Unit 1 Algebra

Complex Numbers

• 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

• Quadratic equations with real coefficients.
• Relations between roots and coefficients.
• Formation of quadratic equations with given roots.
• Symmetric functions of roots.

Sequence and Series

• 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

• Logarithms and their properties.

Permutation and Combination

• Problems on permutations and combinations.

Binomial Theorem

• Binomial theorem for a positive integral index.
• Properties of binomial coefficients.

Matrices and Determinants

• 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, the 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

• Addition and multiplication rules of probability, conditional probability.
• Bayes Theorem, independence of events.
• Computation of probability of events using permutations and combinations.

Unit 2 Trigonometry

Trigonometric Functions

• Trigonometric functions, their periodicity, and graphs, addition and subtraction formulae.
• Formulae involving multiple and submultiple angles.
• The general solution of trigonometric equations.

Inverse Trigonometric Functions

• 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).

Unit 3 Vectors

Properties of Vectors

• The addition of vectors, scalar multiplication.
• Dot and cross products.
• Scalar triple products and their geometrical interpretations.

Unit 4 Differential Calculus

Functions

• 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.
• Even and odd functions, the inverse of a function, continuity of composite functions, intermediate value property of continuous functions.

Limits and Continuity

• 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.

Derivatives

• The derivative of a function, the 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.

Unit 5 Integral calculus

Integration

• 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 Integration

• Application of definite integrals to the determination of Areas involving simple curves.

Differential Equations

• Formation of ordinary differential equations.
• The solution of homogeneous differential equations, separation of variables method.
• Linear first order differential equations.