Handbook of Mathematics

449 Pages·2.882 MB·English

by  Arihant Experts

hand book
KEY NOTES TERMS
DEFINITIONS FORMULAE
Mathematics
Highly Useful for Class XI & XII Students,
Engineering Entrances and Other Competitions
Amit Rastogi
Supported by
Love Agarwal
Brijesh Dwivedi
ARIHANT PRAKASHAN, (SERIES) MEERUT
Arihant Prakashan (Series), Meerut
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PREFACE
Handbook means reference book listing brief facts on a
subject. So, to facilitate the students in this we have
released this Handbook of Mathematics. This book has
been prepared to serve the special purpose of the students,
to rectify any query or any concern point of a particular
subject.
This book will be of highly use whether students are
looking for a quick revision before the board exams or just
before other examinations like Engineering Entrances or
any similar examination, they will find that this handbook
will answer their needs admirably.
This handbook can even be used for revision of a subject
in the time between two shift of the exams, even this
handbook can be used while travelling to Examination
Centre or whenever you have time, less sufficient or more.
The format of this handbook has been developed
particularly so that it can be carried around by the
students conveniently.
The objectives of publishing this handbook are :
— To support students in their revision of a subject just
before an examination.
— To provide a focus to students to clear up their doubts
about particular concepts which were not clear to them
earlier.
— To give confidence to the students just before they
attempt important examinations.
However, we have put our best efforts in preparing this
book, but if any error or what so ever has been skipped
out, we will by heart welcome your suggestions. A part
from all those who helped in the compilation of this book
a special note of thanks goes to Mr. Ashwani of Arihant
Publications.
Authors
CONTENTS
1. Sets and Relations 1-9 ! Cube Roots of Unity
! Set ! nth Roots of Unity
! Subset and Superset ! Geometry of Complex
! Venn Diagram Numbers
! Operations on Sets
4. Theory of Equations and
! Ordered Pair
Inequations 46-61
! Relation
! Polynomial
! Composition of Relation
! Quadratic Equation
2. Functions and Binary ! Quadratic Expression
Operations 10-27 ! Inequality
! Functions ! Linear Inequality
! Equal Functions ! Solution Set
! Real Valued and Real
5. Sequences and Series 62-73
Functions
! Sequence
! Standard Real Functions and
! Series
their Graphs
! Arithmetic Progression (AP)
! Operations on Real
! Geometric Progression (GP)
Functions
! Harmonic Progression (HP)
! Compositions of Two
! Arithmetic-Geometric
Functions
Progression
! Even and Odd Functions
! Binary Operations
6. Permutations and
Combinations 74-84
3. Complex Numbers 28-45
! Fundamental Principles of
! Equality of Complex
Counting
Numbers
! Iota ! Factorial
! Algebra of Complex ! Permutation
Numbers ! Circular Permutation
! Argand Plane and Argument ! Combination
of a Complex Number
7. Binomial Theorem and ! Random Variable
Principle of Mathematical ! Bernoulli Trials and
Induction 85-92 Binomial Distribution
! Binomial Theorem for
11. Trigonometric Functions,
Positive Integer
Identities and Equations
! General Term in a Binomoal
124-140
Expansion
! Middle Term in a Binomial ! Measurement of Angles
Expansion ! Relation Between Degree
! Greatest Term and Radian
! Multinomial Theorem ! Trigonometric Ratios For
! R-f Factor Relations Acute Angle
! Binomial Theorem for Any ! Trigonometric (or Circular)
Index Functions
! Principle of Mathematical
! Graph of Trigonometric
Induction
Functions
! Fundamental Trigonometric
8. Matrices 93-101
Identities
! Matrix
! Trigonometric Functions of
! Algebra of Matrices
Compound Angles
! Transpose of Matrix
! Transformation Formulae
! Symmetric and Skew-
! Trigonometric Functions of
Symmetric Matrices
Multiple Angles
! Elementary Operations
! Trigonometric Periodic
(Transformations of a
Functions
Matrix)
! Conjugate of a Matrix ! Trigonometric Equations
! Rank of a Matrix
12. Solution of
9. Determinants 102-112 Triangles 141-147
! Determinant ! Basic Rules of Triangle
! Minor and Cofactors ! Trigonometrical Ratios of
! Adjoint of a Matrix Half of the Angles of
! Inverse of a Matrix Triangle
! Homogeneous and Non- ! Area of a Triangle
homogeneous System of
13. Heights and
Linear Equations
Distances 148-150
10. Probability 113-123 ! Angle of Elevation
! Experiment ! Angle of Depression
! Algebra of Events
! Baye’s Theorem
!
14. Inverse Trigonometric ! Equation of Tangent
Functions 151-159 ! Equation of Normal
! Domain and Range of ! Pair of Tangents
Inverse Trigonometric ! Common Tangents of Two
Functions Circles
! Graphs of Inverse ! Family of Circles
Triginometric Functions ! Limiting Points
! Elementary Properties of ! Diameter of Circle
Inverse Trigonometric
18. Parabola 190-199
Functions
! Inverse Hyperbolic ! Conic Section
Functions ! General Equation of Conic
! Standard Forms of a
15. Rectangular Axis 160-165 Parabola and Related Terms
! Rectangular Axis ! Equation of Tangent
! Quadrants ! Point of Intersection of Two
! Distance Formulae Tangents
! Section Formulae ! Equation of Normal
! Area of Triangle/ ! Length of Tangent and
Quadrilateral Normal
! Shifting of Origin/Rotation ! Equation of Diameter
of Axes ! Pair of Tangents
! Equation of Locus ! Chor of Contact
16. Straight Line 166-175 19. Ellipse 200-208
! Slope (Gradient) of a Line ! Parametric Equation
! Angle between Two Lines ! Equation of Tangent
! Equation of a Straight Line ! Equation of Normal
! Distance of a Point from a
Line 20. Hyperbola 209-219
! Equation of the Bisectors ! Hyperbola of the Form
! Pair of Lines x2 y2
—–—= 1
a2 b2
17. Circles 176-189 ! Conjugate Hyperbola
! Standard Equation of a –x2 y2
—+—= 1
Circle a2 b2
! Equation of Circle Passing ! Equation of Hyperbola in
Through Three Points Different Forms
! Parametric Equation of ! Tangent Equation of
Circle Hyperbola
! Normal Equation of 24. Indefinite
Hyperbola Integrals 254-268
! Asymptote ! Some Standard Integral
! Rectangular Hyperbola Formulae
! Properties of Integration
21. Limits, Continuity &
! Intergation by Substitution
Differentiability 220-223
! Integration by Parts
! Limit
! Integration by Partial
! Methods of Evaluating
Fractions
Limits
! Integration of Irrational
! Sandwich Theorem
Algebraic Function
! Continuity
! Differentiability 25. Definite Integrals 269-274
! Fundamental Theorems of ! Fundamental Theorem of
Differentiability Calculus
! Properties of Definite
22. Derivatives 234-241
Integral
! Derivatives of Standard ! Integral Function
Functions
! Fundamental Rules for 26. Applications of
Derivatives
Integrals 275-283
! Derivatives of Different
! Area of Curves Given by
Types of Function
Polar Equations
! Differentiation of a
! Area of Curves Given by
Determinant
Parametric Curves
! Successive Differentiations
! Curve Sketching
! Partial Differentiations
! Volume and Surface Area
23. Application of
27. Differential
Derivatives 242-253
Equations 284-290
! Derivatives as the Rate of
Change ! Order and Degree of a
! Tangents and Normals Differential Equation
! Rolle’s Theorem ! Linear and Non-Linear
! Lagrange’s Mean Value Differential Equations
Theorem ! Solution of Differential
! Approximations and Errors Equations
! Increasing Function ! Formation of Differential
Equations
! Maxima and Minima of
Functions
28. Vectors 291-304 31. Mathematical
! Types of Vectors Reasoning 338-342
! Addition of Vectors
! Statement (Proposition)
! Differences (Subtraction) of
! Elementary Logical
Vectors
Connectives or Logical
! Multiplication of a Vector by
Operators
a Scalar
! Truth Value and Truth Table
! Components of a Vector
! Quantifiers
! Vector Joining Two Points
! Validity of Statements
! Section Formulae
! Scalar or Dot Product of 32. Linear Programming
Two Vectors
Problem (LPP) 343-346
! Vector or Cross Product of
! Objective Function
Two Vectors
! Constraints
! Scalar Triple Product
! Non-negative Restrictions
! Vector Triple Product
! Optimal Value
29. Three Dimensional ! Solution of Simultaneous
Geometry 305-319 Linear Inequations
! Graphical Method of Solving
! Coordinate System
a Linear Programming
! Direction Cosines
Problem
! Line in Space
! Different Types of Linear
! Plane
Programming Problems
! Angle Between Two Planes
! Parallelism and
33. Elementary
Perpendicularity of Two
Arithmetic-I 347-367
Planes
! Types of Number System
30. Statistics 320-337 ! Classification of Numbers in
Decimal Number System
! Graphical Representation of
! Test of Divisibility of a
Frequency Distributions
Natural Number
! Measures of Central
! Rule of Determine the Digit
Tendency
at Unit Place
! Arithmetic Mean
! Geometric Mean ! Rational Numbers
! Harmonic Mean ! Irrational Number
! Median ! Real Number
! Mode ! Complex Numbers
! Covariance ! Fraction
! Rank Correlation ! Ascending/Descending
(Spearman’s) Orders in Fraction
! Regression ! Power of Index
! Surds 36. Elementary
! HCF and LCM Algebra 398-406
! Simplification ! Polynomial
! Average ! Synthetic Division Method
! Ratio and Proportion (Horner’s Method)
! Proportion ! Remainder Theorem
! Linear Equations
34. Elementary
! Rational Expression
Arithmetic-II 368-378
37. Logarithms 407-409
! Fundamental Formula
! Speed, Time and Distance ! Types of Logarithms
! Anti Logarithm
! Problem Based on Trains
! Boats and Streams
38. Geometry 410-424
! Pipes and Cisterns
! Triangles
! Clock
! Congruency of Triangles
! Calendar ! Quadrilaterals
! Polygon
35. Elementary
Arithmetic-III 379-397 39. Mensuration 425-439
! Percentage ! Perimeter and Area of Plane
! Profit, Loss and Discount Figure
! Simple Interest ! Surface Area and Volume of
! Compound Interest Solid Figure
! Growth and Depriciation
Some Great Mathematicians &
! Partnership
Their Contributions 440-443
! Share and Debenture
! Alligation or Mixture Appendix I to VII 444-454