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.