Mathematics JEE quick revision notes pdf

CLASS XI - MATHEMATICS
QUICK REVISION NOTES
JEE MAINS

1
OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
Mind map : learning made simple Chapter - 1
Set builder form or Rule Method
Roaster or Tabular form
SETS
S
u
b
s
e
t
Representation of Sets
The number of elements in a finite set is
represented by n (A), known as cardinal
number.
Eg.: A = {a, b, c, d, e} Then, n (A) = 5
A set is collection of well‐defined distinguished objects. The sets are
usually denoted by capital letters A, B, C etc., and the members or
elements of the set are denoted by lower case letters a, b, c etc.., If x is
a member of the set A, we write x ∈A (read as ‘x’ belongs to A) and if
x is not a member of set A, we write x ∉A(read as ‘x’ doesn’t belongs
to A). If x and y both belong to A, we write x, y ∈ A.
Some examples of sets are: A: odd numbers less than 10
N: the set of all rational numbers
B : the vowels in the English alphabates
Q: the set of all rational numbers.
In this form, we first list all the members of the set within braces
(curly brackets) and separate these by commas.
Eg: The set of all natural number less than 10 in this form is written
as: A = {1, 3, 5, 7, 9}
In roaster form, every element of the set is listed only once.
The order in which the elements are listed is immaterial.
Eg. Each of the following sets denotes the same set
{1, 2, 3}, {3, 2, 1}, {1, 3, 2}
In this form, we write a variable (say x) representing any member
of the set followed by property satisfied by each member of the set.
eg.: The set A of all prime number less than 10 in set builder form is
written as
A = {x | x is a prime number less than10}
The symbol “|” stands for the word “such that”. Sometimes, we use
symbol “:” in place of symbol “|”
A set which has no element is called null set. It is denoted by
symbol φ or {}.
E.g: Set of all real numbers whose square is −1.
In set-builder form: {x:x is a real number whose square is −1}
In roaster form: { } or φ
A set which has finite number of elements is called a finite set.
Otherwise, it is called an infinite set.
E.g.: The set of all days in a week is a finite set whereas the set
of all integers, denoted by
{……−2, −1, 0, 1, 2,……} or {x | x is an integer} is an infinite
set.
An empty set φ which has no element is a finite set A is called
empty or void or null set.
Two sets A and B are set to be equal, written as A=B, if every
element of A is in B and every element of B is in A.
e.g.: (i) A = {1, 2, 3,4} and B = {3, 1, 4, 2}, then A =B
(ii) A = {x: x−5 =0} and B = {x: x is an integral positive root
of the equation x2
−2x−15=0}
Then A= B
A set having one element is called singleton set.
e.g.: (i) {0} is a singleton set, whose only member is 0.
(ii) A = {x: 1<x <3, x is a natural number} is a singleton set
which has only one member which is 2.
Two finite sets A and B are said to be equivalent, if
n(A) = n(B). Clearly, equal set are equivalent but equivalent
set need not to be equal.
e.g.: The sets A = {4, 5, 3, 2} and B = {1, 6, 8, 9} are equiva-
lent, but are not equal.
Let A and B be two sets. If every element of A is an
element of B,then A is called a subset of B and
written as A⊂B or B⊃A(read as ‘A’ is contained in
‘B’ or ‘B contains A’). B is called superset of A.
Note:
1. Every set is a subset and superset of itself.
2. If A is not a subset of B, we write A⊄B.
3. The empty set is the subset of every set.
4. If A is a set with n(A) = m, then no. of element
A are 2n and the number of proper subsets
of A are 2
n
Eg. Let A = {3, 4}, then subsets of A are φ, {3}, {4},
{3, 4}. Here, n(A) = 2 and number of subsets
of A = 2
2
=4.
–1

OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
2
Mind map : learning made simple Chapter - 1(a)
If U is a universal set and A is a subset
of U, then complement of A is the set
which contains those elements of U,
which are not present in A and is
denoted by A’ or A
C
. Thus,
Ac = {x: x∈U and x∉A}
e.g.: If U = {1, 2, 3, 4, …}
and A = {2, 4, 6, 8, ….}
then Ac = {1, 3, 5, 7, …}
Properties of complement
Complement law:
(i) A∪A' = U (ii) A∩A'= φ
De morgan’s Law:
(i) (A∪B)' = A'∩B' (ii) (A∩B)' = A'∪B'
Double Complement law:
(A')' =A
Law of empty set and universal set
φ'=U and U' = φ
SETS
A Venn diagram is an illustration of the relationships between
and among sets, groups of objects that share something common.
These diagrams consist of rectangle and closed curves usually
circles
3
7
Eg: 1
2
10
Α
Β
6
4 8
5
9
∪ In the given venn diagram
U={1,2,3,.......10} universe
set of which A={2,4,6,8,10}
and B={4,6} are subsets
and also B⊂A
1. For any set A, we have
(a) A∪A=A, (b) A∩A=A, (c) A∪φ=A, (d) A∩φ=φ, (e) A∪ U=U
(f) A∩U=A, (g) A− φ=A, (h)A−A=φ
2. For any two sets A and B we have
(a) A∪B = B∪A, (b) A∩B= B∩A, (c) A−B⊆ Α, (d) B−A⊆B
3. For any three sets A,B and C, we have
(a) A∪(B∪C)= (A∪B)∪C, (b) A∩(B∩C)= (A∩B)∩C
(c) A∪(B∩C)=(A∪B)∩(A∪C), (d) A∩(B∪C)=(A∩B) ∪(A∩C)
(e) A−(B∪C)=(A−B) )∩(A–C), (f) A–(B∩C)=(A–B)∪(A–C)
The union of two sets A and B, written as A∪B (read as A union
B) is the set of all elements which are either in A or in B in both.
Thus, A∪B = {x: x ∈A or x ∈B}
clearly, x ∈ A∪B ⇒ x ∈A or x ∈ B and x ∉A∪B⇒x ∉A and x ∉B
eg: If A = {a, b, c, d} and B ={c, d, e,f }
then A∪B = {a, b, c, d, e, f}
A B
Α∪Β
The intersection of two sets A and B, written as A∩B (read as
‘A’ intersection ‘B’) is the set consisting of all the common
elements of A and B.
Thus, A∩B = {x : x∈Aand x∈B}
Clearly, x∈A∩B⇒ {x∈A and x∈B} and x∉A∩B ⇒ {x ∉A or x
∉B}.
Eg: If A={a,b,c,d} and B={c,d,e,f}
Then A∩B = {c,d}
A B
Α∩Β
Two sets A and B are said to be disjoint, if A∩B =φ i.e, A and B
have no common element. e.g: if A = {1, 3, 5} and B = {2, 4, 6}
Then, A∩B =φ , so A and B are disjoint.
A B
∪
If A and B are two sets, then their
difference A‐B is defined as:
A–B = {x : x ∈ A and x ∉B}
Similarly, B – A = {x : x ∈B and x ∉A}
Eg. If A={1,2,3,4,5,} and B={1,3,5,7,9}
then
A–B={2,4} and B–A={7,9}
A B
B–A
Α–Β
∪
The symmetric difference of two sets
A and B, denoted by A ∆ B, in defined
as (A ∆ B)=(A–B) ∪ (B – A)
Eg. If A = {1, 2,3,4,5}
and B = {1,3,5,7,9} then
(A ∆ B)= (A–B) ∪ (B – A)
= {2, 4,} ∪ {7, 9}
= {2, 4, 7, 9}
The set containing all objects of element and of
which all other sets are subsets is known as
universal sets and denoted by U.
E.g : For the set of all intergers, the universal
set can be the set of rational numbers or the set
R of real numbers
The set of all subset of a given set
A is called power set of A and
denoted by P(A).
E.g : If A ={1,2,3,}, then P (A)={φ},
{1},{2}, {3},{1,2},{1,3},{2,3},{1,2,3}.
Clearly, if A has n elements, then its
power set P (A) contains exactly
2n elements.
The set of natural numbers
N={1, 2, 3, 4, 5, – – –}
The set of integers
Z={. . . –3, –2, –1, 0, 1, 2, 3,– – –}
Relation among these subsets are
N ⊂ Z ⊂ Q, Q ⊂ R,T ⊂ R, N ⊄T
The set of irrational numbers,
T = {x: x∈R and x∈Q}
The set of rational number
Q={x : x =
p
q
, p, q ∈Z and q ≠ 0}
Let a and b be real numbers with a < b
Interval Notation
(a, b)
[a, b)
(a, b]
[a, b]
(–∞, b)
(–∞, b]
(a, ∞)
[a, ∞)
(–∞, ∞)
Set of Real Numbers
{x|a< x < b}
{x|a ≤ x < b}
{x|a< x ≤ b}
{x|a ≤ x ≤ b}
{x|x < b}
{x|x ≤ b}
{x|x >a}
{x|x >a}
R
Region on the
real number line
Interval Notation
Algebra of sets

3
OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
Relations
&
Functions
F
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c
t
i
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n
s
Algebra
The function f:R → R defined by y=f(x)=x ∀ x ∈R is called
identity function. Domain=R and Range=R
f(x)=log , a>0 , a ≠ 1 Domain=x ∈ (0, ∞) Range=y ∈ R
The function f:R → R defined by y=f(x)=c, ∀ x ∈ R, where
c is a constant is called constant function. Domain=R
and Range ={c}
The function f:R → R defined by f(x)= is called signum
function. It is usually denoted by
y=f(x)=sgn(x) Domain=R and Range ={ 0, -1, 1}
The function f:R → R defined by as the greatest integer
less than or equal to x .It is usually denoted by y=f(x)=[x].
Domain=R and Range=Z(All integers)
f(x)=ax , a>0 , a ≠ 1,
Domain: x∈R: Range:f(x) ⊂ (0,∞)
Identity
function
Constant
function
Modulus
function
Signum
function
Exponential
function
Greatest
integer function
Relatio
n
D
om
ain & range of a Relation
Pictorial representation
of
a relation
I
n
v
e
r
s
e
r
e
l
ation
Definition: A relation ' f ' from a set A to set B is said to be a
function if every element of set A has one
and only one image in set B.
Domain(Input) f y=f(x) Range(Output)
Codomain of ‘f’
A
a b
B
Domain of ‘f ’
Range of ‘f’
Notations:
x
One-One Onto function
Range=Codomain
A B A B
A B
One-One Into function
Many-One Onto function
Range=Codomain
Many One Into function
Range=Codomain
Range ⊂ Codomain
f f
f
A B
f
Given two non empty sets A&B. The cartesian
product A×B is the set of all ordered pairs of
elements from A&B i.e., A×B={(a,b) : a ∈ A ; b ∈ B}.
If n(A)=p and n(B)=q , then n(A×B)=pq
Let A&B be two empty sets. Then any subset ' R ' of
A x B is a relation from A to B.If (a,b) ∈R, then we
write a R b , which is read as ' a is related to b ' by a
relation R, 'b' is also called image of'a' under R.
The total number of relations that can be defined
from a set A to a set B is the number of possible
subsets of A x B. If n(A)=p and n(B)=q, then n
Cartesian product of s
e
t
s
f
Charlss
Divya
b
a
c
P
Ali
Beena
Bindu
Q Given, R={(x,y): x is the first letter
of the name y, x ∈P, y ∈Q}
Then, R={(a,Ali), (b, Beena),
(c, charu)}This is a visual or
pictorial representation of relation
R (called an arrow diagram)is
shown in figure.
If R is a relation from A to B, then the set of first
elements in R is called domain & the set of second
elements in R is called range of R. Symbolically,
Domain of R={x:(x,y) ∈ R}; Range of R={y:(x,y) ∈ R}
The set B is called co-domain of relation R.
Note: the range ⊆ Codomain.
Eg. Given, R={(1,2),(2,3),(3,4),(4,5),(5,6)}.
then Domain of R={1,2,3,4,5}
Range of R={2,3,4,5,6} and
codomain of R={1,2,3,4,5,6}
A B
1
2
3
4
5
6 6
1
2
3
4
5
Let A&B be two sets and R be a relation from set A
to set B.Then inverse of R ,denoted by R-1, is a
relation from B to Aand is defined by
R-1={(b,a) : (a,b) ∈ R}. Clearly, (a,b) ∈R ⇔ (b,a) ∈ R-1
Also, Dom(R)=Range (R-1) and Range(R)=Dom(R-1)
Let f :x → R and g → R be any two real functions
where X⊂R.
Addition: (f+g) x = f(x)+g(x); ∀x∈ R
Subtraction: (f–g) x = f(x)–g(x); ∀ x∈ R
Product: (fg) x = f(x).g(x); ∀ x ∈ R
Quotient: (f/g) (x) = f(x)/g(x); provided
g(x) ≠ 0, ∀ x ∈ R
Log function
1, x > 0
0, x = 0
-1, x < 0
The function f:R → R defined by f(x)=
It is denoted by y=f(x)=|x|. Domain=R and Range =(0, ∞)
is called modulus function.
x; x ≥ 0
−x; x < 0
Even function
Odd function
f(−x)=f(x), ∀x∈ Domain
f(−x)=−f(x), ∀x∈ Domain
Some
standard
real
(A x B)=pq and total number relations is 2
pq
.
of functions
e
K
i
n
d
s
o
f
F
u
n
c
t
i
o
n
s
x

OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
4
Trigonometric
Functions
D
i
f
f
e
r
e
n
c
e
o
f
s
i
n
e
a
n
d
c
o
s
i
n
e
w
i
t
h
t
w
o
v
a
r
i
a
b
l
e
s
Trigonom
etrical Function sum and
M
u
l
t
i
p
l
e
a
n
d
H
a
l
f
A
n
g
l
e
s
Trigonometrical Solution
Principal &
General Solution
of
sin
10
π or sin18°=√5 −1
4
sin15° or sin
12
π =
√3 −1 cos75°or cos5π
12
2√2
=
cos15° or cos
12
π = sin 75°or sin5π
12
√3+1
2√2
=
tan15°=
√3−1
√3+1
=2 − √3= cot 75°
tan75°=
√3+1
= 2+√3= cot 15°
√3−1
cos
5
π or cos36°=
4
√5+1
cos cos
θ= α⇔θ= 2nπ±α, where α∈[0,π]
tan θ= tan α⇔θ= nπ+α, where α∈
sin2θ= sin2α⇔θ= nπ±α,cos2 θ= cos2 α⇔nπ±α
cosθ = 1⇔θ = 2nπ, cos = -1⇔θ = (2n+1)π
sinθ= sinα and cosθ= cosα ⇔θ=2nπ+α, η∈z
sinθ= 0⇔ θ= nπ , cosθ= 0⇔θ=(2n + )
2
π
tan2θ= tan2α⇔θ= nπ±α, sin θ= 1 ⇔θ=(4n+1)2
π
tan θ= 0 ⇔θ= nπ, sin θ= sin α⇔θ=nπ+(–1)
n
α, where α∈ ,
π π
−
2 2



 
 
,
π π
−
2 2
e.g. sinθ= 1
√2
A solution of trigonometric
equation is the value of the
unknown angle that satisfies
the equation.
The equation involving
trigonometric functions of
unknown angles are known
as Trigonometric equations.
e.g cosθ= 0, cos2θ–4cosθ= 1
θ=
⇒ , . . . .
3 9 11
, , ,
4 4 4 4
π π π π
sin2A = 2sinA cosA
= 2cos2A-1
= 1-2sin2A
tan3A =
3
3tan tan
A A
−
2
1 3tan A
−
cos2A = cos2A – sin2A
sin2A = 2tanA
2
1 tan A
+
cos2A =1 tan A
− 2
2
1 tan A
+
sin3A = 3sinA – 4sin3
A
cos3A = 4cos3
A – 3cosA
2cos2
2
θ−1=1−2sin2
2
θ
cosθ
tanθ = 2tan
2
θ/ 1−tan2
2
θ
2
θcos
2
θ
2sin
sin θ =
=
tan2A =
2
1 tan A
−
2tanA
sin(-θ)=-sinθ
cos(-θ)=cosθ
tan(-θ)=-tanθ
cot(-θ)=-cotθ
sec(-θ)=secθ
cosec(-θ)=-cosecθ
sinθ
2
π
θ
 
 
 
=cosθ
cos(( )
π θ =-cosθ
sin( )
π θ =±sinθ
tan( )
π θ = tanθ
cot( )
π θ = cotθ
sec( )
π θ =−secθ
cosec( )
π θ =±cosecθ
+
cos
2
π
θ
 
 
 
=±sinθ
+
tan
2
π
θ
 
 
 
=±cotθ
+
cot
2
π
θ
 
 
 
=±tanθ
+
secθ
2
π
θ
 
 
 
=±cosecθ
+
cosec
2
π
θ
 
 
 
=secθ
+
+
+
+
+ +
+
+
+ sin
3
2
π
θ
 
 
 
=− cosθ
cos
3
2
π
θ
 
 
 
= sinθ
tan
3
2
π
θ
 
 
 
=±cotθ
cot
3
2
π
θ
 
 
 
=±tanθ
+
+ +
+
+
sec
3
2
π
θ = cosecθ
cosec
3
2
π
θ
 
 
=secθ
+
+ +
 
 
 
 
sin( )
2π θ = sinθ
cos( )
2π θ =cosθ
cot( )
2π θ = cotθ
sec( )
2π θ =secθ
cosec( )
2π θ = cosecθ
+
+
+
+ +
tan( )
2π θ = tanθ
+ +
+
+
+
sin2
θ θ
θ
θ = 1, |cosec θ| ≥ 1, θ∈R
+ cos2
= 1, –1 ≤ sinθ ≤ 1, –1 ≤ cosθ ≤ 1 ∀θ∈R
sec2
2 2
θ
θ – cot
− tan 2
= 1, |sec θ| ≥ 1, ∀θ∈R
cosec
If in a circle of radius r, an arc of length l subtends an angle
of θ radians , then l =rθ.
Radian Measure=
180
π
× Degree Measure
Degree Measure=
180
π
× Radian Measure
sin(A±B)=sinA cosB ± cosA sinB
cos(A±B)=cosA cosB sinA sinB
tan(A±B)=
tan tan
1 tan tan
A B
A B
±
cot(A±B)=
cot cot 1
cot cot
A B
A B
±
sin2
A−sin2
B=cos2
B−cos2
A=sin(A+B) sin(A−B)
cos2
A−sin2
B=cos2
B−sin2
A=cos(A+B)cos(A−B)
+
+
+
2sinA sinB=sin(A+B)+sin(A−B)
2cosA sinB=sin(A+B) − sin(A−B)
2cosA cosB=cos(A+B)+cos(A−B)
2sinA sinB=cos(A− B) −cos(A+B)
2
C D
+
 
 
 
cos
2
C D
−
 
 
 
sin C + sin D =2sin
2
C D
+
 
 
 
sin
2
C D
−
 
 
 
sin C– sin D= 2cos
2
C D
+
 
 
 
cos
2
C D
−
 
 
 
cos C+ cos D= 2cos
2
C D
+
 
 
 
sin
2
C D
−
 
 
 
cos C– cos D=–2sin
General solution: The solution consisting of all possible solutions
of a trigonometric equation is called its General
Solution.
(
(

5
OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
Generalisation of Specific Instance
e.g.: Rohit is a man and all men eat food,
therefore, Rohit eats food.
Principle of
Mathematical Induction
Steps
for
Principle
of
Mathematical
Induction
Proof
Step1: Let P(n) be a result or statement formulated in terms of n in a given equation
Step2: Prove that P(1) is true.
Step3: Assume that P(k) is true.
Step4: Using step 3 , prove that P(k+1) is true.
Step5: Thus, P(1) is true and P(k+1) is true whenever P(k) is true
Hence,by the principle of mathematical induction,P(n) is true for all natural numbers n.
Specific Instances to Generalisation
e.g.:Rohit eats food. Vikas eats food. Rohit
and Vikas are men.Then all men eat food.
Statement is true for n=1,n=k and n=k+1,
then, the statement is true for all natural
numbers n.
Deduction
Induction
Ex: Prove that 2n
>n for all positive integer n.
Solution: Step1: Let P(1):2n
>n
Step2: When n=1, 21>1.Hence P(1) is true
Step3: Assume that P(k) is true for any positive integer k, 2k>k
Step4: We shall now prove that P(k+1) is true
Multiplying both sides of step(3) by 2 , we get
2.2k>2 k
⇒2k+1>2k
⇒2k+1= k+k
⇒2k+1> k+1 (since k>1)
Therefore, P(k+1) is true when P(k)
is true Hence, by P.M.I., P(n) is true for every positive integer n .
E
x
a
m
p
l
e

OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
6
Let squaring both sides,
we get (x+iy)2
= a+ib i.e. x2
−y2
=a, 2xy=b
solving these equations, we get square root of z.
,
x iy a ib
+ = +
2 2 2 2
,
a b
i
a b a b
−
+
+ +
z
If z= a+ib is a complex number
(i)Distance of z from origin is called
as modulus of complex number z.
It is denoted by
(ii) Angle θ made by OP with +ve
direction of X-axis is called
argument of z.
2 2
r z a b
= = +
P (a,b)
Re (z)
b
a
O
(0,0)
2
1, 1
i i
= − = −
In general, 4 r
1; 0
; 1
1; 2
; 3
k
r
i r
i =
r
i r
+
=
=
− =
− =
A complex number z=a+ib can be represented by
a unique point P(a,b) in the argand plane
z=a+ib is represented by a point P (a,b)
Im(z)
θ
P (a,b)
Re (z)
r
O
(0,0)
Im(z)
Let: z1
= a+ib and z1
= c+id be two complex numbers,
where a,b,c,d R and
1. Addition:
2. Subtraction:
3. Multiplication:
4. Division:
1
i = −
∈
1 2 ( ) ( ) ( ) (b d)
z z a ib c id a c i
+ = + + + = + + +
1 2 ( ) ( ) ( ) (b d)i
z z a ib c id a c
− = + − + = − + −
1 2
2
( ) ( )
( ) ( )
( ) ( )
( 1)
z z a ib c id
a c id ib c id
ac bd ad bc i
i
. = + +
= + + +
= − + +
= −
∵
0
a ≠
1
2
2 2 2 2
.
z a ib a ib c id
z c id c id c id
ac bd bc ad
i
c d c d
+ + −
= =
+ + −
+ −
   
= +
   
+ +
   
Powers of ‘i’
Argand
P
l
a
n
e
N
u
m
b
e
r
s
A
l
g
e
b
r
a
o
f
C
o
m
p
l
e
x
N
u
m
b
e
r
s
D
e
f
i
n
i
t
i
o
n
o
f
C
o
m
p
l
e
x
Solution of Quadratic Equa
t
i
o
n
Polar Repr
e
s
e
n
t
a
t
i
o
n
Complex Number &
Quadratic Equations
o
f
C
o
m
plex Number
M
o
d
u
l
u
s &
Argument
Let a = r cos
b = r sin
where,
θ
θ
θ
θ
θ
θ
r z
=
and = arg (z)
The argument ‘θ’ of complex
number z = a+ib is called
principal argument of z if
(cos sin )
z a ib r i
θ θ
∴ = + = +
P (a,b)
r sin
r cos
Real (z)
Im (z)
.
π θ π
− < ≤
r
·
General form of quadratic equation
in x is ax2
+bx+c=0,
Where
The solutions of given quadratic equation
, , & 0
a b c ∈R a ≠
are given by
b2
–4ac
2
b
x
a
− ±
=
Note: A polynomial equation has
atleast one root.
A polynomial equation of degree
n has n roots.
A number of the form a+ib, where a,b R and
is called a complex number and denoted by ‘z’.
z= a+ib
∈ 1
i = −
Imaginary part
Real part
Conjugate of a complex number: For a given complex number
z=a+ib, its conjugate is defined as z= a – ib
For a non-zero complex number z=a+ib
there exists a complex number + i
denoted by – or z−l
, called multiplicative inverse of Z
2 2 2 2
a
a ib i i
−b
( ) 1 0 1
a b a b
+ + = + =
+ +
Such that:
( 0, 0),
a b
≠ ≠
Note: If a+ib = c+id
⇔ a = c & b = d
1
z
a
a2
+b2
–b
a2
+b2
( (
b2
–4ac<0
∴

7
OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
Linear
Inequalities
–
Conditions
the
or
or
4

OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
8
P
e
r
m
u
t
a
t
i
o
n
s
o
f
a
l
i
k
e
O
b
j
e
c
t
s
Permutations &
Combinations
1. The no. of circular permutations of ‘n’
distinct objects is (n-1)!
2. If anti-clockwise and clockwise order
of arrangements are not distinct then
the no. of circular permutations of n
1
( 1)!
2
n −
e.g.: arrangements of beads in a necklace,
arrangements of flower in a garland etc.
The no. of permutations of n objects taken all at a time,
where p1
objects are of first kind, p2
objects are of the
second kind, ....pk objects are of the kth
kind and rest,
if any one are all different is
1 2 3
!
! ! ! !
k
n
p p p p
…
Each of the different arrangements which can
be made by taking some or all of a number of
distinct objects is called a permutation.
The no. of permutations of n different things
taken r at a time, where repetition is not allowed,
is denoted by and is given by
!
( )!
n
n r
=
−
F.P
.C. of Multiplication: If an event can occur in
m different ways, following which another event
in n different ways, then the total number of
occurance of the events in the given order is m×n
F.P
.C. of Addition: If there are two events such that
they can performed independently in m and n
ways respectively, then either of the two events
can be performed in (m+n) ways.
r n
2 1
n
+ + = −
… …
Certain
Co
n
d
i
t
i
o
n
s
C
o
m
b
i
n
a
t
i
o
n
s
o
f
C
ounting (F.P.C.)
F
u
n
d
a
m
ental Principle
Permutation
C
i
r
c
u
l
a
r
Perm
utations
R
e
l
a
t
i
o
n
between
!
!( )!
n
r n r
=
−
In particular, r =n
!
1
!0!
n
n
= =
=
+ =
n
Cr
n
Cr
n+1
Cr
n
Cr
n
Cr
n
Cn-r
Cr-1
i.e., Corresponding to each combination of
we have r! permutations, because r objects in
every combination can be rearranged in r! ways
n
Cn
The no. of arrangements of n distinct objects taken r
at a time, so that k particular objects are
(i) Always included:
(ii) Never included:
The no. of selections from ‘n’ different objects, taking
at least one= + +
n
C1
n
C2
n
C3
+..............Cn
=2n
–1
n–k
Cr–k.
r!
n–k
Cr.
r!
n
Cr
,
The no. of all permutations of ‘n’ different objects
taken r at a time:
(i) When a particular object is to be always included
in each arrangement is
(ii) When a particular object is never taken in each
arrangement in
(iii) The no. of permutation of n different things taken r
at a time, where repetition is allowed is (n)r
n-1
P
n-1
Pr
n
Pr
n
Pr
Each of different selections made by taking source or all
of a number of distinct objects or item, irrespective of
their arrangements or order in which they are placed, is
called a combination.
The no. of combinations of n different things taken r at a
time, denoted by is given by
!
; 0
r!( )!
n
n r
= ≤ ≤
−
n
Cr
n
Cr
r n
r
Since,
n
P= C ×r!
n
n n
r r 0<r≤n
distinct items is
∴

9
OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
+b
r
n
y
2
1
1
= + +
n n–1
r
are
r
r
r
r
n
0
x– x y
n n n–1
1
n
C
C
C
–

OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
10
Sequences
and Series
If a, b, c are in G.P
., b is the GM between a & c, b2
= ac,
therefore ; a > 0, c >0
If a, b are two given numbers & a, G1
, G2
, …, Gn
, b are
in G.P
., then G1
, G2
,G3
… Gn
are n GMs between a & b.
b ac
=
( )
1
1
1
n
b
G a
a
+
= ⋅ ( )
2
1
2
n
b
G a
a
+
= ⋅ ( ) 1
n
n
n
b
G a
a
+
= ⋅
......
or
where
2 1
1 2
, , n
n
G ar G ar G ar −
= = … =
( ) 1
1
n
b
r
a
+
=
In a series,where each term is formed by multiplying
the corresponding term of an AP & GP is called
Arithmetico‐Geometric Series.
Eg: 1+3x+5x2
+7x3
+…
Here, 1, 3, 5, … are in AP and 1, x, x2
, … are in GP
.
• If A and G are respectively arithmetic and geometric
means between two positive numbers a and b; then
the quadratic equation have a, b as it’s root is
x2
−2Ax+G2
=0
• Sum of first ‘n’ natural numbers
• Sum of first ‘n’ odd natural numbers
• Sum of squares of first n natural numbers
• Sum of cubes of first n natural numbers
If A and G be the AM & GM between two positive
numbers, then the numbers are 2 2
A A G
± −
•
Let A and G be the AM and GM of two given
positive real numbers a & b, respectively. Then,
2
a b
A
+
= and G ab
=
Thus, we have ,
( )
2
2
2 2
0
2
a b a b ab
A G ab
a b
+ + −
− = − =
−
= ≥
From (1), we obtain the relationship A ≥ G
( )
1
1
1 2 3
2
n
k
n n
k n
=
+
= + + + + =
∑ …
( )( )
2 2 2 2 2
1
1 2 1
1 2 3
6
n
k
n n n
k n
=
+ +
= + + + + =
∑ …
( )
2 2
3 3 3 3 3
1 1
1
1 2 3
2
n n
k k
n n
k n k
= =
 +   
= + + + + = =
   
 
 
∑ ∑
…
2
1
(2 1) 1 3 5 (2 1)
n
k
k n n
=
− = + + + + − =
∑ …
If three numbers are in A.P
., then the middle term is
called AM between the other two, so if a, b, c, are in
A.P
., b is AM of a and c.
AM for any ‘n’ +ve numbers a1
, a2
, a3
, …, an
is
1 2 3 n
a a a +a
AM n
+ + +
=
…
n-Arithmetic Mean between Two Numbers: If a, b are
any two given numbers & a, A1
, A2
, .., An
, b are in
A.P
. then A1
, A2
, …, An
are n AM’s between a & b.
( ) ( )
1 2
2
,
1 1 1
n
b a n b a
b a
A a A a A a
n n n
− −
−
= + = + = +
+ + +
…
or A1
= a+ d, A2
= a+2d, …, An
= a + nd,
where
1
b a
d
n
−
=
+
An A.P
. is a sequence in which terms increase or decrease
regularly by the fixed number (same constant). This fixed
number is called ‘common difference’ of the A.P
. If ‘a’ is the
first term & ‘d’ is the common difference and ‘l’ is the last
term of A.P
., then general term or the nth
term of the A.P
. is
given by an
= a+ (n−1)d.
The sum Sn
of the first n terms of an A.P
. is given by
[2 ( 1) ] [ ]
2 2
n
n n
S a n d a l
= + − = +
Sequence is a function whose domain is the set of N natural
numbers.
Real sequence: A sequence whose range is a subset of R is
called a real sequence
Series: If a1
, a2
, a3
, ...an
, is a sequence, then the expression
a1
+a2
+a3
+…an
+.. is a series. A series is finite or infinite
according to the no. of terms in the corresponding
sequence as finite or infinite.
Progression: Those sequences whose terms follow certain
patterns are called progression.
A sequence is said to be a geometric progression
or G.P
., if the ratio of any term to its preceding
term is same throughout.This constant factor is
called ‘Common ratio‘. Usually we denote the first
term of a G.P
. by ‘a’ and its common ratio by ‘r’.
The gereral or nth
term of G.P
. is given by an
=arn–1
The sum Sn
of the first n terms of G.P
. is given by
( 1)
1
n
n
a r
S
r
−
=
−
(1 )
1
n
a r
r
−
−
or
Sum of infinite terms of G.P
. is given by
; 1
1
n
a
S r
r
= <
−
Definitions
Geometric Mean (GM)
Arithmetic Mean (AM)
Arithmetic Progression (A.P
.)
Special Sequences
Sum
upto terms of some
R
e
l
a
t
i
o
n
s
h
i
p
b
e
t
w
e
e
n
A
M
a
n
d
G
M
b
e
t
w
e
en
two quantities
P
r
o
p
e
r
ties of AM
&
GM
(1)
...............
if r≠1
·
·
P
r
o
g
r
e
s
s
ion
(G.P.)
,

11
OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
1.The image of a point(x1
, y1
) about a line
ax + by+ c =0 is:
2.Similarly, foot of perpendicular from a
point on the line is:
x − x1
a
y − y
1
b
− (ax1 + by1 + c)
a2
+ b2
=
=
x − x1
a
y − y
1
b
=
=
ax1 + by1 + c
a2 + b2
−
Straight
Line
p
oint about a line
R
e
f
l
e
c
t
ion
&
foot of perpendicular a
The length of the
perpendicular from
P(x1
, y1
) on
ax + by +c =0 is:
2 2
ax by c
a b
+ +
+
L
e
n
g
t
h
o
f
t
h
e
p
e
r
pendicular
f
r
o
m
a
p
o
i
n
t
o
n
a
l
i
n
e
If m1
and m2
are the
slopes of two intersecting
lines (m1
m2
≠−1) and θ be the
acute angle between them
2
1
2
1
tan
1
m m
m m
θ
−
=
+
Area of triangle whose vertices are (x1
, y1
), (x2
, y2
) and
(x3
, y3
) is
The distance between the points A(x1
, y1
) and B(x2
, y2
) is
l
i
n
e
s
i
n
t
e
r
m
s
o
f
t
h
e
i
r
s
l
o
p
e
s
A
n
g
l
e
b
e
t
w
e
e
n
t
w
o
s
t
r
a
i
g
h
t
A
r
e
a
o
f
T
r
i
a
n
gle
Distance Formula
S
l
o
p
e
F
o
r
m
u
l
a
Formula
Section
√
√
√ (x
1
–x
2
)2 +(y
1
–y2
)2
1. When two lines are parallel their slopes are equal. Thus, any line parallel to
y = mx+c is of the type y = mx +d , where d is any parameter
2. Two lines ax + by +c=0 and are parallel if
3. The distance between two parallel lines with equations ax +by +c1
= 0
and ax +by +c2
= 0 is 2
1
2 2
c c
a b
−
+
a’x + b’x + c’ = 0 a b c
= ≠
a’ b’ c’
1.When two lines of the slope m1
& m2
are at right
angles, the Product of their slope is −1, i.e. ,
m1
m2
= −1. Thus, any line perpendicular to
y=mx + c is of the form , y = x+d where d is
any parameter.
1
m
−
2.Two lines ax + by +c =0 and are
perpendicular if Thus, any line
perpendicular to ax + by +c =0 is of the form
bx – ay+ k =0, where k is any parameter.
aa’ + bb’ =0.
a’x + b’y + c’=0
1. POINT-SLOPE FORM: y – y1
= m(x – x1
) is the equation of a straight line whose slope is ‘m’ and
passes through the point (x1
, y1
).
2. SLOPE INTERCEPT FORM: y = mx +c is the equation of a straight line whose slope is ‘m’ and
makes an intercept c on the y-axis.
3.TWO POINT FORM: is the equation of a straight line which passes
through (x1
, y1
) & (x2
, y2
).
4. INTERCEPT FORM: is the equation of a straight line which makes intercepts a & b on
x and y axis respectively.
5. NORMAL / PERPENDICULAR FORM: xcosα + ysinα = p (where p > 0,≤0 α < 2π) is the
equation of a straight line where the length of the perpendicular from origin O on the line is p
and this perpendicular makes an angle α with +ve x-axis.
6. GENERAL FORM: ax +by+ c =0 is the equation of a straight line in general form. In this case,
slope of line =
1
x y
a b
+ =
a
b
−
y − y1 =
y2 − y1
x2 − x1
(x −x1)
1
The P(x, y) divided the line joining A(x , y) and B(x , y) in the ratio
m:n, then
Note: • If m/n is +ve, the division is internal, but if m/n is –ve, the
division is external.
• If m = n, then P is the mid-point of the line segment joining
A & B.
x =
mx2
+ nx1
m + n y1
=
my2
+ ny1
m + n
;
1
1
2
x1(y1 − y2) +x2(y3 − y1) + x3 (y1 − y2)
•If θ is the angle at which a straight line is inclined to be+ve
direction of x-axis and 0°≤θ < 180o
, θ 90o
, then the slope
of the line, denoted by m, is defined by m = tan θ. If θ is 90o
,
m doesn’t exist, but the line is parallel to y-axis. If θ=0o
,
then m=0 and the line is parallel to x-axis.
≠
•If A(x1
, y1
) and B(x2
, y2
), x1
≠ x2
are points on straight line,
then the slope m of the line is given by m = (y1
– y2
/ x1
– x2)
P
a
r
a
l
l
e
l
l
i
n
e
s
1
1 2 2
P
e
r
p
e
n
d
i
c
u
l
ar lines
then

OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
12
Conic Sections
An ellipse is the set of all points in a plane, the sum of whose distances from
two fixed points in the plane is constant.
The two fixed points are called the ‘foci’ of the ellipse.
The midpoint of line segment joining foci is called the ‘centre’ of the ellipse.
The line segment through the foci of the ellipse is called’ major axis’.
The line segment through centre & perpendicular to major axis is called
minor axis.
The end point of the major axis are called the vertices of the ellipse.
The equation of ellipse with ‘foci’ on the x-axis is
The eccentricity of an ellipse is the ratio of distances from centre of ellipse to
one of foci and to one of the vertices of ellipse i.e.,
2 2
2 2
1
x y
a b
+ =
Length of the latus rectum of the ellipse
2 2
2 2
1
x y
a b
+ =
2
2b
a
is
is
c
e
a
=
A hyperbola is the set of all points in a plane, the difference of whose
distances from two fixed points in the plane is a constant.
The equation of a hyperbola with foci on the x-axis is
The two fixed points are called the ‘foci’ of the hyperbola.
The mid-point of the line segment joining the foci is called the ‘centre’ of
the hyperbola.
The line through the foci is called ‘transverse axis’.
Line through centre and perpendicular to transverse axis is called
‘conjugate axis’.
Points at which hyperbola intersects transverse axis are called ‘vertices’.
Length of the latus rectum of the hyperbola :
The eccentricity of a hyperbola is the ratio of the distances from the centre
of the hyperbola to one of the foci and to one of the vertices of the hyperbola.
2 2
2 2
1
x y
a b
− =
2 2
2 2
1
x y
a b
− =
2
2b
a
A circle is a set of all points in a plane that are equidistant from
a fixed point in the plane. The fixed point is called the ‘centre’ of the
circle and the distance from the centre to a point on the circle is called
the ‘radius’ of the circle.
The equation of a circle with centre (h, k) and the radius r is
( ) ( )
− + − =
2 2 2
x h y k r
Eg: Find the equation of the circle with centre (−3, 2) and radius 4.
Sol: Here, h = − 3, k = 2 and r = 4
Therefore, the equation of the required circle is ( ) ( )
2 2
3 2 16
x y
+ + − =
A parabola is the set of all points in a plane that are equidistant from a
fixed line and a fixed point in the plane. Fixed line is called ‘directrix’ of
parabola. Fixed point F is called the ‘focus’. A line through focus &
perpendicular to directrix is called ‘axis’. Point of intersection of parabola
with axis is called ‘vertex’.
The equation of parabola with focus at (a, 0), a > 0 and directrix
x= −a is y2
=4ax, where 4a is the length of the latus rectum
Eg: Find the equation of the parabola with vertex at (0, 0) and focus at (0, 2).
Sol: Since, vertex is at (0, 0) and focus is at (0, 2) which lies on y-axis, the
y-axis is the axis of parabola. Therefore, equation of the parabola is of
the form x2
= 4ay. Thus we have x2
= 4(2)y
Circles, ellipses, parabolas and hyperbolas are known as conic sections
because they can be obtained as intersections of plane with a double
napped right circular cone.
E
l
l
i
p
s
e
Hyperbola
.
i.e., x2
= 8y
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•

13
OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
Introduction to
Three Dimensional
Geometry
• In three dimensions, the coordinate axes of a rectangular cartesian
coordinate system are three mutually perpendicular lines. The axes
are called x, y and z axes.
• The three planes determined by the pair of axes are the coordinate
planes, called xy, yz and zx-planes.
• The three coordinate planes divide the space into eight parts known
as octants.
• The coordinates of a point P in 3D Geometry is always written in the
form of triplet like (x,y,z). Here, x, y and z are the distances from yz,
zx and yx planes, respectively.
Eg:
• Any point on x-axis is : (x, 0, 0)
• Any point on y-axis is : (0, y, 0)
• Any point on z-axis is : (0, 0, z)
Distance between two points P(x1
, y1
, z1
) and Q(x2
, y2
, z2
) is given by
( ) ( ) ( )
2 2 2
2 1 2 1 2 1
PQ x x y y z z
= − + − + −
Eg: Find the distance between the points P(1, –3, 4) and (–4, 1, 2).
Sol: The distance PQ between the points P & Q is given by
( ) ( ) ( )
2 2 2
4 1 1 3 2 4
PQ = − − + + + −
25 16 4
= + + 45
= 3 5
= units
D
i
s
t
a
n
c
e
b
e
t
w
e
e
n
T
w
o
P
o
i
n
t
s
Section Formula
C
o
o
r
d
i
n
a
t
e
s
o
f
a
M
i
d
p
o
i
n
t
o
f
a
T
r
i
a
n
g
l
e
C
o
o
r
d
i
n
a
t
e
s
o
f
t
h
e
C
e
n
t
r
o
i
d
I
n
t
r
o
d
u
c
t
i
o
n
The coordinates of the midpoint of the line segment joining two points
P(x1, y1, z1) and Q(x2, y2, z2) are 1 2 1 2 1 2
, ,
2 2 2
x x y y z z
+ + +
 
 
 
Eg: Find the midpoint of the line joining two points P(1, –3, 4) and
Q(–4, 1, 2).
Sol: Coordinates of the midpoint of the line joining the points P & Q are
1 4 3 1 4 2
, ,
2 2 2
− − + +
 
 
 
3
, 1,3
2
−
 
−
 
 
i.e.
.
The coordinates of the point R which divides the line segment joining
two points P(x1
, y1
, z1
) and Q(x2
, y2
, z2
) internally and externally in
the ratio m : n are given by
2 1 2 1 2 1
, ,
mx nx my ny mz nz
m n m n m n
+ + +
 
 
+ + +
 
2 1 2 1 2 1
, ,
mx nx my ny mz nz
m n m n m n
− − −
 
 
− − −
 
&
respectively.
Eg: Find the coordinates of the point which divides the line segment
joining the points (1,–2, 3)and (3, 4, –5) in the ratio 2:3 internally.
Sol : Let P(x, y, z) be the point which divides line segment joining
A (1,–2, 3) and B (3, 4, –5) internally in the ratio 2:3. Therefore,
( ) ( )
2 3 3 1 9
2 3 5
x
+
= =
+
( ) ( )
2 4 3 2 2
2 3 5
y
+ −
= =
+
( ) ( )
2 5 3 3 1
2 3 5
z
− + −
= =
+
1
5
−
Thus, the required point is
9
5
2
5
, ,
( (.
The coordinates of the centroid of the triangle, whose vertices are (x1
, y1
, z1
),
(x2
, y2
, z2
) and (x3
, y3
, z3
) are 1 2 3 1 2 3 1 2 3
, ,
3 3 3
x x x y y y z z z
+ + + + + +
 
 
 
Eg: The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of
A and B are (3, –5, 7) and (–1, 7, –6), respectively, find the coordinates of the
point C.
Sol: Let the coordinates of C be (x, y, z) and the coordinates of the
centroid G be (1, 1, 1). Then 3 1
3
x + − =1, i.e., x=1;
5 7
3
y − +
7 6
3
z + −
=1, i.e., z=2. So, C (x,y,z) = (1,1,2)
= 1, i.e., y=1;

OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
14
Limits and
Derivatives
Let f(x) = an
xn
+ an-1
xn-1
+ – – +a1
x+a0
be a
polynomial function, where ai
s are all real numbers
and an
≠0. Then the derivative function is given by
( )
( )
− −
−
= + − + − −
1 2
1
1
n n
n n
d f x
na x n a x
dx
2 1
2a x a
+ +
The derivative of a function f at a is defined by
( )
( ) ( )
0
lim
h
f a h f a
f a
h
→
+ −
=
′
Eg: Find derivative of ( )
1
f x
x
=
Sol: We have ( )
( ) ( )
0
lim
h
f x h f x
f x
h
→
+ −
′ =
( )
0 0
1
–
1
1
lim lim
h h
x h h
x
h h x x h
→ →
+  
−
= =  
+
 
 
2
1
x
−
=
.
For functions u and v the following holds:
( )
u v u v
′ ′ ′
± = ±
( )
uv u v v u
′ ′ ′
= +
2
u u v uv
v v
′ ′ ′
−
 
=
 
 
provided all are defined
and v ≠ 0
du
u
dx
′ =
Here, and
dv
v
dx
′ =
•
•
•
•
•
•
( ) 1
n n
d
x nx
dx
−
=
( )
sin cos
d
x x
dx
=
( )
cos sin
d
x x
dx
= −
•
•
•
•
•
•
1
lim
n n
n
x a
x a
na
x a
−
→
−
=
−
0
sin
lim 1
x
x
x
→
=
0
1 cos
lim 0
x
x
x
→
−
=
For functions f and g the following holds:
( ) ( ) ( ) ( )
lim lim lim
x a x a x a
f x g x f x g x
→ → →
 
± = ±
 
( ) ( ) ( ) ( )
lim . lim .lim
x a x a x a
f x g x f x g x
→ → →
  =
 
( )
( )
( )
( )
lim
lim
lim
x a
x a
x a
f x
f x
g x g x
→
→
→
 
=
 
 
 
• We say is the expected value of f at x=a given that the
values of f near x to the left of a. This value is called the left
hand limit of f at ‘a’
• We say is the expected value of f at x=a given that the
values of f near x to the right of a. This value is called the
right hand limit of f ‘at a’.
• If the right and left hand limits coincide, we call that common
value as the limit of f(x) at x=a and denoted it by
Eg: Find limit of function f(x)=(x–1)2 at x=1.
Sol: For f(x)=(x–1)2
Left hand limit (LHL) (at x=1)=
and Right hand limit (RHL) (at x=1)=
( )
2
1
lim 1 0
x
x
−
→
− =
( )
2
1
lim 1 0
x
x
+
→
− =
Limit
f
u
n
c
t
i
o
n
at ‘a’
D
e
r
i
v
a
t
i
v
e
of a
f
u
n
c
t
i
o
n
D
e
r
i
v
a
t
i
v
e
o
f
p
o
lynom
ial
D
erivatives
S
om
e Standard L
i
m
i
t
s
S
o
m
e
S
t
a
n
d
a
r
d
A
l
g
e
b
r
a
o
f
L
i
m
i
t
s
of functions
Algebra of derivatives
( )
lim
x a
g x
→
≠ 0
,
( )
2
1
lim 1 0
x
x
→
− =
LHL = RHL
.
..
lim
x a−
→
f(x)
lim
x a+
→
f(x)
lim
x a
→
f(x) .
.
..

15
OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
Mathematical
Reasoning
• The contrapositive of a statement p ⇒ q is the statement
∼q ⇒ ∼p
• The converse of a statement p ⇒ q is the statement
q ⇒ p
Eg: If the physical environment changes, then biological
environmental changes.
• Contrapositive : If the biological environment does not
change then the physical environment does not change.
In this statement, the two important symbols are used.
• The symbol ‘∀’stand for “all values of”;
• The symbol ‘∃ ’ stand for “there exists”
Eg: For every prime number P
, P is an irrational number.
• This means that if S denotes the set of all prime numbers,
then for all the members of P of the set S, P is an irrational
number.
Many mathematical statements are obtained by combining
one or more statements using some connecting words like
“and”, “or” etc. Each statement is called a “Compound
Statement.”
Eg:
‘The sky is blue’ and ‘the grass is green’ is a compound
statement where connecting word is “and”;
the components of Compound Statement are
p : The sky is blue
q : The grass is green
A sentence is called a mathematically acceptable
statement if it is either true or false but not both.
Eg:
• The sum of two positive numbers is positive.
• All prime numbers are odd numbers.
• In above statements, first is ‘true’ and second is ‘false’.
A statement which is formed by changing the true value
of a given statement by using the word like ‘no’, ‘not’ is called
negation of given statement.
• If P is a statement, then negation is denoted by ∼ p
Eg: New Delhi is a city.
The negation of this statement:
It is not the case that New Delhi is a city.
It is false that New Delhi is a city.
New Delhi is not a city.
The following methods are used to
check the validity of statements
(i) Direct Method
(ii) Contrapositive Method
(iii) Method of Contradiction
(iv) Using a Counter example
Q
u
a
n
t
i
f
i
e
r
s
S
t
a
t
em
ent
C
o
m
pound
Sentence
Implications
• Converse: If the biological environment changes then
physical environment changes.
• These are statements with word “if then”, “only if”and “if and only if”.
Eg:
r: if a number is a multiple of 9, then it is multiple of 3.
p: a number is a multiple of 9.
q: a number is a multiple of 3.
Then, if p then q is the same as following:
p implies q is denoted by p ⇒ q, then symbol ⇒ stands for implies,
p is a sufficient condition for q. then symbol ⇒
p only if q
q is a necessary condition for p
∼q implies ∼p
•
•
•
•

OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
16
M
e
d
i
a
n
c
l
a
s
s
V
:
S.D.
Variance of a continuous
frequency distribution
Standard deviation of a
continuous frequency distribution
Median class is the class interval
whose C.F. is just greater than or
equal to , f, h and C are
respectively the lower limit, the
frequency, width of the median
class and C the cumulative
frequency (C.F.) of class just
preceding to median class.
Var (σ2
) = ∑ fi
( xi
– x)2
S.D (σ) =
1
N
1
N
N∑ fi
xi
2
– ( ∑ fi
xi
)2
V
a
r
i
a
n
c
e
&
S
t
a
n
d
a
r
d
D
e
v
i
a
t
i
o
n
of
a
C
o
n
t
i
n
u
o
u
s
f
r
e
q
u
e
n
c
y
d
i
s
t
r
ibution
–

17
OSWAAL
CBSE
Chapterwise
Mind
Maps,
MATHEMATICS,
Class
-
XI
Probability
Events A & B are called mutually exclusive events if
occurance of any one of them excludes occurrance of
other event, i.e. they cannot occur simultaneously.
Eg: A die is thrown. Event A= All even outcomes &
event B = All odd outcomes. Then A & B are
mutually exclusive events, they cannot occur
simultaneously.
Note: Simple events of a sample space are always
mutually exclusive.
Many events that together form sample space are
called exhaustive events.
Eg: A die is thrown. Event A = All even outcomes and
event B = All odd outcomes. Event A & B together
forms exhaustive events as it forms sample space.
It is the set of favourable outcomes. Any subset E of a sample
space S is called an event.
Occurance of event: The event E of a sample space ‘S’ is said to
have occurred if the outcome w of the experiment is such that
w ∈E. If the outcome w is such that w ∉E, we say that event E has
not occurred.
Eg: Event of getting an even number (outcome) in a throw of a die.
An Experiment is called random experiment if it satisfies the
following two conditions:
• It has more than one possible outcome.
• It is not possible to predict the outcome in advance.
Outcome: A possible result of a random experiment is called
its outcome.
Sample Space: Set of all possible outcomes of a random
experiment is called sample space. It is denoted by symbol ‘S’.
Eg: In a toss of a coin, sample space is Head & Tail. i.e., S= {H,T}
Sample Point: Each element of the Sample Space is called a
sample point.
Eg: In a toss of a coin, head is a sample point
Equally Likely Outcomes: All outcome with equal probability.
If A and B are any two events, then
• P(AUB) = P(A) + P(B) – P(A B)
• P(A B) = P(A) + P(B) – P(AUB)
If A and B are mutually exclusive, then P(AUB)
= P(A) + P(B)
Random Experim
en
t
E
v
e
nts
M
u
t
u
a
l
ly
Exclusive
Definition
Events
Exhaustive
A
l
g
ebra of Events
A B S
• Event A or B or (A∪B)
A∪B = {w: w∈A or w∈B}
Event A and B or (A∩B)
A ∩ B = {w: w∈A and w∈B}
•
• Event A but not B or (A–B)
A–B = A ∩ B’
A B
A B
A B
A–B
A∪B
A∩B
• Impossible and Sure Event: The empty set φ is called an Impossible event, where as the whole
sample space ‘S’ is called ‘Sure event’.
Eg: In a rolling of a die, impossible event is that number more than 6 and event of getting number less
than or equal to 6 is sure event.
• Simple Event: If an event has only one sample point of a sample space, it is called a ‘simple event’.
Eg: In rolling of a die, simple event could be the event of getting number 4.
• Compound Event: If an event has more than one sample point, it is called a ‘compound event’.
Eg: In rolling of a die, compound event could be event of getting an even number.
• Complementary Event: Complement event to A = ‘not A’
Eg: If an event A= Event of getting odd number in a throw of a die i.e., {1, 3, 5} then ,
complementary event to A = Event of getting an even number in a throw of a die , i.e. {2, 4, 6}
A–B B–A
Α∩Β
U
U
• Probability of the event ‘not A’
P(A′) = P(not A) = 1–P(A)
Probability is the measure of uncertainty of various
phenomenon, numerically. It can have positive value
from 0 to 1.
No.offavourableoutcomes
Probability =
Totalno.ofoutcomes
Eg: Probability of getting an even no. in a throw of a
die.
Sol. Here, favourable outcomes = {2, 4, 6}
Total no. of outcomes = {1, 2, 3, 4, 5, 6}
Probability =
3 1
6 2
=
A
∩
B
a
n
d
P
(notA)
P
r
o
b
a
b
i
l
i
t
y
of AUB,
Event
A′= {W: W∈ S and W∉A } = S − A (where S is the sample space)

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