## Sunday, November 2, 2008

### IIT JEE MATHEMATICS - Quick Revision Sheet - Section Headings

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
5.7 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.

7.1 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
11.13 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
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
32.5 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

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