Relations & Functions Practice Questions - DPP for JEE

 Page 1


PART-I (Single Correct MCQs)
1. Let R = {(x, y)  :  x, y  N and x
2 
– 4xy + 3y
2
 = 0}, where N is the set of
all natural numbers. Then the relation R is :
(a) reflexive but neither symmetric nor transitive.
(b) symmetric and transitive.
(c) reflexive and symmetric.
(d) reflexive and transitive.
2. Let . Then P is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Anti-symmetric
3. Let f : {x, y, z} ? {1, 2, 3} be a one-one mapping such that only one of
the following three statements is true and remaining two are false : f (x)
? 2, f (y) = 2, f (z) ? 1, then
(a) f (x) > f (y) > f (z)
(b) f (x) < f (y) < f (z)
Page 2


PART-I (Single Correct MCQs)
1. Let R = {(x, y)  :  x, y  N and x
2 
– 4xy + 3y
2
 = 0}, where N is the set of
all natural numbers. Then the relation R is :
(a) reflexive but neither symmetric nor transitive.
(b) symmetric and transitive.
(c) reflexive and symmetric.
(d) reflexive and transitive.
2. Let . Then P is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Anti-symmetric
3. Let f : {x, y, z} ? {1, 2, 3} be a one-one mapping such that only one of
the following three statements is true and remaining two are false : f (x)
? 2, f (y) = 2, f (z) ? 1, then
(a) f (x) > f (y) > f (z)
(b) f (x) < f (y) < f (z)
(c) f (y) < f (x) < f (z)
(d) f (y) < f (z) < f (x)
4. If f (x) =  and
g (x) = 
If domain of g (f (x)) is [–1, 4], then –
(a) a = 0, b > 5
(b) a = 2, b > 7
(c) a = 2, b > 10
(d) a = 0, b ? R
5. Let S be the set of all straight lines in a plane. A relation R is defined on
S by  then R is :
(a) reflexive but neither symmetric nor transitive
(b) symmetric but neither reflexive nor transitive
(c) transitive but neither reflexive nor symmetric
(d) an equivalence relation
6. Let R be a reflexive relation on a finite set A having n-elements, and let
there be m ordered pairs in R. Then
(a)
(b)
(c) m = n
(d) None of these
7. If f : R ? R, f (x) = , then f (x) is
(a) one to one and onto
(b) many to one and onto
(c) one to one and into
(d) many to one and into
Page 3


PART-I (Single Correct MCQs)
1. Let R = {(x, y)  :  x, y  N and x
2 
– 4xy + 3y
2
 = 0}, where N is the set of
all natural numbers. Then the relation R is :
(a) reflexive but neither symmetric nor transitive.
(b) symmetric and transitive.
(c) reflexive and symmetric.
(d) reflexive and transitive.
2. Let . Then P is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Anti-symmetric
3. Let f : {x, y, z} ? {1, 2, 3} be a one-one mapping such that only one of
the following three statements is true and remaining two are false : f (x)
? 2, f (y) = 2, f (z) ? 1, then
(a) f (x) > f (y) > f (z)
(b) f (x) < f (y) < f (z)
(c) f (y) < f (x) < f (z)
(d) f (y) < f (z) < f (x)
4. If f (x) =  and
g (x) = 
If domain of g (f (x)) is [–1, 4], then –
(a) a = 0, b > 5
(b) a = 2, b > 7
(c) a = 2, b > 10
(d) a = 0, b ? R
5. Let S be the set of all straight lines in a plane. A relation R is defined on
S by  then R is :
(a) reflexive but neither symmetric nor transitive
(b) symmetric but neither reflexive nor transitive
(c) transitive but neither reflexive nor symmetric
(d) an equivalence relation
6. Let R be a reflexive relation on a finite set A having n-elements, and let
there be m ordered pairs in R. Then
(a)
(b)
(c) m = n
(d) None of these
7. If f : R ? R, f (x) = , then f (x) is
(a) one to one and onto
(b) many to one and onto
(c) one to one and into
(d) many to one and into
8. If f : B ? A is defined by  and g : A ? B is defined by 
, where A = R –  and
B = R –  and I
A
 is an identity function on A and I
B
 is identity function on
B, then
(a) fog = I
A
 and gof = I
A
(b) fog = I
A
 and gof = I
B
(c) fog = I
B
 and gof = I
B
(d) fog = I
B
 and gof = I
A
9. Let f (x) = [x]
2
 + [x + 1] – 3 where [x] = the greatest integer function.
Then
(a) f (x) is a many-one and into function
(b) f (x) = 0 for infinite number of values of x
(c) f (x) = 0 for only two real values
(d) Both (a) and (b)
10. f (x) = | x – 1 |, f : R
+
 ? R and g (x) = e
x
, g : [–1, 8) ? R. If the
function fog (x) is defined, then its domain and range respectively
are
(a) (0, 8) and [0, 8)
(b) [–1, 8) and [0, 8)
(c) [–1, 8) and 
(d) [–1, 8) and 
11. If X = {x
1
, x
2
, x
3
} and Y = {x
1
, x
2
, x
3
,x
4
,x
5
} then find which is a
reflexive relation of the following ?
(a) R
1 
: {(x
1
, x
1
), (x
2
, x
2
)}
(b) R
1
 : {(x
1
, x
1
), (x
2
, x
2
), (x
3
, x
3
)}
(c) R
3
 : {(x
1
, x
1
), (x
2
, x
2
), (x
1
, x
3
),(x
2
, x
4
)}
Page 4


PART-I (Single Correct MCQs)
1. Let R = {(x, y)  :  x, y  N and x
2 
– 4xy + 3y
2
 = 0}, where N is the set of
all natural numbers. Then the relation R is :
(a) reflexive but neither symmetric nor transitive.
(b) symmetric and transitive.
(c) reflexive and symmetric.
(d) reflexive and transitive.
2. Let . Then P is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Anti-symmetric
3. Let f : {x, y, z} ? {1, 2, 3} be a one-one mapping such that only one of
the following three statements is true and remaining two are false : f (x)
? 2, f (y) = 2, f (z) ? 1, then
(a) f (x) > f (y) > f (z)
(b) f (x) < f (y) < f (z)
(c) f (y) < f (x) < f (z)
(d) f (y) < f (z) < f (x)
4. If f (x) =  and
g (x) = 
If domain of g (f (x)) is [–1, 4], then –
(a) a = 0, b > 5
(b) a = 2, b > 7
(c) a = 2, b > 10
(d) a = 0, b ? R
5. Let S be the set of all straight lines in a plane. A relation R is defined on
S by  then R is :
(a) reflexive but neither symmetric nor transitive
(b) symmetric but neither reflexive nor transitive
(c) transitive but neither reflexive nor symmetric
(d) an equivalence relation
6. Let R be a reflexive relation on a finite set A having n-elements, and let
there be m ordered pairs in R. Then
(a)
(b)
(c) m = n
(d) None of these
7. If f : R ? R, f (x) = , then f (x) is
(a) one to one and onto
(b) many to one and onto
(c) one to one and into
(d) many to one and into
8. If f : B ? A is defined by  and g : A ? B is defined by 
, where A = R –  and
B = R –  and I
A
 is an identity function on A and I
B
 is identity function on
B, then
(a) fog = I
A
 and gof = I
A
(b) fog = I
A
 and gof = I
B
(c) fog = I
B
 and gof = I
B
(d) fog = I
B
 and gof = I
A
9. Let f (x) = [x]
2
 + [x + 1] – 3 where [x] = the greatest integer function.
Then
(a) f (x) is a many-one and into function
(b) f (x) = 0 for infinite number of values of x
(c) f (x) = 0 for only two real values
(d) Both (a) and (b)
10. f (x) = | x – 1 |, f : R
+
 ? R and g (x) = e
x
, g : [–1, 8) ? R. If the
function fog (x) is defined, then its domain and range respectively
are
(a) (0, 8) and [0, 8)
(b) [–1, 8) and [0, 8)
(c) [–1, 8) and 
(d) [–1, 8) and 
11. If X = {x
1
, x
2
, x
3
} and Y = {x
1
, x
2
, x
3
,x
4
,x
5
} then find which is a
reflexive relation of the following ?
(a) R
1 
: {(x
1
, x
1
), (x
2
, x
2
)}
(b) R
1
 : {(x
1
, x
1
), (x
2
, x
2
), (x
3
, x
3
)}
(c) R
3
 : {(x
1
, x
1
), (x
2
, x
2
), (x
1
, x
3
),(x
2
, x
4
)}
(d) R
3
 : {(x
1
, x
1
), (x
2
, x
2
),(x
3
, x
3
),(x
4
, x
4
)}
12. Let R = {(1, 3), (2, 2), (3, 2)} and S = {(2, 1), (3, 2), (2, 3)} be two
relations on set A = {1, 2, 3}. Then RoS =
(a) {(1, 3), (2, 2), (3, 2), (2, 1), (2, 3)}
(b) {(3, 2), (1, 3)}
(c) {(2, 3), (3, 2), (2, 2)}
(d) {(2, 3), (3, 2)}
13. If g(f (x)) = | sin x| and f(g(x)) = (sin )
2
, then
(a) f(x) = sin
2
 x, g(x) = 
(b) f(x) = sin x, g(x) = | x |
(c) f(x) = x
2
, g(x) = sin 
(d) f and g cannot be determined.
14. Let f : be a function defined by , where ,
then
(a) f is one-one onto
(b) f is one-one into
(c) f is many-one onto
(d) f is many-one into
15. If , then  is equal to :
(a)
(b)
(c)
(d) x
16. Which of the function defined below is one-one?
(a) , f (x) = x
2
 – 4x + 3
Page 5


PART-I (Single Correct MCQs)
1. Let R = {(x, y)  :  x, y  N and x
2 
– 4xy + 3y
2
 = 0}, where N is the set of
all natural numbers. Then the relation R is :
(a) reflexive but neither symmetric nor transitive.
(b) symmetric and transitive.
(c) reflexive and symmetric.
(d) reflexive and transitive.
2. Let . Then P is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Anti-symmetric
3. Let f : {x, y, z} ? {1, 2, 3} be a one-one mapping such that only one of
the following three statements is true and remaining two are false : f (x)
? 2, f (y) = 2, f (z) ? 1, then
(a) f (x) > f (y) > f (z)
(b) f (x) < f (y) < f (z)
(c) f (y) < f (x) < f (z)
(d) f (y) < f (z) < f (x)
4. If f (x) =  and
g (x) = 
If domain of g (f (x)) is [–1, 4], then –
(a) a = 0, b > 5
(b) a = 2, b > 7
(c) a = 2, b > 10
(d) a = 0, b ? R
5. Let S be the set of all straight lines in a plane. A relation R is defined on
S by  then R is :
(a) reflexive but neither symmetric nor transitive
(b) symmetric but neither reflexive nor transitive
(c) transitive but neither reflexive nor symmetric
(d) an equivalence relation
6. Let R be a reflexive relation on a finite set A having n-elements, and let
there be m ordered pairs in R. Then
(a)
(b)
(c) m = n
(d) None of these
7. If f : R ? R, f (x) = , then f (x) is
(a) one to one and onto
(b) many to one and onto
(c) one to one and into
(d) many to one and into
8. If f : B ? A is defined by  and g : A ? B is defined by 
, where A = R –  and
B = R –  and I
A
 is an identity function on A and I
B
 is identity function on
B, then
(a) fog = I
A
 and gof = I
A
(b) fog = I
A
 and gof = I
B
(c) fog = I
B
 and gof = I
B
(d) fog = I
B
 and gof = I
A
9. Let f (x) = [x]
2
 + [x + 1] – 3 where [x] = the greatest integer function.
Then
(a) f (x) is a many-one and into function
(b) f (x) = 0 for infinite number of values of x
(c) f (x) = 0 for only two real values
(d) Both (a) and (b)
10. f (x) = | x – 1 |, f : R
+
 ? R and g (x) = e
x
, g : [–1, 8) ? R. If the
function fog (x) is defined, then its domain and range respectively
are
(a) (0, 8) and [0, 8)
(b) [–1, 8) and [0, 8)
(c) [–1, 8) and 
(d) [–1, 8) and 
11. If X = {x
1
, x
2
, x
3
} and Y = {x
1
, x
2
, x
3
,x
4
,x
5
} then find which is a
reflexive relation of the following ?
(a) R
1 
: {(x
1
, x
1
), (x
2
, x
2
)}
(b) R
1
 : {(x
1
, x
1
), (x
2
, x
2
), (x
3
, x
3
)}
(c) R
3
 : {(x
1
, x
1
), (x
2
, x
2
), (x
1
, x
3
),(x
2
, x
4
)}
(d) R
3
 : {(x
1
, x
1
), (x
2
, x
2
),(x
3
, x
3
),(x
4
, x
4
)}
12. Let R = {(1, 3), (2, 2), (3, 2)} and S = {(2, 1), (3, 2), (2, 3)} be two
relations on set A = {1, 2, 3}. Then RoS =
(a) {(1, 3), (2, 2), (3, 2), (2, 1), (2, 3)}
(b) {(3, 2), (1, 3)}
(c) {(2, 3), (3, 2), (2, 2)}
(d) {(2, 3), (3, 2)}
13. If g(f (x)) = | sin x| and f(g(x)) = (sin )
2
, then
(a) f(x) = sin
2
 x, g(x) = 
(b) f(x) = sin x, g(x) = | x |
(c) f(x) = x
2
, g(x) = sin 
(d) f and g cannot be determined.
14. Let f : be a function defined by , where ,
then
(a) f is one-one onto
(b) f is one-one into
(c) f is many-one onto
(d) f is many-one into
15. If , then  is equal to :
(a)
(b)
(c)
(d) x
16. Which of the function defined below is one-one?
(a) , f (x) = x
2
 – 4x + 3
(b) , f (x) = x
2
 + 4x – 5
(c) f : R ? R , f (x) = 
(d) f : R ? R, 
17. The inverse of f (x) =   is
(a) log
10
 
(b) log
10
 
(c) log
10 
(d) log
10
 
18. A function f  from the set of  natural numbers to integers defined by 
 is
(a)  neither one-one nor onto (b) one-one but not onto
(c)  onto but not one-one (d) one-one and onto both
19. If f(x) = sin
2
x + sin
2
 + cos x cos  andg = 1, then
go f(x) =
(a) 1
(b) 0
(c) sin x
(d) None of these