JEE-Main - Grade 12 - Mathematics - Continuity and Differentiability - Introduction and Snapshot of Continuity and Differentiability - Study ref# 911

Introduction
We had learnt to differentiate certain
functions like polynomial functions and trigonometric functions. In this
chapter, we introduce the very important concepts of continuity,
differentiability and relations between them. We... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Introduction and Snapshot of Relations and Functions - Study ref# 899

Introduction
The concept of the term ‘relation’ in
mathematics has been drawn from the meaning of relation in English language,
according to which two objects or quantities are related if there is a
recognizable connection or link between... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Introduction and Snapshot of Relations and Functions - Study ref# 900

Following sections must be covered to
understand the Relations and Functions:
1. Types of Relations
2. Types of Functions
3. Composition of Functions and
Invertible Function
4. Binary Operations... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Target Setup for Relations and Functions - Study ref# 901

Student requires to achieve the following milestones to get better grip over Relation and Function. The milestones / targets are as follows: Starting Point: Days 0Test TypeAppear on TestTarget TestsScorePractice Test000%Chapter T... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Importance and Snapshot of Relations and Functions - Study ref# 605

Introduction:
Much of the mathematics is finding the pattern. In our
daily life, we come across patterns that characterizes relations such as
sister, brother, father, son. In mathematics also, we come across relations
such m number i... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Importance and Snapshot of Relations and Functions - Study ref# 606

Following sections must be covered to understand the
Relations and Functions:
1. Cartesians Products of Sets
2. Relations
3. Functions... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Target Setup for Relations and Functions - Study ref# 607

Student requires to achieve the following milestones to get better grip over Relations and Functions. The milestones / targets are as follows:-Starting Point: Days 0Test TypeAppear on TestTarget TestsScorePractice Test000%Chapter Test... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2390

State whether each of the following statements are true or
false. If the statement is false, rewrite the given statement correctly.
(i) If P={m,n} and Q={n,m} ,then P×Q={(m,n),(n,m)}.
(ii)If A and B are non-empty sets, then A×B is a ... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2391

If A={-1,1}, find A×A×A Answer: Given, A={-1,1}
A×A={-1,1}×{-1,1}
={(-1,-1),(-1,1),(1,-1),(1,1)}
Again , A×A×A={(-1,-1),(-1,1),(1,-1),(1,1)} ×{-1,1}
={(-1,... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2392

Let A={1,2},B={1,2,3,4},C={5,6} and D={5,6,7,8}.Verify
that (i)A×(B∩C)=(A×B)∩(A×C) (ii)A×C is a
subset of B×D. Answer: Given A={1,2,}, B={1,2,3,4},C={5,6} and D={5,6,7,8}
(i) B∩C={1... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2393

The cartesian product A×A
has 9 elements among which are found (-1,0) and (0,1). Find the set A and the
remaining elements of A×.A Answer: Given, (-1,0)ϵ
A×A and (0,1)∈ A×A
A={-1,0,1}
... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2394

Define a relation R on the set N of natural numbers by R=
{(x, y): y=x+5, x is a natural number less than 4, x, y∈ N}. Depict this
relationship using roster form.
Write down the domain and the range. Answer: Since, x is a natural nu... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2395

The figure shows the relationship between the sets P and
Q.
Write this relation
(i)in set builder form (ii)in roster form
What are its domain and range?
P Q
... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2396

Which of the following relations are functions? Give
reasons. If it is a function, determine its domain and range.
(i) {(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)}.
(ii){(2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7)}
(iii){(1,3),(1,5),(2,5)} A... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2397

Find the domain and range of the following real
functions:
(i) (ii) Answer: (i)
Here, f(x)≤0, ∀x∈
R
Domain = The set of all points where the function is defined R.
Range
= The set... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2398

Find the range of each of the following functions:
(i)f(x)=2-3x, x ∈R,
x>0
(ii) f(x)=x2 + 2 , x is a real number.
(iii)f(x)=x , x is a real number. Answer: (i) Let y= f(x)=2-3x
y=2-3x
3x=2-y ... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2399

The relation f is defined by f(x)=
The relation g is defined by g(x)=
Show that f is a function and g is not a function Answer: (a) Giv... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2400

Let f= be
a function from R to R.
Determine the range of f. Answer: Given, f= and
f:R→R
Let y=
f(x)
or y is positive for all values of x.
and ... read more

JEE-Main - Grade 11 - Mathematics - Relations and Functions - Study ref# 2401

Redefine the function f(x)= Answer:
and
... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9346

Show that the relation R in the set R of
real numbers, defined as is
neither reflexive nor symmetric nor transitive.
Note - Here, the result is disproved
using some specific examples. In order to prove a result, we must prove it in
ge... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9347

Check whether the relation R defined in
the set {1,2,3,4,5,6} as is
reflexive, symmetric or transitive. Answer: Let A = {1,2,3,4,5,6}
A relation R is defined on set A is
. Therefore ,... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9348

Check whether the relation R in R defined
by is reflexive,
symmetric or transitive. Answer: Given that
It is observed that as
So, R is not reflexive.
Now,&nb... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9349

Show that the relation R in the set
{1,2,3} given by is
symmetric but neither reflexive, symmetric or transitive. Answer: Given that A = {1,2,3}
A relation R on A is defined as R =
{(1,2),(2,1)}
It is seen that (1,1),(2,2),(3,3) .
... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9350

Show that the relation R in the set A of
all the books in a library of a college, given by R = {(x , y): x and y have
same number of pages}, is an equivalence relation. Answer: Here, set A is the set of all books in
the library of a colle... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9351

Show that the relation R in the set , given by
(i) R = {(a, b): is a multiple of
4}
(ii) R = {(a , b): a = b}, are
equivalence relations. Find the set of all elements re... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9352

Show that the relation R, defined in the
set of A all triangles as R = { is
similar to }, is equivalence
relation. Consider three right angle triangles with sides
3,4,5; ... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9353

Let R is the relation in the set N given
by Choose the
correct answer
(a) (b) (c) ... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9354

Show that the function defined by is one-one,
where is the set of
all non-zero real numbers. Is the result true, if the domain is replaced by N
with co-domain being same as ? Answer: It is given that read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9355

Check the injectivity and surjectivity of
the following function:
(i) given by Answer: is given by
It is see... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9356

Check the injectivity and surjectivity of
the following function:
(iii) given by Answer: is given by .
It is seen that but .
... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9357

Check the injectivity and surjectivity of
the following function:
(v) is given by Answer: is given by
It is seen that for x, y
... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9358

Prove that the greatest integer function , given by , is neither
one-one nor onto, where denotes
the greatest integer less than or equal to . Answer: Here , read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9359

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9360

If show that for all
What is the inverse of ? Answer: It is given that .
Answer: On Z, * is defined by
, define Answer: On Q, * is defined by
be defined as . Find the
function such that
Note- Firstly, we prove one-one and onto,
then we find inverse function, After that we determine the required result... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9364

Given a non-empty set X , let *: be defined as ∀A,B∈ P(X). Show that the empty set is the identity
for the operation * and all the elements A of P(X) are invertible... read more

JEE-Main - Grade 12 - Mathematics - Relations and Functions - Study ref# 9365

Let be the function
defined by ∀ .Find . Answer: Injectivity Let x, y be two arbitrary elements in R. Such that
is defined by write Answer: We have, wh... read more