9th Maths Formula Booklet_Shobhit Nirwan

CLASS 9th
Mathematics
FORMULA
SHEET

-.............
Number System
-................
=>Natural Numbers: All
counting numbers are called natural
numbers.
31,2,3,4
---
3
=>Whole Numbers: Natural numbers
the collection of whole number.
40,1,2,3 ---3
along with a form
4
Every natural number is whole number but every
whole number is notnatural number.
As
O- whole number but not natural no.
=>Integers: The natural numbers, zero and negative of
all natural numbers form the collection of all integers.
S----3,
---
-2,-1,0,1,2,3, 3
-Rational Numbers: The
in the form of
numbers which can be
expressed
where
a, panda are integers and
970,
are called rational numbers.
egi,
*= I=E =5 =y
---
are called equivalent
rational nos.

=>Writing Rational numbers between two given numbers:
Arational number between two rational number
se a and b is
at
ii)Then find rational numbers between a and a +b
I
i.e.
taty)
and so on...
=>Decimal expansion
of a rational number is either
terminating or non-terminating.
7Terminating and non-terminating repeating
are
Rational numbers.
eg.
3.14
3 Rational
1.I Numbers
0
0.88888
---
=Rat. No. I Rat. No: Rational No.
Rat. No
*Iur. No:
Zurational No.
jexcept sat.
when)
no. O
Zur. No. Ivr. No. may be rat.
maybe iur.
=>sab =5a 15;
J =5

-
=>(5a+55)(ja-15) =(xa)- (55)2
=ab
=>(a+55)(a- 15) =(a)2- ((5)2
-
ac- b
=To rationalise the denominator ofthe we
multiply this by where a and b are integer
= Let a is a the real number. (a-0)
m and n are
integers
i am.-an --
---
(am"
!
an
- ---
am
+n
=amn
- am -n
4b>03
am bm (ab)* =
↓----------
I
!

-.......
Polynomials
-.........
=>General form of a
polynomial
of degree'n' is given by
-
---
anIqata,anto
where n is a the
integer and
an, An-, ---, d, do are real no.
->
+ 1x
+c a, b, c-constants
-
e
terms L
x - variable
In any term,a variable part
coefficient
of
x2
So,
No. of terms in above
polynomial
-> 3
coefficient of n2= a
coefficient of
coefficient
25 b
of 20 C
highest degree of - 27
this is called
degree of polynomial.
=>CLASSIFICATION OF POLYNOMIALS
--
On basis On basis
of of
"No. of terms"
-M greeolynomia e
term term Sterm DOP=0 DOP=1 DOP=2 DOP =3
↓
N constant
near Quadratic Cubia
monomial Binomial trinomial Polynomial
#
#

=>constant
polynomial
-
[1,2,3,4,5---I
not
zr Jur
so, a polynomial with degree o is constant
polynomial.
=> zero
polynomial 90 0 xpositive integers
So, degree
-
not defined
=>Remainder theorem: let pin) be any polynomial
of degree
greater
than or equal
to one and 'a' be any real no.
If f(x) is divided by (x-a), then remainder is equal
to p(a)
- Factor theorem: If plu) is a polynomial
of
is any real number then,
i) (n-a) is a
factor of p(u), if p(a)=0
ii) p(a)=0,if (n-a) is a factor of p(n).
and 'a'
=>Algebraic Identities:
I
---
Tab252-
I (a b)*
-
-
=
a = b2 -2ab
I ac- b2 =(a+b)(a
-
I
b)
-
(x+a)(x+b)
=x2 +(a+b)x +ab
2
degree
nx
-Y
1 [a + b = =a2+b2 + 2=2qb+2bc+2a,

---------
(a +b)* =a3+b+3ab(a +b)
I
=a3- b)3
-
b3 (a- 3ab(a-b)
(a+b)(a2 +b2 -ab)
i
as+b*
=
b+ab) (at=
(ab)(a2+
--- ---------------
=(a+b+c)(a2+b2+c2-ab-bc- 93 +b3 +c3-3abc (a)
i
-it_ab + =0,then a+b3133ab
I

-.......
Lines and Angles
...........
=>line with two end
points is called a line segment.
Ag ·B
=>partofa line with one endpoint is called Ray.
A8 7
=>line segment is denoted by , like line segment
by B.
=>If three or more points lie on the same line, they are called
collinear
points. in is 8
otherwise it is called non-collinear points.
AB is denoted
=>
Angle is formed when two rays
originate from the same end points.
7
Angle,
=>Rays making an angle are called arms
of the angle and the end
points are called vertex of the
angle.
=>Acute
angle
7
80xx <982
i
ou >
=>Right Angle
less than 90and more than 80
a
y
=900
7
Y
7
=>Obtuse
Angle
90° < 188
↑
more than 90and less than 188 -
2
7

->StraightAngle
S=1800
S
<
8
7
=>
Reflex Angle
188° ~<368°
7
=Complementary
a
T
Angle
Two
angles whose sum is 90% are
un 7
called
complementaryangles. -b
In both cases -
7a=60
at
hea
= Supplementaryangles
a
=120°
Two
angles whose sum is 188 are -
called
supplementaryangles
=>Adjacent Angles
Two
angles are adjacent if they have
a common
arm and their non-common
b
a
=60:
7b
in both cases
- atb=180
M
-q7b
arms are on differentsides of common >
ab are
arm.
adjacent angles
&
=>Linear Pair
When
of angles
a ray stands on a line and form an
adjacentangle of sum 180
pair
angles.
=>vertically opposite
is called linear
angles (V.0.Al
When two lines intersect each other at a point,
then the opposite angle formed are called V.O.A.
* - they are always equal.
r
[
an
·Ella
a
=
b
c =d

=>
Intersecting and Non-intersecting lines
M 7
C >
intersecting lines
non-intersecting" lines
14-the
lengths of the common perpendicular at different
points on these
parallel lines are same. This equal
lengths is called the distance between parallel
lines
Important Axioms:
Axiom 1) Ifa way stands on a line, then the sum oftwo adjacent
so
angles formed is 1800
Axiom 2)
then the
If the sum of two adjacent angles is 1800
non-common arms of the
angles form a line.
Importantheorems:
Theorem 1)
If
two lines intersect each other, then the
vertically
opposite angles
are equal.
Theorem 2) Lines which are parallel
to the same line are
parallel
to each other.

-...
Triangles
......
= Two
figures are said to be congruent,if they have same
shape
and size.
->Two line
segments
are equal when their
lengths are
equal.
-Two crcles are
congruent when radio are equal.
->SAS congruence rule: Two
triangles are congruent if
two sides and the included angle of one triangle are
equal
to the sides and the included angles of other
triangles.
ASA
=>
conqurence rule: Two A are congruent if two angles
and the included side of one triangle are equal
to
the
angles and the included side of other
=>AAS congruence rule: Two D are
congruent if any two
pairs
are equal.
of
angles
and one
pair of corresponding sides
=>SSS congruence urle: If three
sides of oneI are equal
to the three sides of another A, then the two isare
congruent.
=>RHS congruence rule: If in two
right
and one side of one A are equal to the
hypotenuse
and one side of other , then two are congruent.
the
hypotenuse

=>Angles opposite to equal sides of an isosceles A are equal.
-
BY
If AB =A C
then, LB=L
c
=The sides opposite to equalangles of a B are equal.
of
(B =LC
-
/
then, AB =AC
BD
I c
=>In a A, the
angle opposite to longer side is larger
-
L
=>vice versa, the side opposite to larger angle is larger.
=>Thesumofany two sides of
a 1 is
greater
thatthere
B
C
AB +Al> BC
AB +BC > AC
B + Al > AB

-...
Heron’s Formula
-...........
The area of A when its
height is given=Ixbasex height
but ifthere is no
height?
for example A
triangular park
with three sides
40m, 30m and 20m. How will we calculate?
for that we use Heron's formula:
vi- ---
where a, b and care the sides ofthe triangle
and s is the semiperimeter i.e. = a
for example: sides of A are a
=40m
solution: 3 =4+32
Now,
5-
a
E3-
b
S-c
-
I S =48m
-
=
b24m
c =32m
-
40/m=
=(48 8m
3
=
(48 -24)m =24m
=
=
140-32)m 16m
so
Area of park
asses- you e

-
5
-
-............
Coordinate Geometry
... - ...
y
M
↑
-
5
.
# Quadrant
I
IQuadrant
(-,+) (+, + 1
-
2
-
1
an
negative x-axis
positive
x' n-axis-
11 l 11 I I I I I >x
-4 -3 L 2 3 4 5
--
--2
I Quadrant I
- -3
Qvadvant
-
( -, -
- -4
(+, 7
-2-1· -5
-
e- 3
↓ - y'
·Horizontal line is called n-axis.
·
vertical line is called y-axis.
·Point of intersection of n-axis &y-anis is called origin.
·
Evadvant both - the axes divide the
plain in four equal
parts and each part is called quadrant.
·
coordinate (ordered pair)
-
A coordinate is the mathematical expression for
denoting
the value of X-axis and
y-axis and denoted by (x,y)
wherevalue of his called as abcissa
and, value of is ordinate
y

=>A
linearequationin
-...........
Linear Eqs. in 2 Var.
two variablesaninfinitesolutions.
=An equation ofthe form ax+
by +c
=0, where a, b and c
are real numbers, such that a and b are not both
zero, is called a linear
equation in 2 variables.
=>Every point on the
graph of a linear equation in 2
variables is a solution of the linear equation.
=>And, every solution of the linear equation is a point
on the
graph of the linear equation.
=>Equation of y-axis - x =0
Equation of n-axis - y
=> The
graph a =a is a line parallel
to
y-anis
The
graph y=b is a line parallel tox-anis.
=0