Mathematics
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 skew-symmetric 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, half-angle
formula and the area of a triangle, inverse
trigonometric functions (principal value only).
Analytical geometry (2 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.
Analytical geometry (3 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 one-to-one
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.
