Section Headings - Chapter Wise in R D Sharma

As the examinations are coming closer and closer, one needs to develop an ability to revise or review things in shorter and shorter periods of time.

Goving through each section heading and thinking of what is the important content in the section is one very quick revision exercise. The section headings of each chapter are given at one place for this facilitating this revision exercise.

1. Sets

1.1 Sets

1.2 Description of a set

1.3 Types of sets

1.4 Subsets

1.5 Universal set

1.6 Power set

1.7 Venn diagrams

1.8 Operations on sets

1.9 Laws of algebra of sets

1.10 More results on operations on sets

1.11 Some important results on number of elements in sets

2. Cartesian product of sets and relations

2.1 Cartesian product of sets

2.2 Relations

2.3 Types of relations

2.4 Some results on relations

2.5 Composition of relations

3. Functions

3.1 Function

3.2 Domain, Co-Domain and range of a function

3.3 Description of a function

3.4 Equal function

3.5 Number of functions

3.6 Function as a relation

3.7 Kinds of functions

3.8 Composition of functions

3.9 Properties of composition of functions

3.10 Inverse of an element

3.11 Inverse of a function

3.12 Properties of inverse of a function

4. Binary operations

4.1 Binary operations

4.2 Types of binary operation

4.3 Identity and inverse elements

4.4 Composition table

5. Complex numbers

5.1 Introduction

5.2 Integral powers of IOTA(i)

5.3 Imaginary quantities

5.4 Complex numbers

5.5 Equality of complex numbers

5.6 Addition of complex numbers

5.7 Subtraction of complex numbers

5.8 Multiplication of complex numbers

5.9 Division of complex numbers

5.10 Conjugate of a complex number

5.11 Modulus of a complex number

5.12 reciprocal of a complex number

5.13 Square roots of a complex number

5.14 Representation of a complex number

5.15 Argument or amplitude of a complex number z = x+iy for different signs of x and 5.16 Eulerian form of a complex number

5.17 Geometrical representations of fundamental operations

5.17B Modulus and argument of multiplication of two complex numbers

5.18 Modulus and argument of division of two complex numbers

5.19 Geometrical representation of conjugate of a complex number

5.20 Some important results on modulus and argument

5.21 Geometry of complex numbers

5.22 Affix of a point dividing the line segment joining points having affixes z1 and z2

5.23 Equation of the perpendicular bisector

5.24 Equation of a circle

5.25 Complex number as a rotating arrow in the argand plane

5.25B Some important results

5.26 Some standard loci in the argand plane

5.27 Equation of a straight line

5.28 De-moivere’s theorem

5.29 Roots of a complex number

5.30 Roots of unity

5.31 Cube roots of unity

5.32 Logarithm of a complex number

6. Sequences and series

6.1 Sequence

6.2 Arithmetic progression

6.3 General term of an A.P.

Selection of terms in an A.P.

6.5 Sum of n terms of an A.P.

6.6 Properties of arithmetical progressions

6.7 Insertion of arithmetic means

6.8 Geometric Progression

6.9 The nth or general term of a G.P.

6.10 Selection of terms in G.P.

6.11 Sum of n terms of a G.P.

6.12 Sum of an Infinite G.P.

6.13 Properties of geometric progressions

6.14 Insertion of geometric means between two given numbers

6.15 Some important properties of arithmetic and geometric means between two given quantities

6.16 Arithmetico-geometric sequence

6.17 Sum of n terms of an arithmetico-geometric sequence

6.18 Sum to n terms of some special sequences

6.19 Miscellaneous sequences and series

6.20 Harmonic progression

6.21 Properties of arithmetic, geometric, and harmonic means between two given numbers.

7. Quadratic equations and expressions

7.1 Some definitions and results

7.2 Some results on roots of an equation

7.3 Position of roots of a polynomial equation

7.4 Descartes rule of signs

7.5 Relations between roots and coefficients

7.6 Formation of a polynomial equation from given roots.

7.7 Transformation of equations

7.8 Roots of a quadratic equation with real coefficients

7.9 Quadratic expression and its graph

7.10 Sign of a quadratic expression for real values of the variable

7.11 Solution of inequations

7.12 Position of roots of a quadratic equation

7.13 Common roots

7.14 Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions

7.15 Condition for resolution into linear factors of a quadratic function

7.16 Algebraic interpretation of Rolle’s theorem

8. Permutations and Combinations

8.1 The factorial

8.2 Exponent of prime p in n!

8.3 Fundamental principles of counting

8.4 Permutations

8.5 Permutations under certain conditions

8.6 Permutations of objects not all distinct

8.7 Permutations when objects can repeat

8.8 Circular permutations

8.9 combinations

8.10 Practical problems on combinations

8.11 Mixed problems on permutations and combinations

8.12 Selection of one or more items

8.13 Division of items into groups

8.14 Division of identical objects into groups

8.15 Some important results

9. Binomial theorem

9.1 Introduction

9.2 Binomial theorem for positive integral index.

9.3 Some important conclusions from binomial theorem

9.3B Multinomial theorem

9.4 Middle terms in a binomial expansion

9.5 Some important results

9.6 Some problems on applications of binomial theorem

9.7 Greatest term in the expansion of (x+a)^n

9.8 Properties of the binomial coefficients

9.9 Binomial theorem for any index

10. Exponential and logarithmic series

10.1 The number e

10.2 To prove that e lies between 2 and 3

10.3 To prove that e is an irrational incommensurable number)

10.4 Exponential series

10.5 Exponential theorem

10.6 Some deductions from exponential series

10.7 Some important results

10.8 Logarithmic series

11.Matrices

11.1 Matrix

11.2 Types of matrices

11.3 Equality of matrices

11.4 Algebra of matrices

11.5 Multiplication of a matrix by a scalar (scalar multiplication)

11.6 Subtraction of matrices (definition)

11.7 Multiplication of matrices

11.8 Transpose of a matrix

11.9 Symmetric and skew symmetric matrices

11.10 Determinants

11.11Singular matrix

11.12 Adjoint of a matrix

11.13 Inverse of a matrix

11.14 elementary transformations of elementary operations of a matrix

11.15 Orthogonal matrix

11.16 Submatrix

11.17 Rank of a matrix

11.18 Some theorems on rank of a matrix

11.19 Equivalent matrices

11.20 Echelon form of a matrix

11.21 System of simultaneous linear equations

11.22 Solution of a non-homogeneous system of linear equations

11.23 Rank method

11.24 Solution of a homogeneous system of linear equations

12. Determinants

12.1 Definition

12.2 Singular matrix

12.3 Minors and cofactors

12.4 Properties of determinants

12.5 Evaluation of determinants

12.6 Evaluation of determinants by using factor theorem

12.7 Product of determinants

12.8 Differentiation of determinants

12.9 Applications of determinants to coordinate geometry

12.10 Applications of determinants in solving a system of linear equations

13 Cartesian System of Rectangular Coordinates and straight lines

13.1 Introduction

13.2 Cartesian coordinate system

13.3 Distance between two points

13.4 Area of a triangle

13.5 Section formulae

13.6 Co-ordinates of the centroid, in-centre, and ex-centres of a triangle

13.7 Locus and equation to a locus

13.8 Shifting of origin

13.9 Rotation of axes

13.10 Definition of a straight line

13.11 Slope (gradient) of a line

13.12 Angle between two lines

13.13 Intercepts of a line on the axes

13.14 Equations of lines parallel to the coordinates axes

13.15 Different forms of the equation of a straight line

13.16 Transformation of general equation in different standard forms

13.17 Point of intersection of two lines

13.18 Condition of concurrency of three lines

13.19 Lines parallel and perpendicular to a given line

13.20 Angle between two straight lines when their equations are given

13.21 Distance of a point from a line

13.22 Positions of points relative to a line

13.23 Equations of straight lines passing through a given point and making a given angle with a given line

13.24 Equations of bisectors of the angles between two straight lines

13.25 Some important points of a triangle

13.26 Family of lines through the intersection of two given lines

14. Family of lines

14.1 Introduction

14.2 Joint equation of a pair of straight lines

14.3 Pair of straight lines through the origin

14.4 Angle between pair of lines given by ax² +2hxh+by² = 0

14.5 Bisectors of the angle between the lines given by a homogeneous equation

14.6 General equation of a second degree

14.7 Equations of the bisectors of the angles between the lines represented by the equation ax² +2hxh+by²+2gx+2fy+c = 0

14.8 Lines joining the origin to the points of intersection of a line and curve

Ch.15 Circle

1. Definition

2. Standard equation of a circle

3. Some particular cases of standard equation of a circle

4. General equation of a circle

5. Equation of a circle when the coordinates of end points of a diameter are given

6. Intercepts of the axes

7. Position of a point with respect to a circle

8. Equation of a circle in parametric form

9. Intersection of a straight line and a circle

10. The length of the intercept cut off from a line by a circle

11.Tangent to a circle at a given point

12 Normal to a circle at a given point

13. Length of the tangent from a point to a circle

14. Pair of tangents drawn from a point to given circle

15. Combined equation of pair of tangents

16. Director circle and its equation

17. Chord of contacts of tangents

18. Pole and Polar

19. Equation of the chord bisected at a given point

20. Diameter of a circle – Locus of middle points of parallel chords

21. Common tangents to two circles

22. Common chord of two circles

23. Angle of intersection of two curves and the condition of orthogonality of two circles

24. Radical axis

25. Equation of a circle through the intersection of a circle and line

26. Circle through the intersection of the two circles

27. Coaxial system of circles

Ch. 16. Parabola

1. Conic sections: Definition

2. The parabola

3. Equation of parabola in its standard form

4. Some other standard forms of parabola

5. Position of a point with respect to a parabola

6. Equation of a parabola in parametric form

7. Equation of the chord joining any two points on the parabola

8. Intersection of a straight line and a parabola

9. Equation of tangent in different forms

10. Equation of normal in different forms

11 Number of normals drawn from a point to a parabola

12. Some results in conormal points

13 Number of tangents drawn from a point to a parabola

13a. Equation of the pair of tangents from a point to a parabola

14. Equation of the chord of contacts of tangents to a parabola

15. Equation of the chord bisected at a given point

16. Equation of diameter of a parabola

17. Length of tangent, subtangent, normal and subnormal

18. Pole and Polar

19. Some important results at a glance

Ch. 17. Ellipse

1. Introduction

2. Equation of ellipse in its standard form

3. Second focus and second directrix of the ellipse

4. Vertices, major and minor axes, foci, directrices and centre of the ellipse

5. Ordinate, double ordinate and latus rectum of the ellipse

6. Focal distances of a point on the ellipse

7. Equation of ellipse in other forms

8. Position of a point with respect to an ellipse

9 .Parametric equations and parametric coordinates

10. Equation of the chord joining any two points on an ellipse

11. Condition of a line to be a tangent to an ellipse

12. Equation of tangent in terms of its slope

13. Equation of tangent at a point

14 Number of tangents drawn from a point to an ellipse

15. Equation of normal in different forms

16 Number of normals

17. Properties of eccentric angles of the conormal points

18. Equation of the pair of tangents from a point to an ellipse

19. Equation of the chord of contacts of tangents

19a. Equation of the chord bisected at a given point

20. Equation of diameter of an ellipse

21. Some properties of ellipse

Ch. 18. Hyperbola

1. Introduction

2. Equation of hyperbola in its standard form

3. Second focus and second directrix of the hyperbola

4. Vertices, major and minor axes, foci, directrices and centre of the hyperbola

5. Eccentricity

6. Length of latus rectum

7. Focal distances of a point

8. Conjugate hyperbola

9 .Parametric equations and parametric coordinates

10. Equation of the chord joining any two points on a hyperbola

11. Intersection of a line and a hyperbola

12. Condition of a line to be a tangent to a hyperbola

13. Equation of tangent in different forms

14. Number of tangents drawn from a point to a hyperbola

15. Equation of the pair of tangents from a point to a hyperbola

16. Equation of the chord of contacts of tangents

17. Equation of normal in different forms

18 Number of normals

19. Equation of the chord of a hyperbola bisected at a given point

20 Asymptotes of a hyperbola

21. Rectangular hyperbola

19. Real Functions

19.1 Introduction

19.2 Description of real functions

19.3 Intervals (Closed and open)

19.4 Domains and ranges of real functions

19.5 Some real functions

19.6 Operations on real functions

19.7 Even and odd functions

19.8 Extension of a function

19.9 Periodic function

20. Limits

20.1 Informal approach to limit

20.2 Formal approach to limit

20.3 Evaluation of left hand and right hand limits

20.4 Difference between the value of a function at a point and the limit at a point

20.5 The algebra of limits

20.6 Evaluation of limits

21. Continuity and Differentiability

21.1 Introduction

21.2 Continuity at a point

21.3 Continuity functions

21.4 Continuous functions

21.5 Cauchy’s definition of continuity

21.6Heine’s definition of continuity

21.7 Discontinuous functions

21.8 Properties of continuous functions

21.9 Differentiability at a point

21.10 Relation between continuity and differentiability

21.11 Differentiability in a set

21.12 Some results on differentiability

22. Differentiation

22.1 Differentiation

22.2 Geometrical meaning of derivative at a point

22.3 Differentiation of some standard functions

22.4 Differentiation of a function

22.5 Relation between dy/dx and dx/dy

22.6 Differentiation of implicit functions

22.7 Logarithmic differentiation

22.8 Differentiation of parametric functions

22.9 Differentiation of a function with respect to another function

22.10 Higher order derivatives

23. Tangents , Normals and other applications of derivatives

23.1 Slopes of the tangent and the normal

23.2 Equations of tangent and normal

23.3 Angle of intersection of two curves

23.4 Lengths of tangent, normal, subtangent, and subnormal

23.5 Rolle’s theorem

23.6 Lagrange’s mean value theorem

24. Increasing and decreasing functions

24.1 Some definitions

24.2 Necessary and sufficient conditions for monotonicity of functions

24.3 Properties of monotonic functions

25 Maximum and minimum values

25.1 Maximum and minimum values of a function in its domain

25.2 Local maxima and minima

25.3 Higher derivative test

25.4 Maximum and minimum values in a closed interval

26. Indefinite integrals

27. Definite Integration

27.1 The definite integral

27.2 Evaluation of definite integrals

27.3 Geometric interpretation of definite integral

27.4 Evaluation of integrals by substitution

27.5 Properties of definite integrals

27.6 Integral function

27.7 Summation of series using definite integral as the limit of a sum

27.8 Gamma function

28. Areas of Bounded regions

28.1 Curve sketching

28.2 Sketching of some common curves

28.3 Areas of Bounded regions

29. Differential equations

29.1 Some definitions

29.2 Solution of a differential equation

29.3 Formation of differential equations

29.4 Differential equations of first order and first degree

295 Geometrical interpretation of the Differetial equations of first order and first degree

29.6 Solution of Differential equations of first order and first degree

29.7 Methods of solving first order first degree differential equation

30. VECTORS

1. Introduction

2. Representation of vectors

3. Equality of vectors

4. Types of vectors

5. Parallelogram law of addition of vectors

6. Subtraction of vectors

7. Multiplication of a vector by a scalar

8. Position vector

9. Section formula

10.Linear combination of vectors

11. Collinear and non-collinear vectors

12. Collinear points

13. Components of a vector

14. Components of a vector in three dimensions

15. Collinearity and coplanarity

16. Linear independence and dependence of vectors

17. Angle between two vectors

18. The scalar or dot product

19. Geometrical interpretation of scalar product

20. Properties of scalar product

21. Scalar product in terms of components

22. Angle between two vectors

23. Components of a vector b along perpendicular to vector a

24. Tetrahedron

25. Application of scalar product in mechanics to find the work done

26. Definition of vector product

27. Properties of vector product

28. vector product in terms of components

29. Vectors normal to the plane of two given vectors

30. Some important results

31. Lagrange’s identity

32. Application of vector product in mechanics to find the moment of a force

33. Application of vector product to find the moment of a couple

34. Scalar triple product

35. Properties of scalar triple product

36. Scalar triple product in terms of components

37. Distributivity of cross product over vector addition

38. Volume of a tetrahedron

39. Vector triple product

40. Geometrical applications of vectors

41. Solutions of vector equations

31. THREE DIMENSIONAL GEOMETRY

31.6 Angle between two vectors in terms of their directions cosines and direction ratios

31.7 Straight line in space

31.8 Angle between two lines

31.9 Intersection of two lines

31.10 Perpendiculars distance of a point from a line

31.11 Reflection or image of a point in a straight line

31.12 Shortest distance between two straight lines

31.22 Distance of a point from plane

31.24 Line and a plane

31.25 Angle between a line and plane

31.26 Intersection of a line and a plane

31.27 Condition of coplanarity of two line and equation of the plane containing them

31.28 Image of a point in a plane

PROBABILITY

32.1 Introduction

32.2 Classical approach to probability

32.3 Axiomatic approach to probability

32.4 Addition theorems on probability

32.5 Conditional probability

32.6 Multiplication theorems on probability

32.7 Independent events

32.8 Some solved examples

32.9 the law of total probability

32.10 Baye’s rule

32.11 Random variable and its probability distribution

32.12 Binomial distribution

32.13 Mean and variance of binomial distribution

32.14 aximum value of P(X=r) given values of n and p for a binomial variate X.

TRIGONOMETRY

33. Trigonometric ratios, Identities and Maximum & Minimum Values of Trigonometrical Expressions

33.1 Introduction

33.2 Some basic formulae

33.3 Domain and range of trigonometrical functions

33.4 Sum and difference formulae

33.5 Sum and difference into products

33.6 Product into sum or difference

33.7 T-ratios of the sum of three or more angles

33.8 Values of trigonometrical ratios some important angles and some important results.

33.9 Expressions of sin A/2 and cos a/2 in terms of sin A.

33.10 Maximum andminimum values of trigonometrical functions

Ch.34 properties of Triangles and circles connected with them

34.1 Introduction

34.2 Sine rule

34.3 Cosine formulae

344 Projection formulae

34.5 Trigonometrical ratios of half of the angles of a triangle

34.6 Area of a triangle

34.7 Napier’s analogy

34.8 Circumcircle of a triangle

34.9 Inscribed circle or incircle of a triangle

34.10 Escribed circles of a triangle

34.11 Orthocentre and its distances from the angular points of a triangle

34.12 Regular polygons and radii of the inscribed and circumscribing circles of a regular polygon

34.13 Area of a cyclic quadrilateral

34.14 Ptolemy’s theorem

34.15 Circum-radius of a cyclic quadrilateral

Ch. 35. Trigonometrical equations

35.1 Trigonometrical equations

36. Inverse Trigonometrical functions

36.1 Inverse Trigonometrical functions

36.2 Properties of inverse trigonometrical functions

Ch. 37 Solution of Triangles

37.1 Introduction

37.2 Solution of a right angled triangle

7.3 Solution of a triangle in general

37.4 Some useful results

Ch. 38 Heights and distances

38.1 Angle of elevation and depression of a point

38.2 Some useful results

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