Unit-I: Sets and Functions
1.Sets
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.
2.Relations & Functions
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 (up to R x R x R). 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. Sum, difference, product and quotients of
functions.
3.Trigonometric Functions
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 𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠2𝑥 = 1, for all x. Signs of trigonometric functions. Domain and range of trigonometric functions and their graphs. Expressing 𝑠𝑖𝑛 (𝑥 ± 𝑦) and 𝑐𝑜𝑠 (𝑥 ±𝑦) in terms of 𝑠𝑖𝑛𝑥, 𝑠𝑖𝑛𝑦,𝑐𝑜𝑠𝑥 & 𝑐𝑜𝑠𝑦 and their simple applications. Deducing identities like the following:
tan(𝑥±𝑦)= tan𝑥±tan𝑦
1∓tan𝑥tan𝑦
,cot(𝑥±𝑦)=cot𝑥∓cot𝑦
cot𝑦±cot𝑥
sin𝛼±sin𝛽=2sin1
2
(𝛼±𝛽)cos1
2
(𝛼∓𝛽)
cos𝛼+cos𝛽=2cos1
2
(𝛼+𝛽)cos1
2
(𝛼−𝛽)
cos𝛼−cos𝛽=−2sin1
2
(𝛼+𝛽)sin1
2
(𝛼−𝛽)
Identities related to sin2𝑥,cos2𝑥,tan2𝑥,sin3𝑥,cos3𝑥 and tan3𝑥.
Identities related to sin2𝑥,cos2𝑥,tan2𝑥,sin3𝑥,cos3𝑥 and tan3𝑥.
Identities related to sin2𝑥,cos2𝑥,tan2𝑥,sin3𝑥,cos3𝑥 and tan3𝑥.
Identities related to sin2𝑥,cos2𝑥,tan2𝑥,sin3𝑥,cos3𝑥 and tan3𝑥.
Unit-II: Algebra
1. Complex Numbers and Quadratic Equations
Need for complex numbers, especially √−1, to be motivated by inability to solve some of the quadratic equations. Algebraic properties of complex numbers. Argand plane.
2. Linear Inequalities
Linear inequalities.
Algebraic solutions of linear inequalities in one variable and their
representation on the number line.
3. Permutations and Combinations
Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of Formulae for nr P, nr C and their connections, simple applications.
4. Binomial Theorem
Historical perspective, statement and proof of the binomial theorem for positive integral indices. Pascal’s triangle, simple applications.
5. Sequence and Series
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
Unit-III: Coordinate Geometry
1. Straight Lines
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. Distance of a point from a line.
2.Conic Sections
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.
3. Introduction to Three-dimensional Geometry
Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points
Unit-IV: Calculus
1.Limits and Derivatives
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.
Unit-V Statistics and Probability
1.Statistics
Measures of Dispersion: Range, Mean deviation, variance and standard deviation of ungrouped/grouped data.
2.Probability
Events; occurrence of events, ‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’ and ‘or’ events.