( a ) reflexive and transitive. ( b ) reflexive only. ( c ) an equivalence relation ( d )
reflexive and symmetric only. [ AIEEE 2005 ]. ( 2 ) Let f : ( - 1, 1 ) → B be a function
...
01 - SETS, RELATIONS AND FUNCTIONS
Page 1
( Answers at the end of all questions )
Let R = { ( 3, 3 ) ( 6, 6 ) ( ( 9, 9 ) ( 12, 12 ), ( 6, 12 ) ( 3, 9 ) ( 3, 12 ), ( 3, 6 ) } be a relation on the set A = { 3, 6, 9, 12 }. The relation is
(2)
( b ) reflexive only ( d ) reflexive and symmetric only
Let f : ( - 1, 1 ) → B be a function defined by f ( x ) = one-one and onto when B is the interval π ( a ) 0, 2
π π - 2 , 2
ra
) + f(a - x)
1 - x2
, then f is both
π π ( d ) - , 2 2
(d) f(-x)
( c ) not symmetric
.e
( b ) transitive
The ran e o the function f ( x ) = 7 - x Px - 3
w w w
(7)
2x
[ AIEEE 2005 ]
[ AIEEE 2005 ]
Let R = { ( 1, 3 ), ( 4, 2 ) ( 2, 4 ), ( 2, 3 ), ( 3, 1 ) } be a relation on the set A = { 1, 2, 3, 4 }. The relation R is
( a ) { 1, 2 3 } c ) { 1, 2, 3, 4 }
(6)
(c) f(
xa
(b) f(x)
( a ) a function
(5)
(c)
tan - 1
If a real valued function f ( x ) satisfies the fun tional equation f ( x - y ) = f ( x ) f ( y ) - f ( a - x ) f ( a + y ), whe e ‘a’ is a given constant and f ( 0 ) = 1, then f ( 2a - x ) is equal to (a) -f(x)
(4)
π 0, 2
m
(3)
(b)
[ AIEEE 2005 ]
om
( a ) reflexive and transitive ( c ) an equivalence relation
ce .c
(1)
( b ) [ - 1, 1 ]
[ AIEEE 2004 ]
is
( b ) { 1, 2, 3, 4, 5, 6 } ( d ) { 1, 2, 3, 4, 5 }
If f : R → S, defined by f( x ) = sin x ( a ) [ 0, 3 ]
( d ) reflexive
[ AIEEE 2004 ]
3 cos x + 1 is onto, then the interval of S is
( c ) [ 0, 1 ]
( d ) [ - 1, 3 ]
[ AIEEE 2004 ]
The graph of the function f ( x ) is symmetrical about the line x =2, then (a) f(x + 2) = f(x - 2) (c) f(x) = f(-x)
(b) f(2 + x) = f(2 - x) (d) f(x) = -f(-x)
[ AIEEE 2004 ]
01 - SETS, RELATIONS AND FUNCTIONS
Page 2
( Answers at the end of all questions ) (8)
sin -1 ( x - 3 )
The domain of the function f ( x ) =
is
9 - x2
If f : { 1, 2, 3, …. }
f(x) =
( c ) [ 1, 2 ]
→ { 0, ± 1, ± 2, ….. } is defined by
x if x is even 2 , - ( x - 1 ) , if x is odd 2
( a ) 100
( d ) [ 1, 2 )
( b ) 199
then value of f - ( - 100 ) is 1
( c ) 200
( d ) 201
+ log 10 ( x - x ) is
m
xa
.e
4 - x
[ AIEEE 2003 ]
3
2
( b ) ( - 1, 0 ) ∪ ( 1, 2 ) ( d ) ( - 1 0 ) ∪ ( 1, 2 ) ∪ ( 2, ∞ )
( 11 ) The function f ( x ) = log ( x + ( a ) even function ( c ) periodic function
3
ra
( 10 ) Domain of definition of the function f ( x ) =
( a ) ( 1, 2 ) ( c ) ( 1, 2 ) ∪ ( 2, ∞ )
[ AIEEE 2004 ]
om
(9)
( b ) [ 2, 3 )
ce .c
( a ) [ 2, 3 ]
[ AIEEE 2003 ]
x 2 + 1 ) is a / an
b ) odd function ( d ) none of these
[ AIEEE 2003 ]
w w
( 12 ) The functi n f : R → R defined by f ( x ) = sin x is
w
( a ) into
13
( b ) onto
( c ) one-one
The range of the function f ( x ) = (a) R
( 14 ) If f ( x ) =
(b) R - {-1}
x, 0,
x ∈Q x ∉Q
( a ) one-one, onto ( c ) one-one but not onto
( d ) many-one
2 + x , x ≠ 2 is 2 - x
(c) R - {1}
and
[ AIEEE 2002 ]
g(x) =
0, x,
(d) R - {2}
x ∈Q , x ∉Q
[ AIEEE 2002 ]
then ( f - g ) is
( b ) neither one-one nor onto ( d ) onto but not one-one
[ IIT 2005 ]
01 - SETS, RELATIONS AND FUNCTIONS
Page 3
( Answers at the end of all questions )
( 16 )
π π - 4, 4
(b)
x2 + x + 2
The range of the function f ( x ) =
( a ) [ 1, ∞ ) ( 17 ) f : [ 0, ∞ )
( b ) ( 1,
→
[ 0, ∞ ),
x2 + x + 1
11 ) 7
( c ) ( 1,
f(x) =
( a ) one-one and onto ( c ) onto but not one-one
π 0, 2
(c)
,
(d)
x ∈ (-
7 ] 3
π - 2, 0
x 1+ x
( d ) [ 1,
7 ] 5
[ IIT 2003 ]
is
( b ) one-one bu no onto ( d ) neither one one nor onto
2
[ I T 2004 ]
∞ , ∞ ) is
ce .c
( a ) [ 0, π ]
- 1, then g [ f ( x ) ] will be invertible for the
om
2
If f ( x ) = sin x + cos x and g ( x ) = x domain
[ IIT 2003 ]
ra
( 15 )
x + 1,
(b
x ≥ -1
(d)
1
( x + 1 )2
,
x - 1,
x > -1 x ≥ 0
[ IIT 2002 ]
.e
(c)
x ≥ 0
x - 1,
xa
(a) -
m
( 18 ) If f ( x ) = ( x + 1 ) for x ≥ and g ( x ) is the function whose graph is reflection of the graph of f ( x ) with respect o the line y = x, then g ( x ) equals
( 19 ) If function f : R → R is defined as f ( x ) = 2x + sin x for x ∈ R, then f is
w w
( a ) one on and onto ( b ) one-one but not onto ( c ) ont bu not one-one ( d ) neither one-one nor onto
w
- 1, x < 0 ( 20 ) If g ( x ) = 1 + x - [ x ] and f ( x ) = 0, x = 0 , 1, x > 0 (a) x
(b) 1
(c) f(x)
x +
x2 - 4 2
(b)
x 1 + x2
then for all x, f [ g ( x ) ] =
(d) g(x)
( 21 ) If f : [ 1, ∞ ) → [ 2, ∞ ) is given by f ( x ) = x +
(a)
[ IIT 2002 ]
(c)
x -
[ IIT 2001 ]
1 1 , then f - ( x ) equals x
x2 - 4 2
(d) 1 +
x2 - 4
[ IIT 2001 ]
01 - SETS, RELATIONS AND FUNCTIONS
Page 4
( Answers at the end of all questions )
x 2 + 3x + 2
( b ) ( - 2, ∞ ) ( d ) ( - 3, ∞ ) - { - 1, - 2 }
( a ) R - { - 1, - 2 } ( c ) R - { - 1, - 2, - 3 }
is
[ IIT 2001 ]
om
log 2 ( x + 3 )
( 22 ) The domain of definition of f ( x ) =
( 23 ) If E = { 1, 2, 3, 4 } and F = { 1, 2 }, then the number of onto func ions ( b ) 16
(d) 8
[ IIT 2001 ]
αx , x ≠ - 1, then for which value of α is f [ f ( x ) ] = x ? x + 1
( 24 ) If f ( x ) =
(b) -
2
2
(d) -
(c) 1
[ IIT 2001 ]
ra
(a)
( c ) 12
ce .c
( a ) 14
om E to F is
( 25 ) Let f : R → R be any function. Define g : R → R by g ( x ) = l f ( x ) l for all x. Then g is ( b one-one if f is one-one ( d ) differentiable if f is differentiable
[ IIT 2000 ]
xa
m
( a ) onto if f is onto ( c ) continuous if f is continuous
x
y
( 26 ) The domain of definition of the function y ( x ) as given by the equation 2 + 2 = 2 is (b) 0 ≤ x ≤ 1
(c) -∞ < x ≤ 0
(d) -∞ < x < 1
.e
(a) 0 < x ≤ 1
( 27 ) If the function f : [ 1, ∞ ) → [ 1, ∞ ) is defined by f ( x ) = 2
w w
(
1 2
(c)
1 (1 + 2
- 1)
(b) 1 - 4 log 2 x )
1 (1 + 2
, then f - ( x ) is 1
1 + 4 log 2 x )
( d ) not defined
[ IIT 1999 ]
w
(a)
x(x - 1)
[ IIT 2000 ]
( 28 ) In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The number of newspapers is ( a ) at least 30
( 29 ) If f ( x ) =
( b ) at most 20
x2 - 1 x2 + 1
( c ) exactly 25
( d ) none of these
[ IIT 1998 ]
, for every real number x, then the minimum value of f
( a ) does not exist as f is unbounded ( c ) is not attained even though f is bounded
( b ) is equal to 1 ( d ) is equal to - 1
[ IIT 1998 ]
01 - SETS, RELATIONS AND FUNCTIONS
Page 5
( Answers at the end of all questions ) ( 30 ) If f ( x ) = 3x - 5, then f - ( x ) 1
2
( a ) f ( x ) = sin x,
( c ) f ( x ) = x , g ( x ) = sin
x
g(x) = lxl
( d ) f and g cann t be determined
[ IIT 1998 ]
- 1, ( x ≥ - 1 ), then the se S = { x : f ( x ) = f – 1( x ) } is
-3 + i 0, - 1, 2 ( c ) { 0, - 1 }
3
,
-3 - i 2
3
( b ) { 0, 1, - 1 } ( d ) empty
[ IIT 1995 ]
xa
(a)
ra
2
2
x ) , then
( b ) f ( x ) = sin x
m
( 32 ) If f ( x ) = ( x + 1 )
x
g(x) =
2
[ IIT 1998 ]
ce .c
( 31 ) If g [ f ( x ) ] = l sin x l and f [ g ( x ) ] = ( sin
x+5 3
om
1 ( b ) is given by 3x - 5 ( c ) does not exist because f is not one-one ( d ) does not exist because f is not onto ( a ) is given by
( 33 ) The number log2 7 is
w w
.e
( a ) an integer ( c ) an irrationa number
( b ) a rational number ( d ) a prime number
w
( 34 ) If S is the set of all real x such that
( 35 )
2x - 1 2x 3 + 3x 2 + x
3 (a) -∞, - 2
1 3 (b) - , - 4 2
1 (d) , 3 2
( e ) none of these
If y = f ( x ) =
[ IIT 1990 ]
is positive, then S contains
1 1 (c) - , - 2 4 [ IIT 1986 ]
x + 2 , then x - 1
(a) x = f(y) (b) f(1) = 3 ( d ) f is a rational function of x
( c ) y increases with x for x < 1 [ IIT 1984 ]
01 - SETS, RELATIONS AND FUNCTIONS
Page 6
( Answers at the end of all questions )
( 36 ) Let f ( x ) = l x - 1 l. Then 2
(b) f(x + y) = f(x) + f(y) ( d ) None of these
( a ) ( - 3, - 2 ) excluding - 2.5 ( c ) [ - 2, 1 ] excluding 0
1 + log10 ( 1 - x )
x + 2
s
ce .c
( 37 ) The domain of definition of the function y =
[ IT 1983 ]
om
2
(a) f(x ) = [f(x)] (c) f(lxl) = lf(x)l
( b ) [ 0, 1 ] excluding 0.5 ( d ) None of these
[ IIT 1983 ]
( 38 ) Which of the following functions is periodic ?
xa
m
ra
( a ) f ( x ) = x - [ x ] where [ x ] denotes the largest integer less than or equal to the real number x 1 ( b ) f ( x ) = sin for x ≠ 0, f ( 0 ) = 0 x ( c ) f ( x ) = x cos x ( d ) None of these [ IIT 1983 ]
( 39 ) If X and Y are two se s, then X ∩ ( X ∪ Y ) c) φ
(b) Y
equals
( d ) none of these
[ IIT 1979 ]
.e
(a) X
c
2
w w
( 40 ) Let R be the set of real numbers. If f : R → R is a function defined by f ( x ) = x , then f i ( b ) subjective but not injective ( d ) none of these
w
( a ) inject ve but not subjective ( ) bijective
[ IIT 1979 ]
Answers
1 a
2 d
3 a
4 c
5 a
6 d
7 b
8 b
9 d
10 b
11 a
12 d
13 b
14 a
15 b
16 c
17 b
18 d
19 a
20 b
21 a
22 d
23 a
24 d
25 c
26 d
27 b
28 c
29 d
30 b
31 a
32 c
33 c
34 a,d
35 a,d
36 d
37 c
38 a
39 c
40 d
