Wiley's Mathematics Refresher Course for JEE

Preface

1 Trigonometric Ratios and Identities

1.1 Basic Formulae

1.2 Graphs of Trigonometric Functions

1.3 Points To Ponder

2 Trigonometric Equations

2.1 General Solution  

2.2 Graph of Trigonometric Functions

2.3 Trigonometric Equations (Points to Remember)

2.4 System of Equation

2.5 Key Points to Be Remembered for Solving the Trigonometric Equation  

3 Properties and Solutions of Triangles

3.1 Formulas for Properties of Triangles  

3.2 Trigonometric Ratios of Half-Angles

3.3 Area of a Triangle

3.4 Incircle

3.5 Escribed Circles

3.6 Solution of Triangles

3.7 Regular Polygon

3.8 Pedal Triangle

3.9 Cyclic Quadrilateral

3.10 Theorem of the Medians (Apollonius Theorem)  

4 Quadratic Equations

4.1 Nature of Roots

4.2 Relation Between Roots, Coefficients and Symmetric Functions of Roots  

4.3 To Form an Equation Given the Roots

4.4 Condition for Common Roots

4.5 Quadratic Expression

4.6 Properties of Quadratic Functions

4.7   Graphical Approach to Location of Roots

4.8   Wavy Curve Method (Method of Intervals)  

4.9 Equations of Third and Higher Degrees and the Relation Between Their Roots

4.10 Results on Roots of a Polynomial Equation  

4.11 Equations Reducible to Quadratic Equations  

5 Complex Numbers

5.1 Representation of a Complex Number  

5.2 Properties of Modulus  

5.3 Argument of a Complex Number  

5.4 Conjugate of a Complex Number  

5.5 De Moivre’s Theorem  

5.6 Cube Roots of Unity  

5.7 nth Roots of Unity  

5.8 Coni Method  

5.9 Condition for Four Points to Be Concyclic  

5.10 Complex Number as a Rotating Arrow in the Argand Plane   

5.11 Theory of Equations with Complex Coefficients  

5.12 Logarithm of a Complex Number  

5.13 Section Formula  

5.14 Locus in an Argand Plane  

5.15 Circle   

5.16 Conic Section   

6 Binomial Theorem

6.1 Binomial Expression  

6.2 Binomial Theorem for Positive Index  

6.3 Proof of Binomial Theorem  

6.4 General Term in the Binomial Expansion  

6.5 Middle Terms of the Expansion  

6.6 Binomial Coefficients  

6.7 Greatest Term in the Expansion  

7 Permutations and Combinations

7.1 Introduction  

7.2 Fundamental Principle of Counting  

7.3 Permutations  

7.4 Combinations  

7.5 Divisors of a Given Natural Number  

7.6 Division of Distinct Object into Groups  

7.7 Division of Identical Objects into Groups  

7.8 Arrangements in Groups  

7.9 Method of Inclusion Exclusion  

7.10 Dearrangements   

7.11 Use of Multinomials  

7.12 Use of Multinomials in Solving Linear Equation  

8 Sequence and Series

8.1 Arithmetic Progression (AP)  

8.2 Geometric Progression (GP)  

8.3 Harmonic Progression (HP)

8.4 Arithmetico-Geometric Progression

8.5 Some Important Results

8.6 Inequalities (AM ≥ GM ≥ HM)

8.7 Weighted Means

8.8 Proving Inequalities

8.9 Exponential Series

8.10 Logarithmic Series

8.11 Some Important Results

8.12 Arithmetic Mean of mth Power

9 Straight Lines

9.1 Basic Formulae

9.2 Forms of Straight Lines

9.3 Image Reflection, Foot of Perpendicular, Perpendicular Distance of Point w.r.t. to Line

9.4 Distance Between Two Parallel Lines

9.5 Concurrency of Straight Lines  

9.6 Position of a Point w.r.t. a Line

9.7 Bisector of the Angle Between Two Lines

9.8 Analysis of Three Lines

9.9 Analysis of Quadrilateral

10 Circle

10.1 Basic Formulae

10.2 Equations of Tangent and Normal

10.3 Family of Circles

10.4 External and Internal Contacts of Circles

10.5 Points to Ponder

11 Parabola

11.1 Conic Section

11.2 Section of a Right Circular Cone by Different Planes

11.3 Parabola

11.4 Equations of Tangent in Different Forms

11.5 Equations of Normal in Different Forms

11.6 Position of Point w.r.t Parabola

11.7 Centre of Conic

11.8 Equation of the Pair of Tangents

11.9 Chord of Contact

11.10 Diameter of Parabola

11.11 Important Results

12 Ellipse

12.1 Ellipse

12.2 Position of Point w.r.t. an Ellipse

12.3 Chord with Mid-Point

12.4 Chord of Contact

12.5 Pole and Polar

12.6 Equation of Polar of a Point

12.7 Diameter of an Ellipse

12.8 Equation of Diameter of Ellipse

12.9 Conjugate Diameters of Ellipse

13 Hyperbola

13.1 Hyperbola

13.2 Rectangular Hyperbola

14 Matrices and Determinants

Part A: Matrices

14.1 Types of Matrices

14.2 Trace of a Matrix

14.3 Equality of Matrices

14.4 Algebra of Matrices

14.5 Transpose of a Matrix

14.6 Symmetric and Skew-Symmetric Matrices

14.7 Orthogonal Matrix

14.8 Idempotent Matrix

14.9 Involutory Matrix

14.10 Nilpotent Matrix

14.11 Singular Matrix

14.12 Conjugate of a Matrix

14.13 Transpose Conjugate of a Matrix

14.14 Hermitian and Skew-Hermitian Matrices

14.15 Adjoint of a Square Matrix

14.16 Inverse of a Square Matrix

14.17 Elementary Transformations

14.18 Minor

14.19 Rank of a Matrix

14.20 Solution of a System of Linear Equations by Matrix Method

14.21 The Reflection Matrix

14.22 Rotation Through an Angle q

Part B: Determinants

14.23 Second-Order Determinant

14.24 Third-Order Determinant

14.25 Minors and Cofactors

14.26 Expansion of a Determinant of Order Three

14.27 Properties of Determinants

14.28 Product of Determinants of Same Order

14.29 Solution of Linear Equations by Determinants

15 Functions

15.1 Domain, Co-Domain, Range

15.2 Graphs of Functions

15.3 One-One and Many-One Functions

15.4 Methods to Determine Whether a Function is Onto or Into

15.5 Methods of Finding Inverse of a Function

15.6 Even and Odd Functions

15.7 Periodic Function

15.8 Basic Transformations on Graphs

16 Inverse Trigonometric Functions

16.1 Domain and Range of Inverse Trigonometric Functions

16.2 Graphs of Inverse Trigonometric Functions

16.3 Basic Results

16.4 Special Graphs

16.5 Conversion of Inverse Trigonometric Functions

16.6 Inverse Trigonometric Formulae

17 Limits, Continuity and Differentiability

17.1 Limits

17.2 Continuity

17.3 Differentiability

18 Applications of Derivatives

18.1 Geometrical Interpretation of Derivatives

18.2 Tangent and Normal

18.3 Intermediate Value Theorem

18.4 Rolle’s Theorem

18.5 Lagrange’s Mean Value Theorem

18.6 Monotonicity

18.7 Maxima and Minima

18.8 Application of Derivative in Determining the Nature of Roots of a Cubic Polynomial

19 Indefinite Integration

19.1 Constant of Integration

19.2 Properties of Indefinite Integration

19.3 Basic Formulae (Integration as the Inverse Process of Differentiation)

19.4 Standard Formulae

19.5 Methods of Integration

19.6 Some Formulae

20 Definite Integration

20.1 Definite Integration

20.2 Geometrical Interpretation

20.3 Algorithm to Express the Infinite Series as Definite Integral

20.4 Change of Variables in Definite Integration

20.5 Properties of Definite Integral

20.6 Inequalities

20.7 Average Value of a Function over an Interval

20.8 Integration of Piecewise Continuous Functions

20.9 Wallis Formulae

20.10 Differentiation Under the Integral Sign

20.11 Definite Integral as the Limit of a Sum

20.12 Trapezoidal Rule

20.13 Reduction Formulae for Definite Integration

20.14 Gamma Function

21 Area Under the Curve

21.1 Application of Integration to Areas

21.2 Estimation of Areas

21.3 Area Between Two Curves

21.4 Volumes and Surfaces of Solids of Revolution

22 Differential Equations

22.1 Order and Degree of a Differential Equation

22.2 Formation of Differential Equations  

22.3 Solutions of Differential Equations

22.4 Orthogonal Trajectory

22.5 Geometrical Application of Differential Equations

23 Vectors

23.1 Representation of a Vector

23.2 Position Vector of a Point

23.3 Angle Between Two Vectors

23.4 Addition of Vectors

23.5 Section Formula

23.6 Multiplication of a Vector by a Scalar  

23.7 Geometrical Interpretation

23.8 Vector (or Cross) Product of Two Vectors

23.9 Scalar Triple Product

23.10 Vector Triple Product

23.11 Methods to Prove Collinearity

23.12 Coplanarity

23.13 Vector Equation of a Straight Line

23.14 Angle Between Two Lines

23.15 Shortest Distance Between Two Lines

23.16 Equation of a Plane in the Vector Form

23.17 Angle Between a Line and a Plane

23.18 Angle Between Two Planes

23.19 Miscellaneous Results

24 Three-Dimensional Geometry

24.1 Distance Formula

24.2 Section Formula

24.3 Direction Cosines  

24.4 Direction Ratios

24.5 Vector Equation of a Line Passing Through a Given Point and Parallel to a Given Vector

24.6 Cartesian Equation of a Line Passing Through a Given Point and Given Direction Ratios

24.7 Vector Equation of a Line Passing Through Two Given Points

24.8 Cartesian Equation of a Line Passing Through Two Given Points

24.9 Equation of a Line Passing Through a Point and Perpendicular to Two Vectors

24.10 Angle Between Two Lines

24.11 Intersection of Two Lines

24.12 Shortest Distance Between Two Lines

24.13 Perpendicular Distance of a Point from a Line

24.14 Image of a Point in a Line

24.15 Equation of a Plane Passing Through a Given Point and Normal to a Given Vector

24.16 Equation of a Plane Normal to a Given Vector and at a Given Distance from the Origin

24.17 Equation of a Plane Passing Through a Given Point and Parallel to Two Given Vectors  

24.18 Equation of a Plane Passing Through Three Given Points

24.19 Intercept Form of a Plane

24.20 Equation of a Plane Passing Through Two Points and Parallel to a Vector

24.21 Angle Between Two Planes

24.22 Family of Planes

24.23 Equation of a Plane Containing Two Lines

24.24 Equation of Sphere

25 Probability

25.1 Random Experiment

25.2 Sample Space and Sample Points

25.3 Algebra of Events

25.4 Definition of Probability with Discrete Sample Space

25.5 Basic Theories of Probability

25.6 Conditional Probability

25.7 Multiplication Theorems on Probability

25.8 Total Probability and Baye’s Rule

25.9 Geometrical Method for Probability

25.10 Probability Distribution

25.11 Binomial Probability Distribution

25.12 Binomial Distribution

Practice Questions

Answer Key

Solutions

Appendix

JEE Main Mock Test (with Solutios)

JEE Advanced Mock Test (with Solutions)