Algebra
A.P, G.P., H.P. : Definitions A.P. and G.P.; General term; Summation of first n-terms; A.M. and G.M. definitions of H.P. (only 3 terms)
and H.M. ; Finite arithmetico-geometric series.
Logarithms : Definition ; General properties; Change
of base.
Complex Number : Definition and properties of complex
numbers; Complex conjugate; Triangle inequality; Square root of complex number; Cube roots of unity; D Mover’s theorem
(statement only) and its elementary applications.
Quadratic Equation : Quadratic equations with real
coefficients; Relations between roots and coefficients; Nature of roots; Formation of a quadratic equation, sign and magnitude
of the quadratic expression ax2 + bx + c (a, b, c) are rational numbers and a ≠ 0.
Permutation and Combination: Permutation of n different
things taken r at a time (r £ n).Permutation
of n things not all different. Permutation with repetitions (circular permutation excluded).
Combinations of n different
things taken r at a time (r £n). Combination
of n things not all different. Basic properties.
Problems involving both
permutations and combinations.
Principle of Mathematical Induction: Statement of
the principle. Proof by induction of the sum of squares, sum of cubes of first n natural numbers, divisibility properties
like 22n – 1 is divisible by 3 (n ³ 1), 7 divides 32n + 1 + 2n+2 ( n ³ 1).
Binomial theorem (positive integral index): Statement
of theorem, general term, middle term, equidistant terms, properties of binomial
coefficients.
Infinite series: Binomial theorem for negative and
fractional index. Infinite G.P. series. Exponential and Logarithmic series with
range of validity (statement only), Simple application.
Matrices: Concepts of m ´ n (m £3, n£3) real matrices, operations of addition, scalar multiplication and multiplication
of matrices. Transpose of matrix. Determinant of a square matrix. Properties of determinants (statement only). Minor, fofactor
and adjoint of a matrix, nonsingular matrix. Inverse of a matrix. Finding area of a triangle. Solution of system of linear
equations. ( Not more than 3 variables).
Sets, Relations and Mappings : Idea of sets, subsets,
power set, complement, union, intersection and difference of sets, Venn diagram, De Morgan’s Laws, Inclusion/ Exclusion
formula for two or three finite sets, Cartesian product of sets.
Relation and its properties.
Equivalence relation-definition and elementary examples, mappings, range and domain, injective and bijective mappings, composition
of mappings, Inverse of mapping.
Probability: Classical definition, addition rule,
conditional probability and Bayes’ theorem, independence, multiplication rule.
Trigonometry
Trigonometric ratios, compound
angle , multiple and sub multiple angles, general solution of trigonometric equations. Properties of triangles, inverse trigonometric
functions.
Co-ordinate Geometry of two Dimensions
Basic Ideas: Distance formula, section formula,
area of a triangle, condition of collinearity of three points in a plane.
Polar coordinates transformation
from Cartesian to polar coordinates and vice versa. Parallel transformation of axes, concepts of locus, elementary locus problems.
Straight line: Slope of a line. Equation of lines in different forms, angle between two lines. Condition of perpendicular and parallelism
of two lines. Distance of a point from a line. Distance between two parallel lines. Lines through the point of intersection
of two lines.
Circle: Equation of a circle with a given center
radius. Conditional that a general equation of second degree in x, y may represent a circle. Equation of a circle in terms
of endpoints of a diameter. Parametric equation of a circle. Intersection of a line with a circle. Equation of common chord
of two intersection circles.
Conics: Definition. Directrix. Focus and Ecentricity, classification based on eccentricity.
Parabola: Standard equation, Reduction of the form
x = ay2 + by + c or y = ax2 + bx + c to the standard form y2 = 4ax or x2 = 4ay respectively.
Elementary properties and parametric equation of a parabola.
Ellipse and Hyperbola: Reduction to standard form
of general equation of second degree when xy term is absent. Conjugate hyperbola . Sinple properties Parametric equations.
Location of a point with respect to a conic.
Differential Calculus: Functions, composition of
two functions and inverse of a function, limit, continuity, derivative, chain rule, derivatives of implicit functions and
of functions defined parametrically.
Rolle’s Theorem and
Lagrange’s Mean value theorem (statement only) . Their geometric interpretation and elementary application. L’
Hospital’s rule (statement only) and applications.
Second order derivative
Integral calculus: Integration as a reverse process
differentiation, indefinite integral of standard functions. Integration by parts. Integration by substitution and partial
fraction.
Definite integral as a limit
of a sum with equal subdivisions. Fundamental theorem of integral calculus and its applications. Properties of definite integrals.
Differential Equations: Formulation and solution
of differential equations of the forms.
Application of Calculus: Tangents and normals, conditions
of tangency. Determination of monotonicity and minima. Differential coefficient as a measure of rate.
Motion of a straight line
with constant acceleration.
Geometric interpretation
of definite integral as area, calculation of area bounded by elementary curves and straight lines. Area of the region included
between two elementary curves.
