Summary - Mathematics, physics and chemistry formulas all in one

IIT JEE Formulas

Maths Formulas

Part 1

Circle Formula
The formula for circle are as stated below

Description Formula
2
Area of a Circle ● In terms of radius: π𝑟
π 2
● In terms of diameter: 4
×𝑑


Surface Area of a Circle 2
π𝑟

General Equation of a The general equation of a circle with coordinates of a centre(ℎ, 𝑘),
Circle 2 2
and radius 𝑟 is given as: (𝑥 − ℎ) + (𝑦 − 𝑘) = 𝑟
Standard Equation of a The Standard equation of a circle with centre (𝑎, 𝑏), and radius 𝑟 is
Circle 2 2 2
given as: (𝑥 − 𝑎) + (𝑦 − 𝑏) = 𝑟
Diameter of a Circle 2 × radius

Circumference of a Circle 2π𝑟
Intercepts made by Circle 𝑥2 + 𝑦2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0
2
i. On 𝑥 −axis: 2 𝑔 − 𝑐
2
ii. On 𝑦 −axis: 2 𝑓 − 𝑐

Parametric Equations of 𝑥 = ℎ + 𝑟𝑐𝑜𝑠 θ ; 𝑦 = 𝑘 + 𝑟𝑠𝑖𝑛 θ
a Circle
Tangent 2
● Slope form: 𝑦 = 𝑚𝑥±𝑎 1 + 𝑚
2
● Point form: 𝑥𝑥1 + 𝑦𝑦1 = 𝑎 or 𝑇 = 0
● Parametric form: 𝑥𝑐𝑜𝑠 α + 𝑦𝑠𝑖𝑛 α = 𝑎

Pair of Tangents from a 2
𝑆𝑆1 = 𝑇
Point:

,Length of a Tangent 𝑆1

Director Circle 2 2 2 2 2 2
𝑥 + 𝑦 = 2𝑎 for 𝑥 + 𝑦 = 𝑎

Chord of Contact 𝑇=0
2𝐿𝑅
i. Length of chord of contact= 2 2
𝑅 +𝐿
ii. Area of the triangle formed by the pair of the
3
𝑅𝐿
tangents and its chord of contact = 2 2
𝑅 +𝐿
iii. Tangent of the angle between the pair of tangents
(
from 𝑥1, 𝑦1 = ) ( )
𝐿 −𝑅
2𝑅𝐿
2 2


iv. Equation of the circle circumscribing the triangle
𝑃𝑇1, 𝑇2 is:
(𝑥 − 𝑥1)(𝑥 + 𝑔) + (𝑦 − 𝑦1)(𝑦 + 𝑓) = 0
Condition of 2𝑔1𝑔2 + 2𝑓1𝑓2 = 𝑐1 + 𝑐2
orthogonality of Two
Circles
Radical Axis ( )
𝑆1 − 𝑆2 = 0 i.e. 2 𝑔1 − 𝑔2 𝑥 + 2 𝑓1 − 𝑓2 𝑦 + 𝑐1 − 𝑐2 = 0. ( ) ( )
Family of Circles 𝑆1 + 𝐾𝑆2 = 0, 𝑆 + 𝐾𝐿 = 0


Quadratic Equation Formula
The formula for quadratic equation are as stated below

Description Formula
General form of 2
𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0; where 𝑎, 𝑏, 𝑐 are constants and 𝑎≠0.
Quadratic Equation
Roots of equations −𝑏+ 𝑏 −4𝑎𝑐
2 2
−𝑏− 𝑏 −4𝑎𝑐
α= 2𝑎
, β= 2𝑎
Sum and Product of If α and β are the roots of the quadratic equation
Roots 2
𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0, then
𝑏
Sum of roots, α + β =− 𝑎
𝑐
Product of roots, αβ = 𝑎


Discriminant of 2
The Discriminant of the quadratic equation 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 is
Quadratic equation 2
given by 𝐷 = 𝑏 − 4𝑎𝑐.

Nature of Roots 𝑏
● If 𝐷 = 0, the roots are real and equalα = β =− 2𝑎
.

, ● If 𝐷≠0, The roots are real and unequal.
● If 𝐷 < 0, the roots are imaginary and unequal.
● If 𝐷 > 0 and D is a perfect square, the roots are rational
and unequal.
● If 𝐷 > 0 and 𝐷 is not a perfect square, the roots are
irrational and unequal.

Formation of Quadratic If α and β are the roots of the quadratic equation, then
Equation with given 2
(𝑥 − α)(𝑥 − β) = 0; 𝑥 − (α + β)𝑥 + αβ = 0;
roots 2
● 𝑥 − (𝑆𝑢𝑚 𝑜𝑓 𝑟𝑜𝑜𝑡𝑠)𝑥+ 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑟𝑜𝑜𝑡s=0
Common Roots 2
● If two quadratic equations 𝑎1𝑥 + 𝑏1𝑥 + 𝑐1 = 0 &
2
𝑎2𝑥 + 𝑏2𝑥 + 𝑐2 = 0 have both roots common, then
𝑎1 𝑏1 𝑐1
𝑎2
= 𝑏2
= 𝑐2
.
● If only one root α is common, then
𝑐1𝑎2−𝑐2𝑎1 𝑏1𝑐2−𝑏2𝑐1
α= 𝑎1𝑏2−𝑎2𝑏1
= 𝑐1𝑎2−𝑐2𝑎1


Range of Quadratic 𝑏
● If − 2𝑎
not belong to [𝑥1, 𝑥2] then,
Expression
2
𝑓(𝑥) = 𝑎𝑥 + 𝑏𝑥 + 𝑐 in
[{ ( ) }
𝑓(𝑥)∈ 𝑓 𝑥1 , 𝑓(𝑥2) , 𝑚𝑎𝑥⁡{𝑓 𝑥1 , 𝑓(𝑥2)} ( ) ]
𝑏
restricted domain ● If− 2𝑎
∈[𝑥1, 𝑥2] then,
𝑥∈[𝑥1, 𝑥2]
{( ) ( )
𝑓(𝑥)∈⎡⎢ 𝑓 𝑥1 , 𝑓 𝑥2 , −
⎣
𝐷
4𝑎 }, ( ) ( )
𝑚𝑎𝑥⁡{𝑓 𝑥1 , 𝑓 𝑥2 , −
𝐷
4𝑎
}⎤⎥
⎦
2
Consider the quadratic equation 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0
Roots under special 𝑏
● If 𝑐 = 0, then one root is zero. Other root is− 𝑎
.
cases
● If 𝑏 = 0The roots are equal but in opposite signs.
● If 𝑏 = 𝑐 = 0, then both roots are zero.
● If 𝑎 = 𝑐, then the roots are reciprocal to each other.
● If 𝑎 + 𝑏 + 𝑐 = 0, then one root is 1 and the second root is
𝑐
𝑎
.
● If 𝑎 = 𝑏 = 𝑐 = 0, then the equation will become an
identity and will satisfy every value of 𝑥.

Graph of Quadratic 2
The graph of a quadratic equation 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 is a
equation parabola.
● If 𝑎 > 0, then the graph of a quadratic equation will be
concave upwards.
● If 𝑎 < 0, then the graph of a quadratic equation will be
concave downwards.

, Maximum and Minimum 2
Consider the quadratic expression 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0
value ● If 𝑎 < 0, then the expression has the greatest value at
𝑏 𝐷
𝑥 =− 2𝑎 . The maximum value is − 4𝑎 .
● If 𝑎 > 0, then the expression has the least value at
𝑏 𝐷
𝑥 =− 2𝑎 . The minimum value is − 4𝑎 .

Quadratic Expression in The general form of a quadratic equation in two variables 𝑥 and 𝑦 is
Two Variables 2 2
𝑎𝑥 + 2ℎ𝑥𝑦 + 𝑏𝑦 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐.
To solve the expression into two linear rational factors, the
condition is ∆ = 0
[a h g]
∆= [ h b f ] =0
[g f c]

2 2 2 2
𝑎𝑏𝑐 + 2𝑓𝑔ℎ − 𝑎𝑓 − 𝑏𝑔 − 𝑐ℎ = 0 And ℎ − 𝑎𝑏 > 0. This is
called the Discriminant of the given expression.


Binomial Theorem Formula
Quick formula revision for jee mains and advanced.


Description Formula
Binomial Theorem for 𝑛 𝑛
(𝑥 + 𝑎) =nC0𝑥𝑛𝑎0 + nC1𝑥𝑛−1𝑎 + nC2𝑥𝑛−2𝑎2 + … + nCr𝑥𝑛−𝑟𝑎𝑟 + … + nCn.𝑥𝑎
positive Integral Index General terms = 𝑇𝑟+1 =nCr𝑥𝑛−𝑟𝑎𝑟


Deductions of Binomial 𝑛
● (1 + 𝑥) =nC0+ nC1𝑥 + nC2𝑥2 +nC3𝑥3 + … +nCr𝑥𝑟 + … +nCn𝑥𝑛 which is
Theorem the standard form of binomial expansion.
𝑡ℎ
General Term= (𝑟 + 1) term: 𝑇𝑟+1 =nCr
𝑟 𝑛(𝑛−1)(𝑛−2)……(𝑛−𝑟+1) 𝑟
𝑥 = 𝑟!
.𝑥


● 𝑛 n n
C n
C2 n 3 C
(1 − 𝑥) = 0− 1𝑥 + 2𝑥 − 3𝑥 + … + (− 1)
𝑟nC 𝑟
r𝑥 + … + (− 1)
𝑛n
n𝑥
𝑛C C
𝑡ℎ 𝑟n
General Term= (𝑟 + 1) term: 𝑇𝑟+1 = (− 1) . Cr
𝑟 𝑛(𝑛−1)(𝑛−2)……(𝑛−𝑟+1) 𝑟
𝑥 = 𝑟!
.𝑥



Middle Term in the 𝑡ℎ

expansion of(𝑥 + 𝑎)
𝑛 ● If 𝑛 is even then middle term = ( + 1) term. 𝑛
2
𝑛+1 𝑡ℎ 𝑛+3 𝑡ℎ
● If 𝑛 is odd then middle terms are ( ) and ( ) 2 2
term.
● Binomial coefficients of middle term is the greatest Binomial
coefficients