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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

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


  • 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

Equation of standard parabola:

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


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,




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.



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

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)

Auxiliary circle

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


  • 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


(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 )

Director Circle