1. Applications of Matrices and Determinants
- Adjoint, inverse – properties, computation of inverses, solution of system of linear equations by matrix inversion method.
- Rank of a matrix – lementary transformation on a matrix, consistency of a system of linear equations, Cramer’s rule, non-homogeneous equations, homogeneous linear system and rank method.
2. Complex Numbers
- Complex number system – conjugate, properties, ordered pair representation.
- Modulus – properties, geometrical representation, polar form, principal value, conjugate, sum, difference, product, quotient, vector interpretation, solutions of polynomial equations, De Moivre’s theorem and its applications.
- Roots of a complex number – nth roots, cube roots, fourth roots.
3. Analytical Geometry of two dimensions
- Definition of a conic – general equation of a conic, classification with respect to the general equation of a conic, classification of conics with respect to eccentricity.
- Equations of conic sections (parabola, ellipse and hyperbola) in standard forms and general forms- Directrix, Focus and Latus rectum – parametric form of conics and chords. – Tangents and normals – cartesian form and parametric form- equation of chord of contact of tangents from a point (x1 ,y1 ) to all the above said curves.
- Asymptotes, Rectangular hyperbola – Standard equation of a rectangular hyperbola.
4. Vector Algebra
- Scalar Product – angle between two vectors, properties of scalar product, applications of dot products. vector product, right handed and left handed systems, properties of vector product, applications of cross product.
- Product of three vectors – Scalar triple product, properties of scalar triple product, vector triple product, vector product of four vectors, scalar product of four vectors.
5. Analytical Geometry of Three Dimensions
- Direction cosines – direction ratios – equation of a straight line passing through a given point and parallel to a given line, passing through two given points, angle between two lines.
- Planes – equation of a plane, passing through a given point and perpendicular to a line, given the distance from the origin and unit normal, passing through a given point and parallel to two given lines, passing through two given points and parallel to a given line, passing through three given non-collinear points, passing through the line of intersection of two given planes, the distance between a point and a plane, the plane which contains two given lines (co-planar lines), angle between a line and a plane.
- Skew lines – shortest distance between two lines, condition for two lines to intersect, point of intersection, collinearity of three points.
- Sphere – equation of the sphere whose centre and radius are given, equation of a sphere when the extremities of the diameter are given.
6. Differential Calculus
- Derivative as a rate measurer – rate of change, velocity, acceleration, related rates, derivative as a measure of slope, tangent, normal and angle between curves, maxima and minima.
- Mean value theorem- Rolle’s Theorem, Lagrange Mean Value Theorem, Taylor’s and Maclaurin’s series, L’ Hospital’s Rule, stationary points, increasing, decreasing, maxima, minima, concavity, convexity and points of inflexion.
- Errors and approximations – absolute, relative, percentage errors- curve tracing, partial derivatives, Euler’s theorem.
7. Integral Calculus and its Applications
- Simple definite integrals – fundamental theorems of calculus, properties of definite integrals.
- Reduction formulae – reduction formulae for sin n x dx and cosn x dx , Bernoulli’s formula.
- Area of bounded regions, length of the curve.
8. Differential Equations
- Differential equations – formation of differential equations, order and degree, solving differential equations (1st order), variables separable, homogeneous and linear equations.
- Second order linear differential equations – second order linear differential equations with constant coefficients, finding the particular integral if f (x) = emx, sin mx, cos mx, x, x2.
9. Probability Distributions
- Probability – Axioms – Addition law – Conditional probability – Multiplicative law – Baye’s Theorem- Random variable- probability density function, distribution function, mathematical expectation, variance .
- Theoretical distributions – discrete distributions, Binomial, Poisson distributions- Continuous distributions, Normal distribution.
10. Discrete Mathematics
- Mathematical logic – logical statements, connectives, truth tables, logical equivalence, tautology, contradiction.
- Groups-binary operations, semigroups, monoids, groups, order of a group, order of an element., properties of groups.
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