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 sub-multiple 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:
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 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.
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