Statements of algebraic and order properties of the system of natural numbers, integers, rational numbers and real numbers and simple basic deductions from theses properties.
Complex numbers. Representation of complex numbers as points in a plane. Algebra of complex numbers. Real and imaginary parts. Modulus and argument of a complex number. Conjugate of a complex number, cube roots of unity. Statements of the principal of mathematical induction in respect of natural numbers and simple applications.
Distance formula and section formula. Equation of line in a plane. General equation of first degree. Angle between two lines. Parallel and perpendicular lines.Distance of a point from a line. Family of lines. Equation of a circle. General equation. Equation of tangent and normal to a circle. Radical axis of two circles. Family of circles. Parametric representation of a circle.Conic section. Equations of parabola, ellipse and hyperbola in standard form.
Vector as a directed line segment. Addition of Vector Multiplication of a Vector by a real number. Position Vector of a point. Section formula. Application of Vector to some geometrical results. Scalar and Vector product of two vectors. Scalar triple product, vector triple product.
THREE DIMENSIONAL GEOMETRY
Decomposition of a vector into three non-coplanar directions i, j, k as base in 3-dimensons. Angle between two vectors. Distance between two points. Section formula.Equations of lines and planes in 3-D. Angle between two lines, between a line and a plane as also between two planes. Distance of a point from a line and from a plane. Shortest distance between two lines. Equation of any plane passing through the intersection of two planes. Equation of a sphere in the form (r-C)2=a2, Equation of a sphere with the position Vector On the extremities of diameter.
Angles and their measure in degrees and radians. Trigonometric functions of angles of arbitrary magnitude. Addition formulae. Sine, cosine, and tangent of multiples and sub multiples of angles. Periodicity and graph of sine, cosine and tangent functions. Trigonometrical ratios of related angles. Solutions of simple trigonometric equations.Sine and cosine formulae for triangles and simple cases of solutions of triangles, problems on heights and distances. Inverse trigonometric functions.
Quadratic equations and their solutions. Relationship between the roots and the coefficients. Formation of quadratic equations with given roots. Criteria for the nature of the roots of quadratic equation.
SEQUENCES AND SERIES
AP, GP, n2, n3 and their sums.
EXPONENTIAL AND LOGARITHMIC SERIES.
The infinite series for e, log (1+x), log[1+x)/(1-x)]
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Elementary study of combinations. Values of npr and ncr, simple applications including circular permutations. Binomial theorem for a positive integral index and its proof.Statement only for the binomial theorem for an arbitrary index and its application to approximations. Properties of binomial co-efficient.
MATRICES AND DETERNINANTS
Addition, scalar multiplication and multiplication of matrices, non-computability of matrix multiplication. Singular and non-singular matrices. Linear equations in matrix notations. Determinants: minors and cofactors. Expansion of determinant, properties of elementary transformation of determinants. Application of determinants in solutions of equations. Cramer’s rule. Adjoint and inverse of a matrix and its properties. Applications of matrices in solving simultaneous equations in three variables.
Concept of real function, its domain and range, one-one and inverse functions, composition of functions. Notions of right hand and left hand limits and the limits of a function. Fundamental theorems on limits. Continuity of a function. Properties of continuous functions. Continuity of polynomial, trigonometric, exponential,logarithmic and inverse trigonometric functions. Derivative of a function, its geometrical and physical significance, relationship between continuity and differentiability.
Derivative of sum, difference, product, quotient function and of the functions of a function(chain rule), derivatives of trigonometric functions. Logarithmic and exponential functions. Differentiation of functions expressed in parametric form, derivatives of higher order. Applications of the derivative: increasing and decreasing functions, maxima and minima, Rolle’s and Mean value theorems(without proof).
Order and degree, formation of differential equation, general and particular solution, solution by the method of variables, separable. Homogeneous equations and their solutions. Solution of the linear equation of the first order with constant coefficients. Integration as the inverse of differentiation, indefinite integral, properties of integrals, fundamental integrals involving algebraic trigonometric and exponential functions, integration by substitution and by parts. Definition of definite integrals as the limit of a sum illustrated by simple examples, fundamental theorem of Calculus, evaluation of definite integrals, transformation of definite integrals by substitution, properties of definite integrals. Application to determination of area under plane curves in simple cases.
Population and sample, Measures of central tendency and dispersion, Point and internal estimation(of mean only), Scatter diagrams and Pearson’s correlation coefficients. Calculations of the regression coefficients and the two lines of regression by the method of least squares.
Random experiments and sample space, Events. Probability on a discrete sample space, addition theorem. Conditional Probability, multiplication theorem. Independent events. Random variables (discrete) , Binomial distributions
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