ALGEBRA
a)
Functions: Types of functions –
Definitions - Inverse functions and Theorems - Domain, Range, Inverse of real
valued functions.
b)
Mathematical Induction: Principle
of Mathematical Induction & Theorems - Applications of Mathematical
Induction - Problems on divisibility.
c)
Matrices: Types of matrices - Scalar multiple of a
matrix and multiplication of matrices - Transpose of a matrix - Determinants -
Adjoint and Inverse of a matrix - Consistency and inconsistency of Equations-
Rank of a matrix - Solution of simultaneous linear equations.
d)
Complex Numbers: Complex number
as an ordered pair of real numbers- fundamental operations - Representation of
complex numbers in the form a+ib - Modulus and amplitude of complex numbers –Illustrations
- Geometrical and Polar Representation of complex numbers in Argand
plane-Argand diagram.
e)
De Moivre’s Theorem: De Moivre’s
theorem- Integral and Rational indices - nth roots of unity-Geometrical Interpretations – Illustrations.
f)
Quadratic Expressions: Quadratic
expressions, equations in one variable - Sign of quadratic expressions – Change
in signs – Maximum and minimum values - Quadratic inequations.
g)
Theory of Equations: The relation
between the roots and coefficients in an equation - Solving the equations when
two or more roots of it are connected by certain relation - Equation with real
coefficients, occurrence of complex roots in conjugate pairs and its
consequences - Transformation of equations - Reciprocal Equations.
h)
Permutations and Combinations:
Fundamental Principle of counting – linear and circular permutations-
Permutations of ‘n’ dissimilar things taken ‘r’ at a time - Permutations when
repetitions allowed - Circular permutations - Permutations with constraint repetitions
- Combinations-definitions, certain theorems and their applications.
i)
Binomial Theorem: Binomial theorem for positive
integral index - Binomial theorem for rational Index (without proof) -
Approximations using Binomial theorem.
j)
Partial fractions: Partial fractions of f(x)/g(x)
when g(x) contains non –repeated linear factors -
Partial fractions of f(x)/g(x) where both f(x) and g(x) are polynomials
and when g(x) contains repeated and/or non-repeated linear factors - Partial
fractions of f(x)/g(x) when g(x) contains irreducible factors.
TRIGONOMETRY
a)
Trigonometric Ratios upto
Transformations: Graphs and Periodicity of Trigonometric functions -
Trigonometric ratios and Compound angles - Trigonometric ratios of multiple and
sub- multiple angles - Transformations - Sum and Product rules.
b)
Trigonometric Equations: General
Solution of Trigonometric Equations - Simple Trigonometric Equations –
Solutions.
c)
Inverse Trigonometric Functions: To reduce a
Trigonometric Function into a bijection - Graphs
d)
Hyperbolic Functions: Definition
of Hyperbolic Function – Graphs - Definition of Inverse Hyperbolic Functions –
Graphs - Addition formulae of Hyperbolic Functions.
e)
Properties of Triangles: Relation
between sides and angles of a Triangle - Sine, Cosine, Tangent and Projection
rules - Half angle formulae and areas of a triangle – Incircle and Excircle of
a Triangle.
VECTOR ALGEBRA
a)
Addition of Vectors : Vectors as
a triad of real numbers - Classification of vectors - Addition of vectors -
Scalar multiplication - Angle between two non zero vectors - Linear combination
of vectors - Component of a vector in three dimensions - Vector equations of
line and plane including their Cartesian equivalent forms.
b)
Product of Vectors : Scalar
Product - Geometrical Interpretations - orthogonal projections - Properties of
dot product - Expression of dot product in i, j, k system - Angle between two
vectors - Geometrical Vector methods - Vector equations of plane in normal form
- Angle between two planes - Vector product of two vectors and properties -
Vector product in i, j, k system - Vector Areas - Scalar Triple Product -
Vector equations of plane in different forms, skew lines, shortest distance and
their Cartesian equivalents. Plane through the line of intersection of two
planes, condition for coplanarity of two lines, perpendicular distance of a
point from a plane, angle between line and a plane. Cartesian equivalents of
all these results - Vector Triple Product – Results.
MEASURES OF DISPERSION AND
PROBABILITY
a)
Measures of Dispersion - Range -
Mean deviation - Variance and standard deviation of ungrouped/grouped data -
Coefficient of variation and analysis of frequency distribution with equal means
but different variances.
b)
Probability : Random experiments
and events - Classical definition of probability, Axiomatic approach and
addition theorem of probability - Independent and dependent events -
conditional probability- multiplication theorem and Baye’s theorem and
applications.
c)
Random Variables and Probability Distributions:
Random Variables - Theoretical discrete
distributions – Binomial and Poisson Distributions.
COORDINATE GEOMETRY
a)
Locus: Definition of locus – Illustrations - To
find equations of locus - Problems connected to
it.
b)
Transformation of Axes:
Transformation of axes - Rules, Derivations and Illustrations - Rotation of
axes - Derivations – Illustrations.
c)
The Straight Line: Revision of
fundamental results - Straight line - Normal form – Illustrations - Straight
line - Symmetric form - Straight line - Reduction into various forms -
Intersection of two Straight Lines - Family of straight lines - Concurrent
lines - Condition for Concurrent lines - Angle between two lines - Length of
perpendicular from a point to a Line - Distance between two parallel lines -
Concurrent lines - properties related to a triangle.
d)
Pair of Straight lines: Equations of pair of lines
passing through origin - angle between a pair of
lines -
Condition for perpendicular and coincident lines, bisectors of angles - Pair of
bisectors of angles - Pair of lines - second degree general equation -
Conditions for parallel lines - distance between them, Point of intersection of
pair of lines - Homogenizing a second degree equation with a first degree
equation in x and y.
e)
Circle : Equation of circle
-standard form-centre and radius equation of a circle with a given line segment
as diameter & equation of circle through three non collinear points -
parametric equations of a circle - Position of a point in the plane of a circle
– power of a point-definition of tangent-length of tangent - Position of a
straight line in the plane of a circle-conditions for a line to be tangent –
chord joining two points on a circle – equation of the tangent at a point on
the circle- point of contact-equation of normal - Chord of contact - pole and
polar-conjugate points and conjugate lines - equation of chord in term of its
midpoint - Relative position of two circles- circles touching each other externally,
internally- common tangents –centers of similitude- equation of pair of
tangents from an external point.
f)
System of circles: Angle between
two intersecting circles - Radical axis of two circles-properties- Common chord
and common tangent of two circles – radical centre.
g)
Parabola: Conic sections –Parabola-
equation of parabola in standard form-different forms of parabola- parametric
equations - Equations of tangent and normal at a point on the parabola (
Cartesian and parametric) - conditions for straight line to be a tangent.
h)
Ellipse: Equation of ellipse in
standard form- Parametric equations - Equation of tangent and normal at a point
on the ellipse (Cartesian and parametric) - condition for a straight line to be
a tangent.
i)
Hyperbola: Equation of hyperbola in standard form-
Parametric equations - Equations of tangent and normal at a point on the
hyperbola (Cartesian and parametric) - conditions for a straight line to be a
tangent- Asymptotes.
j)
Three Dimensional Coordinates:
Coordinates - Section formulae - Centroid of a triangle and tetrahedron.
k)
Direction Cosines and Direction Ratios: Direction
Cosines - Direction Ratios.
l)
Plane: Cartesian equation of Plane - Simple
Illustrations.
CALCULUS
a)
Limits and Continuity: Intervals and neighbourhoods
– Limits - Standard Limits – Continuity.
b)
Differentiation: Derivative of a
function - Elementary Properties - Trigonometric, Inverse Trigonometric,
Hyperbolic, Inverse Hyperbolic Function – Derivatives - Methods of
Differentiation - Second Order Derivatives.
c)
Applications of Derivatives: Errors and
approximations - Geometrical Interpretation of a derivative - Equations of
tangents and normals - Lengths of tangent, normal, sub tangent and sub normal -
Angles between two curves and condition for orthogonality of curves -
Derivative as Rate of change - Rolle’s Theorem and Lagrange’s Mean value
theorem without proofs and their geometrical interpretation - Increasing and
decreasing functions - Maxima and Minima.
d)
Integration : Integration as the
inverse process of differentiation- Standard forms -properties of integrals -
Method of substitution- integration of Algebraic, exponential, logarithmic,
trigonometric and inverse trigonometric functions - Integration by parts –
Integration by Partial fractions method - Reduction formulae.
e)
Definite Integrals: Definite
Integral as the limit of sum - Interpretation of Definite Integral as an area -
Fundamental theorem of Integral Calculus (without proof) – Properties -
Reduction formulae - Application of Definite integral to areas.
f)
Differential equations: Formation
of differential equation-Degree and order of an ordinary differential equation
- Solving differential equation by i) Variables separable method, ii)
Homogeneous differential equation, iii) Non - Homogeneous differential
equation, iv) Linear differential equations.
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