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most of the mathematical formulas compiled at one place. easy for engineers in their masters degree courses.

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Page 1: Mathematic Formulas

“Trigonometric Functions”

Page 2: Mathematic Formulas

Trigonometry derived from Greek word

SexaGesimal System:

Degree, Minute and Second

One rotation = 360°

1/2 rotation = 180° is called straight angle.

1/4 rotation = 90° is called right angle.

1 rotation = 360°(1°) one degree = 60’(1’) one minute = 60’’

How to convert degree into radian and radian into degree?

Trigonometric Functions:

Following ratios are called trigonometric functions of angle θ.

Fundamental Identities: Pythagoras Theorem

Trigonometric Identities

Teri → Three

Goni → Angles

Metron → Measurem

ent

Page 3: Mathematic Formulas

Law Of Cosine: Law Of Tangents:

Page 4: Mathematic Formulas

Law Of Sines:

Half Angle Formulas Of Sine, Cosine And Tangent In Terms Of Sides:

Area Of Triangle:

Page 5: Mathematic Formulas

Inverse Trigonometric Functions:

Circles Connected With Triangle:

1. Circum-Circle:

The circle passing through the three vertices of a triangle is called a circum circle.

Its centre is called circum-centre.

Its radius is called circum-radius and is denoted by “R”.

2. In-Circle:

The circle drawn inside a triangle touching its three sides is called its inscribed circle or in-circle.

Its centre is called in-centre.

Its radius is called in-radius and is denoted by “r”.

Page 6: Mathematic Formulas

3. Escribed Circle:

A circle which touches one side of the triangle externally and other two produced sides is called escribed circle, ex-circle or e- circle.

The centre of these circles is called ex-centres.

Their radii are denoted by r1 , r2 , r3.

Page 7: Mathematic Formulas

Formulas Related To Derivatives

Page 8: Mathematic Formulas

Relative Extrema:

A point where function f is neither increasing nor decreasing is called stationary point. A stationary point is called a turning point if it is either a maximum point or minimum point. It is also known as point of inflection.

How to find relative extrema?

Find (a) f /(x) (find first derivative)

(b) f //(x) (find second derivative)

(c) Putting f /(x) =0 and finding values of x.

(d) Putting these values of x into 2nd derivative

Now if f //(x) > 0 f(x) is minimum

If f //(x) < 0 f(x) is maximum.

To find maximum and minimum values of f(x) putting these values of x in given function.

Integration:

Integration is also known as antiderivatives.

∫ f (x )dx =f(x) +c

Where ∫ is the integral sign. f(x) is called integrand. x is called variable of integration. c is arbitrary constant of integration.

The process of finding antiderivatives is called integration.

Page 9: Mathematic Formulas
Page 10: Mathematic Formulas

Some Useful Substitution:

Page 11: Mathematic Formulas

When Above substitutions lead to complicated integrals, it is convenient to make the hyperbolic substitutions.

In integration open following functions as

Formulas Related To Limits:

Page 12: Mathematic Formulas

Continuity Of a Function:

a) Continuous Function : A function f is said to be continuous at a number ‘c’ if and only if the following three conditions are satisfied.

a. f(x) is defined. b. limx→c

f (x ) exists. c. limx→c

f ( x )=f ( c )

b) Discontinuous Function :

If one or more of these three conditions fail to hold at ‘c’ than the function f is said to be discontinuous at ‘c’.

Note: To solve any fraction in limits first look that if it is in form of ( 00 ) ,(∞∞ ) than apply L’Hospital rule to

simplify it. Than proceed further.

Analytic Geometry

Page 13: Mathematic Formulas

In co-ordinate system if (x,y) are the co-ordinates of a point p, than the first number (component) of the ordered pair is called x-coordinate or abscissa of P and second member of the ordered pair is called the y-coordinate or ordinate of P.

First member (x) → abscissa

Second member(y) → ordinate

(x, y) → ordered pair.

Two lines intersecting each other at right angle forms a coordinate system.

Lines are known as coordinate axes.

Horizontal line = x-axis

Vertical line = y-axis

Coordinate axes divide plane into four equal parts called quadrants. They are defined as follows.

Quadrant -1: All points (x, y) with x>0, y>0.

Quadrant - 2 : All points (x, y) with x<0, y>0.

Quadrant -3 : All points (x, y) with x<0, y<0.

Quadrant -4 : All points (x, y) with x>0, y<0.

Distance Formula Midpoint Formula

Page 14: Mathematic Formulas

Point Dividing the Join of Two Points In a Given Ratio

Translation And rotation Of Axes:

Translation Of Axes:

Let x, y are the coordinates of original axes.

X, Y are the coordinates of new axes X = x - h so x = X +h

(h, k) are the points of translation Y = y - k so y = Y +K

Rotation Of A xes :

Let θ be the angle of inclination.

Slop Of Line:

Slope of line is denoted by m.

Slope OF line Joining Two Points:

Two lines l1 and l2 w.r.t slopes m1 and m2 are:

I. Parallel if m1= m2

II. perpendicular iff m1m2 = -1

Derivation Of Standard Forms Of Equations Of Straight Line

Page 15: Mathematic Formulas

Intercepts:

X-intercept: If a line intersects x-axis at (a, o) than “a” is called x-intercept of Line.

Y-intercept: If a line intersects y-axis at (0, b) than “b” is called y-intercept of the line.

1) Slope Intercept Form Of Equation Of a Straight Line

y = mx + c

2) Point Slope Form Of Equation Of Straight Line

y – y1 =m(x- x1)3) Symmetric Form Of Equation Of Straight Line

4) Two Point Form Of Equation Of Straight Line

5) Intercept Form Of Equation Of Straight Line

6) Normal form Of Equation Of Straight Line

A linear Equation In Two variables represents Straight Line

Ax + by + c = 0 This is also known as general equation of straight line.

Point Of Intersection Of two Straight Lines

Page 16: Mathematic Formulas

Let l1 = a1x + b1y + c1 =0

l2 = a2x + b2y + c2 =0

be the two non-parallel lines.

Let P(x1, y1) is the point of intersection of l1 and l2.

a1x + b1y + c1 =0

a2x + b2y + c2 =0

Condition Of Concurrency of Three Straight Lines

Three non-parallel lines

l1 = a1x + b1y + c1 =0

l2 = a2x + b2y + c2 =0

l3 = a3x + b3y + c3 =0

are concurrent iff

This is necessary and sufficient condition of concurrency of given three lines.

Distance Of Point From Line

The distance d from the point p(x1, y1) to the line

l = ax + by +c =0 is

Area Of Triangular Region Whose Vertices Are Given

Where points are P(x1, y1) , Q( x2, y2) and R(x3, y3). If the points are collinear than ∆=0

Page 17: Mathematic Formulas

What is trapezium?

A quadrilateral having two parallel and two non-parallel sides

Area Of Trapezium Region:

12

(sum of parallel sides ) (distance between parallel sides)

Angle Between Two Lines:

Feasible Region:

Such a region (which is restricted to the first quadrant) is referred to as a feasible region.

Each point of the feasible region is called a feasible solution of the system of linear inequalities 9 or for the

set of given constraints).

A set consisting of all the feasible solutions of the system of linear inequalities is called a feasible solution

set.

Objective function:

A function which is to be minimized or maximized is called an objective function.

The feasible solution which maximizes or minimizes the objective function is called the optimal solution.

Page 18: Mathematic Formulas

Conic Section

Equation Of Circle ( Standard Form):

This is equation of circle in standard form.

Equations (a) and (b) are known as parametric equation of circle.

General Form OF Equation OF Circle:

is known as general equation of circle.

is general form of equation of circle with center (-g,-f) and

radius

Parabola:

Let L be a fixed line in a plane and F be a fixed point not on the line L. suppose |PM| denotes the distance of a

point P(x, y) from the line. The set of all points P in the plane such that.

|PF||PM|

=distance of point from fixed pointdistance of point from line L.

=e

|PF||PM|

=e ( a positive Constant)

is called a conic section.

1) If e = 1 than conic is a Parabola.

2) If 0 < e < 1 than conic is Ellipse.

3) If e > 1 than conic is Hyperbola.

Fixed line L is called Directrix.

Fixed point F is called a Focus of Conic.

The number e is called the Eccentricity of the Conic.

Page 19: Mathematic Formulas

Standard Equation Of Parabola

y2 = 4ax

Definations:

a) Axis OF parabola : The line through the focus and perpendicular to the dirextrix is called axis of parabola. In the figure axis is y = 0.

b) Vertex : The point where the axis meets the parabola is called the vertex of the parabola.

c) Chord :A line joining two distinct points on a parabola is called a chord of the parabola.

d) Focal Chhord : A chor passing through the focus of the parabola is called a focal chord of parabola.

e) Latusrectum :The focal chord perpendicular to the axis of the parabola is called latusrectum of the

parabola.

It has an equation x = a.

Standard Form Of Ellipse:

Standard Form Of Hyperbola:

Page 20: Mathematic Formulas

Number Systems

Real Numbers: “Real numbers are the combination of rational and irrational numbers”.

Rational Numbers:

Numbers that can be written in P/Q form are known as rational numbers. Where Q # 0, Rational numbers are

1. Perfect square e.g., (25/100) = (5/10)2

2. Terminating e.g., (6/10) = (0.6)3. Recurring e.g., (1/3) = 0.333333…..

Irrational Numbers:

Numbers that cannot be written in P/Q form are known as irrational numbers. Irrational numbers are

1. Not a perfect square2. Non-terminating3. Non-recurring

Example: π is an irrational number.

Where π is a constant ratio.

Real Numbers: Real numbers are the union of rational and irrational numbers.

R = Q U Q /

Properties Of Real Numbers:

1) Transitive property

x = y and y = z x = z (for equality)

x < y and y < z x < z (for in equality)

2) Trichotomy property For each

So either x < y or x = y or x > y

3) Additive identity The element 0 is called the additive identity for real numbers.

4) Additive inverse a and –a are the additive inverse of each other.

5) Multiplicative identity The number 1 is known as the multiplicative identity for the real numbers.

6) Multiplicative Inverse a and 1/a is the multiplicative inverse of each other.

Complex numbers

Page 21: Mathematic Formulas

Numbers of the form x+ίy, where x, y 𝛆 are called complex numbers.ℝ

Here x is a real part. y is a imaginary part.

ί = and ί2 = -1 → ί read as (iota).

Properties Of The fundamental Operations On Complex numbers

Additive Identity: Additive identity is (0, 0).

Additive inverse:

Every complex number (a, b) has the additive inverse (-a,-b) i.e., (a, b) + (-a,-b) = (0, 0)

Multiplicative Identity:

Multiplicative identity is (1, 0) i.e., (a, b). (1, 0) = (a.1 - b.0, b.1 + a.0) = (a, b)

Multiplicative Inverse: Multiplicative inverse of (a, b) is

Argand Diagram

The figure representing one or more complex numbers on the complex plane is called an argand diagram. Points

on the x-axis represent real numbers whereas Points on the y-axis represent imaginary numbers.

Modulus Of Complex Numbers:

is called modulus of complex number a+ίb.

Modulus of complex number is the distance from the origin of the point representing the number.

Let Z=x+ίy = (x, y) then

A (x+ίy)

2 2x y

Y-axis

O

y

Mx x-axis

Page 22: Mathematic Formulas

Polar Form Of Complex Number

x+ίy = rcosθ +rsinθ is known as polar form of complex number.

Where

De Movire’s Theorem:

(Cosθ+ίsinθ) n = Cosnθ + ίSinnθ

Theorem is used to find real and imaginary part of complex number.

Sets,Functions And Groups

Set: set is described as a “Well defined collection of distinct objects”

By distinct objects we mean objects no two of which are identical (Same). The objects in a set are called members or elements. Capital letters A, B, C, X, Y, Z etc., are used as names of set. Small letters a, b, c, x, y, z etc., are used as members of sets.

How to describe a set:

There are three different ways of describing a set.

A (x+ίy)

2 2x y

Y-axis

O

y = rsinθ

Mx= rcosθ x-axis

θ

Page 23: Mathematic Formulas

1. Descriptive Method :

A set may be described in words. For instance, the set of all vowels of the English alphabets.

2. The Tabular Method :

A set may be described by listing its elements within brackets. If A is a set mentioned above, then

A = {a, e, I, o, u}

3. Set-Builder Method :

It is sometimes more convenient or useful to employ the method of set-builder notation in specifying sets. For e.g. A={x/x is a vowel of English alphabets}

Some Important Sets

N= set of all natural numbers = {1, 2, 3…}W= set of all whole numbers = {0, 1, 2, 3….}Z = set of all integers = {0, ±1, ±2, ±3…}Z/ =set of all negative integers = {-1, -2, -3 …}O =set of all odd integers = {±1, ±3, ±5….}E = set of all even integers = {0, ±2, ±4…..}Q= set of all rational numbers ={x/x=p/q where p, q ∈ℝ}Q/=set of all irrational numbers. {x/x # p/q where p, q ∈ ℝ}ℝ = set of all real numbers QUQ/Equal sets:

Two sets are said to be equal A=B if and only if they have the same elements that is, if and only if every element of each set is an element of other set.

Example: sets {1, 2, 3} and {3, 1, 2} are equal sets.

Equivalent Set:

Two sets are said to be equivalent if a (1-1) correspondence can be established between them.

Set A and B are said to be equivalent set.

A = {Bilal, Ahsan, Ali}

↕ ↕ ↕

B= {Fatima, Anum, Sadia}

Singleton Set: A set having only one element is called a singleton set.

Empty set or Null set: A set with no element (zero number of elements) is called empty set or null set.

Example: the set of odd integers between 2 and 4 is a singleton i.e., {3} and the set of even integers between the same numbers is the empty set.

Page 24: Mathematic Formulas

Finite Set: If a set having fixed number of elements then the set is known as finite set.

Example: A= {1, 3, 5…….999}

Infinite Set:

A set having infinite number of elements is known as infinite set. Example: A= {1, 2, 3………}

Disjoint Set:

If the intersection of two sets is the empty set then the sets are said to be disjoint set.

Example:

S1= set of odd natural numbers

S2= set of even natural numbers

S1 and S2 are known as disjoint sets.

Overlapping sets:

If the intersection of two sets is non- empty but neither is the subset of other, the sets are known as overlapping sets. At one element is common.

Subset:

If every element of set A is an element of Set B, then A is a subset of B.

Symbolically A B (A is a subset of B)

Also subset is defined as

A B if x A ∈ ⇒

x B∈

Proper Subset:

If A is a subset of B and B contains at least one element which is not an element of A, then A is a proper subset

of b. Symbolically A B

Improper Subset:

If A is a subset of B and A=B then A is an improper subset of B.

From this definition it follows that every set A is an improper subset of itself. Empty set is a subset of every set.

Power set:

The power set of a set S denoted by P(S) is the set containing all the possible subset of S.

Example: A= {a, b}

P (A)= { {}, {a}, {b}, {a,b}}

Formula to find power set is nP(S) = 2m

De Morgan’s Theorem:

(AUB)/ = A/ ∩ B/

Page 25: Mathematic Formulas

(A∩B ¿/ = A/ U B/

Tautologies

1. Tautology : A statement which is true for all possible values of the variables involved in it is called a tautology.

2. Absurdity : A statement which is always false is called absurdity or a contradiction.3. Contingency : A statement which can be true or false depending upon the truth values of the variables

involved in it is called contingency.

Groupoid: A groupoid is a non-empty set on which a binary operation * is defined.

Semi – Group:

A non-empty set S is semi-group if

It is closed w.r.t an operation * and The operation * is associative.

Monoid:

A semi-group having identity is called a monoid.

A monoid is a set S:

Which is closed w.r.t some operation * The operation * associative. It has an identity.

Matrices And DeterminantsMatrix: A rectangular array of numbers enclosed by a pair of brackets such as

are matrices.

Rows: The horizontal lines of numbers are called rows.

Columns: The vertical lines of numbers are called column.

Entries Or Elements: The number used in rows or columns are said to be the entries or elements of the matrix.

Order Of Matrix:

m

×

n = 3

×

3 is a order of a matrix.

Page 26: Mathematic Formulas

Type Of Matrices

1. Row Matrix Or Row Vector : A matrix which has only one row is called row matrix or row vector.

is row matrix.

2. Column Matrix : A matrix which has only one column is known as column matrix or column vector. For example

is a column matrix.

3. Rectangular Matrix : If m # n, then the matrix is known as rectangular matrix. A matrix having number of rows not equal to number of columns is known as rectangular matrix.

For example are rectangular matrices.

4. Square matrix : A matrix in which number of rows is equal to number of columns called square matrix.

For example are square matrices.

Principal diagonal: Let A=[a ij ] be a square matrix of order n, then the entries a11, a22, a33…….ann from the principal diagonal.

For example

So the entries a11 , a11 , a11 form the PRINCIPAL DIAGONAL.

Principal diagonal of a square matrix is also known as LEADING DIAGONAL or MAIN DIAGONAL.

And the entries a14 , a23 , a32 form the SECONDARY DIAGONAL.

5. Diagonal Matrix : Let A=[a ij ] be a square matrix of order n. then if some elements of the principal

diagonal of A may be zero but not all, then the matrix A is called a diagonal matrix. The matrices below are diagonal matrices.

Page 27: Mathematic Formulas

,

6. Scalar Matrix : A matrix is known as scalar matrix if the elements in the principal diagonal are same.

For example and are scalar matrices.

7. Unit Matrix or Identity matrix : A matrix is said to be unit matrix if the elements of the principal diagonal are 1.

For example is a unit matrix.

8. Null Matrix Or Zero Matrix : A square or rectangular matrix whose each element is zero is called null or zero matrix.

For example , , are null or zero matrices.

9. Equal Matrices : Two matrices of the same order are said to be equal if their corresponding entries are equal.

Let and A and B are equal matrices.

Transpose Of a Matrix: Converting rows of a matrix into column is known as transpose of a matrix. It is denoted by AT.

Let

Multiplication Of Matrices: Two matrices are said to be comfortable for multiplication if the number of columns of A is equal to the number of rows of B.

Scalar Multiplication: If we multiply a scalar number with a matrix, then whole matrix will be multiplied by that number. For example

Page 28: Mathematic Formulas

In determinants: In determinants if we multiply it with a scalar number then we have to multiply number with a single row or column.

Determinants: To find determinant of 2×2 matrix.

= (1) (4)-(3) (2 =4-6 = -2

Singular Matrix:

A square matrix A is said to be singular if

Non-Singular matrix:

A square matrix A is said to non-singular if

Adjoint of a 2×2 Matrix:

Adjoint of the matrix A is find as

Inverse Of a 2×2 matrix:

AA-1 = A-1A=I and (AT)T =A

Generally multiplication of matrices is not commutated. AB ≠ BA

Minor Of an Element: Let us consider matrix of order 3×

3. Then minor of an element aij denoted by Mij

Let Minor of A12 =M12 =

Cofactor Of an element: The cofactor of an element aij denoted by Aij is defined as

Aij = (-1) i+j Mij Where Mij is the minor of an element aij

For example:

Page 29: Mathematic Formulas

A12 = (-1)1+2 M12

Properties Of Determinant:

1. For a square matrix A, |A|=|AT|

2. If in a square matrix A, two rows or two columns are interchanged, the determinant of resulting matrix is -|A|

.3. If a square matrix A has two identical rows or columns, then |A|=0.

4. If all the entries of a row (or column) of a square matrix A are zero, then |A|=0.

5. If the entries of row ( or column) of a square matrix A are multiplied by a number K ∈R , then the

determinant of resulting matrix is K|A|.

6. If each entry of row (or column) of a square matrix consists of two terms, then its determinant can be written as the sum of two determinants.

7. If the matrix is in triangular form, then the value of its determinant is the product of the entries on its main diagonal.

Upper Triangular Matrix:

A square matrix is said to be upper triangular if all elements below the principal diagonal are zero. That is

is a upper triangular matrix.

Lower Triangular Matrix:

A square matrix is called lower triangular matrix if all the elements above principal diagonal are zero.

is lower triangular matrix.

Triangular Matrix: A square matrix is said to be triangular matrix if it is upper triangular or lower

triangular.

Symmetric Matrix: If AT = A then matrix A is called symmetric matrix.

Skew Symmetric Matrix: If AT = -A then matrix A is called skew symmetric matrix.

Hermitian matrix: If ( A )T=¿ A then matrix is called hermitian matrix.

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Skew Hermitian matrix: If ( ´ A )T=¿ -A then matrix is called skew hermitian matrix.

Echelon Form Of a Matrix:

A m×n matrix A is called in (row) echelon form if

In each successive non-zero row, the number of zeros before the leading entry is greater than the number of such zeros in the preceding row.

Every non-zero row in A precedes every zero ow if any. The first non-zero entry in each row is 1.

Reduced Echelon Form Of A matrix: An m× n matrix A is said to be in reduced (row) echelon form if it is in (row) echelon form and if the first non-zero entry ( or leading entry0 in R1 lies in C1 , then all the entries of C1 are zero.

Rank Of A Matrix: Rank of a matrix is number of non-zero rows in a matrix when it is in reduced echelon form.

System Of Linear Equations:Non-homogeneous linear equation:

a1x + b1y + c1z = K1

a2x + b2y + c2z = K2

a3x + b3y + c3z = K3

is called a system of non-homogeneous linear equations in three variables x, y, z. if constants K1, K2, K3 are not all zero.

Homogeneous linear equations:a1x + b1y + c1z = 0 -------------------(1)a2x + b2y + c2z = 0 ------------------- (2)a3x + b3y + c3z = 0 ------------------- (3)

is known as homogeneous linear equations.The solution set (0, 0, 0) of the above homogeneous equation (1), (2), (3) is called trivial solution.Any other solution of equations (1), (2), (3) other than the trivial solution is known as non-trivial solution.

Cramer’s Rule 3x1+x2-x3 = -4 x1+x2+x3 = -4 -x1+2x2-x3 = 1How to use Cramer’s rule?Solution:

First find ∆ =

Now x1 = x2 = x3 =

Quadratic Equations

Quadratic Equation:

A quadratic equation in x is an equation that can be written in the form

ax2 + bx + c = 0

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Where a, b, c are real numbers and a≠ 0. Another name for quadratic equation in x is 2nd degree polynomial in x.

Example:

a) x2 -7x +10 = 0 a=1, b= -7, c=10b) 3x2 –x = 0 a=3, b= -1, c= 0c) x2=0 a=1, b= 0, c=0

Solution Of Quadratic Equation:

There are three basic techniques for solving a quadratic equation.

1) By factorization.2) By completing squares.3) By applying quadratic formula.

The solutions of an equation are also known called its roots.

Quadratic Formula:

Solution Of Equations Reducible To The Quadratic Equation

Type-1: ax2n + bxn +c =0

Solution: put xn = y and x2n = y2 and

Get the equation reduced in quadratic equation in y.

Type-2: Equation of the form

(x+a)(x+b)(x+c)(x+d) =K

Where a+b= c+d

Here multiplying the equations we get equation in quadratic form.

Type-3 (Exponentials Equations):

Equations, in which the variable occurs in exponent, are called exponential equations.

Example: 22x +3.2x+2 +32 = 0

Here we put 2x = y and 22x = y2

We get equation reduced in quadratic equation.

Type-4 (Reciprocal Equations):

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An equation, which remains unchanged when x is replaced by 1/x, is called reciprocal equation.

Example: x4 -3x3 +4x2 -3x +1 =0 We divide equation by x4.

Type-5 (Radical Equations):

Equations involving radical expressions of the variable are called radical equations.

Example:

(a) 3x2 +15x -2 =2

Putting =y

x2 +5x+1= y2

Extraneous roots:

A root that doesn’t satisfy the equation are known as extraneous roots.

(b) Equation Of The Form:

We solve this type of equation by squaring both sides.

(c) Equation Of The Form:

To solve this equation First make factors, than taking common and than squaring both sides.

Three Cube Roots Of Unity

Let x3 = 1

(x3 -1) = 0 now (x-1)(x2+x+1) =0

x-1=0, x2+x+ =0

So 1, , are three cube roots of unity.

Page 33: Mathematic Formulas

So now and

Properties Of cube Roots Of Unity:

Each complex cube root of unity is square of the other. The sum of all three cube roots of unity is zero.

1+ ω + ω2= 0 The product of all the three cube roots of unity is 1.

(1)(ω)(ω2) =0 ω3= 1

Four Fourth Of Unity

Let x4 = 1

(x4 -1) = 0

(x2 – 1)( x2 +1) =0

x2 – 1=0, x2 +1 =0

x2 =1, x2 =1

x= , x =

So four fourth roots of unity are +1, -1, + , -

Properties OF Four Fourth Roots Of Unity:

Sum of all the four fourth roots of unity is 0.

+1 + (-1) + + (- ) =0 The real four fourth roots of unity are additive inverse of each other.

+1,-1 are real four fourth roots of unity.+1 + (-1) = 0 = -1 +1

Both the complex fourth roots of unity are conjugate of each other. , -

Product of all the four fourth roots of unity is -1.

(1)(-1)( )( - ) =-1

Relationships Between The Roots And The Coefficients Of A Quadratic Equation

Let α ,β be the roots of equation ax2+bx+c=0 than

and

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and

Formation Of Equation Whose Roots Are Given

(x- α)(x-β) =0 has the roots α and β

x2 – (α+β)x +αβ=0

x2 – Sx +p =0

Where s= sum of the root and P= product of the roots

Nature Of The Roots Of Quadratic Equation

We know that

The nature of the roots of an equation depends on the value of expression b2 - 4ac, which is known as discriminante.

Case-1:

If b2 – 4ac =0 Then the roots are real and real and repeated equal.

Case-2:

If b2 – 4ac < 0 then will be imaginary. So roots are complex/imaginary and distinct/unequal.

Case-3:

If b2 – 4ac > 0 then will be real. So the roots are real and distinct/unequal.

Case-4:

However if b2 – 4ac is a perfect square then will be rational, and the roots are rational otherwise

irrational.

Sequences And Series

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Sequences also called progressions are used to represent ordered lists of numbers.An infinite series has no last term.

Arithmetic Progression:

A sequence is said to be arithmetic if the difference between two consecutive numbers is same.

Difference of two consecutive terms of A.P called common difference and is represented by “d”.

d= an – an-1

Rule For The nth Term Of A.P:

an = a1 + (n-1) d

Arithmetic Mean: Sum Of First n Terms Of An A.M:

Geometric Progression:

The series is said to be geometric if the common ratio of consecutive terms is same.

The quotient is known as common ratio and is denoted by “r”.

Rule For nth Term Of G.P: Geometric Mean:

Sum Of n Terms Of Geometric Series:

The Infinite geometric Series:

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Harmonic Progression:

A sequence of numbers is known as harmonic sequence if the reciprocals of its terms are in A.P.

The sequence is a H.P since their reciprocals 1,3,5,7 are in A.P.

Harmonic Mean:

To Find Formulas for The Sums

Permutation, combination And Probability

Factorial:

Factorial sign n! Or

0! =1

n! = n (n-1) (n-2)……….3.2.1

n! = n (n-1)!

Permutation:

An ordering arrangement of the n objects is called permutation of the objects.

Where n = number of objects r = arrangement of objects.

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Combination:

Probability:

Probability is the numerical evaluation of a chance that a particular event occurs.

Formulas For Addition Of Probabilities:

P(AUB) = P(A) + P(B) when A and B are disjoint .

P(A B) = P(A) +p(B) – P(A∩B) when A and b are overlapping sets.

Mathematical Induction And Binomial Theorem

There is no integer n for which 3n is even.

Binomial Theorem:

Where a and x are real numbers.

are binomial coefficients.

The following points can be observed in the expansion (a+x)n.

1. The number of terms in the expansion is one greater than the index.2. The sum of exponents of a and x in each term of the expansion is equal to its index.3. The exponent of “a” decreases from index to zero.4. The exponent of “x” increases from o to index.

5. The coefficients of the terms equidistant from beginning and end of the expansion are equal as 6. The sum of coefficients in the binomial expansion equals to 2n.7. The sum of odd coefficients of a binomial expansion equals to the sum of its even coefficients.

To Find General term:

Binomial Theorem When Index n Is A Negative Or A Fraction

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Is also known as Binomial Series.