Algebra:
Algebra of complex numbers, addition,
multiplication, conjugation, polar representation,
properties of modulus and principal argument,
triangle inequality, cube roots of unity, geometric
interpretations.
Quadratic
equations with real coefficients, relations between
roots and coefficients, formation of quadratic
equations with given roots, symmetric functions of
roots.
Arithmetic,
geometric and harmonic progressions, arithmetic,
geometric and harmonic means, sums of finite
arithmetic and geometric progressions, infinite
geometric series, sums of squares and cubes of the
first n natural numbers.
Logarithms and their properties.
Permutations and
combinations, Binomial theorem for a positive
integral index, properties of binomial coefficients.
Matrices as a rectangular
array of real numbers, equality of matrices,
addition, multiplication by a scalar and product of
matrices, transpose of a matrix, determinant of a
square matrix of order up to three, inverse of a
square matrix of order up to three, properties of
these matrix operations, diagonal, symmetric and
skewsymmetric matrices and their properties,
solutions of simultaneous linear equations in two or
three variables.
Addition and multiplication
rules of probability, conditional probability, Bayes
Theorem, independence of events, computation of
probability of events using permutations and
combinations.
Trigonometry:
Trigonometric functions, their periodicity and
graphs, addition and subtraction formulae, formulae
involving multiple and submultiple angles, general
solution of trigonometric equations.
Relations between sides and
angles of a triangle, sine rule, cosine rule,
halfangle formula and the area of a triangle,
inverse trigonometric functions (principal value
only).
Analytical
geometry:
Two dimensions:
Cartesian coordinates, distance between two points,
section formulae, shift of origin.
Equation of a
straight line in various forms, angle between two
lines, distance of a point from a line; Lines
through the point of intersection of two given
lines, equation of the bisector of the angle between
two lines, concurrency of lines; Centroid,
orthocentre, incentre and circumcentre of a
triangle.
Equation of a circle in various forms, equations of tangent,
normal and chord.
Parametric
equations of a circle, intersection of a circle with
a straight line or a circle, equation of a circle
through the points of intersection of two circles
and those of a circle and a straight line.
Equations of a
parabola, ellipse and hyperbola in standard form,
their foci, directrices and eccentricity, parametric
equations, equations of tangent and normal.
Locus Problems.
Three Dimensions:
Direction cosines and direction ratios, equation of
a straight line in space, equation of a plane,
distance of a point from a plane.
Differential
Calculus:
Real valued functions of a real variable, into,
onto and onetoone functions, sum, difference,
product and quotient of two functions, composite
functions, absolute value, polynomial, rational,
trigonometric, exponential and logarithmic
functions.
Limit and
continuity of a function, limit and continuity of
the sum, difference, product and quotient of two
functions, L’Hospital rule of evaluation of limits
of functions.
Even and odd
functions, inverse of a function, continuity of
composite functions, intermediate value property of
continuous functions.
Derivative of a
function, derivative of the sum, difference, product
and quotient of two functions, chain rule,
derivatives of polynomial, rational, trigonometric,
inverse trigonometric, exponential and logarithmic
functions.
Derivatives of
implicit functions, derivatives up to order two,
geometrical interpretation of the derivative,
tangents and normals, increasing and decreasing
functions, maximum and minimum values of a function,
Rolle’s Theorem and Lagrange’s Mean Value Theorem.
Integral Calculus:
Integration as the inverse process of
differentiation, indefinite integrals of standard
functions, definite integrals and their properties,
Fundamental Theorem of Integral Calculus.
Integration by
parts, integration by the methods of substitution
and partial fractions, application of definite
integrals to the determination of areas involving
simple curves.
Formation of
ordinary differential equations, solution of
homogeneous differential equations, separation of
variables method, linear first order differential
equations.
Vectors:
Addition of vectors, scalar multiplication, dot and
cross products, scalar triple products and their
geometrical interpretations.
