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Exploring Secrets of Math Formulae

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MATHEMATICSFORMULAE EXPLORER

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This is dedicated to my parents –

Mrs. S. Geethabai

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MATHEMATICS FORMULAE EXPLORER

CONTENTS

S. No. Topics PageNo.

1.2.3.4.5.6.7.8.9.

10.11.12.13.14.15.16.17.18.19.20.

21.22.23.24.25.26.27.28.29.30.31.32.

33.34.

AlgebraAnalytical Geometry3D- Analytical GeometryBoundary Value ProblemsCoordinate GeometryCommercial ArithmeticComplex NumbersData AnalysisDeterminantsDifferential CalculusDifferential EquationsDiscrete MathematicsFourier SeriesFourier TransformGraphsIntegral CalculusLaplace TransformMatricesMeasurementMensuration

Multiple IntegralsNumber WorkNumbers and OperationsOrdinary Differential EquationsPartial Differential EquationsProbabilityPure ArithmeticsSetsStatisticsTablesTheoretical GeometryTrignometry

Vector AlgebraZ-Transform

004007020023028031035040044048051057062068070071074076079087

092093094099102108117118120122123131

139143

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1. Algebra

Expansion and Factorisation

H.C.F x L.C.M of two expression =Product of the two expressions

Equation

Two expression connected by a sign of equality is

is consistent equation, if the equality holds for some value of the

variable/unknown an inconsistent equation, if the equality holds for no value of the

variable/unknown

an identical equation, if the equality holds for any value of thevariable/unknown

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Simultaneous Equation

Simultaneous Equation of the type is consistent and has only one set of solution if

is consistent and has no solution if

have infinite number of solutions if

Laws of Indices

….

• √ •

• ,

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Logarithms

• 0, 0

• 0,

[ in all cases from third formulae, a > 0, b > 0 & a, b , 0,]

Some standard forms of the Binomial Expansion

! ! ….

! ! ….

! ! ….

! ! ….

….

….

….

….

. . . ….

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2. Analytical Geometry

Introduction

‘Geometry’ is the study of Points, Lines, Curves, Surfaces, etc and theirproperties. In the 17 th century AD, the methods of Algebra were applied in thestudy of Geometry and thereby ‘Analytical Geometry’ emerged out. Therenowned French philosopher and Mathematician Rane Descartes (1596-1650)showed how the methods of Algebra could be applied to the study ofGeometry.

Locus

The path traced by a point when it moves according to specified geometricalconditions is called the Locus of the point.

Straight LinesA straight line is the simplest geometrical curve. Every straight line isassociated with an equation.

• Slope-Intercept Form : y = mx + c

• Point –Slope Form : y-y 1 = m(x – x 1)

• Two Point Form :

• Intercept Form : , where ‘a’ and ‘b’ are x and y intercepts.

• Normal Form :

• General Form : ax + by + c = 0

Length of the Perpendicular

• The length of the perpendicular from the point (x 1, y 1) to the line

ax+by+c=0 is

• The length of the perpendicular from the Origin to the line ax+by+c=0 is

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Slope of an equation ax + by + c = 0

For ax + by + c = 0, Slope m =

Angle between two straight line

If is the angle between the two straight lines, then

Condition for Parallel and Perpendicular

• If the two straight lines are Parallel, then their slopes are equal. i.e., m 1=m 2

• If the two straight lines are Perpendicular, then the product of theirslopes is -1. i.e., m 1 x m 2= -1

Condition for Concurrent

The condition for three straight lines to be concurrent is ,

if

Equation of the Straight line passing through the intersection of the two lines

• represents a straightline passing through the intersection of the straight lines

and .

Pair of Straight Lines

• Combined equation of the pair of straight lines isax 2 +2hxy+by 2 +2gx+2fy+c=0, where a, b, c, f, g, h are constants.

• Pair of straight lines passing through the origin is ax 2 +2hxy+by 2 =0

• The Straight line is ( i ) Real and Distinct if h 2 > ab ( ii ) Coincident if h 2 = ab ( iii ) Imaginary if h 2 < ab

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Slopes of pair of straight line

• Sum of the slopes of pair of straight lines, m 1+m 2 =

• Product of the slopes of pair of straight lines, m 1m 2 =

Angle between the pair of Straight line

• Angle between the pair of straight lines passing through the o rigin is

• If the straight lines are parallel, then h 2 = ab

• If the straight lines are perpendicular, thenc

Condition to represent a pair of straight line

• The condition for a general second degree equationax 2 +2hxy+by 2 +2gx+2fy+c= 0 represent a pair of straight lines is abc+2fgh-af 2-bg 2-ch 2 = 0.

CircleDefinition

A circle is the locus of a point which moves in such a way that its distancefrom a fixed point is always constant. The fixed point is called the Centre of theCircle and the constant distance is called the Radius of the circle.

• The equation of circle when the centre is (h, k) and radius ‘r’ is(x – h) 2 + (y – k) 2 = r 2

• If the centre is origin, equation of circle is x 2+y2 = r 2

• The equation of circle, if the end points of a diameter are given by(x – x 1) (x – x 2) + (y – y 1) ( y – y 2) = 0

• The General equation of the circle is x 2 + y 2 + 2gx + 2fy + c = 0 withcentre is (-g, -f) and radius is

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Tangent to the Circle

• Equation of the tangent to a circle at a point (x 1, y 1) isxx 1 + yy 1 + g(x + x 1) + f(y + y 1) + c = 0

• Length of the tangent to the circle from a point (x 1, y 1) is

• If PT 2 = 0, then the point is on the Circle.

• If PT 2 > 0, then the point is outside the Circ le.

• If PT 2 < 0, then the point is inside the Circle.

• Condition for the line y = mx + c to be a tangent to the circle x 2 + y 2 = a 2 is c 2 = a 2 (1 + m 2 )

• Point of contact of the tangent y = mx + c to be a tangent to the circle

x 2 + y 2 = a 2 is ,

• Equation of any tangent to a circle if of the form √

• Two tangent can be drawn from a point to a circle ism 2 (x 2 – a 2 ) – 2mxy +(y 2 – a 2 ) = 0 .

This is a Quadratic equation in ‘m’. Thus ‘m’ has two values. But ‘m’ is

the slope of the tangent. Thus, two tangents can be drawn from a pointto a circle.

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Family of Circles

Concentric Circles

Two (or) more circles having the same centre are called Concentric Circle.

Circles Touching Internally or Externally

Two circles may touch each other either internally or externally. Let C 1, C 2 bethe centres of the circles and r 1, r 2 be their radii and P, the point of contact.

• Two circle touch externally, if C 1C2 = r 1 + r 2

• Two circle touch internally, if C 1C2 = r 1 - r 2

Orthogonal CirclesTwo circles are said to be Orthogonal if the tangent at their point ofintersection are at right angles.

Condition for Orthogonal

• Condition for two circles to cut orthogonal is 2g 1g 2 + 2f 1f 2 = c 1+c 2

Conic

Definition

A conic is the locus of a point which moves in a plane, so that its distancefrom a fixed point bears a constant ratio to its distance from a fixed straightline. The fixed point is called focus, the fixed straight line is called directrixand the constant ratio is called eccentricity, which is denoted by ‘e’.

Classification with respect to the General Equation of a Conic

The equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 represents either a (non-degenerate) conic or a degenerate conic. If it is a conic, then it is

• a Parabola if B 2- 4AC = 0

• an Ellipse if B 2- 4AC < 0

• a Parabola if B 2- 4AC > 0

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Parabola ( y 2 = 4ax )Definition

The locus of a point whose distance from a fixed point is equal to the distancefrom a fixed line is called a Parabola. i.e., Parabola is a conic whose eccentricityis 1.

Definitions

• The fixed point used to draw the parabola is called the Focus F. Here,the focus is F(a,o).

• The fixed line used to draw a parabola is called the Directrix of theparabola. Here, the equation of the directrix is x = - a

• The axis of the parabola is the axis of symmetry. The curve y 2 = 4ax issymmetrical about x-axis and hence x-axis or y = 0 is the axis of the

parabola y2

= 4ax. Note that the axis of the parabola passes through thefocus and perpendicular to the directrix.

• The point of intersection of the parabola and its axis is called its Vertex.Here, the vertex is V(0,0).

• The Focal Distance is the distance between a point on the parabola andits focus.

• A chord which passes through the focus of the parabola is called theFocal Chord of the parabola

• Latus Rectum is a focal chord perpendicular to the axis of the parabola.Here, the equation of the latus rectum is x = a.

• End points of Latus Rectum is L (a, 2a) and L /(a, -2a)

• Length of Latus Rectum = 4a. Length of Semi-Latus Rectum is 2a.

General form of the standard equation of a Parabola

The General form of the standard equation of the parabola is•

(open rightwards)

• ( open leftwards)

• (open upwards )

• (open downwads)

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Ellipse

Definition

The locus of a point in a plane whose distance from a fixed point bears aconstant ratio, less than one to its distance from a fixed line is called Ellipse.

Definitions

Focus : The fixed point is called focus, denoted as F(ae,0)

Directrix : The fixed line is called directrix l of the ellipse and its eq uationis

Major axis : The line segment AA / is called the major axis and the length of themajor axis is 2a. The equation of the major axis is y = 0.

Minor axis : The line segment BB / is called the minor axis and the length of theminor axis is 2b. The equation of the minor axis is x = 0.

Centre : The point of intersection of the major axis and minor axis of theellipse is called the Centre of the Ellipse.

Vertices : The points of intersection of the ellipse and its major axis are calledits vertices.

Focal Distance : The focal distance with respect to any point P on the ellise isthe distance of P from the referred focus.

Focal Chord : A chord which passes through the focus of the ellipse is calledthe focal chord of the ellipse.

Latus Rectum : It is the focal distance perpendicular to the major axis of theEllipse. The equation of the latus rectum are x = + ae, x = - ae.

Eccentricity :

End Points of Latus Rectum are

,and other latus rectum are

,.

Length of the Latus Rectum are

Special Property : Thanks to the symmetry about the origin, it permits thesecond Focus F2(-ae,0) and the second directrix x = -

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General forms of Standard Ellipses

The General forms of Standard Equation of Ellipses, if the centre C(h,k) is

a > b

Focal Property of an Ellipse

The sum of the focal distances of any point on an elli pse is constant and isequal to the length of the major axis.

Hyperbola

Definition

The locus of a point in a plane whose distance from a fixed point bears aconstant ratio, greater than one to its distance from a fixed line is calledHyperbola.

Definitions

Focus : The fixed point is called focus, denoted as F(ae,0)

Directrix : The fixed line is called directrix l of the hyperbola and its equationis

Transverse axis : The line segment AA / joining the vertices is called thetransverse axis and the length of the transverse axis is 2a. The equation of thetransverse axis is y = 0.

Conjugate axis : The line segment joining the points B(0, b) and B / (0, -b) iscalled the conjugate axis and the length of the conjugate axis is 2b. Theequation of the conjugate axis is x = 0.

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Centre : The point of intersection of the transverse and conjugate axes of thehyperbola is called the Centre of the Hyperbola.

Vertices : The points of intersection of the hyperbola and its transverse axisare called its vertices.

Latus Rectum : It is the focal chord perpendicular to the transverse axis of theHyperbola. The equation of the latus rectum are x = + ae, x = - ae.

Eccentricity :

End Points of Latus Rectum are

,and other latus rectum are

,.

Length of the Latus Rectum are

The other form of the Hyperbola

If the transverse axis is along y-axis and the conjugate axis is along x-axis,

then the equation of the hyperbola is of the form

For this type of hyperbola, we have the following points.

• Center is C(0,0) • Vertices A(0, a) and A / (0, -a) • Foci are F(0, ae) and F(0, -ae) • Equation of transverse axis is x = 0 • Equation of conjugate axis is y = 0 •

End points of conjugate axis is (b, 0) and (-b, 0) • Equations of Latus rectum is

• Equations of directrices is

• End points of Latus rectum is , , ,

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Parametric form of Conics

Conic Parametricequations

Parameter Range ofparameter

Any point onthe conic

Parabola x = at 2 y = 2at

t ∞ ∞ ‘t’ or(at 2, 2at)

Ellipse x = a cos ,y = b sin 2 or

(acos ,

.

t ∞ ∞ ‘t’ or

, .Hyperbola x = a sec ,y = b tan 2 or

(a sec ,

Equation of Chord

Conic Equation of Chord joining (x 1, y 1) and (x 2, y 2)

Parabola

Ellipse

Hyperbola

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Equation of Tangent and Normal

Conic Equation of Tangents at (x 1, y 1) Equation of Normal at (x 1, y 1)

Parabola

Ellipse

Hyperbola

Equation of Chord and Tangent at Parametric Form

Conic Equation of Chord atParametric Form

Equation of Tangents atParametric Form

Parabola Chord joining the points

& is

at ‘t’ is yt = x + at

2

Ellipse Chord joining the points & is

at ‘ ’ is

Hyperbola Chord joining the points

&

is

at ‘ ’ is

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Results connected with Conics

Conic Condition thaty=mx+c may

be a tangent tothe conic

Point of Contact Equation of anytangent is of the form

Parabola ,

Ellipse ,

where

Hyperbola ,

where

Asymptotes

Definition

An asymptote to a curve is the tangent to the curve such that the point ofcontact is at infinity. In particular, the asymptote touches the curve at∞ ∞.

Results regarding Asymptotes

• The equations of the asymptotes to the hyperbola is

• The combined equation of asymptotes is

i.e.,

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• The asymptotes pass through the centre C(0,0) of the hyperbola.

• The slopes of asymptotes are. i.e., the transverse axis and

conjugate axis bisect angles between the asymptotes.

• If is the angle between the asymptotes then the slope of is

. • Angle between the asymptotes is

• Angle between the asymptotes is

Rectangular Hyperbola ( xy = c 2 where )

Definition

A hyperbola is said to be a rectangular hyperbola if its asymptotes are at rightangles.

Results

• Eccentricity of the Rectangular Hyperbola is √ andalso b 2 = a 2(e 2-1)

• The Vertices of the rectangular hyperbola are √ ,√ and √ ,√ • The foci are (a, a) and (-a, -a)• The equation of the transverse axis is y = x and the conjugate axis is

y = - x .•

If the centre of the rectangular hyperbola is (h, k) then (x – h) ( y – k) = c 2

• The parametric equation of the rectangular hyperbola xy = c 2 is x = ct,y =

• Equation of the tangent at (x 1, y 1) to the rectangular hyperbola xy = c 2 isxy 1+yx 1 = 2c 2

• Equation of the tangent at ‘t’ is x + yt 2 = 2ct • Equation of normal at (x 1, y 1) to the rectangular hyperbola xy = c 2 is

xx 1- yy 1 = x 12- y1

2 • Equation of normal at ‘t’ is y - xt 2 = ct 3 • Two tangents and four normals can be drawn from a point to a

rectangular hyperbola.

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3. Three Dimensional Analytical Geometry

The equation of the Sphere whose centre is (a, b, c) and radius ‘r’ is

The equation of the Sphere has the following three characteristics.• It is of second degree equation in x, y, z• The coefficients of x 2, y 2, z 2 are equal• The terms xy, yz and zx are absent

If the coefficients of x 2, y 2, z 2 are each unity, then the coordinates of the centreof the Sphere are

,

and square of the radius is equal to the sum of the squares of the coordinatesof the centre minus the constant term.

The equation of a Sphere whose centre is (x 1, y 1, z 1) is

Equation of a Sphere with the extremities of diameter at the points (x 1, y 1, z 1)and (x 2, y 2, z 2) is

Two Spheres S 1 and S 2 whose radii are r 1 and r 2 touch externally if the distancebetween their centres is equal to the sum of their radii.

d = r 1 + r 2

The point of contact is the point which divides internally the line joining thecentres in the ratio of the radii.

Two Spheres S 1 and S 2 whose radii are r 1 and r 2 touch internally if the distancebetween their centres is equal to the difference of their radii.

d = r 1 ~r2

The point of contact is the point which divides externally the line joining thecentres in the ratio of the radii.

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To find the condition that the plane Ax + By + Cz + D = 0 may touch the Sphereis

Condition for the two Spheres to cut Orthogonally

and is

The General Equation of a Right Circular Cylinder

If the axis of the required cylinder is and radius is ‘r’ then theequation of a circular cylinder is

– The equation of the Cylinder whose generators intersect the curve

and parallel to the line is

The equation of the cylinder whose generators touch the sphere

and are parallel to the line is

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The equation of the Cone whose vertex is at the point

, , and whose

generators intersect the conic is

The equation of the Right Circular Cone whose vertex is at , , and its axis

at the line and whose semi-vertical angle is

The equations of the enveloping cone whose vertex is at , , and whosegenerators touch the sphere is

The equation of the tangent plane at the point (x 1, y 1, z 1) to the cone

is

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4. Boundary Value Problems

VIBRATION OF STRING

Equation andConditions

BoundaryConditions

CorrectSolution

Most General Solution

Initial velocityzero.

, ,

, ,

,,

,

,

,

Initial velocityis given

, ,

, ,

,

,

, ,

ONE DIMENSIONAL HEAT FLOW EQUATION

Equation andConditions

BoundaryConditions

CorrectSolution

Most General Solution

Beginning point‘A’ and Endingpoint ‘B’ are atzero temperature

, ,

, ,

,

,

,

Beginning point‘A’ is at zerotemperature andEnding point ‘B’is at non-zero temperature k.

, ,

, ,

, , ,

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SQUARE PLATE - Condition I

Conditions of Square Plate Boundary Conditions

f(x)

0o

,

• , • , • ,

Correct Solution Most General Solution

, ,

SQUARE PLATE - Condition II

Conditions of Square Plate Boundary Conditions

0o

f(x)

,

• , • , • ,

Correct Solution Most General Solution

, ,

y = a

0o 0o x=0 x=a

= 0

y = a

0o 0o x=0 x=a

y = 0

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SQUARE PLATE - Condition III

Conditions of Square Plate Boundary Conditions

0o

0o

,

• , • , • ,

Correct Solution Most General Solution

, ,

RECTANGULAR PLATE - Condition I

Conditions of Rectangular Plate Boundary Conditions

f(x)

0o

• ,

• , • , • ,

Correct Solution Most General Solution

, ,

y = a

0o f(x) x=0 x=a

y = 0

y = b

0o 0o x=0 x=a

y = 0

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RECTANGULAR PLATE - Condition II

Conditions of Rectangular Plate Boundary Conditions

0o

f(x)

,

• , • , • ,

Correct Solution Most General Solution

, ,

RECTANGULAR PLATE - Condition III

Conditions of Rectangular Plate Boundary Conditions

0o

0o

• ,

• , • , •

, Correct Solution Most General Solution

, ,

y = b

0o 0o x=0 x=a

y = 0

y = b

0o f(y) x=0 x=a

y = 0

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RECTANGULAR PLATE – Infinite Plate - Condition I

Conditions of Rectangular Plate- Infinite Plate

Boundary Conditions

f(x)

,

• , • ,∞ • ,

Correct Solution Most General Solution

, ,

RECTANGULAR PLATE – Infinite Plate - Condition II

Conditions of Rectangular Plate- Infinite Plate

Boundary Conditions

0o

0o

• ,

• , • ∞, •

, Correct Solution Most General Solution

, ,

0o 0o x=0 x=l

y = 0

y = l

f(y)

y = 0

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5. Co ordinate Geometry

Introduction

The Modern terms Co-ordinates, abscissa and ordinate were contributed byGerman Mathematician Gottfried Wilhelm Von Neibliz in 1692. Rene Descartesinvented co-ordinate geometry.

Distance Formula between two points A(x 1, y 1) and B(x 2, y 2)

,

Mid-Point Formula between two points A(x 1, y 1) and B(x 2, y 2)

,

Centroid Formula between three points A(x 1, y 1), B(x 2, y 2) and C(x 3, y 3)

,

Area of the Triangle from the given three points A(x 1, y 1), B(x 2, y 2) and C(x 3, y 3)

Condition for the three points A(x 1, y 1), B(x 2, y 2) and C(x 3, y 3) to be Collinear

Area of the Parallelogram from the given four points A(x 1, y 1), B(x 2, y 2),C(x 3, y 3) and D(x 4, y 4)

Slope (or) Gradient of the Line

If is the angle of inclination, then Slope, m = tan

Slope of the line joining two points A(x 1, y 1) and B(x 2, y 2)

Slope, m =

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Slope of the line ax+by+c = 0

Slope =

Equation of Straight Line with Slope m and y-intercept c

Equation of straight line is y = m x + c

Equation of Straight Line with Slope m and point A(x 1, y 1)

Equation of straight line is y – y 1 = m (x – x 1)

Equation of Straight Line with Slope m and joining two p oints A(x 1, y 1) andB(x 2, y 2)

Equation of straight line is

Equation of Straight Line with x intercept a and y intercept b

Equation of straight line is

Condition for two lines to be Parallel

Two lines are Parallel, then their slopes are equal. i.e., m 1 = m 2

Condition for two lines to be Perpendicular

Two lines are Perpendicular, then their product of their slopes gives -1i.e., m 1 x m 2 = -1

Equation of Straight Lines with different cases

• Any line parallel to ax + by + c = 0 is ax + by + k = 0 (differ only byconstant)

• Any line parallel to x-axis is y=k ( k is constant)

• Any line parallel to y-axis is x = c ( c is constant)

• The line which is perpendicular to the line ax + by + c = 0 is of the formbx – ay + k = 0

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Circumcentre, Centroid and Orthocentre

Circumcentre : The perpendicular bisector of the sides of a triangle areconcurrent. The point of concurrence is called circumcentre.

Centroid of a triangle : The medians of a triangle meet at a point. This point isknown as centroid.

Orthocentre of a triangle : The altitudes of a triangle meet at a point. Thi s pointis called Orthocentre.

Slope of both axes

• The Slope of x-axis = 0

• The Slope of y-axis = not defined

Concurrency of Three Lines

Condition that the lines , andmay be concurrent if,

Intersection of Two Straight Lines

The two lines if not parallel in a plane intersect in a unique point.

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6. Commercial ArithmeticBasic Definitions

Percentage

% %

Profit and Loss

Profit = Selling Price (S.P) – Cost Price (C.P)

Loss = Cost Price (C.P) – Selling Price (S.P)

Selling Price = Cost Price + Profit

Selling Price = Cost Price - Loss

Cost Price = Selling Price – Profit

Cost Price = Selling Price + Loss

Profit (in percent) = x 100 = . . x100

Loss (in percent) = x 100 = . . x100

Selling Price = Cost Price + x% of Cost Price, , if Profit is x%.

Cost Price = Selling Price x , if Profit is x%.

Selling Price = Cost Price - x% of Cost Price, , if Loss is x%.

Cost Price = Selling Price x , if Loss is x%.

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Discount and Market Price

Discount = Marked Price – Actual Selling Price

Discount in Percent = Marked Price –

x 100

Actual Selling Price = Marked Price – Discount

= Marked Price - % x Marked Price

Marked Price = %

Successive (2 nd ) discount is calculated on the balance after deduction ofthe first discount from the marked price and so on.

Simple Interest

• Simple Interest (S.I) = = PNi, where P is the Principal, N is the Period

in years and R% is the rate of interest for 1 year. = interest forunit principal for one year

• Amount (A) = Principal + Interest

• Interest = Amount - Principal

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Compound Interest (C.I)

• Compound Interest (C.I) = , where P is the Principal, N isthe Period in years and R% is the rate percent annually.

• Amount, A =

• Principal = Amount – Compound Interest

• Difference between C.I and S.I for 2 years =

• Difference between C.I and S.I for 3 years =

Recurring Deposit (R.D)

Recurring Deposit is a special type of deposit in which a person deposits afixed sum every month over a period of years and receives a large sum at theend of the specified number of years. Since the deposit is made month aftermonth, it is called Recurring Deposit. Recurring Deposits are also known asCumulative Term Deposits. The amount deposited every month is called theMonthly Deposit.

Total Interest = , where N = ,

P be the Monthly Instalments,

R % be the rate of Interest and

‘n’ be the number of monthly instalments.

Amount Due = Amount Deposited + Total Interest

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Fixed Deposit

Fixed Deposit are deposits for a fixed period of time and the depositor canwithdraw his money only after the expiry of the fixed period. It is also knownas Term Deposits. However, in the case of necessity, the depositor can get hisfixed deposit terminated earlier to get a loan from the bank under term s laiddown by the bank. There are two types of fixed deposits, namely

Short Term Deposits Long Term Deposits

Short Term Fixed Deposits are accepted by the banks for a short periodranging from 46 days to one year. The interest paid on this deposit is SimpleInterest.

Long Term Fixed Deposits are accepted by the banks for a period of one yearor more. The interest paid on this type of deposit is Compound Interest.

Quarterly Interest =

Half Yearly Interest =

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7. Complex Numbers

The Complex Number System

• A Complex number is of the form a+ib , where ‘a’ and ‘b’ are real

numbers and ‘I’ is called the imaginary unit, having the property i 2

= -1 .

• If z = a+ib then ‘a’ is called the real part of z, denoted by Re(z) an d ‘b’ is

called the imaginary part of z and is denoted by Im(z).

• If z = a+ib is a complex number then the negative of z is denoted by –z

and it is defined as –z = -a + i (-b).

• Basic Algebraic Operations with Complex Numbers

(a + ib) + (c + id) = (a + c) + i (b + d)

(a + ib) - (c + id) = (a - c) + i (b - d)

(a + ib) (c + id) = (ac - bd) + i (ad + bc)

• If z = a + i b , then the conjugate of z is denoted by and

is defined by .

• Properties of Complex Numbers

. . .,

Z is real the imaginary part is zero

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Conjugate of the sum is the sum of their conjugates

Conjugate of the product of two complex numbers is the

product of their conjugates

The conjugate of the quotient of two complex numbers is

the quotient of their conjugates.

• The Modulus (or) Absolute value of z = a+ib is de noted by

||is defined

by √ • The Amplitude (or) Argument of z = a+ib is denoted by arg z or arg z is

defined by

• It is obvious that

|| ||. Also,

|| √

• ||and || • The Modulus of a product of two complex numbers is equal to the

product of their moduli. | | | ||| • The above result can be extended to any finite number of complex

numbers. i.e.,

| ….| | |||||…| |

• The Modulus of a quotient of two complex numbers is equal to the

quotient of their moduli. | || |

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• Triangle Inequality

The Modules of sum of two complex numbers is always less than

or equal to the sum of their moduli. | | | | | | | | | | | | | …| | | | | | |

• The Modulus of the difference of two complex num bers is always greater

than or equal to the difference of their moduli.

| | | | | |

• Polar form of a Complex Number

• For any two complex numbers | | | |||

• The above result can be extended to any finite number of complex

numbers .

| …….| | |||……..| |

………. …….. • The Exponential form of a Complex Number is known as Euler’s

Formula and is defined by

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General Rule for determining the argument

Let z = a + ib where a, b R . Take |||| In First Quadrant,

In Second Quadrant,

In Third Quadrant, In Fourth Quadrant,

Both cos and sin are positive.

Z lies in the first quadrant.

Sin is positive and cos is negative.

Z lies in the second quadrant.

Both cos and sin are negative.

Z lies in the third quadrant.

Sin is negative and cos is positive.

Z lies in the fourth quadrant.

TheoremFor any polynomial equation P(x) = 0 with real coefficients, imaginary(complex) roots occur in conjugate pairs.

De Moivre’s TheoremFor any rational number n, is the value or one of the valuesof

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8. Data Analysis

Statistics is the study of the methods of collecting, organizing and analyzingquantitative data, and drawing conclusions. The data are collected on samplesfrom various populations of people, animals and things by different methodssuch as observations, interviews, etc. Statistics is used in almost every fieldsuch as business, education, science, psychology, research, etc.

The word ‘data’ is the plural form of datum , which means facts and figures.

Data

Data represent factual information (in the form of measurements or statistics)which is used as a basis for reasoning, discussion or calculation. Data areclassified as either Primary or Secondary.

Primary Data

Primary data are the data which are collected directly for a specific purpose forthe first time and they are original in character.Examples : Questionnaires, Interviews, etc.,

Secondary Data

Secondary data are data already collected, analyzed and presented in writtenform ready for people to use.Examples : Government reports, books, articles, maps, etc.,

Types of Data

Data can be qualitative or quantitative. Names of persons, marital status, etc.,are examples of qualitative data.

Quantitative Data

Quantitative data are measurements expressed in terms of numbers. Incomeof individuals, production of a car company, exports in units of a garmentcompany, marks of students, etc., are all quantitative data.

Quantitative data can further be classified as continuous data and discretedata.

Continuous Data : Takes numerical values within a certain range.Example : Height of a person.

Discontinuous (or) Discrete Data : Takes only whole-number values.Example : The number of boys in each class can be expressed only in wholenumbers.

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Displaying Data

Tables, Charts and Graphs are examples of visual representation of data.Graphs or Charts show the relationship between changing things and are usedto make facts clearer and more understandable.

Line Graph

A Line Graph is used to show continuous data. The dependent data is p lottedalong the y-axis and the independent data along the x-axis.

Multiple-Line Graph

A multiple-line graph can effectively compare similar data over the sameperiod of time.

Pie Chart

A pie chart is a circular chart divided into segments. Each segment illustratesrelative magnitudes or frequencies. It shows the component parts of a whole.A pie chart uses percentages to compare information since they are theeasiest way to represent a whole (100%). In a Pie chart, the arc length, centralangle and area of each segment is proportional to the quantity it represents.

Exploded Pie Chart

A chart with one or more segments separated from the rest of the disc iscalled an exploded pie chart.

Formation of Frequency Tables

Classification and Tabulation

Collection of data in the form of numbers alone will not help us to makedecisions or form conclusions. Since just a huge collection of numbers doesnot have any meaning, it is necessary to classify the numbers as values andpictures before presentation.

Classification is the process of grouping data according to their commoncharacteristics.

Tabulation is the process of arranging the classified data in tabular form.

Notes

• The number of times a particular observation or a variable ‘x’ occurs in adata set is called its frequency which is denoted by ‘f’.

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• Frequency distributions show the actual number of observations fallingin each range of observations.

• In a continuous distribution the data are obtained by measurement.

• The vertical bar ‘|’ which represents each occurrence of a variable ‘x’ orobservation is called a tally mark.

• The mid-value of a class interval is called its class mark.

• Class boundaries are actual or true limits of a class interval in a groupeddistribution table and are continuous.

Measures of Central Tendency

The classification and tabulation of statistical data is a process of condensingthe entire data. The graphs / charts give a visual presentation and make the

comparisons easier. But for analysis of given numerical data, somedescription of the given data is needed. The statistical average is a numericalvalue around which the greatest proportion of the data concentrates. Forexample, if we say in a class of 40 students, the mathematics marks vary from40 to 95, but most of them secured 70 marks then 70 is the statistical averagemarks of the class. Such values are called measures of central tendency. Thethree important measures of central tendency are

• Arithmetic mean (or) Average• Median• Mode

Arithmetic Mean (A.M)

The Arithmetic Mean of a collection of data is a measure of central tendencyand it helps in interpreting the data. The arithmetic mean (or) AM is commonlyknown as the mean or the average of a given set of data.

Arithmetic Mean (A.M) of Ungrouped Data

The formula used is . , ∑

Median of Ungrouped DataMedian is the middle value or the mean of the middle two values, when a set ofobserved data is arranged in numerical order.

Median divides the distribution into two equal halves such that there are asmany observations less than it as there are greater than it.

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In a set of N observations, when N is odd, the observation of

arranged data in the numerical order is the median.

In a set of N observations, when N is even, the average of observation

and observation of the arranged data in numerical order is the

median.

Mode of Ungrouped Data

Mode is the data which occurs most frequently in the given set of observations(data). It is possible to have more than one mode.

Range of Ungrouped Data

The difference between the highest and lowest values of the observed data iscalled the Range.

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9. Determinants

Singular / Non Singular

A Square Matrix A is said to be Singular if | | . Otherwise it is said to beNon-Singular.

Adjoint of A

Let A = [ a ij ] be a square matrix of order n. Let A ij be the cofactor of a ij. Theadjoint of A is nothing but the transpose of the cofactor matrix [A ij ] of A.

Theorem

If A is a Square matrix of order n, then

A (Adjoint A) =

| |In = (adjoint A) A

where I n is the identity matrix of order n.

Theorem

If a matrix A possesses an inverse then it must be unique.

Theorem

If A is a non singular matrix, there exists an inverse which is given by

| |

Reversal Law for Inverses

If A, B are any two non-singular matrices of the same order, then AB is alsonon-singular and

Reversal Law for Transposes

If A and B are matrices conformable to multiplication, then

Inverses and Transposes

For any non-singular matrix A,

Matrix Inversion Method

For a system of n linear non-homogeneous equations in ‘n’ unknowns isrepresented by AX = B, then its unique solution is given by X = A -1B.

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Properties of Determinants

• The Value of a determinant is unaltered by interchanging its rows and

columns

• If any two rows (columns) of a determinant are interchang ed the

determinant changes its sign but its numerical value is unaltered.

• If two rows (columns) of a determinant are identical then the value of the

determinant is zero.

• If every element in a row (or column) of a determinant is multiplied by a

constant “K” then the value of the determinant is multiplied by K.

• If every element in any row (column) can be expressed as the sum of

two quantities then given determinant can be expressed as the sum of

two determinants of the same order with the elements of the remaining

rows (columns) of both being the same.

• A determinant is unaltered when to each element of any row (column) is

added to those of several other rows (columns) multiplied respectively

by constant factors.

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Rank of a Matrix

The matrix A is said to be of rank r, if

• A has atleast one minor of order r which does not vanish

• Every minor of A of order (r+1) and higher order vanishes

In other words, the rank of a matrix is the order of any highest order nonvanishing minor of the matrix.

The rank of A is denoted by .

The rank of an m x n matrix A cannot exceed the minimum of m and n.i.e., ,.

Elementary Transformation on a Matrix

Let A be an mxn matrix. An elementary row (column) operation on A is of anyone of the following three types.

• The interchange of any two I th and j th rows (columns). i.e., • Multiplication of a I th row (column) by a non zero constant C. i.e.,

• Addition of any multiple of one row (column) with any other row(column).i.e.,

Echelon Form

A matrix A (of order m x n) is said to be in Echelon form (Triangular form) if• Every row of A which has all its entries 0 occurs below every row which

has a non-zero entry.

• The first non zero entry in each non zero row is 1.

• The Number of zeros before the first non zero element in a row is lessthan the number of such zeros in the next row.

Note :• Any matrix can be brought to Echelon matrix form.• The Rank of a matrix in Echelon form is equal to number of non zero

rows of the matrix.

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Cramer’s Rule Method ( Determinant Rule)

For a system of non-homogeneous equation with 3 unknowns, the system isConsistent and has Unique Solution, if

∆ .

Solution is ∆∆, ∆∆ ∆∆. Consistency for a given System of Equations by using Rank Method

• ,

,

• ,

Consistency for a System of Homogeneous Equation

A System of Homogeneous equations is always consistent.

,

• ,

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10. Differential Calculus

Derivatives of Standard Functions :

.

.

.

.

.

.

.

.

.

.

. ,

. ,

. √

. √

.

.

. √

. √

.

.

.

.

.

.

. √

.

.

.

. √

. √

. / /

. / / Seven Indeterminant Forms

,, 0x ∞,∞ ∞,1,∞,0 Maclaurin’s Series

/ ! // ! /// !

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Curvature of CurveThe rate of bending of a curve in any interval is called the Curvature of the curve inthat interval.

Cartesian Curve y = f(x) Polar Curve r = f( ϴϴϴϴ)

1

1

Sin Ψ =

cos Ψ =

tan Ψ =

1

sin =

cos =

tan =

p= r sin Radius of Curvature

The reciprocal of theCurvature of a curve atany point is called theRadius of Curvature atthe point and is denot edby /

Parametric Form

Let x=f(t) and y=g(t) be theparametric equations ofthe given curve.

/ / / /// ///

Implicit Form

Let f(x,y)=o be the implicitform of the given curve.

/

Polar Form

Let r = f(ϴϴϴϴ) be the givencurve in polar coordinates.

/

Centre of Curvature inthe Cartesian Form

, ,

where

,

Circle of Curvature

The equation of the circleof curvature is

Local Maxima and Minima for functions of one variable

Given y=f(x) , (i) if f/(c)=0 and f//(c)> 0, then f has a local minimum at c.(ii) if f/(c)=0 and f//(c)< 0, then f has a local maximum at c.

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Maxima and Minima for functions oftwo variables

Necessary Condition

Let fx(a,b)=0 and fy(a,b)=0

Sufficient Condition

If fx(a,b)=0, f y(a,b)=0 and fxx(a,b)=A,fxy(a,b)=B, f yy(a,b)=C then

i) f(a,b) is maximum value if AC-B 2 > 0 and A<0 (or B<0 )

ii) f(a,b) is minimum value if AC-B 2 > 0 and A>0 (or B>0 )

iii) f(a,b) is not an extremum if AC-B 2 < 0and

iv) If AC-B 2 > 0, the test is inconclusive.

Stationary Value

A function f(x,y) at (a,b) or f(a,b) is saidto be a Stationary Value of f(x,y) iffx(a,b)=0 and fy(a,b)=0 .

Method of Lagrangian Multiplier

To find the maximum and minimumvalues of f(x,y,z) where x,y,z are subjectto a constraint equation g(x,y,z)=o, wedefine a function

F(x,y,z) = f(x,y,z) + λ g(x,y,z) ,

where λ is called Lagrange M ultiplierwhich is independent of x,y,z,

The necessary condition for a maximumor minimum are

, ,

Solving the above equations for fourunknowns λ , x, y, z, we obtain the point(x,y,z) . The point may be a maxima,minima or neither which is decided by thephysical consideration.

Jacobians

If u1, u 2, u 3, …….u n are functions of n

variables x 1, x2, x3, …xn, then theJacobian of the transformation from x 1,x2, x3, …x n to u 1, u 2, u 3, …….u n isdefined by

, ,……,,….., ,……

Properties of Jacobian

1. If u and v are the functions of x and y,

then ,, ,,1.

2. If u,v are the functions of x,y and x,yare themselves functions of r,s then

then ,, ,, ,,.

3. If u,v,w are functionally dependentfunction of three independent variables

x,y,z then , ,, ,0

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11. Differential Equations

Definition

An equation involving one dependent variable and its derivatives with respect

to one or more independent variables is called a Differential Equation.

Differential Equation are of two types namely

• Ordinary Differential Equations• Partial Differential Equations

Definition

An Ordinary Differential Equation is a differen tial equation in which a singleindependent variable enters either explicitly or implicitly.

Order and Degree of a Differential Equation

Definition

The Order of a differential equation is the order of the highest order derivativeoccurring in it. The degree of the differential equation is the degree of thehighest order derivative which occurs in it, after the differential equation hasmade free from radicals and fractions as far as the derivatives are concerned.

Differential Equations of First Order and First Degree

For the solutions of first order and first degree equations, we shall consideronly certain special types of equations of the first order and first degree. They

are• Variable Separable• Homogeneous• Linear

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Variable Separable

Variables of a differential equation are to be rearranged in the form

The above equation can be rewritten as

The solution is

Homogeneous Equations

Definition

A differential equation of first order and first degree is said to be

homogeneous if it can be put in the form , ,

Solving this, by putting y = vx , we get the solution.

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Linear Differential Equation

Definition

A first order differential equation is said to be linear in y, if the power of theterms and y are unity.

A differential equation of order one satisfying the above condition c an always

be put in the form ,where P and Q are function of x only.Similarly a first order linear differential equation in x will be of the form

where P and Q are functions of y only.

The solution of the equation which is linear in y is given as

where is known as an integrating factor and itis denoted by I.F.

Similarly, the solution of the equation which is linear in x is given as

where is known as an integrating factor and it

is denoted by I.F.

We frequently use the following properties of Logarithmic and Exponentialfunctions

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SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANTCOEFFICIENTS

A General Second Order non-homogeneous linear differential equation withconstant coefficients is of the form

// /

where a 0, a 1, a 2 are constants a 0 0, and X is a function of x. The equation

// /

,

is known as a homogeneous linear second order differential equation withconstant coefficients.

Theorem : If is a root of , then is a solution of // / .

Definition : The equation is called the characteristicequation of // / , .

General Solution : The General Solution of a linear equation of second orderwith constant co-efficient consists of two parts namely the ComplementaryFunction (C. F) and the Particular Integral (P.I).

Method of finding Complementary Function (C.F)

Let , be the two roots of then the solution of// / , is

,

, ,

where A and B are arbitrary constants.

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Method of finding Particular Integral (P. I )

• Suppose X is of the form , where ‘a’ is a constant.

Formula 1

P. I = ,

Formula 2

P. I =

Formula 3

P. I =

,

• When X is of the form sin ax (or) cos ax

Formula 1

P. I =

Formula 2

Sometimes we cannot form . Then we shall try to get ,.Multiplying and Dividing by the conjugate of the demoninator and get thesolution.

Formula 3

If ,P. I

If ,P. I

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• Suppose X is of the form

Working Rule : Take the Particular Integral as

. , , Since Particular Integral is also a solution of (aD 2+bD+c ) y = f(x), take according as f(x) = x or x 2. BySubstituting y value and comparing the like terms, one can find c 0, c 1 and c 2.

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12. Discrete Mathematics

Discrete Mathematics deals with several selected topics in Mathematics thatare essential to the study of many Computer Science areas. Among manytopics, only two topics, namely ‘Mathematical Logic’ and ‘Groups’ have beenintroduced here. These topics will be very much helpful to the students in

certain practical applications related to Computer Science.

Mathematical Logic

Logic deals with all types of reasoning’s. These reasoning’s may be legalarguments or mathematical proofs or conclusions in a scientific theory.

Logic is widely used in many branches of sciences and social sciences. It isthe theoretical basis for many areas of Computer Science such as DigitalLogic, Circuit Design, Automata Theory and Artificial Intelligence.

Logic Statement (or) Proposition

A statement or a proposition is a sentence which is either true or false but notboth.

Truth Value of a Statement

The truth or falsity of a statement is called its truth value. If a statement is true,we say that its truth value is TRUE or T and if it is false, we say that its truthvalue is FALSE or F.

Simple Statements

A Statement is said to be Simple if it cannot be broken into two or morestatements.

Compound Statements

If a statement is the combination of two or more simple statements, then it issaid to be a Compound Statement.

Basis Logical Connectives

Three basis connectives are Conjunction which corresponds to the Englishword ‘and’ denoted by the symbol , Disjunction which corresponds to theword ‘or’ denoted by the symbol and Negation which corresponds to theword ‘not’ denoted by the symbol

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Truth Tables

A table that shows the relation between the truth values of a compoundstatement and the truth values of its sub-statements is called the truth table. Atruth table consists of rows and columns. The initial columns are filled with thepossible truth values of the sub-statements and the last column is filled withthe truth values of the compound statement on the basis of the truth values ofthe sub-statements written in the initial columns. If the compound statement ismade up of n sub-statements, then its truth table will contain 2 n rows.

Logical Equivalence

Two compound statements A and B are said to be logically equivalent orsimply equivalent, if they have identical last columns in their truth tables.

Negation of a Negation

Negation of a Negation of a statement is the statement itself. Equivalently wewrite ( p) p.

Conditional and Bi-Conditional Statements

In Mathematics, we frequently come across statements of the form “If p thenq”. Such statements are called Conditional statements or implications anddenoted by and read as ‘p implies q’.

If p and q are two statements, then the compound statement is called a Bi-Conditional statement and is denoted by .

TRUTH TABLEp q p q p T T T T T TT F F T F FF T F T T FF F F F T T

Tautologies and Contradiction

A statement is said to be a Tautology, if the last column of its truth tablecontains only T. In other word, it is true for all logical possibilities.

A statement is said to be a Contradiction, if the last column of its truth tablecontains only F. In other word, it is false for all logical possibilities.

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Groups

Binary Operation

A binary operation * on a non-empty set S is a rule, which associates to eachordered pair (a, b) of elements a, b in S an element a*b in S .

Multiplication table for a Binary Operation

Any binary operation * on a finite set S = {a 1, a 2 , a 3 , …., a n } can be describe d bymeans of multiplication table. This table consists of ‘n’ rows and ‘n’ columns.Place each element of S at the head of one row and one column, usually takingthem in the same order for columns as for rows. The operator * is placed at theleft hand top corner. The nxn=n 2 spaces can be filled by writing a i * aj in thespace common to the i th row and the j th column of the table.

List of Symbols used

- for every

- belongs to

- there exists

- such that

- implies

DefinitionA non-empty set G, together with an operation * i.e,, (G, *) is said to be a Group if it satisfies the following axioms.

• Closure axiom : a,b G, a * b G

• Associative axiom : a, b, c G, (a * b) * c = a * (b * c)

• Identity axiom : a G, e G, such that a * e = e * a = a

• Inverse axiom : a, e G, a -1 G, such that a * a -1 = a -1 * a = e

Here e is called the identity element of G and a -1 is called the inverse of a in G .

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Definition ( Commutative Property )

A binary operation * on a set S is said to be commutative, if a * b = b * a,a, b S.

Definition

If a Group satisfies the Commutative property then it is called an Abelian Group (or) a Commutative Group , otherwise it is called a non-abelian group .

Order of a Group

The Order of a Group is defined as the number of distinct elements in theunderlying set.

If the number of elements is finite , then the group is called a finite group and ifthe number of elements is infinite then the group is called an infinite group .The Order of a group G is denoted by o(G).

Definition ( Semi-Group )

A non-empty set S, together with an operation * i.e,, (S, *) is said to be a Semi- Group if it satisfies the following axioms.

• Closure axiom : a,b S, a * b S

• Associative axiom : a, b, c S, (a * b) * c = a * (b * c)

Definition ( Monoid )

A non-empty set M, together with an operation * i.e,, (M, *) is said to be aMonoid if it satisfies the following axioms.

• Closure axiom : a,b M, a * b M

• Associative axiom : a, b, c M, (a * b) * c = a * (b * c)

• Identity axiom : a M, e M, such that a * e = e * a = a

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Properties of Groups

• The identity element of a group is unique.

• The inverse of each element of a group is unique.

• Let G be a group. Then for all a, b, c G

o a * b = a * c b = c

o b * a = c * a b = c

• In a group G, , for every a G

• Reversal Law : Let G be a group. a, b G. Then

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13. Fourier Series

Results :

Fourier Series of f(x)

Interval f(x) a 0 a n b n

(-l, l)

,

,

,

(0, 2l)

,

,

,

,

,

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Half Range Fourier Sine Series of f(x)

Interval f(x) a 0 a n b n

(0, l) 0 0

(0, )

0 0

Half Range Fourier Cosine Series of f(x)

Interval f(x) a 0 a n b n

(0, l)

0

(0, ) 0

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COMPLEX FORM OF FOURIER SERIESInterval Complex form of Fourier Series Fourier Coefficients

(-l, l)

(0, 2l)

,

,

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Parsevals Identity

Interval Parsevals Identity a 0 a n b n

(-l, l) ∑

(0, 2l) ∑

, ∑

,

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Parsevals Identity for Half Range Fourier Sine Series

IntervalParsevals Identity forHalf Range Fourier Sine Series a 0 a n b n

(0, l) ∑ 0 0

(0, ) ∑ 0 0

Parsevals Identity for Half Range Fourier Cosine Series

Interval Parsevals Identity forHalf Range Fourier Cosine Series

a 0 a n b n

(0, l) ∑ 0

(0, ) ∑ 0

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HARMONIC ANALYSISInterval Fourier Series a 0 a n b n

(0, 2l) , , (0, 2 ) , ,

Harmonic Analysis for Half Range Fourier Sine Series

Interval Harmonic Analysis forHalf Range Fourier Cosine Series

a 0 a n b n

(0, l)

0 0 ,

(0, ) 0 0 ,

Harmonic Analysis for Half RangeFourier Cosine Series of f(x)

Interval Harmonic Analysis forHalf Range Fourier Sine Series

a 0 a n b n

(0, l) ,

, 0

(0, ) , , 0

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14. Fourier Transforms

Fourier Transform Fourier Sine Transform Fourier Cosine Transform

--- ---

/ /

. .

--- ---

--- ---

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Convolution of two functions

Convolution theorem for Fourier Transforms

.

Parsevals Identity

| | | | , where

Parsevals Identity for Fourier Sine Transforms

| | | | , where

Parsevals Identity for Fourier Cosine Transforms

| | | |, where

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15. GraphsQuadratic Graph

To draw a straight line, two points are sufficient but to graph of a quadratic,more numbers of point required.

For each value of x, the equation y = ax 2 + bx + c gives the correspondingvalue of y. The set of all such ordered pairs (x,y) which defines the graph iscalled quadratic graph.

Quadratic Polynomials

A polynomial with degree 2 is called quadratic polynomial. The gen eral form ofa quadratic polynomial is y = ax 2 + bx + c , where a, b, c are real numbers suchthat a 0, and x is a variable.

Value of a Quadratic Polynomial

Let y = ax 2

+ bx + c be a quadratic polynomial and let a be a real number. Theny = ax 2 + bx + c is known as the value of the quadratic polynomial y = f(x) andit is denoted by y = f( ).

i.e., f( ) = a 2 + b + c

Solving Quadratic Equation by Graphical Method

In Algebra, we have solved the quadratic equation byalgebraic method. Now we are going to solve this quadratic equation byGraphical Method.

Type – I

First draw the graph of the equation y = ax 2 + bx + c

Here y = 0 is the equation of x=axis

Get the points of intersection of the curve y = ax 2 + bx + c with x-axis.

The x- coordinates of the intersecting points will give the roots of the givenequation.

Type – II

Split the quadratic equation into two equations representing a parabola and astraight line. Draw their graphs.

The x- coordinates of the points of intersection of the parabola and the straightline will give the roots of the given quadratic equation.

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16. Integral CalculusIntegrals of Standard Functions :

. ,

. ,

.

. ,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. , (x>a)

.√

. √

.√ √

.√ √

. √ √

. √

. √ √

. ,

.√

.√

.

.

.√

.√

.√

.√

.

.

.√

.√

.

.

.√ √

.

.

.

.

…………….., if n is even.

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. …… ……………..

only if both m and n are even.

. , ,

. , ,

.

.

. . / // ///.

Double Integral in Cartesian CoordinatesDouble Integral over region R may beevaluated by two successive integrations. If Ais described as

, , ,

Double Integral in Polar Coordinates

I = , Triple Integration in Cartesian C oordinates

1) I =

, ,

2)

represents the Volume of the

Region R.Area of Bounded Regions

Volume of Solids of Revolution

,

Properties of the Definte Integrals

.

.

.

.

.

.

. , , . , ,

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Length of a Curve , / /

(Parametric Form)

Surface Area of a Solid , / /

(Parametric Form)

Gamma Function

, 0

!,

/√

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17. Laplace TransformBasic Definitions:

Definition of Laplace Transform

, 0.

Inverse Laplace Transform

Linearity

First Shifting Theorem

t - shifting Second Shifting Theorem

Differentiation of Function

/

// /

Integration of the Function

Differentiation of Transform

L { t. f(t) } = /(s)

Integration of Transform

Convolution

..

f periodic with period p

Scaling

, 0

Initial Value

Final Value

Table of Laplace Transform

! ( n=1,2,3…)

√ √

√ /

! ( n=1,2,3…)

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,

(a 2≠ b2)

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Multiplication of Matrices :

Two matrices A and B are conformable for the product AB only, if the number ofColumn in A (Pre-Multiplier) is the same as the number of rows in B (Post-Multiplier).

A B = AB

[mxn] [nxp] [mxp]

Matrix Multiplication is always associative :

If A,B,C are mxn, nxp, pxq matrices, then (AB)C = A(BC) .

Multiplication of a matrix by a Unit Matrix :

If A is a Square Matrix of order n and I is the Unit Matrix of same order n, then

A.I = I.A = A.

Note : AB=0 (NULL) does not necessarily imply A=0 (or) B=0 (or) both A,B=0.

Properties of Matrix Addition :• Matrix Addition is commutative if both are same order. A+B = B+A .• It is also associative . A+(B+C) = (A+B)+C .• Additive Identity : A Null matrix of same order is the identity matrix.• Additive Inverse : For matrix [A], additive inverse is [-A]. [A] + [-A] = 0 .

Properties of Matrix Multiplication :• Matrix Multiplication is not commutative .• Matrix Multiplication is distributive over matrix addition.• A,B,C is of order mxn, nxp and nxp, then A(B+C) = AB + AC. • A,B,C is of order mxn, mxn, and nxp, then (A+B)C = AC + BC.

Characteristic Equation :The equation is said to be Characteristic Equation of thetransformation or the Characteristic Equation of the matrix A .

Eigen Values :

To solve the characteristic equation, we get characteristic roots. They are calledEigen Values .

Eigen Vectors :To find the eigen vectors, solve (A- I)=0 for the different values of .

Cayley-Hamilton Theorem :Every Square Matrix satisfies its own characteristic equation.

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Properties of Eigen values and Eigen Vectors:• Sum of the eigen values is equal to the sum of the main diagonal

elements.• Product of the eigen values is equal to its determinant value.• The eigen values of A and A T are the same.• The characteristic roots of a triangular matrix are just the diagonal

elements of the matrix.• If is a eigen value of a matrix A, then 1/ is the eigen value of A -1.• If is an eigen value of an orthogonal matrix, 1/ is also its eigen v alue.• If , ,,…are eigen values of a matrix A, then A m has the eigen

values , , ,….• The eigen values of a real symmetric matrix are real numbers.• The eigen values corresponding to distinct eigen values of a real

symmetric matrix are orthogonal.• The similar matrices have same eigen values.• If a real symmetric matrix of order Z has equal eigen values then the

matrix is a scalar matrix.• The eigen vector X of a matrix A is not unique.• If A and B are nxn matrices and B is a non-singular matrix then A and

B-1AB have same eigen values.

Diagonalisation of a Matrix :

If a Square matrix A of order n has ‘n’ linearly independent eigen vectors, thena matrix P can be found such that P -1AP is a diagonal matrix.

Fundamental theorem on Quadratic Form :

Any Quadratic form may be reduced to Canonical form by means of a non-singular transformation.

Quadratic Form :A homogeneous polynomial of the second degree in any number of variablescalled a Quadratic Form. The matrix corresponding to the Quadratic form in

three variables is

Nature of Quadratic Form:Let

/ be the given quadratic form in the variables x 1,x 2,x3,….x n.

i.e., / ….. Let the rank of A be r, then / contains only ‘r’ terms. The number ofpositive terms in the above equation of / is called the index of thequadratic form and it is denoted by ‘s’. The difference between the number ofpositive terms and the negative term is called the Signature of the quadraticform. Signature = 2s-r, where ‘s’ is equal to the number of positive terms and‘r’ is equal to the rank of A.

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19. Measurements

Denominate Number

A Denominate Number is one that refers to a unit of measurement which hasbeen established by law or by general usage. Examples are 2 inches, 8pounds, 3 seconds, etc.,

Compound Denominate Number

A compound Denominate Number is one that consists of two or m ore units ofthe same kind. Examples are 4 foot 3 inches, 3 hours 15 minutes, 1 pound 14ounces, 5 rupees 30 paise etc.,

Denominate numbers are used to express measurements of many kinds, suchas• Linear (Length)

• Square (Area)

• Cubic (Volume)

• Weight (pounds)

• Time (Seconds)

• Angular (degrees)

This classification is by no means complete. Systems of currency (dollars andcents, pounds sterling and pence, etc.,) would, for instance, be considereddenominate numbers, and the various foreign systems of weights andmeasures would, of course, come under the same head, though they arebeyond the scope of this webpage.

To gain facility in working out arithmetic problems involving denominatenumbers it is necessary to know the most common tables of measures, suchas are given here for reference. Note the abbreviation used, since these are inaccordance with the manner in which the values are usually written.

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TABLE OF MEASUREMENTS

1. Length or Linear Measure

Linear units are used to measure distances along straight lines.

U.S. (or) English System

12 inches (in. or “) = 1 foot (ft. or ‘)

3 feet or 36 inches = 1 yard (yd.)

5 ½ yards or 16 ½ feet = 1 rod (rd.)

220 yards or ½ mile = 1 furlong (fur.)

320 rods or 8 furlongs = 1 mile (mi.)

1,760 yards = 1 mile

5, 280 feet = 1 mile

Nautical Measure

6080 feet = 1 English nautical mile

1.15 land miles = 1 English nautical mile

60 nautical mile = 1 degree of arc

(at the equator)

360 degrees of arc = circumference of earth atEquator

1 fathom = 6 ft. (of depth)

1 hand = 4 in.

Metric System

Unit Metres

1 millimetre (mm.) = 0.001 = 0.03937 in.

10 millimetres = 1 centimetre (cm.) = 0.01 = 0.3937 in.

10 centimetres = 1 decimetre (dm.) = 0.1 = 3.937 in.

10 decimetres = 1 Metre (M.) = 1 = 39.3707 in.

10 metres = 1 dekametre (Dm.) = 10 = 32.809 ft.

10 dekametres = 1 hectometre (Hm.) = 100 = 328.09 ft.

10 hectometres = 1 kilometre (Km.) = 1000 = 0.52137 mile

10 kilometres = 1 myriametre (Mm.) = 10000 = 6.2137 miles

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3. Cubic Measure

This is used to measure the volume or amount of space within the boundariesof three-dimensional figures. It is sometimes referred to as Capacity .

CUBIC OR VOLUME MEASURE

1728 cubic inches (cu.in.) = 1 cubic foot (cu.ft.)27 cubic feet = 1 cubic yard (cu.yd.)1 cubic yard = 1 load of sand or dirt128 cubic feet = 1 chord of wood (cd.)24 ¾ cubic feet = 1 perch of stone (pch.)

LIQUID MEASURE OF CAPACITY

4 gills (gi.) = 1 pint (pt.)2 pints = 1 quart (qt.)4 quarts = 1 gallon (gal.)

The imperial gallon is used in the United Kingdom.

1 Imperial gallon = 1.20094 U.S. gallon

APOTHECARIES LIQUID MEASURE

60 drops or minims = 1 fluid drachm8 fluid drachms = 1 fluid ounce

DRY MEASURE OF CAPACITY

2 pints (pt.) = 1 quart (qt.)8 quarts = 1 peck (pk.)4 pecks = 1 bushel (bu.)

METRIC MEASURE OF CAPACITY

1000 cubic millimetres (cu.mm) = 1 cubic centimetre (c.c)

1000 cubic centimetres = 1 cubic decimetre (cu. dm.)1000 cubic decimeters = 1 cubic metre (cu. m.)10 centilitres (cl.) = 1 decilitre (dl.)10 decilitres = 1 litre (l.) = 1 cubic metre10 cubic litres = 1 dekalitre (Dt.)10 dekalitres = 1 hectolitre (Hl.)

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4. Measures of Weight

These are used to determine the quantity of matter a body contains. Fourscales of weight are used.

• Tray – for weighting gold, silver and other precious metals.

• Apothecaries – used by chemists for weighting chemicals.

• Avoirdupois - used for all general purposes.

• Metric – used in scientific work.

AVOIRDUPOIS WEIGHT

16 drams (dr. ) = 1 ounce (oz.)16 ounces = 1 pound (lb.)7000 grains (gr.) = 1 pound14 lb. = 1 stone (st.)2 st. = 1 quarter (qtr.)112 lb. = 1 cwt.2240 lb. = 1 ton

TROY WEIGHT

24 grains (gr.) = 1 pennyweight (dwt.)20 pennyweights = 1 ounce (oz.)12 ounces = 1 pound (lb.)5760 grains = 1 pound3 grains = 1 carat (kt.)

The carat, as defined in the table, is used to weigh diamonds. The same term isused to indicate the purity of gold. In this case, a carat means a twenty-fourthpart. Thus, 14 kt. Gold means that 14 parts are pure gold and that 10 parts are of other metals.

APOTHECARIES WEIGHT

20 grains (gr.) = 1 scruple (sc.)

3 scruples = 1 drachm (dr.)8 drams = 1 ounce (oz. )12 ounces = 1 pound (lb. )5760 grains = 1 pound

The above table is now obsolete but is given for historical interest.

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METRIC WEIGHT

10 milligrams (mg. ) = 1 centigram (cg. )10 centigrams = 1 decigram (dg. )10 decigrams = 1 gram (g. )10 grams = 1 dekagram (Dg.)10 dekagrams = 1 hectogram (Hg.)10 hectograms = 1 kilogram (Kg. )10 kilograms = 1 myriagram (Mg.)10 myriagrams = 1 quintal (Q. )10 quintals = 1 tonne (T.)

5. MEASURE OF TIME

60 seconds (sec. ) = 1 minute (min. or ‘)60 minutes = 1 hour (hr.)24 hours = 1 day (da. )7 days = 1 weel (wk. )2 weeks = 1 fortnight365 days = 1 common year366 days = 1 leap year12 calendar months = 1 year10 years = 1 decade100 years = 1 century (C.)

6. ANGULAR OR CIRCULAR MEASURE

Angular ( L) or Circular ( 0 ) Measure

60 seconds ( ‘’ ) = 1 minute ( ‘ )60 minutes = 1 degree( 0)90 degrees = 1 right angle (L) or 1 Quadrant360 angle degrees = 4 right angles360 arc degrees = 1 Circumference (0)

7. MONEY

UNITED STATES MONEY

10 mills (m. ) = 1 cent (c., or ct. )10 cents = 1 dime (d.)10 dimes = 1 dollar ($)10 dollars = 1 eagle (E. )

ENGLISH MONEY

2 half pennies = 1 penny (d. )12 pence = 1 shilling (s. )20 shillings = 1 pound ( ₤)21 shillings = 1 guinea

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THE METRIC SYSTEM

Since the metric system is based on decimal values, all ordinary arithmeticaloperations may be performed by simply moving the decimal point. The metricsystem is a system of related weights and measures. The metre is the basisfrom which all other units are derived. The unit of capacity, the litre, is thevolume of 1 Kg. (1,000 g.) of water, and thus is represented by a 1,000 c.c. Theunit of capacity, the litre and its derivates are used for both dry and liquidmeasure.

In the following tables whenever the metric equivalents of standard measuresare given, metric equivalents of other denominations may be found by simplymoving the decimal point to the right or the left as may be necessary.

EQUIVALENT VALUES

LINEAR MEASURE

1 inch = 2.5400 centimetres

1 foot = 0.3048 metre

1 yard = 0.9144 metre

1 rod = 5.0292 metres

1 mile = 1.6093 kilometres

1 centimetre = 0.3937 inch1 decimetre = 3.9379 inches

1 decimetre = 0.3281 foot

1 metre = 39.3700 inches

1 metre = 3.2808 feet

1 metre = 1.0936 yards

1 kilometre = 3280.83 feet

1 kilometre = 1093.611 yards

1 kilometre = 198.838 rods1 kilometre = 0.62137 mile

SQUARE MEASURE

1 sq. inch = 6.4516 sq. centimeters1 sq. foot = 0.0929 sq. metre1 sq. yard = 0.8361 sq. metre1 sq. rod = 25.2930 sq. metres1 acre = 4046.8730 sq. metres1 acre = 0.404687 hectare1 sq. mile = 258.9998 hectares1 sq. mile = 2.5900 kilometres

1 sq. centimetre = 0.15550 sq. inch1 sq. decimetre = 15.5000 sq. inches1 sq. metre = 15550.0000 sq. inches1 sq. metre = 10.7640 sq. feet1 sq. metre = 1.1960 sq. yards1 hectare = 2.4710 acres1 hectare = 395.3670 sq. rods1 hectare = 24.7104 sq. chains1 sq. kilometre = 247.1040 acres1 sq. kilometer = 0.3861 sq. mile

The hectare is the unit of land

measure.

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CUBIC MEASURE

1 cu. inch = 16.3872 cu. centimetres1 cu. foot = 28.3170 cu. decimeters1 cu. yard = 0.7645 cu. metre1 cord = 3.624 cu. metres

1 cu. centimetre = 0.0610 cu. inch1 cu. decimeter = 0.0353 cu. foot1 cu. metre = 1.3079 cu. yards1 cu. metre = 0.2759 cord

The cubic metre when used for measuring wood is called a ster .

CAPACITY

1 gallon U.S. = 3.7853 litres

1 gallon U.K. = 4.546 litres

WEIGHT

1 grain = 0.0648 gram

1 ounce troy = 31.103 grams1 pound troy = 0.3732 kilogram1 ounce avoirdupois = 28.350 grams1 pound avoirdupois = 0.4536 kilogram1 ton = 1.0160 tonne1 gram = 15.4324 grains1 gram = 0.0322 ounce troy1 gram = 0.0353 ounce avoirdupois1 kilogram = 2.6792 pounds troy1 kilogram = 2.2046 pounds avoirdupois1 tonne = 0.9842 ton1 tonne = 2,204.6223 pounds avoirdupois

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20. Mensuration

Square

If one side of a square is ‘a’, then its

• Area = a2

• Perimeter = 4a

• Diagonal = √ Rectangle

If the length and breadth of a rectangle are ‘a’ and ‘b’ res pectively, then

• Area = a x b

• Perimeter = 2(a+b)

• Diagonal = √

Parallelogram

If one side = ‘a’ and height = ‘h’, then

• Area = a x h

Rhombus

If d 1, d 2 be the diagonals of a rhombus, then

• Area =

• Side, a =

• Perimeter =

Trapezium

Area =

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Triangle

• Area = , where a,b,c are the

sides and

• Area of the equilateral triangle of side ‘a’ = , its Perimeter = 3a, its

altitude = √ .

Circle

If r is the radius of a Circle, then

• Area =

• Perimeter = 2 r

• Diameter =

Cube

If ‘a’ is side of a Cube, then

• Volume =

• Total Surfa ce =

• Diagonal = √

Cuboid

If length, breadth and height of a Cuboid are a, b, c respectively, then

• Volume =

• Total Surface Area =

• Diagonal = √

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Right Circular Cone

• Slant Height,

• Volume of Cone = cubic un its

• Curved Surface Area = square units

• Total Surface Area = square units

Hollow Cone made from a sector of radius ‘r’ and central angle ,

• Radius of Cone =

• Radius of Sector = Slant height of Cone

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Sphere

• Volume of Sphere = cubic Units

• Surface Area of Sphere = sq uare units

Hemisphere

• Volume of Hemi Sphere = cubic Units

• Curved Surface Area of Hemi Sphere = square units

• Total Surface Area = Cursed Surface Area + Area of Circular Base= + =

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21. Multiple Integrals

1. Change of Order of Integration

If the limits of the inner integral is a function of x ( or function of y), thenthe first integration should be with respect to y (or with respect to x).

Draw the region of integration by using the given limits.

If the integration is first with respect to x keeping y as a constant , thenconsider the horizontal strip and find the new limits accordingly.

If the integration is first with respect to y keeping x as a constant, thenconsider the vertical strip and find the new limits accordingly.

After finding the new limits, evaluate the inner integral first and then theouter integral.

• In evaluating double integrals by changing Cartesian to polarcoordinates, put x = r cos , y = r sin and dxdy = r. dr. d in the givenintegral and then find the new limits for and r and then evaluate.

• To change the three dimensional Cartesians to Cylindrical Coordinates,we have to put x = r cos , y = r sin , z = z.

• To change the three dimensional Cartesians to spherical polarcoordinates, we have to put x = r sin . Cos , y = r sin sin , z = r cos

.

• / ……, …… ,

• The area included between the curves y = f 1(x) and y = f 2(x) and theordinates x = a and x = b is given by

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22. Number Work

Arithmetic Progression (A. P)

• The first term of an Arithmetic Progression = a

• Common Difference = d

• n th term, t n = a + (n-1)d

• Number of terms of an Arithmetic Progression,

• Sum of ‘n’ terms,

• , if first and last term are known

Geometric Progression (G. P)

• The first term of an Geometric Progression = a

• Common Ratio = r

• n th term, t n = a r n-1

• Sum of ‘n’ terms , 1, 1

• Sum to infinity terms,

Special Series

∑ …

• ∑ …….

• ∑ …….

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23. Number & Operations

The Number System of Algebra

ELEMENTARY MATHEMATICS

ELEMENTARY MATHEMATICS is concerned mainly with certain elementscalled Numbers and with certain operations defined on them.

The unending set of symbols 1,2,3,4,5,6,7,8,9,10,11,12,13,…. used in c ountingare called Natural numbers .

In adding two of these numbers, say, 5 and 7, we begin with 5 ( or with 7) andcount to the right seven (or five) numbers to get 12. The sum of the two naturalnumbers is a natural number, i.e., the sum of the tow members of the above setis a member of the set. In subtracting 5 from 7, we begin with 7 and count tothe left five numbers to 2. It is clear, however, that 7 cannot be subtracted from5, since they are only four numbers to the left of 5.

INTEGERS

INTEGERS : In order that subtraction be always possible, it is necessary toincrease our set of numbers. We prefix each natural number with a + sign (inpractice, it is more convenient not to write the sign) to form the positiveintegers, we prefix each natural number with a – sign ( the sign must always bewritten ) to form the negative integers, and we create a new symbol 0, readzero. On the set of integers … -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ….. theoperations of addition and subtraction are possible without exception.

To add two integers such as +7 and -5, we begin with +7 and count to the leftfive numbers to +2, or we begin with -5 and count to the right (indicated by thesign +7) seven numbers to +2.

To subtract +7 and -5, we begin with -5 and count to the left (opposite to thedirection indicated by +7) seven numbers to -12. To subtract -5 from +7, webegin with +7 and count to the right (opposite to the direction indicated by -5)five numbers to +12.

If one is to operate easily with integers, it is necessary to avoid the process ofcounting. To do this, we memorize an addition table and establish certain rulesof procedure. Now, we may state

Rule 1 : To add two numbers having like signs, add their numerical values andprefix their common sign.

Rule 2 : To add two numbers having unlike signs, subtract the smallernumerical value from the larger and prefix the sign of the number having thelarger numerical value.

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Rule 3 : To subtract a number, change its sign and add.

Rule 4 : To multiply or divide two numbers (never divide by 0!), multiply ordivide the numerical values, prefixing + sign if the two numbers have likesigns and a – sign if the two numbers have unlike signs.

Every positive integer m is divisible by

. A positive integer m > 1 is

called a Prime if its only factors or divisors are . Otherwise, m iscalled Composite . For example 2, 7, 19 are primes, while 6 = 2.3, 18 = 2.3.3 and30 = 2.3.5 are composites. In these examples, the composite numbers havebeen expressed as products of prime factors, i.e., factors which are primenumbers. Clearly, if m = r.s.t is such a factorization of m, then –m = (-1).r.s.t issuch a factorization of m.

THE RATIONAL NUMBERS

The set of rational numbers consists of all numbers of the form , where mand n are integers. Thus, the rational numbers include the integers andcommon fractions. Every rational number has an infinitude of representations,

for example, the integer 1 may be represented by ,,,… and the fraction

may be represented by ,, ,…A fraction is said to be expressed in

lowest terms by the representation , where m and n have no common prime

factor. The most useful rule concerning rational numbers is, therefore

Rule 5 : The value of a rational number is unchanged if both the numerator anddenominator are multiplied or divided by the same nonzero number.

If two rational numbers have representations and , where n is a positive

integer, then > if r > s and < if r < s. Thus, in comparing two rational

numbers it is necessary to express them with the same denominator. Of themany denominators (positive integers) there is always a least one, called the

least common denominator. For the fractions and , the least common

denominator is 15. We conclude that < since = < = .

Rule 6 : The sum (difference) of two rational numbers expressed with thesame denominator is a rational number whose denominator is the commondenominator and whose numerators is the sum (difference) of the numerators.

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Rule 7 : The product of two or more rational numbers is a rational numberwhose numerator is the product of the numerators and whose denominator isthe product of the denominators of the several factors.

Rule 8 : The quotient of two rational numbers can be evaluated by the use ofRule 5 with the least common denominator of the two numbers as themultiplier.

DECIMALS

In writing numbers, we use a positional system, that is, the value given anyparticular digit depends upon its position in the sequence. For example, in 423the positional value of the digit 4 is 4(100). While in 234 the positional value ofthe digit 4 is 4(1). Since the positional value of a digit involves the number 10,this system of notation is called the decimal system. In this system, 4238.75means

4(1000) + 2(100) + 3(10) + 8(1) + 7( ) + 5( )

PERCENTAGE

The symbol %, read percent, means per hundred. Thus 5% is equivalent to

or 0.05.

• Any number, when expressed in decimal notation, can be written as apercent by multiplying by 100 and adding the symbol %.

• Conversely, any percentage may be expressed in decimal form bydropping the symbol % and dividing by 100.

• When using percentages, express the percent as a decimal and , whenpossible, as a simple fraction.

THE IRRATIONAL NUMBERS

The existence of numbers other than the rational numbers may be inferredfrom either of the following considerations.

We may conceive of a non repeating decimal constructed in endless time bysetting down a succession of digits chosen at random

The length of the diagonal of a square of side 1 is not a rational number. i.e.,

there exists no rational number a such that a 2 = 2. Numbers such as

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√ ,√ ,√ √ √ are called irrational numbers. The firstthree of these are called radicals. The radical √ is said to be of order n, n iscalled the index , and a is called the radicand .

THE REAL NUMBERS

The set of real numbers consists of the rational and irrational numbers. Thereal numbers may be ordered by comparing their decimal representations.

• We assume that the totality of real numbers may be placed in one-to-onecorrespondence with the totality of points on a straight line.

• The number associated with a point on the line, called the coordinate ofthe point, gives its distance and direction from the point (called theorigin) associated with the number 0. If a point A has coordinate a, weshall speak of it as the point A(a).

• The directed distance from point A(a) to point B(b) on the real numberscale is given by AB = b – a. the midpoint of the segment AB hascoordinate .

THE COMPLEX NUMBERS

In the set of real numbers, there is no number whose square is -1. If there is to

be such a number, say, √ , then by definition √

Note carefully that

√ √ √ √ is incorrect. In

order to avoid this error, the symbol i with the following properties is used.

If a > 0, √ √ , Then √ √ √ √ √ .

Also, √ √ √ √ .√ √

Numbers of the form a + bi, where a and b are real numbers, are calledComplex Numbers . In the complex number a + bi , a is called the real part andbi is called as the imaginary part . Numbers of the form ci, where c is real, arecalled imaginary numbers or sometimes pure imaginary numbers.

The complex number a + bi is a real number when b = 0 and a pure imaginary number when a = 0 .

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Only the following operations will be considered here.

• To add (subtract) two complex numbers, add (subtract) the real partsand add (subtract) the pure imaginary parts, i.e., (a + ib) + (c + id) = ( a + c) + (b+d)i

• To multiply two complex numbers, form the product treating I as anordinary number and then replace i 2 by -1.i.e., (a + ib) (c + id) = ( ac - bd) + (bc+ad)i

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24. Ordinary Differential Equations

General form of a Linear Differential Equation of the n th order with constantcoefficients is

…….. , where

, , ,.. are constants.

The Solution of above equation consists of• Complementary Function (C.F)• Particular Integral (P.I)

Also,

Complementary Function

An auxillary equation is given by ….

Solving this, we get , , ,……,′ ′ .

Nature of the Roots

Roots of an Auxillary Equation Complementary Function (C.F)

Roots are Real and Distinct.

Roots are m 1, m 2 where m 1 m 2.

Roots are Real and Equal.

Roots are m 1, m 2 where m 1 m 2.

Roots are imaginary.

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To find Particular Integral

Particular Integral P. I =

X Particular Integral

X, where X is afunction of x.

P. I = Pf 1 + Qf 2

where P =

Q =

P. I = ,

= / , ,/

= // , / ,//

P. I =

By expanding ,we get a solution.

Sin x (or) cos x P. I =

Replacing D 2 by –

P. I =

P. I =

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Homogeneous Equation of Euler Type

, z = log x

xD = D /

x2D2 = D / (D/ - 1)

x3D3 = D / (D/ - 1) (D / - 2)

x4D4 = D / (D/ - 1) (D / - 2) (D / - 3)

Some Standard Binomial Expansion

…….

…….

…….

…….

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25. Partial Differential Equation

Let us assume that z will always represent a function of x and y. i.e., z = f(x,y) where x and y are two independent variables and z is a dependent variable.

Notations

, , , ,

• If the number of constants to be eliminated is equal to the number ofindependent variables then the required Partial Differential Equation willbe of First Order.

• If the number of constants to be eliminated is more than the number ofindependent variables then the required Partial Differential Equation willbe of Second Order or higher order.

• If the number of functions to be eliminated is one, then the requiredPartial Differential Equation will be of first order otherwise it will be ofsecond order or higher order.

• Eliminating фfrom ф , gives a Partial Differential Equation

To Solve f(p, q) = 0

Let z = ax + by + c be the solution.

Then p = a, q = b, we get f(a,b)=0

Solving, we get ф

The Complete Integral is z = ax +

фy + c . There is no singular integral.

• The complete integral of Partial Differential Equation of the type

Z = px + qy + f(p, q) is z= ax + by + f(a,b)

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To Solve f(z, p, q) = 0

Let z=f(x + ay) be the solution.

Put u = x + ay, then z = f(u).

,

Solving ,, which is an ordinary differential equation, w e getthe required solution.

To Solve f 1(x, p) = f 2(y, q)

Let f 1(x, p) = f 2(y, q) = k

p = F 1(x, k)

q = F 2(y, k)

Then , ,To Solve F(x mp, y nq) = 0 and F(z, x mp, y nq) = 0

If , ,

Xm

p = (1 – m)P

Ynq = (1 – n)Q, where ,

Solution is F ( P, Q ) = 0F (z, P, Q) = 0

If , ,

put log x = X and log y = Y

xp = P and yq = Q

Solution is F ( P, Q ) = 0F (z, P, Q) = 0

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Lagranges Linear Equation

The standard form is Pp + Qq = R where P, Q and R are functions of x, yand z.

The subsidiary equation is

Choose any three multiplier l, m, n such that

,

i.e., = 0

Solving, we get u(x, y, z ) = c 1

Similarly choose another set of three multipliers l /, m /, n / such that

/ / // / / , / / /

Solving, we get v(x, y, z ) = c 2

The Solution is given by ф (u, v) = 0

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HOMOGENEOUS LINEAR PARTIAL DIFFERENTIAL EQUATI ON

The general form of Linear Partial Differential Equation is

… ,

The Solution is z = Complementary Function (C.F) + Particular Integral (P.I)

To find Complementary Function (C.F)

Auxillary Equation is ……

This equation has n roots say

, , ,…..

Case (i)

If the roots are real ( or imaginary) and different say ..,then the C.F is

Case (ii)

If any two roots are equal say …then the C.F is

……

Case (ii)

If any three roots are equal say …thenthe C.F is

……

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To find Particular Integral ( P. I )

If F(x, y) = e ax+by , then

Rule 1

.ф ,/ ф , ф,

Ifф , , . Rule 2

If F(x, y) = sin (mx+ny) or cos(mx+ny), then

.ф ,/

Replace D 2 by –m 2, D /2 by –n 2 and DD / by –mn in ф , provided thedenominator is not equal to zero. If the denominator is zero, then refer tocase (iv).

Rule 3

If F(x, y) = x myn then

.ф ,/ ф ,/

Expand ф ,/ by using Binomial theorem and then operate on x myn.

Note 1

,means integrate f(x,y) with respect to ‘x’ one time assuming ‘y’ as a

constant. / ,means integrate f(x, y) with respect to ‘y’ one timeassuming ‘x’ as a constant.

Note 2

In x myn , if m < n, then try to write ф ,/ as ф/ and if n < m, write ф ,/

as ф/

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Rule 4

If F(x, y) is any other function, resolve ф ,/ into linear factors say

/ /

….. , . / / ,

Now, , ,

Note :

If the denominator is zero in Rule 1 and Rule 2, then apply rule 4 to findParticular Integral.

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26. Probability

Random Experiment

Any experiment whose outcomes cannot be predicted in advance or determinein advance is a random experiment.

Trial

Each performance of the random experiment is called a trial.

Sample Space

The set of all possible outcomes of a random experiment is called a samplespace and is denoted by S.

Sample Point

Each element of the sample space is called a sample point.Event

An event is a subset of a sample space.

Equally Likely Events

Two or more events are said to be equally likely if each one of them has anequal chance of occurring. In tossing a coin, getting a head and getting a tailare equally likely events.

Mutually Exclusive Events

Two events A and B are said to be mutually exclusive events if the occurrenceof any one of them excludes the occurrence of the other event. i.e., they cannotoccur simultaneously.

Favourable Events or Cases

The number of outcomes of cases which entail the occurrence of the event inan experiment are called favourable events or favourable cases.

Probability

Let A be any event and the number of outcomes of an experiment favourableto the occurrence of A be ‘m’ and let ‘n’ be the total number of outcomes whichare all equally likely. Then the probability of occurrence denoted by P(A) isdefined as

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Definition (Conditional Probability)

The conditional Probability of an event B, assuming that the event A hasalready happened and is denoted by P(B/A) and defined as

, provided P(A) 0

Similarly

, provided P(B) 0

Theorem (Multiplication theorem on probability)

The Probability of the simultaneous happe ning of two events A and B is givenby

.

Definition

Two events A and B are independent if .

Baye’s Theorem

Suppose A 1, A2, A3, …. A n are mutually exclusive and exhaustive events suchthat (A i) > 0 for i = 1,2,3,4,…,n. Let B be any event with P(B) > 0, then

.

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Points to remember

• Number of outcomes which are not favourable to the event A = n – m .Probability of non-occurrence of A denoted by A / is given by

P(A / ) + P(A) = 1

• If P(A) = 0 , then A is an impossible event. i.e., Probability of an impossibleevent is zero. That is P(A) = 0 .

• Probability of the sure event is 1. That is P(S) = 1. S is called sure event.

Addition Theorem on Probability

• If A and B are any two events then P(AUB) = P(A) + P(B) – P (A B)

• If A and B are mutually exclusive events, then P(AUB) = P(A) + P(B)

Definition - Random Variable

If S is a sample space with a probability measure and X is a real valuedfunction defined over the elements of S, then X is called a Random Variable .

Types of Random Variable

• Discrete Random Variable

• Continuous Random Variable

Discrete Random Variable

Definition : Discrete Random Variable

If a random variable takes only afinite or a countable number of

values, it is called a Discrete RandomVariable.

ExampleNumber of Aces when ten cards aredrawn from a well shuffled pack of 52cards.

Continuous Random Variable

Definition : Continuous Random Variable

A Random variable X is said to becontinuous if it can take all possible

values between certain given limits. i.e., Xis said to be continuous if its valuescannot be put in 1-1 correspondence withN, the set of Natural Numbers.

ExampleThe Life length in hours of a certain lightbulb.

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Discrete Random Variable

Probability Mass Function (p.m.f)

• p(x) is non-negative for all real

x.• ∑ , where p i is the

probability at X = x i

Also,

Distribution Function(Cumulative Distribution Function)

The distribution function of a randomvariable X is defined as

∞ ∞

Properties of Distribution Fun ction

• F(x) is a non-decreasingfunction of x.

• , ∞ ∞

• ∞

• ∞

Continuous Random Variable

Probability Density Function (p.d.f)

> 0

for all real X.•

Cumulative Distribution Function

If X is a continuous random variable,the function given by

∞∞

where f(t) is the value of theprobability density function of X at t iscalled the distribution function orcumulative distribution of X.

Properties of Distribution Function

• F(x) is a non-decreasingfunction of x.

• , ∞

• ∞

• ∞

• F /(x) = f(x)

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Discrete Random Variable

Mathematical Expectation

Expectation of a Discrete RandomVariable

Definition

If X denotes a discrete randomvariable which can assume the valuesx1, x 2, ….,x n with respectiveprobabilities p 1,, p 2, ….p n then themathematical expectation of X,denoted by E(X) is defined by

Hence the mathematical expectationE(X) of a random variable is simplythe arithmetic mean.

Result

If is a function of the randomvariable X, then

Properties

• E (Constant) = Constant

• E (cX) = cE(X)

• E (aX+b) = aE(X)+b

• Var (X c) = Var(X)

• Var (aX) = a 2 Var(X)

• Var (Constant) = o

Continuous Random Variable

Mathematical Expectation

Expectation of a Continuous RandomVariable

Definition

Let X be a continuous randomvariable with probability densityfunction f(x). Then the mathematicalexpectation of X is defined as

NoteIf is function such that is arandom variable and existsthen

Properties

• E (Constant) = Constant

• E (cX) = cE(X)

• E (aX+b) = aE(X)+b

• Var (X c) = Var(X)

• Var (aX) = a 2 Var(X)

• Var (Constant) = o

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Moments

Expected values of a function of a random variable X is used for calculatingthe moments. The two types of moments are

• Moments about the origin• Moments about the mean which are called Central Moments.

Moments about the origin

If X is a discrete random variable for each positive integer r ( r = 1,2,3…) the r th moment

μ/

First Moment : μ/ ∑

Second Moment : μ/ ∑

Moments about the Mean : (Central Moments)

For each positive integer n, (n=1,2,3,…) the n th central moment of the discreterandom variable is

μ

The algebraic sum of the deviations about the arithmetic mean is always zero.i.e., μ

Second moment about the mean is called the variance of the random variableX. i.e.,

μ

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Theoretical Distributions

The values of random variables may be distributed according to some definiteprobability distribution is called theoretical distribution. Theoreticaldistribution are based on expectations on the basis of previous experience.

Discrete Distributions

Definition of Binomial Distribution

A random variable X is said to follow Binomial distribution if its probabilitymass function is given by

, , ,

Constants of Binomial Distribution

• Mean = np

• Variance = npq

Standard Deviation =

√ =

• p+q = 1

~. denotes that the random variable X follows Binomial Distributionwith parameters n and p .

In Binomial Distribution, mean is always greater than the variance.

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Poisson Distribution

Poisson Distribution is a limiting case of Binomial Distribution under thefollowing conditions.

• n th number of trials is indefinitely large. i.e.,

• p the constant probability of success in each trial is very small i.e.,

• np = ,is finite where is a positive real number. When an eventoccurs rarely, the distribution of such an event may be assumed tofollow a Poisson Distribution.

Definition

A Random Variable X is said to have a Poisson Distribution if the probabilitymass function of X is

!, , , ,

The mean of the Poisson Distribution is , and the Variance is also .

The parameter of the Poisson Distribution is .

Examples of Poisson Distribution

• The number of printing errors at each page of a book by a goodpublication.

• The number of telephone calls received at a telephone exchange in agiven time interval.

• The number of defective articles in a packet of 100, produced by a goodindustry.

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Continuous Distribution Function

Normal DistributionDefinitionA continuous random variable X is said to follow a normal distribution withparameters µ and , if the distribution function is

√ ,∞ ∞ ,∞ ∞ 0.

~ ,denotes that the random variable X follows Normal Distribution withmean and standard deviation .The Normal Distribution is also calledGaussian Distribution.

Constants of Normal Distribution• Mean = • Variance = 2 • Standard Deviation =

Properties of Normal Distribution• The normal curve is bell shaped.• It is symmetrical about the line X= . i.e., about the mean line.• Mean = Median = Mode = • The height of the normal curve is maximum at X = and √ is the

maximum height.• It has only one mode at X= . The normal curve is unimodal.• The normal curve is asymptotic to the base line.• The points of inflection are at X= .• Since the curve is symmetrical about X= , the skewness is zero.• A normal distribution is a close approximation to the binomial

distribution when n, the number of trials is very large and p theprobability of success is close to ½. i.e., neither p nor q is so small.

• It is also a limiting form of Poisson Distribution i.e., ∞, PoissonDistribution tends to normal distribution.

• Area Property : . 2. 3.

Standard Normal DistributionA random variable X is called a Standard Normal Variate if its mean is zero andits standard deviation is unity. i.e., N(0,1). The formula that enables to changefrom the x-scale to the z-scale and vice versa is .

.

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27. Pure Arithmetic

Basic Definitions

Uniform Speed, Time and Distance

Relative Speed

Relative Speed = Sum of the speeds of two bodies when they aremoving along straight path in the OPPOSITE DIRECTION.

Relative Speed = Difference of the speeds of two bodies when they aremoving along straight path in the SAME DIRECTION.

Average Speed

Average Speed = [not average of speeds]

Resultant Speed

• Resultant [ or effective ] speed of a boat = Speed of the boat in stillwater – Speed of the stream, when the boat is moving up stream

• Resultant speed of the boat = Speed of the boat in still water + Speed ofthe stream, when the boat is moving down stream.

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28. SetsSymbolsN : Set of Natural Numbers = { 1, 2, 3, 4, …}

W : Set of Whole Numbers = { 0, 1, 2, 3, 4, ….}

I+ or Z + : Set of Positive integers = { 1, 2, 3, 4, …}

I- or Z - : Set of Negative integers = { -1, -2, -3, -4, …}

I or Z : Set of integers = { -3, -2, -1, 0, 1, 2, 3, …}

Q : Set of Rational Numbers = { x : x = when p I, q I, q 0}

Q / : Set of Irrational Numbers

R : Set of Real Numbers

: Belongs to

: Does not belongs to

: Subset A B means A is a proper subset of B

⊇⊇⊇⊇: Subset A ⊇⊇⊇⊇B means A contains B

: Proper Subset A B means A is a proper subset of B

: Superset A B means A is a superset of B. i.e., A Properly contains B

: Difference : A B means a set containing all elements of A which arenot elements of B

: Universal Set

/ (or) ‘ (or) - : Complement A / or A ’ or A - means complement set of A, is theset of elements of U which do not belong to A

: Union (or) join : A B means a set of elements which belong eitherto A or to V or to both

: Intersection (or) meet : A B means a set of elements which belongto both A and B

N(A) : Cardinal No. of A : means the number of elements in the set A

P(A) : Power set of A

A x B : A x B = Cartesian product of A and B

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Operation on Sets

Union of Sets : A B = /

Intersection of Sets : A B =

/

Complement of a Set : A / or A c or /

Set Difference : A – B = /

Properties of Union

Set Union is Commutative. i.e., A B = B A

Set Union is Associative. i.e., A

(B

C) = (A

B)

C

Properties of Intersection

Set Intersection is Commutative. i.e., A B = B A

Set Intersection is Associative. i.e., A (B C) = (A B) C

Properties of Set Difference

Set Difference is not Commutative. i.e., A B B A

Set Difference is not Associative. i.e., A (B C) (A B) C

Distributive Property

Union is distributed over intersection. i.e., A (B C) = (A B) (A C)

Intersection is distributed over union. i.e., A (B C) = (A B) (A C)

Other Laws

nABnBCnACnABC

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29. Statistics

Class Boundaries

• Lower Class Boundary = Lower Class Limit -

• Upper Class Boundary = Upper Class Limit +

where d is the common difference between the upperclass limit of a class and the lower class limit of thenext class.

Class Mark

Class Mark =

= Width or Size of a Class

Width or Size of a Class = Upper Class Boundary – Lower Class Boundary

• Relative Frequency =

• Percentage Frequency = Range

Range = Difference between the maximum values and minimum valuesof a set of observations.

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Arithmetic Mean

∑and ∑ ∑

Arithmetic Mean = ∑ ∑ , where C is common interval.

Rang e= Maximum Value – Minimum Value

Standard Deviation = ∑,

= ∑ ∑,

Standard Deviation for Disordered Series

∑ ∑ ,

∑ ∑ ∑ ∑

Variance =

Standard Deviation of first n natural number = • The Standard Deviation of a series remain unchanged when each value

is added (or) subtracted by the same quantity.

• The Standard Deviation of a series gets multiplied (or) divided by thesame quantity k, if each value is multiplied (or) divided by k.

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31. Theoretical GeometryDefinition

The Geometry which deals with properties and characters of variousgeometrical shapes with axioms / theorems without accurate measurements isknown as Theoretical Geometry.

History

Geometry was developed by the Egyptians more than 1000 years before C hristto help them mark out of their fields after the floods from the Nile, but wasabstracted by the Greeks into logical system of proofs many centuries later.For measurements, the length of line and sizes of angles were needed. Forlogical system of proofs, basic postulates or axioms were necessary. Now thestudy of Geometry is useful in our daily life in many ways.

Axioms (or) Postulates

Some Geometrical statements are accepted and they are without any proof.Such statements are called Axioms. An axiom is a self-evident truth.

Let us learn some important axioms.

Axiom – 1

Given any two distinct points in a plane, there exists one and only one linepassing through them.

Axiom – 2

Two distinct lines cannot have more than one point in common.

Axiom – 3

Given a line and a point not on the line, there is one and only one line whichpasses through the given point and is parallel to the given line.

Complementary Angles

Two angles are complementary if their sum is 90 o.

Supplementary Angles

Two angles are said to be supplementary if their sum is 180 o .

Adjacent Angles

Two angles are adjacent angles if both angles have a common vertex.

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Linear pair

Two adjacent angles form a linear pair if the two non-common arms are in astraight line.

Theorems

A theorem is a geometrical statement which needs to be proved. To prove atheorem, the following five important steps are followed.

• Draw the figure• Write all the data• Write what is to be proved, using letters of figures• Write the construction, if necessary which will help to prove t he theorem• Write the proof with statements and reasons

Theorem

If a ray stands on a line, then the sum of the two adjacent angles so formed is180 o .

Theorem

If the sum of two adjacent angles is 180 o, then their outer arms are in the samestraight line.

Corollary : Corollary is also a Geometrical Statement which can be provedfrom the theorem.

Corollary

If two straight line intersect each other, the sum of the four angles so formed isequal to 360 o (or) 4 right angles.

Corollary

If any number of straight lines meet at a point, the sum of all the angles soformed is equal to 360 o (or) 4 right angles.

Corollary

If from a given point on a line, any number of rays are drawn on the same sideof it, the sum of all the angles so formed is equal to two right angles (180 o)

Theorem

If two lines intersect, the vertically opposite angles so formed are equal.

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Parallel Lines

Two (or) more lines are said to be parallel to each other, if they are in the sameplane and do not intersect when produced on either side. i.e., distance betweenthem remains same.

Transversal

A straight line which intersect two or more lines at distinct points is called atransversal. When a transversal intersect two lines, four pairs of angles areformed.

Playfair’s Axiom

Lines which are parallel to the same line are parallel to each other.

Theorem

If a transversal intersects two parallel lines then the pair of correspondingangles are equal.

Converse of the above theorem

If a transversal intersects two straight lines such that a pair of correspondingangles are equal, then the two lines are parallel.

Theorem

If a transversal intersects two parallel line then• Each pair of alternate angles are equal•

The interior angles on the same side of the transversal aresupplementary

Theorem

The sum of three angles of a triangle is 180 o .

Theorem

If a side of a triangle is produced, the exterior angle so formed is equal to thesum of the two interior opposite angles.

Important Notes• In a triangle, the sum of any sides is always greater than the third side.• Every triangle should have atleast two acute angles.• The sum of the angles of a triangle is 180 o or 2 right angles.• The sum of the angles of an n-sided polygon is (2n-4) right angles.• In any right angled triangle, the square on the hypotenuse is equal to the

sum of the squares in the other two sides (Pythagorus theorem)

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Angles

Triangle

In a triangle ABC, A, B, C are the vertical angles and sides BC, CA, AB aredenoted by a, b, c respectively. Then

• A+B+C =

• a + b > c, a + c > b, b + c > a

• ~ , ~ ,~

• Exterior angle = Sum of two opposite interior angles

Polygon

• Sum of interior angles of a polygon of n sides = (2n – 4) right angles= (2n -4) x

• Sum of exterior angles of a convex polygon =

• Each angle of a regular polygon of n sides = • Number of sides of a regular polygon each

having internal angle =

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Theorem

If two chord of a circle intersect inside the circle (or outside) when producedthe rectangle formed by the two segments of one chord is equal in area to therectangle formed by the two segments of the other chord, then

PA x PB = PC x PD

Secant Theorem

If PAB is a secant to a circle intersecting it at A and B and PT is a tangent at T,then

PA x PB = PT 2

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Basic Proportionality Theorem (or) Tales Theorem

If the straight line is drawn parallel to one side of a triangle it cuts the othertwo sides proportionally.

Angle Bisector Theorem

If the vertical angle of a triangle is bisected internally (or) externally, thebisector divides the base internally (or) externally in to two segments whichhave the same ratio as the order of two sides of the triangle.

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Theorem [AAA – Similarity]

If two triangles are equiangular to one another, then the two triangles aresimilar.

Theorem [SAS – Similarity]

If two triangles have one angle of the one equal to one angle of the other andthe sides about the equal angles proportional, then two triangles are simila r.

Theorem [SSS – Similarity]

If two triangles have their corresponding sides proportional then the twotriangles are similar.

Areas of Similar Triangle

Similar triangles are to one another as the squares on their correspondingsides (or) the ratio of the areas of two similar triangles is equal to the ratio ofthe squares of their corresponding sides.

Similar Triangles

In

If D, E are midpoints of AB, AC in triangle ABC,

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Right angled triangle ABC, Right angled at A

If a perpendicular is drawn from the vertex of the right angle of a right triangleto the hypotenuse, the triangle on each side of the perpendicular are similar ttothe whole (original) triangle and to each other.

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32. Trigonometry

Trigonometry is that branch of mathematics which deals with the study of therelationship between the sides and angles of triangle.

Basic Definitions:

Inter Relations

Identities

Radian Measure

Trigonometric Ratios for certainstandard angles

sin √ √

cos √ √

tan

√ ∞

sec

√ √

cosec ∞ √ √

cot ∞ √ √

Also,

sin

cos

tan

cosec

sec

cot

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Allied Angles

Trigonometrical ratio’s of 90 , 180 , 270 , 360 in terms of thoseof can be found easily by the following rule known as A-S-T-C rule.

When the angle is 90 (or) 270 , the trigonometrical ratio changesfrom sine to cosine, tan to cot, sec to cosec and vice versa.

When the angle is 180 (or) 360 , the trigonometrical ratio rem ainsthe same. i.e., sin --> sin , cos --> cos , etc.,

In each case the sign (+) or (-) is premultiplied by the A-S-T-C quadrantrule.

S AII (90 – 180) I (0 – 90 )

T CIII (180 – 270) IV (270 – 360 )

A : all ratio’s are positive in the I QuadrantS : sine is positive in the II QuadrantT : tan is positive in the III QuadrantC : cos is positive in the IV Quadrant

Compound Angle Formulae

To convert product in to sum ordifference formulae

Formulae for A, 2A & 3A angles

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Formula to convert a su m or difference into product :

Hyperbolic Functions

If

is called

as the Exponential Function.

Hyperbolic Functions are definedin terms of exponential functionas below :

Hyperbolic Identities

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Relationship between Trigonometricand Hyperbolic Functions :

= ]

= =

Standard Results

,

/

∞/

General Solutions of

,

General Solutions of

,

General Solutions of

,

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Properties of Triangles

Consider a triangle ABC. It has three angles A, B and C. The sides opposite tothe angles A, B, C are denoted by the corresponding small letters a, b, crespectively. Thus a = BC, b = CA, c = AB . We can establish number offormulae connecting these three angles and sides.

Sine Formula

In any triangle ABC, ,

Napier’s Formulae

In any triangle ABC,

Cosine Formulae

In any triangle ABC, the following results are true with usual notation.

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Projection Formulae

In any triangle ABC,

Sub – Multiple (half) angle formulae

In any triangle ABC, the following results are true .

where s =

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Area Formulae (

In any triangle ABC,

• ∆

• ∆

• ∆

• ∆

• ∆

Are true with the usual notations and these are called Area formulae.

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33. Vector AlgebraBasic DefinitionsDot and Cross Product

Let and be any two vectorssubtending an angle , between them.

Also, let

represent the unit vectorperpendicular to the plane containing

the vectors and . (i.e.,)Perpendicular to both and . Then,we have

. ||

|| where ,

, forms a right handled

system.

Angle Between the vectors and If is the angle between them, then

. ||

Further, the vectors are Perpendic ular,if .

.

Also, to , .

Angle Between the vectors and using Cross Product

If is the angle between them, then

||

Further, the vectors are Parallel, if .

Also, to ,

Properties

. . and

.   .   .

     

; ;

.    . . 

Analytic Expressions for the Dot andCross Product

If and

then

.and

Scalar Trible Product (or) BoxProduct

. denoted by [ ] iscalled as the scalar triple product (or)

the box product of the vectors

,, .Properties

. = . . = . = . [ ] is equal to the value

of the coefficient determinant

of the vectors

,, .

If any two vectors are identicalin a box product then the valueis equal to zero. In such a case,we say that the vectors areCoplanar.

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Vector Triple Product

= . .

=

.

. Scalar Product of Four Vectors

. = . . . .

Vector Product of Four Vectors

= [ ] - [ ]

= [

] - [

]

Equation of a Straight Line

Equation of straight line passingthrough a point and parallel to

is

=

Equation of a Plane

Equation of plane pass ing througha point and parallel to and is

= + t

where

and are scalars.

Gradient

=

Directional Derivative =

.||

Unit Tangent Vector =

Normal Derivative = | | Unit Normal Vector | | Angle between the Surfaces is givenby,

cos = .| || |

Divergence

Divergence = .

=

. = 0 is Solenoidal.

Curl

Curl = =

= 0 is Irrotational.

Laplace Operator

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Line Integral

StatementThe Line Integral along the curve C is denoted by

. ., if c is a closed curve.

Surface Integral

StatementThe Surface Integral of is defined to be

. ..

Volume Integral

Statement

Volume Integral of F(x,y,z) over a region enclosing a volume V is given by

, , , ,

Green’s Theorem in a Plane

Statement

If u, v, , are continuous and one-valued functions in the Region R enclosedby the curve C, then

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Gauss Divergence Theorem

Statement

The Surface Integral of the normal component of a vector function F over aclosed surface S enclosing volume V is equal to the volume Integral of thedivergence of F taken throughout the volume V is represented by,

. .

Stoke’s Theorem

Statement

The Surface Integral of the normal component of the Curl of a vector function Fover an open surface S is equal to the line integral of the tangential componentof F around the closed curve C bounding S is given by ,

. .

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34. Z - Transforms

PROPERTIES OF UNILATERAL Z-TRANSFORM

Property DiscreteSequence

Z-Transform

Linearity

Frequency shifting

Time shifting

m 0

,

Scaling inZ-domain (or)

Multiplicationby

DifferentiationIn Z-domain

Time reversal

(Bilateral)

Convolution

Initial Value Theorem

Final Value Theorem ∞

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TABLE OF Z-TRANSFORMS

! !

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,,

1

Solving Simultaneous Equations with given initial conditions

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Inverse Z-Transform