none of these 4. If f(x) = , 1 4 5 g and 3 x cos . x cos 3 x sin x sin 2 2 = π + + π + + then (gof)(x) = (a) 1 (b) -1 (c) x (d) none of these 5. Let [x] denote the greatest integer x . The domain of definition of function 2 ] x [ x 4 ) x ( f 2 + - = is (a) ] 2 , 1 [ ) 2 , ( - - -∞ (b) [0, 2] (c) [-1, 2] (d) (0, 2) 6. The domain of definition of the function - = 4 x x 5 log ) x ( f 2 10 is (a) [1, 4] (b) (1, 4) (c) (0, 5) (d) [0, 5] 7. The range of the function x 3 cos 2 1 ) x ( f - = is (a) - 0 , 3 1 (b) R (c) 1 , 3 1 (d) none of these 8. The domain of definition of the function x | x | 1 ) x ( f - = is (a) R (b) (0, ) (c) (- ,0) (d) none of these LEVEL - 1 (Objective) FUNCTION

description

Apex institute for IIT-JEE is the institution of making IITians in the Ghaziabad. It is the Institute in Indirapuram to making IITians (Eng..).Its mission is to upgrade the teaching profession by providing high quality education and training to students who will be the industry's future engineers.ObjectivesThe Institute performs the five basic functions of teaching, fulfilling the following objectives:1. Highly experienced & highly qualified recruitments and specialized training of faculty members.2. Providing study material (specially in house designed) as per requirement3. Teaching methodology and conduct periodical exams.4. Framing of test papers based on Latest Examination pattern.5. Course Co-ordination.

Transcript of Function ASSIGNMENT FOR IIT-JEE

Page 1: Function ASSIGNMENT FOR IIT-JEE

1. The period of the function, f(x) = [sin 3x] + |cos 6x| is : ( [.] denotes the greatest integer less than or equal to x)

(a) π (b) 3

2π(c) 2π (d)

2

π

2. The function f(x) = [ ]

sec−

1 x

x x , where [x] denotes the greatest integer less than or equal to x is defined for all

x belonging to :

(a) R (b) R - {(-1, 1) ∪ {n : n ∈ I)} (c) R′ - (0, 1) (d) R′ - {n : n ∈N}

3. The function f(x) = )x(log 2x is defined for x belonging to :

(a) (- ∞ , 0) (b) (1,∞ ) (c) (0, ∞ ) (d) none of these

4. If f(x) = ,14

5gand

3xcos.xcos

3xsinxsin 22 =

π++

π++ then (gof)(x) =

(a) 1 (b) -1 (c) x (d) none of these

5. Let [x] denote the greatest integer x≤ . The domain of definition of function 2]x[

x4)x(f

2

+−= is

(a) ]2,1[)2,( −∪−−∞ (b) [0, 2] (c) [-1, 2] (d) (0, 2)

6. The domain of definition of the function

−=4

xx5log)x(f

2

10 is

(a) [1, 4] (b) (1, 4) (c) (0, 5) (d) [0, 5]

7. The range of the function x3cos2

1)x(f

−= is

(a)

− 0,

3

1(b) R (c)

1,3

1(d) none of these

8. The domain of definition of the function x|x|

1)x(f

−= is

(a) R (b) (0,∞ ) (c) (-∞ ,0) (d) none of these

LEVEL - 1 (Objective)

FUNCTION

Page 2: Function ASSIGNMENT FOR IIT-JEE

9. The function )1xxlog()x(f 2 ++= is

(a) an even function (b) an odd function (c) periodic function(d) none of these

10. The function ( ))1xxlog(cos)x(f 2 ++= is

(a) even (b) odd (c) constant (d) none of these

11. Thr period of the function f(x) = sin4x+ cos4x is

(a) π (b) π /2 (c) 2π (d) none of these

12. Which of the following functions is an odd function

(a) f(x) = constant (b) f(x) = sinx + cosx (c) ( ))1xxlog(sin)x(f 2 ++= (d) f(x) = 1 + x + 2x3

13. The domain of definition of the functionf(x) = 7-xPx-3

is

(a) [3, 7] (b) {3,4,5,6,7} (c) {3,4,5} (d) none of these

14. The function xtanxx

xcosxsin)x(f 2

44

++= is

(a) even (b) odd (c) periodic with periodπ (d) periodic with period 2π15. Range of f(x) = |x| + |x+1| is

(a) [0, ∞ ) (b) (0, ∞ ) (c) (1, ∞ ) (d) [1, ∞ )

16. Range of tan-1x - cot-1x is

(a) (0,π) (b)

ππ−

2,

2

3(c)

1,3

2(d)

1,3

2

17. The range of x|x| − is

(a) (0,∞ ) (b) [0,∞ ) (c) (-∞ ,0) (d) (-∞ ,0]

18. If

+−=

110

110x)x(f

x2

x22 then ‘f’ is

(a) an even function (b) an odd function (c) neither even nor odd (d) cannot be determined

19. The domain of the function f(x) = )x1(log

1

10 − + )2x( + is

(a) [-3, -2] excluding (-2.5) (b) [0, 1] excluding 0.5

(c) [-2, 1], excluding 0 (d) None of these

20. The period of xcos]x[xxcos 24

e π+−+π is ([.] denotes the greatest integer function)

(a) 2 (b) 1 (c) 0 (d) -1

21. The domain of the function f(x) =

+++ −−

x2

x1sin)xcos(sin)x(logsin

21

21

(a) {x : 1 < x < 2} (b) {1} (c) Not defined for any value x (d) {-1, 1}

Page 3: Function ASSIGNMENT FOR IIT-JEE

22. If [.] denotes the greatest integer function then the domain of the real valued function is|2xx|log 2]2/1x[ −−+

(a)

∞,2

3(b) ),2(2,

2

3 ∞∪

(c) ),2(2,2

1 ∞∪

(d) None of these

23. The domain of the function f(x) = 6 )1x(2

)2x(3

2x 25284 −−

−−+ is

(a) (0, 1) (b) [3, ∞ ) (c) (1, 0) (d) none

24. A function whose graph is symmetrical about the y-axis is given by

(a) )1xx(log)x(f 2e ++= (b) f(x + y) = f(x) + f(y) for all x, y ∈ R

(c) f(x) = cos x + sin x (d) none of these

25. If

∞∈+−= ,2

1x,1xx)x(f 2 then value of ‘x’ satisfying f(x) = f -1(x) is

(a) 1 (b) 2 (c) 2

1(d) none of these

26. If

−−=

5x

1xlog)x(f 4.0 and g(x) = x2 - 36, then D

f/g is

(a) }6{~)0,( −−∞ (b) }6,1{~),0( ∞

(c) }6{~),1( ∞ (d) }6{~),1[ ∞

27. The domain of the real-valued function f(x) = loge |log

e x| is

(a) ),1()1,0( ∞∪ (b) ),0( ∞ (c) ),e( ∞ (d) ),1( ∞

28. If f(x) and g(x) are two functions of ‘x’ such that f(x) + g(x) = ex and f(x) - g(x) = e-x, then

(a) f(x) is odd, g(x) is odd (b) f(x) is even, g(x) is even

(c) f(x) is even, g(x) is odd (d) f(x) is odd, g(x) is even

29. The inverse of the function )1a,0a()1xx(logy 2a ≠>++= is

(a) )aa(2

1 xx −− (b) not defined for all x

(c) defined for only positive x (d) none of these

30. Let f(x) = cos p x, where p = [a] = the greatest integer less than or equal to ‘a’. If the period of f(x) is π, then

(a) ]5,4[a∈ (b) ]5,4[a = (c) )5,4[a∈ (d) none of these

31. If f(x) = ,e |xn|cos........|x2|cos|x|cos]x[x π++π+π+− then period of f(x) is

(a) 1 (b) n

1(c)

1n

1n2

2

+−

(d) n......3.2.1

1

Page 4: Function ASSIGNMENT FOR IIT-JEE

32. Let

−+

−+

−= −−−

4

|x|2tan

4

|x|2cos

4

|x|2sin)x(f 111 , then Df is

(a) [-6, 6] (b) ),6[ ∞ (c) [-6, 3] (d) [-3, 6]

33. Identify the statement(s) which is/are incorrect ?

(a) The function f(x) = cos(cos-1 x) is neither odd nor even

(b) The fundamental period of f(x) = cos(sin x) + cos(cos x) is π(c) The range of the function f(x) = cos (3 sin x) is [- 1, 1]

(d) None of these

34. Let 9

xcos4)x(f2

2 π−= . Then

(a) ]1,1[R,,,3

D ff −=

∞π= (b) ]2,2[R,,,

3D ff −=

∞π=

(c) ]4,4[Rand,33

,D ff −=

∞π∪

π−∞−= (d) ]4,0(R,

3,D ff =

π−∞−=

35. The range of the function ),1(x),xxxesin()x(f 2]x[ ∞−∈−+= where [x] denotes the greatest integer function is:

(a) φ (b) [0, 1] (c) [-1, 1] (d) R

36. The domain of f(x) = log2 log

3 π/4log (tan-1 x)-1 =

(a) R (b) ),/4( ∞π (c) (0, 1) (d) None of these

37. If f(x) is a periodic function of the period ‘λ ’ then f(λx + a), where a is a constant, is a periodic function of the

period

(a) 1 (b) λ (c) a

λ(d) none of these

38. If ]1x[

1

|x|

11)xx(cos)x(f

221

−+

−+−= − then domain of f(x) is (where [.] is the greatest integer)

(a)

+2

51,2 (b)

+2

51,2 (c)

−−2

21,2 (d) none of these

39. Let f(x) = sin x, g(x) = ln |x|. If the ranges of the composite functions fog and gof are R1 and R

2 respectively, then

(a) )0,(R),1,1(R 21 −∞=−= (b) ]1,1[R],0,(R 21 −=−∞=

(c) ]0,(R),1,1(R 21 −∞=−= (d) ]0,(R],1,1[R 21 −∞=−=

40. If x1

x1log)x(f

−+= , then

(a) f(x) is even (b) f(x1).f(x

2) = f(x

1 + x

2) (c) )xx(f

)x(f

)x(f21

2

1 −= (d) f(x) is odd

Page 5: Function ASSIGNMENT FOR IIT-JEE

1. Find the value of x for which, 0)4x(

)2x()1x)(1x2()x(f

4

32

>−

−−−=

2. Find the values of x for which 0)1x(

)4x()3x()x1()2x()x(f

22

≤+

−−−−= .

3. Solve 12x

x|3x| >+

++.

4. Find the domain of the function;

−+−=

|xsin|log

3)23x8x(loglog)x(f

2

2|xsin|

5. Find domain for

+−=

5x

1xlog)x(f 4.0 .

6. Find domain for xlog3

|x|21cosy |1x|

1−

− +

−= .

7. Find domain for

−π=

]1x[cos]x[sin)x(f where [ ] denotes greatest integer function.

8. Find the range for x]x[1

]x[xy

+−−= where [ ] denotes greatest integer function.

9. Find domain and range of the function y = loge(3x2 - 4x + 5).

10. If f is an even function, find the real values of x satisfying the equation

++=

2x

1xf)x(f .

11. Find whether the given function is even or odd function, where

21x

)xtanx(sinx)x(f

ππ++= .where [ ] denotes

greatest integer function.

LEV EL - 2 (Subjective)

3

Page 6: Function ASSIGNMENT FOR IIT-JEE

12. If ),2[),1[:f ∞→∞ is given by x

1x)x(f += then find )x(f 1− . (assume bijective).

13. Let ),4/3[),2/1[:f ∞→∞ , where f(x) = x2 - x + 1. Find the inverse of f(x).

14. ,x]x[

1)x(f

−= where [ ] denotes greatest integral function less than or equals to x. Then find domain of f(x).

15. Find the domain of )13x7x(log

1)x(f

22/1 +−

=

16. Find the domain of single valued function y = f(x) given by the equation 10x + 10y = 10.

17. Let

π∈

2,0x , then find the solution of the function

xtanlog

1)x(f

xsin−= .

18. Find the range of log3(log

1/2(x2 + 4x + 4)).

19. Find the domain & range of : 22

x9

sin3)x(f −π= .

20. Find the inverse of following functions:

(i) ]3,3[x),3/x(sin)x(f 1 −∈= − [assuming bijective]

(ii) ]3,1[x),1x3x(ln)x(f 2 ∈++= . [assuming bijective]

21.

≤≤≤≤−−

=1x0,x

0x1,1x)x(f 2

and g(x) = sinx. Find h(x) = f(|g(x)|) + |f(g(x))|.

22. If ,x]cos[x]cos[)x(f 22 π−+π= where [x] stands for the greatest integer function, then evaluate

)4/(fand)(f),(f),2/(f ππ−ππ .

23. A cubic expression f(x) satisfies the condition

=

+

x

1f)x(f

x

1f)x(f , then prove that f(x) = 1 + x3or1 - x3.

If f(3) = 28. Then prove that f(2) = 9.

24. Let f(x) be a polynomial function satisfying, Ry,x2)xy(f)y(f)x(f)y(f)x(f ∈∀−++= . If f(2) = 5 then

prove that f(5) = 26.

25. If for non-zero x, 5x

1

x

1bf)x(af −=

+ where ba ≠ then find f(x).

Page 7: Function ASSIGNMENT FOR IIT-JEE

1. The domain of the function 1x)x1log()x(f 2 −+−= is

(a) [-1, 1] (b) ),1( ∞ (c) (0, 1) (d) ]1,( −−∞

2. The range of the function 2

2

x

x1)x(f

+= is equal to

(a) [0, 1] (b) (0, 1) (c) ),1( ∞ (d) ),1[ ∞

3. The curves 2x3xyand2|x|3|x|y 2323 ++=++= have the same graph for

(a) x > 0 (b) 0x ≥ (c) all x except 0 (d) all x

4. Domain of the function 73x

12

x

1)x(f xsin

2

1

+−

++=−

is

(a) φ (b) R - {0} (c) R (d) None of these

5. The domain of definition of the function )1x(loge3y 1x2

−= − is

(a) ),1( ∞ (b) ),1[ ∞ (c) R ~ {1} (d) ),1()1,( ∞∪−−∞

6. The range of the function f(x) = cos [x], where 2

x2

π<<π− , is

(a) {-1, 1, 0} (b) {cos 1, 1, cos 2} (c) }1,1cos,1{cos − (d) none of these

7. If b2 - 4ac = 0 and a > 0, then domain of the function y = log (ax3 + (a + b)x2 + (b + c)x + c) is

(a)

a2

b~R

2

(b)

−≥∪

− }1x|x{

a2

b~R

(c)

−−∞∪

− ]1,(

a2

b~R (d) none of these

8. Which of the following functions is an even function?

(a) xx

xx

aa

aa)x(f −

−+= (b)

1a

1a)x(f x

x

−+= (c)

1a

1ax)x(f x

x

+−= (d) ( )1xxlog)x(f 2

2 ++=

9. If (log3 x) (log

x 2x) (log

2x y) = log

x x2, then y is equal to

(a) 9 (b) 18 (c) 27 (d) 81

LEVEL - 3(Questions asked from previous Engineering Exams)

Page 8: Function ASSIGNMENT FOR IIT-JEE

10. The range of the function 1xx

1xx)x(f

2

2

+++−= is

(a) R (b) ),3[ ∞ (c)

3,3

1(d) none of these

11. The domain of the function 2xx22)x(f −−= is

(a) 3x3 ≤≤− (b) 31x31 +−≤≤−− (c) 2x2 ≤≤− (d) none of these

12. If the domain of the function ),,(is7x6x)x(f 2 ∞−∞+−= then range of the function is

(a) ),( ∞−∞ (b) ),2[ ∞− (c) (-2, 3) (d) )2,( −−∞

13. The domain of the function

= −

2

xlogsin)x(f

2

21 is

(a) 1x1 ≤≤− (b) 1x0 ≤≤ (c) 2x1 ≤≤ (d) none of these

14. If f(x) = cos (log x) then

+

− )xy(f

y

xf

2

1)y(f)x(f has the value

(a) 1 (b) 2

1(c) -2 (d) 0

15. The inverse of the function 2ee

ee)x(f

xx

xx

++−= −

is given by

(a) 2/1

e 1x

2xlog

−−

(b) 3/1

e x3

1xlog

−−

(c) 2/1

e x2

xlog

− (d) 2

e 1x

1xlog

+−

16. The range of the function for real x of x3sin2

1y

−= is

(a) 1y3

1 ≤≤ (b) 1y3

1 <≤− (c) 1y3

1 >>− (d) 1y3

1 >>

17. The domain of the function 2xx2

]x[

1 −+ , where [ . ] denotes greatest integer function.

(a) [1, 2] (b) [0, 2] (c) [0, 1) (d) [1, 2)

18. Domain of sin-1(log3(x/3)) is

(a) [1, 9] (b) [-1, 9] (c) [-9, 1] (d) [-9, -1]

19. The range of f(x) = 7-xPx - 3

is

(a) {1, 2, 3} (b) {1, 2, 3, 4, 5, 6} (c) {1, 2, 3, 4} (d) {1, 2, 3, 4, 5}

Page 9: Function ASSIGNMENT FOR IIT-JEE

20. The value of Zn∈ for which the function

=

nx

sin

nxsin)x(f has π4 as its period is

(a) 2 (b) 3 (c) 5 (d) 4

21. The domain of the function f(x) = log2(log

3(log

4 x)) is

(a) x < 4 (b) x > 4 (c) 0 < x < 2 (d) 2 < x < 4

22. If f(x) is an odd periodic with period 2, then f(4) is

(a) 0 (b) 2 (c) 4 (d) -4

23. If f(x) = 1 + xα , ( 0≠α ) is the inverse of itself then the value of α is

(a) -2 (b) -1 (c) 0 (d) 2

24. If g(x) be a function defined on [-1, 1] and if the area of the equilateral triangle with two of its vertices at (0, 0)

and (x, g(x))) is 4

3, then the function is

(a) 2x1)x(g −±= (b) 2x1)x(g −−= (c) 2x1)x(g −= (d) g(x) = 2x1+

25. Which of the following functions is periodic

(a) f(x) = x - [x] (b)

=

≠=0x,0

0x,x

1sinx)x(f (c) f(x) = x cos x (d) none of these

26.2x

)3x)(1x()x(f

−−+= is real valued in the domain

(a) ),3[]1,( ∞∪−−∞ (b) ]3,2(]1,( ∪−−∞ (c) ),3[)2,1[ ∞∪− (d) none of these

27. The domain of definition of the function y(x) given by the equation 2x + 2y = 2 is

(a) 1x0 ≤< (b) 1x0 ≤≤ (c) 0x ≤<∞− (d) 1x <<∞−

28. If the function ),1[),1[:f ∞→∞ is defined by f(x) = 2x(x - 1) then f-1(x) is

(a) )1x(x

2

1−

(b) ( )xlog4112

12++ (c) ( )xlog411

2

12+− (d) not defined

29. If log0.3

(x - 1) < log0.09

(x - 1), then x lies in the interval

(a) ),2( ∞ (b) (1, 2) (c) (-2, -1) (d) none of these

30. If g(f(x)) = |sin x| and f(g(x)) = 2)x(sin , then

(a) x)x(g,xsin)x(f 2 == (b) |x|)x(g,xsin)x(f ==

(c) xsin)x(g,x)x(f 2 == (d) f and g cannot be determined

Page 10: Function ASSIGNMENT FOR IIT-JEE

31. Let

>=<−

=−+=0x,1

0x,0

0x,1

)x(fand]x[x1)x(g . Then for all x, f(g(x)) is equal to

(a) x (b) 1 (c) f(x) (d) g(x).

32. The domain of definition of 2x3x

)3x(log)x(f 2

2

+++= is

(a) R ~ {-1, -2} (b) ),2( ∞− (c) R ~ {-1, -2, -3} (d) }2,1{~),3( −−∞−

33. If 1x,1x

x)x(f −≠

+α= , then for what value of α is f(f(x)) = x ?

(a) 2 (b) 2− (c) 1 (d) -1

34. The set of all real numbers x for which x2 - |x + 2| + x > 0, is

(a) ),2()2,( ∞∪−−∞ (b) ),2()2,( ∞∪−−∞

(c) ),1()1,( ∞∪−−∞ (d) ( )∞,2

35. Range of the function Rx,1xx

2xx)x(f

2

2

∈++++= is

(a) ),1( ∞ (b)

7

11,1 (c)

3

7,1 (d)

5

7,1

Page 11: Function ASSIGNMENT FOR IIT-JEE

ANSWER KEY

1. b

2. b

3. b

4. a

5. a

6. a

7. c

8. c

9. b

10. a

11. b

12. c

13. c

14. b

15. d

16. b

17. b

18. b

19. d

20. b

21. b

22. b

23. b

24. d

25. a

26. c

27. a

28. c

29. a

30. c

31. a

32. a

33. a

34. c

35. c

36. c

37. a

38. a

39. d

40. d

LEVEL - 1 (Objective)

Page 12: Function ASSIGNMENT FOR IIT-JEE

ANSWER KEY

1. }4{~),2()2/1,(x ∞∪−∞∈

2. ),3[]1,1(x ∞∪−∈

3. ),1()2,5(x ∞−∪−−∈

4.

π∪

ππ∪π∈ 5,

2

3

2

3,),3()x(f

5. ),1()x(f ∞∈

6. ]2,1()1,0(x ∪∈

7. R ~ [1, 2)

8. Range =

2

1,0

9. Range is

,3

11log

10.

−−+−−−+−=

2

53,

2

53,

2

51,

2

51x

11. f(x) is an odd function (if π≠ nx ) and f(x) isan even function if ( π= nx ).

12.2

4xx 2 −+

13.4

3x

2

1 −+

14. φ

15. (3, 4)

16. )1,(x −∞∈

17.

ππ

2,

4

18. RRange∈

19.

ππ−∈

3,

3Domain ,

2

33,0Range

20. (i) 3 sin x (ii) 2

e53 x+±−

21.

≤≤<<−+−1x0,xsin2

0x1,1xsinxsin2

2

22. 2

1

4f,0)(f,0)(f,1

2f =

π=π−=π−=

π

23.

24.

25.)ba(

5bx

x

a

ba

1)x(f 22 +

−=

LEVEL -2 (Subjective)

Page 13: Function ASSIGNMENT FOR IIT-JEE

1. d

2. c

3. b

4. a

5. a

6. b

7. c

8. c

9. a

10. c

11. b

12. b

13. d

14. d

15. b

16. a

17. a

18. a

19. a

20. a

21. b

22. a

23. b

24. b

25. a

26. c

27. d

28. b

29. a

30. a

31. b

32. d

33. d

34. b

35. c

LEVEL - 3 (Questions asked from previous Engineering Exams)

ANSWER KEY