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RELATIONS AND FUNCTIONS
JEE (Main/Advance) MATHS
Stage – I
Q.1) The binary operation * : R × R → R is defined as 𝑎 ∗ 𝑏 = 2𝑎 + 𝑏. Find (2 ∗ 3) ∗ 4
Q.2) Let * be a binary operation on set of integers I, defined by 𝑎 ∗ 𝑏 = 2𝑎 + 𝑏 − 3. Find the value
of (3 ∗ 4)
Q.3) State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be
transitive
Q.4) Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B.
State whether f is one-one or not.
Q.5) Give an example to show that the relation R in the set of natural numbers, defined by
R = {(x, y), x, y € N, x ≤ y 2 } is not transitive.
Q.6) Write the number of all one-one functions from the set A = {a, b, c} to itself.
Q.7) If the function f : R → R, defined by f(x) = 3x-4 is invertible find 𝑓 −1 .
Stage – II
Q.8) If f : R → R is defined by f(x) = 3x+2, find f(f(x)).
Q.9) What is the range of the function f(x)=
|x−1|
(x−1)
?
Q.10) If f : R → R is defined by f(x) = (3 − x 3 )1/3 , then find the fof(x).
Q.11) Let * be a binary operation on set Q of rational numbers defined as 𝑎 ∗ 𝑏 =
𝑎𝑏
5
. Write the
identity for *, if any.
Q.12) If f : R → R is defined by f(x) =
3𝑥+5
2
is an invertible function, find 𝑓 −1 .
Q.13) Let * be a operation on N given by a ∗ b = H. C. F. (a, b), a, b € N. Write the value of 22 ∗ 4.
Q.14) If f(x) = x+7 and g(x) = x-7, x € R, find (fog) (7).
Classes for JEE(Main & Advanced) / PMT / NEET & NTSE / NSTSE / OLYMPIAD and FOUNDATION batches in
PHYSICS ,CHEMISTRY & MATHEMATICS for 9th to 12th CBSE & MP Board students
Address – Agnihotri Engg. & GATE Classes (AEGC) , Sherpura Vidisha 07592 – 408822 , 7415712500/4
Stage – III
x−2
Q.15) Let A=R-{3} and B=R-{1}. Consider the function f : A → B defined by f(x)= (
). Show
x−3
that f is one-one and onto and hence find 𝑓 −1 .
x + 1, if x is odd
Q.16) Show that f : N → N, given by f(x)= {
is both one-one and onto.
x − 1, if x is even
Q.17) Consider the binary operation * : R × R → R and o : R × R → R defined as 𝑎 ∗ 𝑏 = |𝑎 − 𝑏| and
a o b = a for all a, b € R. Show that ‘*’ is commutative but not associative, ‘o’ is associative but not
commutative.
Q.18) If f : R → R be the function defined by f(x) = 4x 3 +7, show that f is a bijection.
n + 1, if n is even
Q.19) Show that f : W → W, given by f(n)= {
, is bijective function.
n − 1, if n is odd
Q.20) Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a ∗ b = min. {a, b}. Write
the operation table of the operation.
Q.21) Let f : R → R be defined as f(x) = 10x+7. Find the function g : R → R such that gof = fog = IR
Q.22) A binary operation * on the set {0, 1, 2, 3, 4, 5} is defined as :
a + b, if a + b < 6
a∗b= {
a + b − 6, if a + b ≥ 6
Show that zero is the identity for this operation and each element ‘a’ of the set is invertible with 6-a,
being the inverse of ‘a’.
Q.23) Let N be the set of all natural numbers and R be the relation in N × N defined by (a, b) R (c, d)
if ad = bc. Show that R is an equivalence relation.
Q.24) Show that the function f : R → R be defined by f(x) = 2x 3 − 7, for x € R is bijective.
Q.25) Show that the relation R in the set A = {x : x € Z, 0 ≤ x ≤12} given by
R = {(a, b) : |a − b| is divisible by 4}is an equivalence relation. Find the set of all elements related
to 1.
Q.26) Show that the function f : R → R given f(x) = ax+b, where a, b € R, a ≠ 0 is a bijection.
Q.27) Let f : X → Y be a function. Define a relation R on X given by R = {(a, b) : f(a) = f(b)}. Show
that R is an equivalence relation on X.
Stage – IV
Q.28) Let Z be the set of all integers and R be the relation on Z defined as R = {(a, b) : a, b € Z, and
(a-b) is divisible by n}. Prove that R is an equivalence relation.
Q.29) Show that the relation S defined on the set N × N by (a, b) S (c, d) => a+d = b+c is an
equivalence relation.
Classes for JEE(Main & Advanced) / PMT / NEET & NTSE / NSTSE / OLYMPIAD and FOUNDATION batches in
PHYSICS ,CHEMISTRY & MATHEMATICS for 9th to 12th CBSE & MP Board students
Address – Agnihotri Engg. & GATE Classes (AEGC) , Sherpura Vidisha 07592 – 408822 , 7415712500/4
Q.30) Let * be a binary operation on Q defined by ∗ 𝑏 =
3𝑎𝑏
5
. Show that * is commutative as well
as associative. Also find its identity element, if it exists.
Q.31) Show that the relation S in the set R of real numbers, defined as
S = {(a, b) : a, b € R and a ≤ b3 } is neither reflexive, nor symmetric nor transitive.
Q.32) If the function f : R → R is given by f(x) =
𝑥+3
find i) fog and ii) gof. Is 𝑓 −1 = g ?
2
and g : R → R is given by g(x) = 2x-3,
Q.33) If the function f : R → R is given by f(x) = x 2 + 3x + 1 and g : R → R is given by g(x) = 2x-3
find i) fog and ii) gof.
Q.34) Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |𝑎 − 𝑏| is even}, is
an equivalence relation.
n+1
, if n is odd
Q.35) Let f : N → N be defined by f(n) = { n2
∀𝑛 ∈ 𝑁 . Find whether the function f is
, if n is even
2
bijective.
Q.36) (i) Is the binary operation *, defined on set N, given by a ∗ b =
a+b
2
, for all a, b ∈ 𝑁,
commutative?
(ii) Is the above binary operation * associative?
Questions asked in JEE and other competitive exams
Q.37) Find domain of 2𝑥 + 2𝑦 = 2
a) (-∞, 1)
JEE 2001
b) (1, ∞)
c) (-1, 1)
d) none
Q.38) q(x) = 1 + {x}, f(x) = sgn(x) where {x}=fractional part function. Find f{g(x)} ∀ x
a) 2
b) 1
c) 0
d) ½
Q.39) a R b iff (a+b) is even integer is
JEE 2000
a) Reflexive & symmetric
b) Reflexive & transitive
c) Symmetric & transitive
d) Equivalence relation
Q.40) Let f(x) = (𝑥 + 1)2 where x ≥ -1. If g(x) is a function such that its graph is the relation of the
graph of f(x) with respect to y =x then g(x) = ?
a) −√𝑥 − 1, x ≥ 0
b)
Q.41) If f : R+ → R and f(x) = x−
a) Injective
1
x
1
(𝑥+1)2
, x > -1
c) √𝑥 + 1, x ≥ -1
then f(x) is
b) Bijective
c) Disjunctive
d) none
c) Disjunctive
d) none
Q.42) If f : R+ → R and f(x) = x 2 then f(x) is
a) Injective
b) Bijective
d) √𝑥 − 1, x ≥ 0
Q.43) If f : R+ → R and f(x) = x 3 then f(x) is
a) Injective
b) Bijective
c) Disjunctive
d) none
Classes for JEE(Main & Advanced) / PMT / NEET & NTSE / NSTSE / OLYMPIAD and FOUNDATION batches in
PHYSICS ,CHEMISTRY & MATHEMATICS for 9th to 12th CBSE & MP Board students
Address – Agnihotri Engg. & GATE Classes (AEGC) , Sherpura Vidisha 07592 – 408822 , 7415712500/4