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Mathematics includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis). Mathematics is widely used in science for modeling phenomena. This enables the extraction of quantitative predictions from experimental laws. For example, the movement of planets can be predicted with high accuracy using Newton's law of gravitation combined with mathematical computation.
Sets and their representations, Empty set, Finite and Infinite sets, Equal sets, Subsets, Subsets of a set of real numbers especially intervals (with notations). Universal set. Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement.
Ordered pairs. Cartesian product of sets. Number of elements in the Cartesian product of two finite sets. Cartesian product of the set of reals with itself ( R x R only).Definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorial representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs.
Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity π ππ2π₯ + πππ 2π₯ = 1, for all x. Signs of trigonometric functions. Domain and range of trigonometric functions and their graphs. Expressing π ππ (π₯ Β± π¦) and πππ (π₯ Β± π¦) in terms of π πππ₯, π πππ¦, πππ π₯ & πππ π¦ and their simple applications. Deducing identities like the following: tan(π₯ Β± π¦) = tan π₯Β±tan π¦ 1βtan π₯ tan π¦ , cot(π₯ Β± π¦) = cotπ₯βcotπ¦ cotπ¦Β±cotπ₯ sin πΌ Β± sin π½ = 2 sin 1 2 (πΌ Β± π½) cos 1 2 (πΌ β π½) cos πΌ + cos π½ = 2 cos 1 2 (πΌ + π½) cos 1 2 (πΌ β π½) cos πΌ β cos π½ = β2 sin 1 2 (πΌ + π½) sin 1 2 (πΌ β π½) Identities related to sin 2π₯ , cos 2π₯ ,tan 2π₯ , sin 3π₯ , cos 3π₯ and tan 3οΏ½
Need for complex numbers, especiallyββ1, to be motivated by inability to solve some of the quardratic equations. Algebraic properties of complex numbers. Argand plane.
Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line.
Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of Formulae for n Pr , nCr and their connections, simple applications.
Historical perspective, statement and proof of the binomial theorem for positive integral indices. Pascalβs triangle, simple applications.
Sequence and Series. Arithmetic Mean (A.M.) Geometric Progression (G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M
Brief recall of two dimensional geometry from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point -slope form, slope-intercept form, two-point form, intercept form and normal form. General equation of a line. Distance of a point from a line.
Sections of a cone: circles, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.
Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points.
Derivative introduced as rate of change both as that of distance function and geometrically. Intuitive idea of limit. Limits of polynomials and rational functions trigonometric, exponential and logarithmic functions. Definition of derivative relate it to scope of tangent of the curve, derivative of sum, difference, product and quotient of functions of polynomial and trigonometric functions.
Measures of Dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data.
Events; occurrence of events, βnotβ, βandβ and βorβ events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories of earlier classes. Probability of an event, probability of βnotβ, βandβ and βorβ events.
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