Maths Formulas For JEE Mains - Get Complete List of Important Formulas

JEE Main and JEE Advanced are the two mountains one needs to climb to get to a top technical institute like the IITs or the NITs. Mathematics is a crucial but a haunting subject for many of the JEE aspirants, with a ton of formulas and brain-aching concepts, it ranks at the top of the difficulty level for IIT JEE. To help aspirants Testbook brings a collection of well-organized Maths Formulas For JEE Mains. Now no need to scratch your head to recall one, get all the formulas at your fingertips. For your convenience Maths Formulas for JEE Mains PDF is also available in this article. 

What is the importance of Maths Formulas For JEE Mains?

In the JEE Main, there are instances when you get a direct formula-based question, the only regret some aspirants have is that they were not able to recall the formula.

Apart from that, having all the formulas at your fingertips is an advantage and will lead you through this cut-throat competition.

Every question you encounter during those 3 hours in the examination hall will be based on some formula or concept, and every formula has a concept behind it. Knowing the concept is the same as learning the formulas, if you know the concept, you can produce your own formula - that is the beauty of Science.

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Overview of the Maths Formulas For JEE Mains

Mathematics is a subject that can only be understood through its application in the real world. There are many formulas in each topic in Mathematics, and remembering each and every one is very difficult, hence you need an organised collection of all the important formulas at your study desk.

The entire syllabus of the JEE Main Mathematics is divided into 22 topics, these topics are -

• Logarithms
• Trigonometry
• Inverse Trigonometric Functions
• Quadratic Equations
• Sequence and Series
• Binomial Theorem
• Complex Numbers
• Matrices
• Determinants
• Properties and Solutions of Tringle
• Straight Lines
• Circles
• Parabola
• Ellipse
• Hyperbola
• Limits, Continuity and Differentiability
• Differential Calculus
• Tangents and Normals
• Integral Calculus
• Vectors and 3-D Geometry
• Probability and Statistics
• Mathematical Reasoning

This breakdown of the Mathematics syllabus will keep the article clean and visible and will be easy to navigate through this chaos.

We suggest you to not mug up the formulas without understanding but apply the same in practice problems, this will help you to keep all these formulas within your grip.

Download JEE Main Maths Important Formulas PDF

Topic - 1: Logarithms

• (x*y)=x+y
• (x/y)=x-y
• xy=y*x
• ab=1/ba
• ba=ca/cbBase Change
• aax=x

Topic - 2: Trigonometry

• Radians=180o=200 Grade
• 2+2=1
• 2=1+2
• 2=1+2
• 0 in the first and second quadrant.
• 0 in the first and fourth quadrant.
• 0 in the first and third quadrant.
• ( 2n+ )= and [ (2n+1)+ ]= -
• ( 2n+ )= and [ (2n+1)+ ]= -
• ( 2n- )= - and [ (2n+1)- ]=
• [ (/2) ]=
• [ (/2) ]=
• (n*)=0 and [ (2n+1)/2 ]=0
• ()=**
• ()=**
• ()=[ ]/[ 1* ]
• =2*[ ()/2 ]*[ ()/2 ]
• +=2*[ (+)/2 ]*[ (-)/2 ]
• -=2*[ (+)/2 ]*[ (-)/2 ]
• 2**=(+)+(-)
• 2**=(+)+(-)
• 2**=(-)-(+)
• (2*)=2**
• (2*)=2-2=2*2-1=1-2*2
• (3*)=3*-4*2
• (3*)=4*3-3*
• (3*)=( 3*-3 )/(1-3*2)
• 2-2=(+)*(-)=2-2
• 2-2=(+)*(-)
• k=0n-1(+k*) =[ +{ (n-1)/2 } * ](/2) *(n/2)
• k=0n-1(+k*) =[ +{ (n-1)/2 } * ](/2) *(n/2)
• A=BA=n+(-1)n*B
• A=BA=2nB
• A=BA=n+B

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Topic - 3: Inverse Trigonometric Function

• Domain and Range of Inverse Trigonometric Functions -
  • x[ -1, 1 ] and x[ -/2, /2 ]
  • x[ -1, 1 ] and x[ 0, ]
  • x(-, ) and x( -/2, /2 )
• Properties of Inverse Trigonometric Functions -
  • (x)=x
  • (x)=x
  • (x)=x
  • -1x=(1/x) for |x|1
  • -1x=(1/x) for |x1|
  • -1x=-1(1/x) for x>0 and -1x=+-1x for x<0
  • -1(-x)= --1x
  • -1(-x)=--1x
  • -1(-x)= --1x
  • -1(-x)=--1x
  • -1x+-1x=/2
  • -1x+-1x=/2
  • -1x+-1x=/2
  • -1x+-1y=-1[ (x+y)(1-xy) ] when x>0, y>0 and 0<xy<1
  • -1x--1y=-1[ (x-y)(1+xy) ] when x>0, y>0
  • -1x--1y=-1[ x1-y2-y1-x2 ] for x>0, y>0 and x2+y2<1
  • -1x--1y=-1[ xy-1-x21-y2 ] for x<y
  • -1[ 2x/(1+x2) ]=2-1x when |x|1
  • -1[ (1-x2)/(1+x2) ]=2-1x when x0
  • -1[ 2x/(1-x2) ]=2-1x when |x|<1
  • -1(3x-4x3)=3-1x when |x|1/2
  • -1(4x3-3x)=3-1x when 1/2x1
  • -1( 2x1-x2 )=2-1x when |x|1/2
  • -1( 2x2-1 )=2-1x when 0x1

Topic - 4: Quadratic Equation

• ax2+bx+c=0x=(1/2a)(-bD) where D=b2-4ac
• If and are the roots of f(x)=ax2+bx+c, then += -b/a and *=c/a
• D>0 Roots are real
• D=0 Roots are real and identical
• D<0 Roots are Imaginary
• If a+ib is one root then a-ib must be the other root.
• Roots of ax2+bx+c=0 and x2+x+=0 then a=b=c
• If a , b0 then a+b2*a*b
• If f(x)=ax2+bx+c and a>0 then f(x) ( 4ac-b2 )/4a
• If f(x)=ax2+bx+c and a<0 then f(x) ( 4ac-b2 )/4a
• f(x,y)=ax2+by2+2hxy+2gx+2fy+c can be resolved into two linear factors if and only if abc+2fgh-af2-bg2-ch2=0

Topic - 5: Sequence and Series

• If the first term is a and common difference is d then the nth term of AP is an=a+(n-1)*d
• k=1na+(k-1)*d=n2[ 2a+(n-1)d ]=n2(a+an)
• If an=An+B then a=A+B and d=B
• If Sn=An2+Bn then a=A+B and d=2*A
• The Arithmetic Mean of a and b is (a+b) / 2
• If a , A1 , A2 , A3 , ... , An , b are in AP with Ak (k=1,2,3,...,n) is the Arithmetic Mean, then the common difference is d=(b-a)/(n+1)
• If the first term is a and the common ratio is r then the nth term of GP is an=a*rn-1
• k=1na*rk-1=a * (rn-1)r-1 where r1
• k=1a*rk-1=a1-r where |r|<1
• The Geometric Mean of a and b is a*b
• If a , G1 , G2 , G3 , ... , Gn , b are in GP with Gk (k=1,2,3,...,n) are the Geometirc Mean, then the common ratio is r=(b/a)1/(n+1)
• If a1 , a2 , a3 , ... , an>0 are terms in GP then a1 , a2 , a3 , ... , an are in AP
• If the first term is a and the common difference is d then the nth term of HP is an=1 / [ a+(n-1)*d ]
• The Harmonic Mean of a and b is 2ab/(a+b)
• AMGMHM
• k=1nk=n(n+1)2, k=1nk2=n(n+1)(2n+1)6 and k=1nk3=n2(n+1)24

Topic - 6: Binomial Theorem

• (x+y)n = nC0xn + nC1xn-1y + nC2xn-2y2 +...+ nCnyn
• The (r+1)th term of (x+y)n is Tr+1 = nCrxn-ryr
• Cx=Cyx=y or x+y=n
• Cr-1+Cr = n+1Cr
• k=0nCkk+1=2n+1-1n+1 and k=0n(-1)kCkk+1=1n+1
• C0+C1+C2+...+Cn=2n and C02+C12+C22+...+Cn2=2nCn
• The greatest coefficient of (x+y)n is Cn/2 is n is even or C(n-1)/2 and C(n+1)/2 if n is odd.
• (1+x)k=1+kx+k(k+1)x2/2!+k(k+1)(k+2)x3/3!+... if kR
• (1-x)-1=1+x+x2+x3+...
• (1+x)-1=1-x+x2-x3+...
• ex= k=0xkk! and (1+x)= k=1(-1)k+1xkk where x -1

Topic - 7: Complex Numbers

• If z=a+ib then z=a-ib
• z+z=2*Re(z), z-z=2*i Im(z) and zz=a2+b2
• |z|=a2+b2 and Arg(z)=-1(y/x)
• Arg(z1*z2)=Arg(z1)+Arg(z2) , Arg(z1/z2)=Arg(z1)-Arg(z2) and Arg(zn)=n*Arg(z)
• z=r(+i)=rei where r is the magnitude and is the argument.
• Euler’s Formula is ei=+i
• 2=ei+e-i and 2=ei-e-i
• If f(+i)=a+ib then f(-i)=a-ib
• The triangle inequalities are | z1+z2 || z1 |+| z2 | and | z1-z2 || | z1 |-| z2 | |
• If z=r(+i) then zn=rn[ (n)+i(n) ], this is De Moiver’s Theorem
• Cube Root of Unity is 1, and 2 where = -1 + i32 and 2= -1 - i32
• 1**2=1 and 1++2=0
• a2+b2+c2-ab-bc-ca=(a+b+c2)(a+b2+c)
• a3+b3=(a+b)(a+b2)(a2+b)
• x2+x+1=(x-)(x-2)
• If z=r(+i) then z1/n=r1/n[ {(+2k)/n}+i{(+2k)/n} ]
• Equation of a Circle with center z0 and radius r is | z-z0 |=r

Topic - 8: Matrices

• Sum of diagonal elements or the Trace of a Matrix is tr(A)=i=1naii.
• Two Matrices [aij]=[bij] are equal then aij=bij.
• [aij]+[bij]=[aij+bij] and k*[aij]=[k*aij].
• Define Amn=[aij] and Bnp=[bij], then AB= [k=1naikbkj].
• Cayley-Hamilton’s Theorem: | A-I |=0 where is the Eigenvalue.
• Commutative Property of Matrix Addition A+B=B+A.
• Associative Property of Matrix Addition (A+B)+C=A+(B+C).
• Associative Property of Matrix Multiplication (AB)C=A(BC).
• Distributive Property of Matrices A(B+C)=AB+AC and (B+C)A=BA+CA.
• Define n, mN then AnAm=An+m, (An)m=Anm and In=I where I is the Identity Matrix.
• Orthogonal Matrix A-1=AT or ATA=I.
• Symmetric Matrix AT=A and Skew-Symmetric Matrix AT= -A.
• The diagonal elements of a Skew-Symmetric Matrix are 0, that is aii=0.
• A(adjA)=(adjA)A=|A| In| adjA |=| A |n-1.
• adj(adjA)=| A |n-2A and | adj(adjA) |=| A |(n-1)(n-1).
• adj(AB)=(adjB)(adjA) and adj(kA)=kn-1adjA where k is a constant.
• The Inverse of a Matrix is A-1=1| A | adjA, the inverse exists only when | A |0.
• Inverse of a Matrix Product is (AB)-1=B-1A-1.
• The solution to the system of equations AX=B is X=adjA| A |B where | A |0.
• If | A |0 but (adjA)B=0 then the solution is x=y=z=0.
• If | A |=0 and (adjA)B=0 then the system of equations has Infinite Solutions.
• If | A |=0 and (adjA)B0 then there is NO Solution.

Topic - 9: Determinants

• For an nn Determinant, if ak is the k th element in ith row with Cofactor AK, then the value of hte Determinant is k=1nakAk.
• The value of the Determinant does not change if the rows and columns are interchanged.
• Interchanging the i th row with (i1) th row will change the sign of the Determinant.
• If any two rows (or columns) are proportional then the Determinant evaluates to Zero.

• Row Operation .
• If the Determinant is then its derivative is written as
• Multiplication of Determinants | [ aij ] |*| [ bij ] |=| [ aij*bij ] |.
• Determinant of a Skew-Symmetric Matrix is Zero.
• If we have akx+bky+ck=dk k=1,2,3 then the solution using the Cramer’s Rule is x=1 / , y=2 / and z=3 / where 1, 2, 3 and are as shown

Topi - 10: Properties and Solutions of a Triangle

• The Law of Sine is aA=bB=cC
• The Law of Cosine is A= b2+c2-a22bc
• The Law of Tangent is [ (A-B)/2 ]= a-ba+b(A/2)
• The Projection Formula is a=bC+cB
• (A/2)= (s-b)(s-c)bc , (A/2)= s(s-a)bc and (A/2)= s(s-a) where s= a+b+c2
• Area of a Triangle is =s(s-a)(s-b)(s-c) where s= a+b+c2
• Area of an Equilateral Triangle is eq= 34l2 where l is the length of each side of the triangle.
• Area of a Triangle given an angle is =12bcA=12caB=12abC
• Radius of the Circumcenter is R= a2A=b2B=c2C=abc4
• Radius of Incircle is r=/s=(s-a)(A/2)=(s-b)(B/2)=(s-c)(C/2)
• Radius of Excircle is r1= s-a , r2= s-b and r3= s-c
• Length of Median from angle A is mA=122b2+2c2-a2
• Length of Angle Bisector of A is bA= 2bcb+c(A/2)
• Length of Altitude from A is hA= aB + C
• mA2+bA2+hA2=34( a2+b2+c2 )
• For an Isosceles Triangle aB=bA
• r= a2 (/n) and R= a2 (/n) for a Regular Polygon with n sides and side length a
• Area of a Cyclic Quadrilateral with sides a, b , c and d is =(s-a)(s-b)(s-c)(s-d) where s=(a+b+c+d)/2
• Ptolemy’s Theorem is AC*BD=a*c+b*d

Topic - 11: Straight Lines

• x=r and y=r where r=x2+y2 and =-1(y/x)
• Distance Formula in 2-D is d=(x1-x2)2+(y1-y2)2
• Coordinates for Internal Division is ( mx2+nx1m+n, my2+ny1m+n )
• Coordinates for External Division is ( mx2-nx1m-n, my2-ny1m-n )
• Coordinates of the Centroid is ( x1+x2+x33, y1+y2+y33 )
• Coordinates of Incenter is ( ax1+bx2+cx3a+b+c, ay1+by2+cy3a+b+c )
• Area of a Triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is

Latex Code:

• Equation of x-axis is y=0 and the Equation of y-axis is x=0
• Slope of a line is given as m= y2-y1x2-x1
• Slope Intercept Form of a Line: y=mx+c
• Point Slope Form of a Line: y-y1=m(x-x1)
• Two Point Form of a Line: y-y1= y2-y1x2-x1 (x-x1)
• Intercept Form of a Line: xa+yb=1
• Normal Form of a Line: x+y=p
• Parametric Form of a Line: x=h+r and y=k+r
• General Form of a Line: ax+by+c=0
• Angle between two lines is =-1( m1-m21+m1m2 )
• Lines are Parallel if and only if a1a2=b1b2c1c2
• Lines are Perpendicular if and only if a1a2+b1b2=0
• Lines are Coincident if and only if a1a2=b1b2=c1c2
• Lines are Intersecting if and only if a1a2b1b2
• Length of Perpendicular from (x1,y1) to the line ax+by+c=0 is
• Distance between two Parallel Lines is
• Line parallel to ax+by+c=0 has the equation ax+by+k=0 where ck
• Line perpendicular to ax+by+c=0 is bx-ay+=0
• Three lines of the form ax+by+c=0 are concurrent if and only if
• Linear Translation of axes by (, ): xnew=x- and ynew=y-
• Rotation of axes by angle anti-clockwise: xnew=x+y and ynew= -x+y
• Equation of the line passing through the intersection of a1x+b1y+c1=0 and a2x+b2y+c2=0 is a1x+b1y+c1+(a2x+b2y+c2)=0
• ax2+2hxy+by2+2gx+2fy+c=0 is the equation for pair of straight lines if and only if or abc+2fgh-af2-bg2-ch2=0
• For a pair of straight line ax2+2hxy+by2+2gx+2fy+c=0, angle between them is =-1( 2h2-aba+b ). Lines are Perpendicular if a+b=0
• Equation of the Angle Bisectors of ax2+2hxy+by2=0 is x2-y2a-b=xyh
• Pair of Straight Lines perpendicular to ax2+2hxy+by2=0 is bx2-2hxy+ay2=0
• Coordinates of the foot of the perpendicular (h, k) from (x1, y1) to the line ax+by+c=0 is h-x1a=k-y1b= -ax1+by1+ca2+b2
• Coordinate (h, k) of the image of a point (x1, y1) over the line ax+by+c=0 is h-x1a=k-y1b= -2 * (ax1+by1+c)a2+b2

Topic - 12: Circle

• Standard Equation of a Circle is (x-h)2+(y-k)2=r2 where the centre is (h, k) with Radius r.
• General Equation of a Circle is x2+y2+2gx+2fy+c=0 where the coordinates of the centre is (-g, -f) with Radius g2+f2-c.
• If g2+f2-c=0 then the Circle is actually the point (-g, -f).
• If g2+f2-c<0 then the Circle is Imaginary.
• If the end-points of the Diameter are (x1, y1) and (x2, y2), then the Equation of the Circle is (x-x1)(x-x2)+(y-y1)(y-y2)=0.
• The parametric form of (x-h)2+(y-k)2=r2 is x=h+r and y=k+r.
• The parametric form of x2+y2+2gx+2fy+c=0 is x= -g+g2+f2-c and y= -k+g2+f2-c.
• The point (r, r) is simply written as point .
• The equation of a line joining the points and of the circle x2+y2=a2 is x{(+)/2}+y{(+)/2}=a{(-)/2}.
• Define C(x, y)=x2+y2+2gx+2fy+c and (a,b) is any arbitrary point, then -
  • If C(a, b)=0 The point is on the boundary of the Circle.
  • If C(a, b)<0 The point lies inside the Circle.
  • If C(a, b)>0 The point lies outside the Circle.
• For x2+y2=r2, the Equation of the Tangent at (a, b) is ax+by=r2.
• For x2+y2=r2, the Equation of the Tangent at (r, r) is x+y=r.
• Two Tangents at points and of the Circle x2+y2=r2 intersects each other at the point

• The Equation of Tangent for the Circle x2+y2+2gx+2fy+c=0 at the point (a, b) is ax+by+g(x+a)+f(y+b)+c=0.
• Equation of the Tangent on the Circle x2+y2=r2 with Slope m is y=mxr1+m2.
• Equation of the Tangent on the Circle (x-h)2+(y-k)2=r2 with Slope m is written as y-k=(x-h)r1+m2.
• The length of Tangent from (a, b) to the Circle C(x, y) is C(a, b).
• From an external point (a, b) two Tangents can be drawn to the Circle C(x, y), the Equation of the pair of Tangents is C(x, y)C(a, b)=T2(x, y) where T(x, y)=ax+by-r2
• The Equation of the Normal to the Circle with center (-g, -f) and on the point (a, b) is written as y-b=[ (b+f)/(a+g) ](x-a).
• The Equation of the Chord of Contact is T(x, y)=0.
• If the midpoint of the Chord is (a, b), then the Equation of Chord is T(x, y)=S(a, b).
• If C1(x, y)=0 and C2(x, y)=0 are two given Circles, then the family of Circles passing through the point of intersection of the Circles is C1(x, y)+C2(x, y)=0.
• The angle between two Circles with Radius r1 and r2 and d being the distance between Radii is given by =-1[ (r12+r22-d)/2r1r2 ].
• Radical axis of two Circles is C1(x, y)=C2(x, y).

Topic - 13: Parabola

• Eccentricity of a Parabola is 1.
• There are four types of Parabolas - Upward, Downward, Leftward and Rightward openings.
• Equation of Rightward Opening Parabola is y2=4ax where a is the distance between the Center and the Vertex of Parabola.
• Equation of Leftward Opening Parabola is y2= -4ax where a is the distance between the Center and the Vertex of Parabola.
• Equation of Upward Opening Parabola is x2=4ay where a is the distance between the Center and the Vertex of Parabola.
• Equation of Downward Opening Parabola is x2= -4ay where a is the distance between the Center and the Vertex of Parabola.
• Length of Latus Rectum of y2= 4ax or x2= 4ay is 4a.
• Parametric representation of y2=4ax is x=at2 and y=2at.
• Define P(x, y)=y2-4ax and any arbitrary point (a, b), then -
  • P(a, b)=0 The point is on the boundary of the Parabola.
  • P(a, b)<0 The point is inside the Parabola.
  • P(a, b)>0 The point is outside the Parabola.
• Equation of a Chord joining the points t1 and t2 is (t1+t2)y=2x+2at1t2.
• Equation of Tangent to y2=4ax with Slope m is y=mx+(a/m).
• Equation of Tangent to x2=4ay with Slope m is y=mx-am2.
• Equation of Tangent to y2=4ax at the point (a, b) is by=2a(x+a).
• Equation of Tangent to y2=4ax at the point t is ty=x+at2.
• Equation of Normal to y2=4cx at the point (a, b) is y-b=(-b/2c)(x-a).
• Equation of Normal to y2=4ax with Slope m is y=mx-2am-am3.
• Equation of Normal to y2=4ax at the point t is y+tx=2at+at2.
• Define P(x, y)=y2-4cx and T(x, y)=by-2c(x+a), the Equation for the pair of Tangents drawn from the point (a, b) is P(x, y)P(a, b)=T2(x, y).
• The Equation for the Chord of Contact is T(x, y)=0 where the Tangents to the Parabola is drawn from an external point (a, b).
• Equation of the Chord with midpoint (a, b) is T(x, y)=P(a, b).
• Length of Co-Tangent is 2at2 and the length of Co-Normal is 2a.

Topic - 14: Ellipse

• Equation of a standard Horizontal Ellipse is x2a2+y2b2=1 where a>b.
• The Eccentricity of the Horizontal Ellipse is e=1-b2a2(0,1).
• Coordinates of the Foci of a Horizontal Ellipse are (ae, 0).
• Coordinates of the Vertices of a Horizontal Ellipse are (a, 0).
• Coordinates of the Co-Vertices of a Horizontal Ellipse are (0, b)
• The equation of Directrix of a Horizontal Ellipse are x= a/e.
• Length of the Latus Rectum of a Horizontal Ellipse is 2b2/a.
• Equation of the Latus Rectum of a Horizontal Ellipse is x= ae.
• Equation of a standard Vertical Ellipse is x2a2+y2b2=1 where a<b.
• The eccentricity of the Vertical Ellipse is e=1-a2b2.
• Coordinates of the Foci of a Vertical Ellipse are (0, be).
• Coordinates of the Vertices of a Vertical Ellipse are (0, b).
• Coordinates of the Co-Vertices of a Vertical Ellipse are (a, 0).
• The equation of Directrix of a Vertical Ellipse are y= b/e.
• Length of the Latus Rectum of a Vertical Ellipse is 2a2/b.
• Equation of the Latus Rectum of a Vertical Ellipse is y= be.
• Define E(x, y)=x2a2+y2b2-1 and an arbitrary point (a, b), then
  • E>0 The point is outside the Ellipse.
  • E=0 The point is on the boundary of the Ellipse.
  • E<0 The point is inside the Ellipse.
• Parametric Form of x2a2+y2b2=1 is x=a and y=b.
• The Equation of a Chord joining two points and of x2a2+y2b2=1 is xa+2+yb+2=-2.
• The Equation of the Tangent to x2a2+y2b2=1 with Slope m is y=mxa2m2+b2.
• The Equation of the Tangent to x2a2+y2b2=1 at (p, q) is pxa2+qyb2=1.
• The Equation of the Tangent to x2a2+y2b2=1 at the point is xa+yb=1.
• The Equation of the Normal to x2a2+y2b2=1 at the point (p, q) is a2px-b2q =a2-b2.
• The Equation of the Normal to x2a2+y2b2=1 with Slope m is y=mx(a2-b2)ma2+b2m.
• The Equation of the Normal to x2a2+y2b2=1 at the point is ax-by=a2-b2.
• The Equation for the Chord of Contact for x2a2+y2b2=1 is T(x, y)=0.
• The Equation for the pair of Tangents to x2a2+y2b2=1 is E(x, y)E(p, q)=T2(x, y) where the Tangents are drawn from the external point (p, q).
• The Equation of the Director Circle of x2a2+y2b2=1 is x2+y2=a2+b2.
• The Equation of a Chord in x2a2+y2b2=1 with mid-point at (p, q) is T(x, y)=E(p, q).
• The point of intersection of two Tangents at the points and is

Topic - 15: Hyperbola

• The Equation of a standard Horizontal Hyperbola is x2a2-y2b2=1.
• The Eccentricity of a Horizontal Hyperbola is e=1+b2a2.
• The Coordinates of the Foci of a Horizontal Hyperbola are (ae, 0).
• The Coordinates of the Vertices of a Horizontal Hyperbola are (a, 0).
• The Equation of the Directrix for a Horizontal Hyperbola are x= a/e.
• The Equation of the Latus Rectum is x= ae.
• The Length of the Latus Rectum is 2b2/a.
• The Equation of the Conjugate Hyperbola is -x2a2+y2b2=1.
• If the Eccentricities of Horizontal and Vertical Hyperbolas are e1 and e2, then 1e12+1e22=1.
• The Equation of a Rectangular Hyperbola is x2-y2=a2 or xy=c2.
• The Eccentricity of a Rectangular Hyperbola is 2.
• The Parametric Form of x2a2-y2b2=1 is x=a and y=b.
• Define H(x, y)=x2a2-y2b2-1 and an arbitrary point (p, q), then
  • H(p, q)<0 The point lies inside the Hyperbola.
  • H(p, q)=0 The point lies on the boundary of the Hyperbola.
  • H(p, q)>0 The point lies outside the Hyperbola.
• The Equation of the Tangent to x2a2-y2b2=1 at the point (p, q) is pxa2-qyb2=1.
• The Equation of the Tangent to x2a2-y2b2=1 with Slope m is y=mxa2m2-b2.
• The Equation of the Tangent to x2a2-y2b2=1 at the point is xa-yb=1.
• The Equation of the Normal to x2a2-y2b2=1 at the point (p, q) is a2px+b2qy =a2+b2.
• The Equation of the Normal to x2a2-y2b2=1 with Slope m is y=mx m(a2+b2)a2-b2m2.
• The Equation of the Normal to x2a2-y2b2=1 at the point is ax+by =a2+b2.
• The Equation for the Chord of Contact is T(x, y)=0.
• The Equation for the pairs of Tangents is H(x, y)H(p, q)=T2(x, y) where the Tangents to the Hyperbola is drawn from an external point (p, q).
• The Equation of the Chord with midpoint (p, q) is T(x, y)=H(p, q).
• The Equation of the pair of Asymptotes of x2a2-y2b2=1 is x2a2-y2b2=0.
• The Equation of the Tangent to a Rectangular Hyperbola at (p, q) is xp+yq=1.
• The Parametric Form of Rectangular Hyperbola xy=c2 is x=ct and y=c/t where t0.
• The Equation of Normal to the Rectangular Hyperbola is y-(c/t)=t2(x-ct).

Topic - 16: Limits, Continuity and Differentiability

• Sum Rule: xp[ f(x)+g(x) ]=xpf(x)+xpg(x)
• Difference Rule: xp[ f(x)-g(x) ]=xpf(x)-xpg(x)
• Product Rule: xpf(x)g(x)=xpf(x)xpg(x)
• Quotient Rule: xp[ f(x)g(x) ]=xpf(x)xpg(x) provided xpg(x)0
• Scalar Multiple Rule: xpk*f(x)=k*xpf(x)
• Power Rule: xp[ f(x) ]k=k[ xpf(x) ]
• Composite Function Rule: xpf(g(x))=f(xpg(x)) provided f(x) exists at xpg(x)
• Important Limit Properties -
  • x0[ xx ]=1=x0[ xx ]=x0[ xx ]=x0[ xx ]
  • xp[ ( xn-pn )( x-p ) ]=npn-1
  • x0[ (1+x)x ]=1
  • x0[ ( ex-1 )x ]=1 and x0[ ( ax-1 )x ]=a where a>0
  • x0(1+x)1/x=e and x(1+1/x)x=e
  • If xpf(x)=1 and xpg(x)= then xp[ f(x) ]g(x)=exp[ xpg(x){ f(x)-1 } ]
• L’ Hospital Rule: xp[ f(x)g(x) ]=xp[ f'(x)-g'(x) ]
• Missing Point Discontinuity: xpf(x) exists, f(p) undefined
• Isolated Point Discontinuity: xpf(x)f(p)
• Finite Type Discontinuity: Left-Hand LimitRight-Hand Limit
• Infinite Type Discontinuity: Either Left-Hand Limit or Right-Hand Limit
• Oscillatory Type Discontinuity: Limit oscillates between two finite numbers.
• Intermediate Value Theorem: If f(x) is continuous in the interval [ a, b ] then f(c)=K for some

c(a, b) and K( f(a), f(b) ).

• Right-hand Derivative: f(p+)=h0[ { f(p+h)-f(p) }h ]
• Left-Hand Derivative: f(p-)=h0[ { f(p)-f(p-h) }h ]
• A function is differentiable only if f(p+)=f(p-) at a point x=p

Topic - 17: Differential Calculus

• Sum Rule: (f+g)'(x)=f'(x)+g'(x)
• Difference Rule: (f-g)'(x)=f'(x)-g'(x)
• Product Rule: (fg)'(x)=(f'g)(x)+(fg')(x)
• Quotient Rule: (fg)(x)=[ (gf')(x)-(fg')(x) ]g2(x)
• Scalar Multiple Rule: (k*f(x))'=k*f'(x)
• Chain Rule: [ f{ g(x) } ]'=f'{ g(x) }*g'(x)
• Derivative of some common functions -
  • ( Constant )'=0
  • ( xn )'=nxn-1
  • ( ex )'=ex and ( ax )'=axa where a>0
  • (x)'=1/x
  • (px)'=1/(xp)
  • ( x )'=x and ( x )'= -x
  • ( x )'=2x and ( x )'= -2x
  • ( x )'=xx and ( x )'= -xx
  • ( x )'=1/1-x2 where | x |<1
  • ( x )'= -1/1-x2 where | x |<1
  • ( x )'=1/(1+x2)
• Logarithmic Differentiation: y=[ f(x) ]g(x) then y'=[ f(x) ]g(x)[ { g(x)f(x) }+g'(x)f(x) ]

provided f(x)>0 xDomain

• If f(x,y)=0 then dydx= -df/dxdf/dy
• If x=1(t) and y=2(t) then dydx=dy/dtdx/dt
• If g(x)=f-1(x) or g(f(x))=x then f'(x)g'[ f(x) ]=1
• Maxima/Minima are of two types - Local and Global
• f'(p)=0 and f''(p)<0 then f(x) is maximum in the neighborhood of x=p
• f'(p)=0 and f''(p)>0 then f(x) is minimum in the neighborhood of x=p

Topic - 18: Tangents and Normals

• Tangent on y=f(x) at the point x=p has the Slope
• Normal on y=f(x) at the point x=p has the Slope
• Length of Tangent to the curve y=f(x) at the point (x1, y1) is
• Length of Sub-Tangent to the curve y=f(x) at the point ( x1, y1 ) is
• Length of Normal to the curve y=f(x) at the point ( x1, y1 ) is

• Length of Subnormal to the curve y=f(x) at the point ( x1, y1 ) is

Topic - 19: Integral Calculus

• Indefinite Integral of some common functions -
  • (ax+b)n dx=(ax+b)n+1a(n+1)+C where n -1
  • 1ax+b dx=| ax+b |a+C
  • (ax+b) dx=(ax+b)a+C
  • pax+b dx=pax+bap+C where p>0
  • (ax+b) dx= -(ax+b)a+C
  • (ax+b) dx=(ax+b)a+C
  • (ax+b) dx=| (ax+b) |a+C
  • (ax+b) dx=| (ax+b) |a+C
  • 2(ax+b) dx=(ax+b)a+C
  • 2(ax+b) dx= -(ax+b)a+C
  • (ax+b)(ax+b) dx= -(ax+b)a+C
  • (ax+b) dx=| (ax+b)+(ax+b) |a+C
  • (ax+b) dx=| (x+b)-(ax+b) |a+C
  • 1a2-x2 dx=-1xa+C
  • 1a2+x2 dx=1a-1xa+C
  • 1xx2-1 dx=1a-1xa+C
  • 1a2+x2 dx=-1xa+C
  • 1x2-a2 dx=-1xa+C
  • 1a2-x2 dx=12a|a+xa-x|+C
  • 1x2-a2 dx=12a|x-ax+a|+C
  • a2-x2 dx=x2a2-x2+a22-1x+C
  • x2+a2 dx=x2x2+a2+a22-1xa+C
  • x2-a2 dx=x2x2-a2+a22-1xa+C
  • eax(bx) dx=eaxa2+b2[a(bx)-b(bx)]+C
  • eax(bx) dx=eaxa2+b2[a(bx)+b(bx)]+C
• ex[f(x)+f'(x)]=exf(x)+C
• ef(x)[1+xf'(x)]=xef(x)+C
• f(x)g(x) dx=f(x)g(x) dx-[ddxf(x)g(x) dx] dx
• abf(x) dx=abf(y) dy
• abf(x) dx= -baf(x) dx
• abf(x) dx=acf(x) dx+cbf(x) dx
• -aaf(x) dx=0 if f(x) is an Odd Function
• -aaf(x) dx=20af(x) dx if f(x) is an Even Function
• King Property abf(x) dx=abf(a+b-x) dx
• Queen Property 02af(x) dx=20af(x) dx if f(x) is an Even Function otherwise 0
• 0nTf(x) dx=n0Tf(x) dx when f(T+x)=f(x)
• xT+xf(x) dx=0Tf(x) dx when f(T+x)=f(x)
• a+nTb+nTf(x) dx=abf(x) dx when f(T+x)=f(x)
• mTnTf(x) dx=(n-m)0Tf(x) dx when f(T+x)=f(x)
• Walli’s Formula 0/2nxnx dx=[ (n-1)(n-3)...(1 or 2) ][ (m-1)(m-3)...(1 or 2) ]K(m+n)(m+n-2)(m+n-4)...(1 or 2) where K=/2 when m, n are Even or K=1 otherwise
• Leibniz-Newton Formula ddxg(x)h(x)f(x) dx=f[h(x)]h'(x)-f[g(x)]g'(x)
• Limit of Riemann Sum (Left Rule) abf(x) dx=(b-a)n1nk=0n-1f[a+k(b-an)]
• Limit of Riemann Sum (Right Rule) abf(x) dx=(b-a)n1nk=1nf[a+k(b-an)]
• Important Integrals -
  • 0/2(x) dx=0/2(x) dx= -22
  • ab|x| dx=b-a2
  • ab|x|x dx=|b|-|a|
• Area under two curves from x=a to x=b is ab[f(x)-g(x)] dx

Topic - 20: Vectors and 3-D Geometry

• Internal Section Formula using position vectors is r=(na+mb)(n+m)
• Vector Equation of a Line is r=a+tb
• Three points are collinear if ax+by+cz=0 and x+y+z=0
• Projection of a on b is ab
• | a+b+c |2=a2+b2+c2+2( ab+bc+ca)
• Lagrange’s Identity | ab |2=a2b2-( ab )2
• Area of a Triangle is the magnitude of (1/2)[ ab+bc+ca ]
• Area of a Quadrilateral with diagonals d1 and d2 is A=(1/2)| d1d2 |
• Shortest Distance between Skew Lines is |(b-a)(pq)| pq ||
• Distance between parallel lines is |b(a1-a2)b|
• Scalar Triple Product
• Volume of a Tetrahedron is [ a b c ]/6
• Volume of a Parallelepiped is [ a b c ]
• [ a-b b-c c-a ]=0 and [ a+b b+c c+a ]=2[ a b c ]
• [ a b c ]2=[ ab bc ca ]
• Vector Triple Product is a( bc )=(ac)b-(ab)c
• Distance Formula in Space is d=x2+y2+z2
• Centroid of a Triangle is (x3,y3,z3)
• Equation of a Straight Line in symmetrical form is x-x1l=y-y1m=z-z1n
• Equation of a Line through two points is x-x1x2-x1=y-y1y2-y1=z-z1z2-z1
• Vector Equation of a Plane is (r-a)n=0
• Equation of a Plane in Intercept Form is xa+yb+zc=1
• Equation of a plane in Normal Form is lx+my+nz=p
• Angle between two planes with normal vectors n1 and n2 is n1n2n1 * n2
• Angle between a Line and a Plane is ndn * d
• Foot of the Perpendicular of a Point on a Plane is given by xa=yb=zc= -ax1+by1+cz1+da2+b2+c2
• Reflection of a Point on a Plane is given by xa=yb=zc= -2(ax1+by1+cz1+da2+b2+c2)
• Two lines are coplanar if and only if
• Perpendicular Distance of a Point from a Plane is |ax1+by1+cz1a2+b2+c2|
• Distance between Parallel Planes is |d1-d2a2+b2+c2|

Topic - 21: Probability and Statistics

• Probability=Number of favorable outcomesNumber of total outcomes
• P(A)+P(A)=1
• P(AB)=P(A)+P(B)-P(AB)
• Conditional Probability P(A|B)=P(AB)P(B)
• Multiplication Theorem P(AB)=P(A)P(B|A)=P(B)P(A|B)
• Bayes’ Theorem P(B|A)=P(B)P(A|B)P(A)
• Binomial Distribution P(X=x)=nxpxqn-x
• Mean and Variance of Binomial Distribution are np and npq respectively
• Arithmetic Mean is x=k=1nxkn for ungrouped discrete data
• Arithmetic Mean is x=k=1nfkxkk=1nfk when frequency is given
• Mean by Shortcut Method is x=A+k=1nfkdkk=1nfk where dk=xk-A and A is the Assumed Mean
• Mean by Step Deviation Method is x=A+(k=1nfkukk=1nfk)h where uk=dkh
• Weighted Mean is x=k=1nwkxkk=1nwk
• Combined Mean is given by x=k=1Nnkxkk=1Nnk
• Median for ungrouped distribution is
• Median of Grouped Distribution is l+(N/2)-Ffh
• Mode of a Grouped Frequency Distribution is l+f0-f12f0-f1-f2h
• Mode=3*Median-2*Mean
• Mean Deviation is k=1n| xk-x |n
• Variance for discrete distribution is k=1nxk2n-Mean2
• Variance for frequency distribution is k=1nfkxk2N-Mean2
• Standard Deviation is the Principal Square Root of Variance
• Mean Square Deviation is S2=k=1n(x-A)2n or S2=k=1nfk(x-A)2k=1nfk

Topic - 22: Mathematical Reasoning

• AND () is called the Conjunction Operator.
• OR () is called the Disjunction Operator.
• NOT () is called the Negation Operator.
• IMPLIES () is called the Conditional Operator.
• IFF () is called the Bi-Conditional Operator.
• The Truth Table for Conjunction Operator is 

• The Truth Table for Disjunction Operator is 

• The Truth Table for Negation Operator is 

• The Truth Table for the Conditional Operator is 

• The Truth Table for bi-Conditional Operator is 

• The Bi-Conditional Operator is Equivalent to (p q) (q p).
• Two compound statements are logically equivalent if both have the same Truth Table.
• Tautology is the statement that is always true.
• Fallacy is the statement that is always false.
• If the statement is p q then the Converse is q p.
• If the statement is p q then the Inverse is p q.
• If the statement is p q then the Contrapositive is q p.
• (p q) p q and (p q) p q.

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