NUMBER SYSTEMS & ALGEBRA (30 Marks)
Euclid's division lemma, Fundamental Theorem of Arithmetic - statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of results - irrationality of √2,√3, √5, decimal expansions of rational numbers in terms of terminating/non-terminating recurring decimals.
Zeros of a polynomial. Relationship between zeros and coefficients of a polynomial with particular reference to quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients.
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
Pair of linear equations in two variables. Geometric representation of different possibilities of solutions inconsistency. Algebraic conditions for number of solutions. Solution of pair of linear equations in two variables algebraically- by substitution, by elimination and by cross multiplication. Simple situational problems must be included.
Standard form of a quadratic equation ax2 + bx + c = 0, (a1 0). Solution of the quadratic equations (only real roots) by factorization and by completing the square, i.e. by using quadratic formula. Relationship between discriminant and nature of roots. Problems related today to day activities to be incorporated.
Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms.
TRIGONOMETRY & COORDINATE GEOMETRY (30 Marks)
INTRODUCTION TO TRIGONOMETRY
Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0o & 90o. Values (with proofs) of the trigonometric ratios of 30o, 45o & 60o. Relationships between the ratios.
Proof and applications of the identity sin2 A + cos2 A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.
HEIGHTS AND DISTANCES
Simple and believable problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation/depression should be only 30o, 45o, 60o.
LINES (In two-dimensions)
Review the concepts of coordinate geometry done earlier including graphs of linear equations. Awareness of geometrical representation of quadratic polynomials. Distance between two points and section formula (internal). Area of a triangle.
GEOMETRY & MEASUREMENT (30 Marks)
Definitions, examples, counter examples of similar triangles
AREAS RELATED TO CIRCLES
Motivate the area of a circle; area of sectors and segments of a circle. Problems based on areas and perimeter / circumference of the above said plane figures. (In calculating area of segment of a circle, problems should be restricted to central angle of 60o, 90o & 120o only. Plane figures involving triangles, simple quadrilaterals and circle should be taken.)
SURFACE AREAS AND VOLUMES
STATISTICS AND PROBABILITY (10 Marks)
Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph.
Classical definition of probability. Connection with probability as given in class IX. Simple problem on single events, not using set notation.