JEE Exam » JEE Important Formulas » JEE Maths Formulas- Part 2

# JEE Maths Formulas- Part 2

In this article we will go through maths quick formula revision for JEE 2022. Find the important formulas for Vectors, Parabola, Definite Integration and Ellipse.

## Vectors Formula

The formula for vectors are as stated below

DescriptionFormula
Position Vector of a Point

If a and b are positive vectors of two points A and B, then

AB =b- a

• Distance Formula: Distance between the two points A( a ) and B( b) is

AB= |( a ) ⃗-b ⃗ |

• Section Formula: r ⃗=(n ( a ) ⃗+m b ⃗ )/(m+n)      Midpoint of AB=(( a ) ⃗+b ⃗)/2
Scalar Product of Two vectors

a .b= a b cosθ , where a , b are the magnitude of a and b respectively and θ is the angle between a and b

• i.i=j.j=k.k=1; i.j=j.k=k.i=0 , projection of a  on b= a . b /b
• If ( a ) ⃗=a_1 i+a_2 j+a_3 k & ( b ) ⃗=b_1 i+b_2 j+b_3 k then a ⃗.( b ) ⃗=a_1 b_1+a_2 b_2+a_3 b_3
• The angle ∅ between a & b is given by ∅ =〖cos〗^(-1) (a ⃗.( b ) ⃗)/| ( a ) ⃗ || ( b ) ⃗ | ,0≤∅≤π.
• a . b =0 a Perpendicular to  b   (a≠0, b ≠0)
Vector Product of Two vectors
• If a & b are two vectors and is the angle between them then
• a. b =absinθ n , where n is the unit vector perpendicular to both a & b such that a,b& n form a right handed screw system
• Geometrically a b=area of the parallelogram whose two adjacents sides are represented by a & b
• i.i=j.j=k.k=0;  i.j=k,  j.k=i,  k.i=j
•  a ⃗=a_1 i ̂+a_2 j ̂+a_3 k ̂ & b ⃗=b_1 i ̂+b_2 j ̂+b_3 k ̂ then a ⃗× b ⃗= =  b= o↔  a and b are parallel (collinear) (a≠0, b≠0) i.e. a=K b where K is a scalar. ±( a ⃗× b ⃗)/| a ⃗× b ⃗ |
• Unit vector perpendicular to the plane of a & b is n= ABC=1/2 [a ⃗×b ⃗+b ⃗×c ⃗+c ⃗×a ⃗ ]
• If a,b & c are the position vectors of 3 points A, B & C then the vector area of triangle ABC=1/2ab+bc+ca. The points A, B & C are collinear if
• a ⃗×b ⃗+b ⃗×c ⃗+c ⃗×a ⃗=0 ⃗
• Area of any quadrilateral whose diagonal vectors are (d_1 ) ⃗ & (d_2 ) ⃗ is given by 1/2 |(d_1 ) ⃗×(d_2 ) ⃗ |
• Lagrange’sIdentity:(a ⃗×b ⃗ )^2=|a ⃗ |^2 |b ⃗ |^2-(a ⃗.b ⃗ )^2=[((a) ⃗×a ⃗ ) (a ⃗×(b)) ⃗ ( b ⃗×(a)) ⃗ ( b ⃗×(b)) ⃗]
Scalar Triple Product
• The scalar triple product of three vectors a,b & c  is defined as:

a ⃗×b ⃗.c ⃗=|a ⃗ ||b ⃗ ||c ⃗ | □sin sin θ □cos cos ∅

• Volume of tetrahedron V=[a ⃗.b ⃗.c ⃗ ]
• In a scalar triple product the position of dot and cross can be interchanged i.e.

a ⃗.(b ⃗×c ⃗ )=(a ⃗×b ⃗ ).c ⃗ Or [a ⃗ b ⃗ c ⃗ ]=[b ⃗ c ⃗ a ⃗ ]=[c ⃗ a ⃗ b ⃗ ] a ⃗.(b ⃗×c ⃗ )=-a ⃗.(c ⃗×b ⃗) i.e. [a ⃗ b ⃗ c ⃗ ]=-[a ⃗ c ⃗ b ⃗]

• If a ⃗=a_1 i+a_2 j+a_3 k; b ⃗=b_1 i+b_2 j+b_3 k & c ⃗=c_1 i+c_2 j+c_3 k then

a.b.c=

• If a,b,c are coplanar
[a ⃗ b ⃗ c ⃗ ]
• Volume of tetrahedron OABC with O as origin & A(a), B(b) and C(c) be the vertices = |1/6[a ⃗ b ⃗ c ⃗]|
• The position vector of the centroid of a tetrahedron if the pv’s of its vertices are a,b,c & d are given by 1/4[a ⃗+b ⃗+c ⃗+d ⃗]
Vector Triple Product

a ⃗×(b ⃗×c ⃗ )=(a ⃗.c ⃗ ) b ⃗-(a ⃗.b ⃗ ) c ⃗, (a ⃗×b ⃗ )×c ⃗=(a ⃗.c ⃗ ) b ⃗-(b ⃗.c ⃗)a ⃗

In general: (a ⃗×b ⃗ )×c ⃗≠a ⃗×(b ⃗×c ⃗ )

## Parabola formula

The formula for parabola are as stated below

DescriptionFormula
Equation of standard parabola:

The equation of parabola with focus at (a,0), a>0 and directrix x = -a is given as

y^2=4ax

When vertex is (0, 0) then axis is given as

y = 0

Length of latus rectum is equals to 4a

Ends of the latus rectum are L(a, 2a) and L’(a, -2a).

Parametric representation

The point (x,y1) lies outside, on or inside the parabola which is given as y = 4ax

Therefore, equation of parabola now becomes,

〖y_1〗^2-4ax≥0

Or

〖y_1〗^2-4ax<0

Line and a parabola

Length of the chord intercepted by the parabolay^2=4ax  on the line y = mx+c is given as

4/m^2 (√(a(1+m^2 )(a-mc) )

Tangents to the parabola

Tangent of the parabola y^2=4axis given as T = 0

y=mx+am , m≠0 is the tangent of parabola y^2=4ax at (a/m^2 ,2a/m)

Normal to the parabola y2=4ax

Normal to the paraboly^2=4ax is given as

y-y_1=(-y_1)/2a (x-x_1 )

A chord with a given middle point

The equation of the chord of parabola y^2=4axwith midpoint (x1, y1) is given as T = S1.

Here,

S_1=y_1-4ax

## Definite Integration Formula

The formula for definite integration are as stated below

Description Formula
Definite Integral as Limit Sum

∫_a^b f(x)dx=∑_(r=1)^n hf(a+rh)

Here h=(b-a)/n

is the length of each subinterval

Definite Integral Formula Using the Fundamental theorem of calculus∫_a^b f(x)dx=F(b)-F(a), where F^’ (x)=f(x)
Properties of Definite Integral

∫_a^b f(x).dx=∫_a^b f(t).dt

∫_a^b f(x).dx=-∫_b^a f(x).dx∫_a^b cf(x).dx=c∫_a^b f(x).dx∫_a^b f(x)±g(x).dx=∫_a^b f(x).dx±∫_a^b g(x).dx∫_a^b f(x).dx=∫_a^c f(x).dx+∫_c^b f(x).dx∫_a^b f(x).dx=∫_a^b f(a+b-x).dx∫_0^a f(x).dx=∫_0^a f(a-x).dt

∫_0^2a f(x).dx=2∫_0^a f(x).dx f(2a-x)=f(x)

∫_0^2a f(x).dx=0∫_(-a)^a f(x).dx=2∫_0^a f(x).dxf(-x)=f(x)
Definite Integrals involving Rational or irrational Expression

∫_a^∞ dx/(x^2+a^2 )=π/2a

• ∫_a^∞ (x^m dx)/(x^n+a^n )=(πa^(m-n+1))/(n ((m+1)π)/n) ),0<m+1<n
• ∫_a^∞ (x^(p-1) dx)/(1+x)=π/(□sin sin (pπ) ),0<p<1
• ∫_a^∞ dx/√(a^2-x^2 )=π/2
• ∫_a^∞ √(a^2-x^2 ) dx=(πa^2)/4
Definite Integrals involving Trigonometric Functions
• ∫_0^π mx )nx )dx={0 if m≠n π/2 if m=n m,n positive integers
• ∫_0^π mx )nx )dx={ 0 if m≠n π/2 if m=n m,n positive integers
• ∫_0^π mx )nx )dx={ 0 if m+n even 2m/(m^2-n^2 ) if m=n odd m,n integers
• ∫_0^(π/2) x dx=∫_0^(π/2) x dx=π/4
• ∫_0^(π/2) x dx =∫_0^(π/2) dx =(1.3.5…….2m-1)/(2.4.6……2m).π/2,m=1,2,…
• ∫_0^(π/2) x dx =∫_0^(π/2) dx =(2.4.6….2m)/(1.3.5…2m+1),m=1,2,…
If f(x) is a periodic function with period T
• ∫_0^nT f(x)dx=n∫_0^T f(x)dx,n∈z,∫_a^(a+nT) f(x)dx=n∫_0^T f(x)dx,n∈z, a∈R
• ∫_mT^nT f(x)dx=(n-m)∫_0^T f(x)dx,m,n∈z,∫_nT^(a+nT) f(x)dx=∫_0^a f(x)dx,n∈z, a∈R
• ∫_(a+nT)^(b+nT) f(x)dx=∫_a^a f(x)dx,n∈z, a,b∈R
Leibnitz TheoremIf F(x)=∫_(g(x))^(h(x)) f(t)dt, then (dF(x))/dx=h^’ (x)f(h(x))-g^’ (x)f(g(x))

## Ellipse Formula

The formula for ellipse are as stated below

DescriptionFormula
Standard Equation

x^2/a^2 +y^2/b^2 =1, where

• Eccentricity: e=√(1-b^2/a^2 ) (0<e<1)  Directrices: x=±a/e
• Foci: S=±a e,0. Length of major axes =2a and minor axes =2b
• Vertices: A’=-a,0 & A=a,0.
• Latus Rectum: = (2b^2)/a=2a(1-e^2)

x^2+y^2 = a^2

# Parametric Representation

x=a cos θ & y=b sin θ

# Position of a Point w.r.t. an Ellipse

The point P(x_1,y_1) lies outside, inside or on the ellipse according as;

(x_1^2)/a^2 + (y_1^2)/b^2 -1><or=0

# Line and an Ellipse

The line y=mx+c meets the ellipse
x^2/a^2 +y^2/b^2 =1in two points real, coincident or imaginary according as c^2is < =or > a^2 m^2+b^2

# Tangents

• Slope form: y=mx±√(a^2 m^2+b^2 )
,  point form: 〖xx〗_(1 )/a^2 +〖yy〗_1/b^2 =1
• Parametric form: xcosθ/a+ysinθ/b=1

# Normal

(a^2 x)/x_1 – (b^2 y)/y_1 =a^2-b^2, ax.secθ-by.cosecθ=(a^2-b^2 ), y=mx-(〖(a〗^2-b^2)m)/(√(a^2+b^2 ) m^2 )

x^2+y^2=a^2+b^2