| Section |
Topics |
| Algebra |
Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations. Statement of the fundamental theorem of algebra, Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots. Arithmetic and geometric progressions, arithmetic, and geometric means, sums of finite arithmetic and geometric progressions, infinite geometric series, the sum of the first n natural numbers, 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 |
Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, elementary row, and column transformations, determinant of a square matrix of order up to three, adjoint of a matrix, 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. |
| Probability |
Random experiment, sample space, different types of events (impossible, simple, compound), addition and multiplication rules of probability, conditional probability, independence of events, total probability, Bayes Theorem, computation of probability of events using permutations and combinations. The measure of central tendency and dispersion, mean, median, mode, mean deviation, standard deviation, and variance of grouped and ungrouped data, analysis of the frequency distribution with the same mean but different variance, random variable, mean, and variance of the random variable. |
| Trigonometry |
Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles, and general solution of trigonometric equations. Inverse trigonometric functions (principal value only) and their elementary properties. |
| Differential Calculus |
Limit of a function at a real number, 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. Continuity of composite functions, the intermediate value property of continuous functions. Derivative of a function, a derivative of the sum, difference, product, and quotient of two functions, chain rule, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential, and logarithmic functions. Tangents and normals, increasing and decreasing functions, derivatives of order two, maximum and minimum values of a function, Rolle’s theorem and Lagrange’s mean value theorem, geometric interpretation of the two theorems, derivatives up to order two of implicit functions, geometric interpretation of derivatives. |
| Integral Calculus |
Integration as the inverse process of differentiation, indefinite integrals of standard functions, definite integrals as the limit of sums, definite integrals, and their properties, fundamental theorem of integral calculus. Integration by parts, integration by the methods of substitution and partial fractions, and application of definite integrals to the determination of areas bounded by simple curves. Formation of ordinary differential equations, solution of homogeneous differential equations of the first order and first degree, separation of variables method, linear first order differential equations. |
