worksheet of functions

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Final Step BY ABJ (Function)

LEVEL -1

1. Find the domain of definition of the following functions :

(i) f(x) =

2

1x3sin3x21 1 .

(ii) f(x) = xx

2cossin

1 .

(iii) f(x) =

4

]x[32sin 1

, where [.] denotes greatest integral function .

(iv) f(x) = log10

(1 - log10

(x2 - 5x + 16)) .

(v) f(x) = )1x2(cos

3log1

x2

.

(vi) f(x) = )x2(coslogx .

(vii) f(x) = logx sin x .

(viii) f(x) = 143

123

2

2

xxxx .

(ix) f(x) =

xx 3 x 2

logx 1 x

. .

(x) f(x) = x64xlog10 .

2. Find the range of the following functions :

(i) f(x) = 3x2x

1x2

.

(ii) f(x) = 2x9 .

(iii) f(x) = 3x1|,2x||1x| .

(iv) f(x) = 1xxsin 21 .

(v) f(x) = 2

21

xx 1

.

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3. Find the domain and range of the following functions :

(i) y = 3xcosxsin2log5

.

(ii) f(x) = 2x3x

4x5x2

2

.

(iii) f(x) = )xcos(sin .

(iv) y = 3logxtanlogsec xtan31 .

(v) y = xe

,1 [x]

find range only for x 0 , where [.] denotes greatest integer function .

4. Functional Equation :

(i) If for non-zero x, a f(x) +b f 5x

1

x

1

, where ba , then find f(x).

(ii) If f(x + y) = f(x) . f(y) for all real x, y and 0)0(f , then prove that function

f(x) = 2)}x(f{1

)x(f

is even function.

(iii) A function f, defined for all x, y R is such that f(1) = 2 and f(x + y) - kxy = f(x) + 2y2,

where k is some constant. Find f(x) & show that; 0yxforkyx

1f)yx(f

.

(iv) Find a function RR:f satisfying f(x) f(y) - f(xy) = x + y for all Ry,x .

(v) If function f(x) is satisfying 2f(sin x) + f(cos x) = x for all Rx then express f(sin x) as apolynomial in x.

(vi) If f(x) = )0a(aa

ax

x

, evaluate

1n2

1r n2

rf2 .

5. Identical Function :

Find for what values of x, the following functions be identical.

(a) f(x) = n (x - 1) - n (x - 2) and g (x) = x 1

nx 2

.

(b) f(x) = xlog

1)x(g,elog

ex .

(c) f(x) = sin(cos-1 x); g(x) = cos(sin-1 x) .

(d) f(x) = sec2 x � tan2 x; g(x) = cosec2 x - cot2 x .

(e) f(x) = x)x(g,x

x 2

.




6. Periodic Function :

(i) Find the period of the following functions :

(a) f(x) = sin(x + sin x) (b) f(x) = sin (cos x ) + cos (sin x)

(c) f(x) = sin4 x + cos4 x (d) f(x) = x7

2sinx

5

3cos

(e) f(x) = n1n32 2

xtan

2

xsin..........

2

xtan

2

xsin

2

xtanxsin

(ii) Prove that the following function are not periodic :

(a) f(x) = x + x sin x (b) f(x) = cos x2

(iii) Let f(x) = 3|xsin||xcos|2 2 . If 2

is the fundamental period of f(x), find .

7. Types Of Function :

(i) Let A = R � {3}, B = R � {1} and BA:f defined by f(x) = 3x

2x

. Is �f � bijective ?

Give reasons.

(ii) Determine the kind of mapping of BA:f where f(x) = x2 � 2x + 2, )2,1(A and

]5,1[B .

8. Composite Function :

(i) Let ]1,0[]1,0[:f , where f(x) = x1

x1

and ]1,0[]1,0[:g , where g(x) = 4x(1 � x).

Determine the following :(a) fog(x) (b) gof(x)

(ii) Let f(x) = x1

1

. Let f

2(x) denotes f [f(x)] and f

3(x) denotes f [f{f(x)}]. Find f

3n(x), where

n is a natural number. Also state the domain of this composite function.

(iii) If f(x) = � 1 + |x � 2|, 4x0 and g(x) = 2 � | x |, 3x1 . Then find fog (x) andgof (x). Draw rough sketch of the graphs of fog (x) and gof (x).

(iv) Find fog(x) if f(x) = [x] + {x}2 and g(x) = [x] + x and also find the range of fog(x).




9. Inverse Function :

(i) Find the inverse of the following function :

(a) R:f (�1, 1), f(x) = 2x1

|x|x

. (b) R:f R, f(x) =

x4

4x1

.2

,x1x,x

x

2.

(ii) Let RR:g be given by g(x) = 3 + 4x. If gn(x) = n times

gog.......og(x) ,show that

gn(x) = (4n -1) + 4n x. If g-n(x) denotes the inverse of gn(x), prove that the above formulahold for all negative integers.

(iii) Show that the function }0{R}0{R:f given by f(x) = x

k, where k is a non-zero

real number, is inverse of itself.

(iv) Show that the function RR:f given by f(x) = 1a,0a,1xxlog 2a is

invertible and find its inverse.

(v) Prove that the inverse of the fractional function

c

aR

c

dR:f ,

f(x) = 0bcaddcx

bax

is also a linear fractional function. Under what condition f(x)

coincide with its inverse.

10. Draw the graph of each of the following functions :

(i) y = |1 � | x2 � 2 || (ii) |y| = 2y � x

(iii) y = [sin x] (iv) y = sin x + cos x

(v) y = ex + e�x (vi) y = ||x|n|

11. Check Whether The Following Functions Are Even Or Odd Or Neither Even Nor Odd :

(i) f(x) = 2x x (ii) f(x) = 3/123/12 1x1x

(iii) f(x) = 22 xx1x1x (iv) f(x) = | x3 |

(v) f(x) = sin x + cos x

12. A is a point on the circumference of a circle. AB and AC divide the area of the circle into threeequal part. If the angle BAC is the root of the equation, f(x) = 0 then find one such f(x).




LEVEL -2

1. Find the domain of definition of the following functions :

(i) f(x) = 6]x[]x[

12

, where [.] denotes greatest integral function .

(ii) f(x) = 1xxxx 4912 .

(iii) f(x) = 8x2x

)1x(log2

3.0

.

(iv) f(x) =

xsin22

1xsin2e1xcos3xcos2n)xsin(cos

1cos2 .

(v) f(x) =

x2

x1sin)xcos(sin)x(logsin

21

21 .

(vi) f(x) = 1x

)!1x2(5

]2/x[

3 21 xcos

, where [ ] denotes greatest integer function .

(vii) f(x) = 1x2

5xlog1xsin4 xsin

2

.

(viii) f(x) = 2

(x 2)6 x 2(x 1)34 8 52 2 .

(ix) f(x) =

2

23

xx

xx.

(x) f(x) = xxn12

xtane 12

xsin 1

, where [.] denotes greatest integer function

2. Find the range of the following functions :

(i) f(x) =

4x8x5logtan 2

5

41 .

(ii) f(x) = x

|x|logcos ]x[

1, where [.] denotes greatest integer function.

(iii) f(x) = n(cos(sin x)) .

(iv) f(x) = tan x cot x .

(v) f(x) = 1 1 1sin x cos x tan x .




3. Find the domain and range of the following functions :

(i) f(x) = x64xlog2 .

(ii) y = sec-1 (2x - x2) .

(iii) y = ..........xcos1xcos1 .

(iv) y = 1 2n sin x x 1 , where [.] denotes greatest integer function.

(v) xsinlogy x

, where [ . ] denotes the greatest integer function.

4. Functional Equation :(i) If for all real values of u and v, 2f(u)cos v = f(u + v) + f(u - v), prove that, for all real values

of x

(a) f(x) + f(-x) = 2a cos x .

(b) 0)x(fxf .

(c) xsinb2)x(fxf . Deduce that f(x) = a cos x - b sin x, where a, b arearbitrary constants.

(iii) Let f and g be real valued functions such that f(x + y) + f(x - y) = 2 f (x) . g (y) Ry,x and

f :R[-1,1] onto function then prove that Ry1|)y(g| .

(iv) Let f(x) + f(y) = 22 x1yy1xf . Prove that f(4x3 - 3x) + 3f(x) = 0,

2

1,

2

1x also prove that f(x) = 0,

1,2

1

2

1,1x .

(v) Consider a real valued function f(x) satisfying, 2 f (x y) = (f (x))y + (f(y))x for all Ry,x

and f(1) = a where 1a . Prove that aa)i(f1a 1nn

1i

.

5. Periodic Function :

(i) Prove that if the graph of the function, y = f(x), defined throughout the number scale, issymmetrical about two lines x = a and x = b,(a < b), then the function is a periodic one.

orIf f(a - x) = f(a + x) and f(b - x) = f(b + x) for all real x, where a, b (a < b) are constants,then prove that f(x) is a periodic function.




(ii) (a) Find out the integral values of n if 3 is a period of the function :f(x) = cos nx . sin (5/n)x.

(b) Find the value of In for which the function f(x) = )n/xsin(

nxsin has 4 as its

period.

(iii) Let �f� be a real valued function defined for all real numbers x such that the equation

)x(f3)1x(f)1x(f holds for all x. Prove that the function f is periodic.

(iv) Let �f� be a real valued function defined for all real numbers x such that for some positive

constant �a� the equation 2xf)x(f2

1)ax(f holds for all x. Prove that the

function f is periodic.

6. Types Of Function :(i) Find the set of values of a for which the function RR:f given by

f(x) = x3 + (a + 2)x2 + 3ax + 5 is one - one.

(ii) Classify the following functions as injective, surjective both or none :(a) RR:f , f(x) = x3 � 6x2 + 13 x � 6 .

(b) f : R,2

1

, f(x) = (x2 + x + 5) (x2 + x � 3) .

(iii) Prove that R)1,1(:f defined by f(x) =

x, 1 x 0

1 xx

, 0 x 11 x

is a bijective function.

(iv) Let f : X Y be a function defined by f(x) = cxcosb4

xsina

. If f is both

one-one and onto, find sets X and Y.

(v) Check whether R),3()2,(:f , f(x) = 652 xxn is a bijective function

or not. If not then choose a suitable longest domain and co-domain for which the abovefunction becomes bijective.

7. Composite Function :

(i) Let f(x) = Rx,x2x2 and g(x) = f(f(x) - 1) + f(5 - f(x)). Show that, Rx0)x(g .

(ii) If f(x) =

x

2xlog2log 10

x100 ; g(x) = {x}, where {x} denotes the fractional part of x.

If the function fog(x) exists, then find the maximum possible range of g(x).




(iii) Draw the graph of f(x) =

1x;)1x(

1x;1x2 and g(x) =

2x

4;2xtan

4x

2;xtan

and

hence find fog(x) and gof(x).

(iv) If f(x) =

0x,x3

0x,x12

2

and g(x) =

0x2/,ecxcos

2/x0,xtan then find the fog(x).

8. Inverse Function :

(i) Compute the inverse of the function }2{R}1{R:f , f(x) = 1x

x

2 .

Also find domain and range of f-1.

(ii) Find the minimum value of �a� and �b� for which f(x) = xx ; ,b,a be an

invertible function.

(iii) Check whether RR:f , f(x) = xx (where [.] and {.} represents greatest

integral and fractional part function respectively) is an invertible or not, if yes then find itsinverse. Also solve the equation f(x) = f -1(x).

(iv) Let f(x) = x2 + 3x - 3, 0x . n points x1, x

2, .........x

n are so chosen on the x-axis that :

(a)

n

1i

n

1iii

1 xn

1f)x(f

n

1.

(b)

n

1i

n

1iii

1 x)x(f , where f -1 denotes the inverse of f. Find the A.M. of xi �s.

9. Draw the graph of each of the following functions :(i) y = x2 � 3| x | (ii) y = 1 + | sin x |

(iii) | y | = log x (iv) y = max {1 � x, 1 + x, 2}

(v) y = min {| x |, | x � 2 |, 2 � | x � 1 | } (vi) y = 1x0,e1,2

3,emin xx

(vii) y = sgn (x � | x | )

10. f(x) =

1|x||,x|x

1|x|,2

xtanx 4

. Prove that f(x) is an odd function.

11. Find the integral solutions to the equation [x] [y] = x + y. Show that all the non-integralsolutions lie on exactly two lines. Determine these lines.




SET - I

1. If f(x + 2y, x � 2y) = xy, then f(x, y) equals

(A)8

yx 22 (B)

4

yx 22

(C)4

yx 22 (D)

2

yx 22

2. Let RR:f be a function such that f(x) =

cQx,5x

Qx,5x, then

(A) f is one�one and onto (B) f is one�one and into(C) f is many one and into (D) f is many one and onto

3. If the function f : R A given by f (x) = 1x

x2

2

is a surjection, then A is

(A) R (B) [0, 1](C) (0, 1] (D) [0, 1)

4. The value of the function f(x) = 3 sin

22

x16 lies in the interval

(A)

4,

4(B)

2

3,0

(C) (� 3, 3) (D) none of these

5. f(x) is an odd function and g(x) is neither odd nor even , then

(A) f(x) + g(x) is neither even nor odd (B) f(x) + g(x) is even(C) f(x) + g(x) is odd (D) none of these

6. If f(x) = cos�1

4

|x|2 + [log (3 � x)]�1, then its domain is

(A) [� 2, 6] (B) [� 6, 2) (2, 3)(C) [� 6, 2] (D) [� 2, 2))(2, 3]

7. Let f(x) = 6x5x

10x7x2

2

, then the range of f(x) is

(A) R (B) R � {1}(C) R � {3) (D) none of these

8. The function f : R R defined byf(x) = (x � 1) (x -� 2)(x � 3) is(A) one-one but not onto (B) onto but not one-one(C) both one-one and onto (D) neither one-one nor onto




9. The domain of the function f(x) 1 | x || x | 2

is

(A) , 1 1, (B) , 2 2,

(C) 2, 1 1, 2 (D) none of these

10. If f(x) = (ax2 + b)3, then the function g such that f(g(x)) = g (f(x)) is

(A) g(x) = 2/13/1

a

xb

(B) g(x) = 32 bax

1

(C) g(x) = (ax2 + b)1/3 (D) g(x) =

2/13/1

a

bx

11. If 21/ 2log x 5x 7 0 , then exhaustive range of values of x is

(A) , 2 3, (B) (2, 3)

(C) , 1 1, 2 2, (D) none of these

12. If f(x) = ,x1

x1

then domain of f-1 (x) is

(A) R (B) R � {�1}(C) (� , �1) (D) (�1, )

13. If f(x) = 2

x

1 x, then (fof of) (x) is

(A) 2

3x

1 x(B) 2

x

1 3x

(C) 2

3x

1 x(D) none of these

14. If 3f(x) � f

x

1= log x4, then f(e�x) is

(A) 1+ x (B)x

1

(C) x (D) � x

15. The value of the parameter , for which the function f(x) = 1 + x, 0 is the inverse of itself, is -(A) � 2 (B) � 1(C) 1 (D) 2




16. The number of roots of the equation,

2

3,inx

2xcot is ,

(A) 3 (B) 2(C) 1 (D) infinite

17. ),6(R:f , f(x) = x2 � (a � 3)x + a + 6, then the values of 'a' for which the function isonto(A) (1, 9) (B) [1, 9](C) {1, 9} (D) none of these

18. The number of solutions of the equation )1xsec(2x2

x1sin

21

is / are

(A) 1 (B) 2(C) 3 (D) infinite

19. Let f(x) = cot(5 3x) (cot(5) cot(3x)) cot 3x 1 , then domain is

(A)

3

nR , In (B) In,

6)1n2(

(C)

3

5n,

3

nR , In (D)

3

5nR , In

20. The domain of ; f(x) = }x{log)xcos(sin x ; {.} denote the fractional part, is

(A) ,1 (B) ,12,0

(C) }1{2

,0

(D) (0, 1)

21. The complete set of values of 'a' for which 0aex4 x2 has only one real solutionwhich is positive, is

(A) 22

1, e

e

(B) 22

1e ,

e

(C) 22

1, e

e

(D) none of these

22. Let f(x) = 2x x

2x x

9.3 6.3 4

9.3 6.3 4

, then range of f(x) is

(A)

3,3

1(B)

1,3

1

(C) [0, 2] (D) none of these




23. The solution set of the equation 2

1xxsin)1x(xtan 211 is

(A) (-1, 0) (B) [-1, 0](C) {-1, 0} (D) none of these

24. Let f(x) = ]x[|1x|

1

, [.] denotes the greatest integer function, then domain of f(x) is

(A) (-1, 1) (B) )1,(

(C) )1,( (D) none of these

25. If f(x) =

x

2xln2log

xe2 and g(x) = {x} then range of g(x) for the existence of f(g(x)) is

(A)

2e

1~

e

1,0 (B)

2e

1~

e

2,0

(C)

2e

1~

e

3,0 (D) none of these

26. Period of function f(x) = ])x3[sin|x3sin|x3(sin3

1 where [.] denotes the greatest integer

function is

(A) 3

(B)

3

2

(C) 3

4(D)

27. Fundamental period of the function, f(x) = cos(tan x + cot x) . cos(tan x - cot x), is

(A) 4

(B)

2

(C) (D) 2

28. The domain of the function f(x) = )}x{sin(}x{sin

1

where {.} denotes fractional part, is

(A) ],0[ (B) In,2

nR

(C) ),0( (D) none of these

29. Let f(x) = xcos12xsin5

1

, then range of f(x) is

(A)

13

1,

13

1 (B) R

(C)

,

13

1

13

1, (D) none of these




30. Let f(x) = xcot1

xcos

xtan1

xsin22

then range of f(x) is

(A) [-1, 0] (B) [0, 1](C) [-1, 1] (D) none of these

SET - II

1. If log3 (x2 � 6x + 11) < 1, then the exhaustive range of values of x is

(A) (� , 2)(4, ) (B) (2, 4)(C) (� , 1)(1, 3) (4, ) (D) none of these

2. Complete solution set of the inequality x(ex � 1)(x + 2)(x � 3)2 0 is(A) [�2, 3] (B) (� 2, 0](C) (� , � 2]{0, 3} (D) ( � , � 2)[0, 3]

3. Total number of positive real values of x satisfying 2[x] = x + {x}, where [] and {} denotes thegreatest integer function and fractional part respectively, is(A) 2 (B) 1(C) 0 (D) 3

4. Solution set of 2( )x

xlog x 0

| x |

is

(A) ( , 0) (1, 2) (B) ( ,1) (2, )

(C) ( , �1) (0,1) (D) ( , � 2] (0,1)

5. Period of f(x) = sin�1(sin x) is(A) 2 (B)

(C)2

(D) none of these

6. f:[2, ) (� , 4], where f(x) = x(4 � x) then f�1 (x) is

(A) 2 � x4 (B) 2 + x4

(C) � 2 + x4 (D) � 2 � x4

7. Which of the following is not an odd function?(A) g(x) � g( � x) (B) (g(x) � g(�x))3

(C) log

1xx

1xx2

24

(D) xg(x). g(�x) + tan (sinx)




8. If f(x) + 2f(1 � x) = x2 + 1 x R then f(x) is

(A) 3x4x3

1 2 (B)3

2(x2 + 4x � 3)

(C)3

1(x2 � 4x + 3) (D)

3

2(x2 � 4x + 3)

9. The domain of the function x|x|

1y

(A) )0,( (B) ]0,(

(C) ]1,( (D) ),(

10. Period of the function 3 5x x

sin cos2 2 is

(A) 2 (B) 10(C) 8 (D) 5

11. If f(x) = )xncos( , then 1 xf (x) f (y) f f x.y

2 y

has the value

(A) �2 (B) �1

(C) 0 (D) 1

12. If �f� is a real valued function not identically zero, satisfying f(x + y) + f(x � y) = 2f(x). f(y) Ry,x ,then f(x) is(A) odd (B) even(C) neither even nor odd (D) none of these

13. If f: R R where f(x) = ax + cos x is an invertible function then(A) a (� 2, 1][1, 2) (B) a [� 2, 2](C) a (� , � 1][1, ) (D) a [� 1, 1]

14. Total number of solution of 2x + 3x + 4x � 5x = 0 is / are(A) 1 (B) 2(C) 3 (D) none of these

15. Total number of solutions of the equation sin x = |x|n are

(A) 8 (B) 10(C) 9 (D) 6

16. If |sin x + cos x| = |sin x| + |cos x|, then x belongs to the quadrant,(A) I or III (B) II or IV(C) I or II (D) III or IV




17. If II:f be defined by f(x) = [x + 1], where [.] denotes the greatest integer function, then f -1(x)is equal to(A) x - 1 (B) [x + 1]

(C) ]1x[

1

(D) 1x

1

18. The domain of f(x) = 2x1]x[ ; where [.] denotes the greatest integer function is

(A) ),1[)2,( (B) ,12,

(C) ,13, (D) , 3 1,

19. The domain of the function f(x) = cos-1(sin x2) is

(A) ),1( (B) ),0(

(C) (-1, 1) (D) ( , )

20. If RR:f be a function such that f(x) = x3 + x2 + 3x + sin x. Then(A) f is one-one into (B) f is one-one and onto(C) f is many one and into (D) f is many one and onto

21. On [0, 1], f(x) is defined as f(x) = irrationalisxif,x1

rationalisxif,x. Then for all ))x(f(f],1,0[x is

(A) constant (B) 1 + x(C) x (D) none of these

22. The domain of the function sec-1[x2 - x + 1], is given by [where [.] is greatest integer function]

(A) ),( (B) ),1[]0,(

(C) ),2[ (D) none of these

23. The domain and range of the function f(x) = log2

2

23xcosxsin are given by

(A) Df = ]2,1[R),,( f (B) D

f = ]1,1[R),,0()0,( f

(C) Df = ]2,1[R),,0( f (D) none of these

24. Let )2/,0[R:f (where R is the set of real numbers) be a function defined byf(x) = tan-1 (x2 + x + a). If f is onto then a equals(A) 0 (B) 1(C) 1/2 (D) 1/4

25. The image of the interval [-1, 3] under the mapping RR:f given by f(x) = 4x3 - 12x is




(A) [8, 72] (B) [-8, 72](C) [0, 8] (D) none of these

26. If BA:f is a bijection and AB:g is the inverse of f. then fog is equal to(A) I

A(B) I

B

(C) f (D) g

27. The function NN:f ( N is the set of natural numbers) defined by f(n) = 2n + 3 is(A) surjective (B) not surjective(C) bijective (D) none of these

28. Let f(1) = 1 and f(n) =

1n

1r

)r(f2 . Then

m

1r

)n(f is equal to

(A) 3m � 1 (B) 3m

(C) 3m - 1 (D) none of these

29. Let f be a function satisfying f(x + y) = f(x) . f(y) for all Ry,x .If f(1) = 3 then

n

1r

)r(f is equal to

(A) )13(2

3 n (B) )1n(n2

3

(C) 3n + 1 - 3 (D) none of these

30. If RR:g,RR:f be two given functions then h(x) = 2 min (f (x) - g(x), 0) equals(A) f(x) + g(x) - | g(x) - f(x) | (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) |

SET - III

I. Fill in the blanks :

1. The number of solutions of 2x � x2 + 1 = 0 is ..............

2. The inverse of the function f : [0, ) [2, ) = x2 + 2 is ..............

3. The period of the function f(x) = |xcos||xsin|

|xcosxsin|

is ..............

4. Solution of the equation x + [y] + {z} = 3.1{x} + y + [z] = 4.3[x] + {y} + z = 5.4

( where [.] denotes the greatest integer function {.} denotes fractional part ) is ..............

5. The domain and the range of f(x) = sin�1x + cos�1x + tan�1 x + sec�1 x + cot�1 x + cosec�1 x is




..............

6. The integral values of 'n' for which f (x) =

n

x3sin.

3

nxcos is periodic with period 6 is

..............

7. The number of solution of 0)1x(cos3x2 is ..............

8. The domain of f(x) =

21

1xx3

1

is ..............

9. The period of f(x) = x7cosx13cos

x7sinx13sin

is ..............

10. If f(x + y) = Ry,x3

)y(f)x(f

, f is even/odd/even as well as odd/neither..

II. Condition and result :Each question has a conditional statement followed by a result statements.If condition result, then condition is sufficient andIf result condition, then condition is necessaryIf condition is necessary as well as sufficient for the result, mark (A) on OMR sheet.If condition is necessary but not sufficient for the result, mark (B) on OMR sheet.If condition is sufficient but not necessary for the result, mark (C) on OMR sheet.If neither necessary nor sufficient for the result, mark (D) on OMR sheet.Consider the following example :Condition : a > 0, b > 0Result : a + b > 0Here, if a > 0 and b > 0, then it always implies that a + b is positive but if a + b is positive, thena and b both need not to be positive. So condition implies result but result does not always impliescondition hence condition is sufficient but not necessary for the result to be hold. So answer is �C�.

11. Condition : f(x) is periodic

Result : f(x) is many-one

12. Condition : f (x) is positive in the domain of f(x)

Result : f(x) is one-one

13. Condition : g(x) is a periodic function and for every function f(x), f (g(x)) is well defined.

Result : f (g(x)) is periodic for any function f(x)

14. Condition : f(x) and g(x) both are even functions

Result : f(x) + g(x) is an even function




15. Condition : RR:g,RR:f both are one to one function.

Result : f(x) + g(x) is one to one function.

16. Condition : RR:g,RR:f , f(x) and g(x) both are odd functions

Result : f(x) � g(x) is an odd function

17. Condition : f(x) and g(x) both are periodic functions

Result : f (x) . g(x) is a periodic function.

III. More than one correct :18. Which of the following homogeneous functions are of degree one ?

(A) x y2 2 + x (B) x y

x y

3 3 1 2

1 2

/

/( )

(C) ln x ln y (D) x xy y2 22 3

19. Which of the following pairs of functions are not identical ?

(A) f(x) = x2 and g(x) = x2

(B) f(x) = sec (sec1 x) and g(x) = cosec (cosec1 x)

(C) f(x) = 1 2

2

cos x and g(x) = cos x

(D) f(x) = tan1 x + cot1 x & g(x) = 2

20. Which of following pairs of functions are identical(A) f(x) = e n x sec1

& g(x) = sec1 x(B) f(x) = tan (tan1 x) & g(x) = cot (cot1 x)(C) f(x) = sgn (x) & g(x) = sgn (sgn (x))(D) f(x) = cot2 x . cos2 x & g(x) = cot2 x cos2 x

21. The period of the function f(x) = sin4 3x + cos4 3x is(A) /6 (B) /3(C) /2 (D) /12

22. If f(x) is a polynomial function satisfying the condition f(x) . f(1/x) = f(x) + f(1/x) and f(2) =9 then(A) 2 f(4) = 3 f(6) (B) 14 f(1) = f(3)(C) 9 f(3) = 2 f(5) (D) f(10) = f(11)

23. D [ 1, 1] is the domain of the following functions, state which of them has the inverse.




(A) f(x) = x2 (B) g(x) = x3

(C) h(x) = sin 2x (D) k(x) = sin (x/2)

24. Let f : I R (where I is the set of positive integers) be a function defined by, f(x) = x ,then f is(A) one one (B) many one(C) onto (D) into

25. The domain of definition of the function, f(x) = 2 2

2 2 3

4 2 4 3tanlog[ tan ] x x

x x

x x

where [ ] denotes

the greatest integer function is n n

1

4

1

2, ; n I then

(A) n = 0 (B) n 4(C) n 4 (D) none of these

26. The period of the function,f(x) = x + a [x + b] + sin x + cos 2x + sin 3x + cos 4x + ...... + sin (2n 1) x + cos 2 nxfor every a, b R is(A) 2 (B) 4(C) 1 (D) 0

W I Read the following passage and answer the question from 27 to 30 :Mr. X is a teacher of mathematics. His students want to know the ages of his son's S

1 and S

2.

He told that their ages are 'a' and 'b' respectively such that f(x + y) � axy = f(x) + by2

Ry,x after some time students said that information is insufficient, please give moreinformation . Teacher says that f

(1) = 8 and f

(2) = 32 .

27. The age of S1 will be

(A) 4 (B) 8(C) 16 (D) 32

28. The age of S2 will be

(A) 4 (B) 8(C) 16 (D) 32

29. The function f(x) is(A) even (B) odd(C) neither even nor odd (D) periodic as well as odd

30. The function RR:f , then function will be(A) one one onto (B) one one into(C) many one onto (D) many one into

W II. Read the following passage and answer the questions from 31 to 35 :If domain of a function is non-symmetrical about x = 0, then it can not be categorized as evenor odd function. In certain cases domain of such a function can be extended to make thefunction even or odd.Let a function be defined on certain domain which is entirely non-negative (or non positive).Thedomain of f(x) can be extended to the set X = { x : x domain of f (x)} in two ways :




(i) Even extension : The even extension is obtained by defining a new function f(-x) for x X , such that f(-x) = f(x).

(ii) Odd extension : The odd extension is obtained by defining a new function f(-x) for x X , such that f(-x)=-f(x).

Example :If f(x) = sin3 x + 3x2 � x + 5, 0x1 .

then odd extension g(x) of f(x) is given by

g(x) = 3 2

3 2

f (x) sin x 3x x 5, 1 x 0

f ( x) [ sin x 3x x 5], 0 x 1

=

1x0,5xx3xsin

0x1,5xx3xsin23

23

Consider

4x3,ee

3x1,xsinx)x(f

xx

2

1 ,

0x,x1xnx2

x4,xtanx)x(f 22

and

2x1,xsinx

1x1,xtanx)x(f 23 .

31. Even extension of f1(x) is

(A)

1x3,xsinx

3x4,ee2

xx

(B)

1x3,xsinx

3x4,ee2

xx

(C)

1x3,xsinx

3x4,ee2

xx

(D) none of these

32. Odd extension of f1(x) is

(A)

1x3,xsinx

3x4,ee2

xx

(B)

1x3,xsinx

3x4,ee2

xx

(C)

1x3,xsinx

3x4,ee2

xx

(D) none of these

33. Even extension of f2(x) is

(A) 22x n x 1 x , 0 x

x tan x, x 4

(B)

22x n x 1 x , 0 x

x tan x, x 4

(C)

4x,xtanx

x0,x1xnx2 2(D) none of these




34. Odd extension of f2(x) is

(A) 22x n x 1 x , 0 x

x tan x, x 4

(B) 22x n x 1 x , 0 x

x tan x, x 4

(C) 22x n x 1 x , 0 x

x tan x, x 4

(D) none of these

35. Domain of f3(x) can be extended to make it

(A) both even and odd function (B) only odd function(C) only even function (D) neither even nor odd function




LEVEL -1 ANSWER

1.(i)

2

1,

3

1(ii) R x x n n I : ,l q (iii) )3,0[x

(iv) )3,2(x (v) 1 1

0, , 12 2

x

(vi) 2x,Nx:x1,4

3

4

1,0

(vii) 1k2xK2 but 1x where K is non-negative integer

(viii) FHGIKJ , ,

1

32 g (ix) 1 2 3, ,b g b g (x) ]6,4[x

2.(i) 2 2

,4 4

(ii) [0, 3] (iii) ]5,1[Range (iv)

2,

3 (v) 1,

3.(i) f fD : x R, R :[0, 2] (ii) Df : R � {1, 2}, R

f : R � {1, 3} (iii) f fD : R, R : cos1, 1

(iv) f

3 5D : x 2n , 2n (2n ) 2n x | x 2n or 2n , n I

2 2 4 4

f

2R : ,

3 3 2

(v) D

f : R � [�1, 0), Range )0xfor( : ,1

4.(i) ba

5bx

x

a

ba

122

(iii) f(x) = 2x2 (iv) x + 1 (v)

6x

(vi) 2n - 1

5.(i) (a) ,2 (b) ,11,0 (c) [-1, 1] (d) 2

nR

, where n is an integer. (e) R - {0}

6.(i)(a) 2 (b) 2 (c) 2

(d) 70 (e) n2 (iii) 2 7. (ii) many one into

8.(i)(a) ]1,0[x,1x4x4

1x4x42

2

(b) 1x0,)x1(

)x1(x82

(ii) f3n

(x) = x, Domain = R - {0, 1}

(iii) fog (x) =(1 x); 1 x 0;

x 1; 0 x 2;

gof(x) =

4x3;x5

3x2;1x

2x1;x3

1x0;1x




(iv) fog(x) = x; Range is ,

9.(i)(a)

)1,0(x,x1

x

)0,1(x,x1

x

(b)

x16,xlog

16x1,x

1x,x

2

(iv)2

aa)x(f

xx1

(v) a + d = 0

10. (i) (ii)

(iii)

(iv)

(v) (vi)

11. (i) even (ii) even (iii) even (iv) even (v) neither even nor odd

12.(i) f(x) = 3

xxsin

LEVEL -2

1. (i) ( , 2) [4, )x (ii) ),(x (iii) 4,2x

(iv)

In,

6n2x|x (v) {1} (vi) {-1/2}




(vii)

6

5n2,

6n24,

6

11

6

7x and x 2n , n I, n 0, 1

2

(viii) ,3 (ix) 3 2 0 2 2 3, , ,b g b (x) (-2, 2) - {-1, 0, 1}

2.(i) f fD :x R, R : ,2 4

(ii)

2(iii) {0} (iv) ,2 (v)

3,

4 4

3.(i) Df : [4, 6], R

f :

1,2

1(ii) f fD : , 1 2 1 1 2, ,R : 0 ,

2

(iii) f fD : x R, R :[0, 2] (iv) f fD : 1, 0 , R :{0} (v) f

n N

D : (2n , (2n 1) )

, fR : , 0

5.(ii) (a) 3, 15 (b) 2

6.(i) a [1, 4] (ii) (a) bijective (b) Injective but not surjective

(iv)

2,

2X , Y = c , c , where

a

2batan 1 , ab2ba 22

(v) Domain = 2, , Co-domain= ,

7.(ii) 122 10,1010,0 (iii) fog(x) =

2x)3(1tan;23xtan

)3(1tanx4

;1xtan

4x

2;1xtan

and

gof(x) =

21x

41;21xtan

41x1;1xtan

1x14

;2)1xtan(

14

x12

;)1xtan(

2

2(iv) fog(x) =

2/x0;xsec

0x2/;xcot22

2

8.(i) f�1(x) = 1xlog

xlog

2

2

, Domain : R+ � {0}, Range : R � {1} (ii)

e/eb,e

a 11

(iii) f -1(x) = [x] + {x}2, solution is all integer (iv)(b)

n

1ii 1x

n

1




9. (i)

(ii)

(iii)

(iv)

(v)




(vi)

(vii)

11. (i) x + y = 6, x + y = 0

SET - I1. (A) 2. (C) 3. (D) 4. (B) 5. (D) 6. (B) 7. (C)8. (B) 9. (C) 10. (D) 11. (B) 12. (B) 13. (B) 14. (D)15.(B) 16. (A) 17. (D) 18. (A) 19. (B) 20. (D) 21. (D)22. (D) 23. (C) 24. (B) 25. (A) 26. (B) 27. (B) 28. (B)29. (C) 30. (C)

SET - II

1. (B) 2. (C) 3. (B) 4. (D) 5. (A) 6. (B) 7. (C)8. (C) 9. (A) 10. (A) 11. (C) 12. (B) 13. (C) 14. (A)15.(D) 16. (A) 17. (A) 18. (D) 19. (D) 20. (B) 21. (C)22. (B) 23. (A) 24. (D) 25. (B) 26. (B) 27. (B) 28. (C)29. (A) 30. (D)

SET - III1. 3 2. 2x 3. 4. x = 2, y = 1.3, z = 3.1

5. Df : (�1, 1), R

f :

2

3,

26. n = 1 7. 5 8.

8

311,1 9.

10

10. both even and odd function11. (C) 12. (D) 13. (A) 14. (C) 15. (D) 16. (C) 17. (D)18. (ABD) 19. (AC) 20. (BCD) 21. (ABC) 22. (BC) 23. (BD) 24. (AD)25. (ABC) 26. (AB) 27. (C) 28. (B) 29. (A) 30. (D) 31. (A)32. (B) 33. (B) 34. (A) 35. (B)


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