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
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.
Analytical geometry (3 dimensions):
Direction cosines and direction ratios, equation of a straight line in space, equation
of a lane, distance of a point from a plane.
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.
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, pplication 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
Addition of vectors, scalar multiplication, dot and cross products, scalar triple
products and their geometrical interpretations.