|Unit 1 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.
|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 and their properties.
|Permutation and Combination|
- Problems on permutations and combinations.
- 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.
- 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, 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|
- 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.
- 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 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.
- Formation of ordinary differential equations.
- The solution of homogeneous differential equations, separation of variables method.
- Linear first-order differential equations.