MATHEMATICS
"Mathematics should be visualized as the vehicle to train a child to think, reason, analyse and articulate logically. Apart from being a specific subject, it should be treated as a concomitant to any subject involving analysis and reasoning.
With the introduction of computers in schools, educational computing and emergence of learning through understanding of cause - effect relationships and the interplay of variables, the teaching of Mathematics will be suitably redesigned to bring it in line with modem technological devices." - From National Policy on Education, 1986.
Objectives The teaching and learning of Mathematics at secondary stage should enable the pupil to consolidate the mathematical knowledge and skills acquired at the Upper Primary stage and to:
- acquire knowledge and understanding of the terms, symbols concepts, principles, process, proofs etc. pertaining to secondary stage.
- develop mastery of basic algebraic skills;
- develop drawing skills;
- apply mathematical knowledge and skills to solve real-life mathematical problems, by developing abilities to analyse, to see interrelationship involved, to think and reason;
- develop the ability to articulate logically;
- develop skill in the use of mathematical tables as aids for computational work.
- develop ability to write/interpret algorithms for problem solving;
- develop necessary skill to work with modern technological devices such as calculators, computers, etc. where available and develop understanding of the cause effect relationship and the interplay of variables;
- develop interest in mathematics and participate in mathematical competitions and other mathematics club activities in the school;
- develop appreciation for mathematics as a problem-solving tool in various fields, for its beautiful structures and patterns etc.; and
- develop reverence and respect towards great mathematicians particularly towards the Indian mathematicians for their contributions to the fields of mathematics, astronomy etc.
CLASS IX
One Paper Time: 3 hours Marks: 100
Unitwise weightage
Units Marks
1. Algebra 30
2. Logarithm 16
3. Trigonometry 10
4. Geometry 30
5. Statistics 8
6. Computing (I) 6
44
Unit 1 : Algebra 63 pds.
Number Systems
Introduction of irrational numbers as non-terminating and non-repeating decimals.
Surds and rationalization of surds-real numbers and their properties.
Sets and their representation, finite and infinite sets, subset, empty set, universal set,
complement of a set, union and intersection of sets, examples of sets from number systems; Venn
Diagram and applications.
Functions
Concept of a function as a mapping. Constant linear functions and their graphs in rectangular
coordinate systems.
Polynomials
Remainder theorem (without proof) and its application in the factorization of polynomials of
degree. not more than four. Reveiw of factorization of algebraic expressions done in the earlier
classes, factorize the polynomial ax2 + bx + c, a # 0 (by breaking the middleterm).
Linear Equations in One Variable
Solution of linear equations and application to problems of commercial mathematics,
mensuration etc. studied in earlier classes.
Unit 2: Logarithm 31 pds.
Meaning of logarithm of a number to a given base : common logarithms (base 10),
characteristic and mantissa; meaning of antilogarithm, laws of logarithms, computations using
logarithmic tables (mastery level to be achieved by au).
Compound interest, population growth, depreciation of value of articles and solution of
problems using logarithmic tables for computational work. and
Areas off rectangle, square, triangle, rhombus, trapezium, parallelogramm etc.
Unit 3 : Trigonometry 22 pds.
Trigonometric ratios of an acute angle A, (or the other angle) of a right triangle :
sin A = opposit side/hypotenuse, cos A = adjacent side/hypotenuse
tan A = sin A/cos A, cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A
Trigonometric Ratios of 0o, 30o, 45o, 60o, 90o : Results for trigonometric ratios of 30o, 45o,
60o to be arrived at through geometrical proofs, results for trigonometric ratios of 0o and 90o to
be given as axioms, simple applications of these trigonometric ratios, for solving problems such as
heights and distances.
Unit 4 : Geometry 74 pds.
In the teaching of Geometry at the Secondary level, the emphasis should be to make the
pupil understand and appreciate the nature and method of a deductive proof. While the proofs
of all the prepositions listed below are to be taught. In order to reduce the load on students, only
45
the proofs of the star-marked propositions may be asked in the Examination. In order to achieve the objectives of teaching geometry, the solving of riders (Exercises) covering all the propositions should be taught and tested.
Lines and Angles 1. Two distinct lines cannot have more than one point in common.
2. Two lines which are both parallel to the same line, are parallel to each other.
3. If two lines intersect then the vertically opposite angles are equal.
*4. If a transversal intersects two parallel lines, then each pair of alternate interior angles are equal.
*5. If a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.
*6. If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, then the two lines are parallel.
*7. If a transversal intersects two lines in such a way that a pair of consecutive interior angles are supplementary, then the two lines are parallel.
*8. The sum of the three angles of a triangle is 180o.
9. If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two. interior opposite angles.
Congruence of Triangles *1. Angles opposite to two equal sides of a triangle are equal.
2. Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other traingle.
*3. If two angles of a triangle are equal, then the sides opposite to them are also equal.
4. Two triangles are congruent if the three sides of one triangle are equal to the corresponding three sides of the other triangle.
5. Two triangles are congruent if any two sides and the included angle of the one triangle are equal to the corresponding side and the included angle of the other triangle.
6. Two right triangles are congruent if the hypotenuse and one side of one triangle are respectively, equal to the hypotenuse and the corresponding side of the other triangle.
Inequalities of a Triangle 1. If two sides of a triangle are unequal, the longer side has greater angle opposite to it.
2. In a triangle, the greater angle has the longer side opposite to it.
3. The sum of any two sides of a triangle is greater than its third side.
4. Of all the line segments that can be drawn to a given line, from a point not lying on it, the perpendicular line - segment ii the shortest.
*5. The locus of a point equidistant from two fixed points, is the perpendicular bisector of the segment joining the two points.
*6. The locus of a point equidistant from two intersecting lines is the pair of bisectors, of the angles formed by the given lines.
7. The angle bisectors of a triangle pass through the same point.
8. The perpendicular bisectors of the sides of a triangle pass through the same point.
9. In a triangle, the, three altitudes pass through the same point.
*10. The medians of a triangle pass through the same point which divides each of the medians in the ratio 2 : 1.
46
Parallelograms 1. In a parallelogram, opposite sides are equal.
2. The opposite angles of a parallelogram are equal.
3. The two diagonals of a parallelogram bisect each other.
4. A quadrilateral is a parallelogram if its opposite sides are equal.
5. A quadrilateral is a parallelogram if opposite angles are equal.
6. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
7. A quadrilateral is a parallelogram, if one pair of opposite sides are equal and parallel.
8. The diagonals of a rectangle are of equal length.
9. If the two diagonals of a parallelogram are equal, it is a rectangle.
10. The diagonals of a rhombus are perpendicular to each other.
11. If the diagonals of a parallelogram are perpendicular, then it is a rhombus.
12. The diagonals of a square are equal and perpendicular to each other.
13. If in a parallelogram, the diagonals are equal and perpendicular, then it is a square.
* 14. In a triangle, the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is half of it.
15. The line drawn through the mid-point of one side of a triangle, parallel to another side (bisects the third side) intersects the third side at its mid-point.
* 16. If there are three or more parallel lines and the intercepts made by them on one transversal are equal the corresponding intercepts on any other transversal are also equal.
Area 1. A diagonal of a parallelogram divides it into two triangles of equal area.
2. Parallelograms on the same base and between the same parallels are equal in area.
3. Triangles on the same base and between the same parallels are equal in area.
4. Triangles having equal areas have their corresponding altitudes equal.
Constructions 1. Construction of regular pentagons, hexagons and octagons inscribed and circumscribed in a circle.
2. Construction of a triangle equal in area, to a given quadrilateral.
Units 5 : Statistics 10 pds.
Drawing of histograms.
Class marks, cumulative frequency, cumulative frequency tables, drawing of frequency
polygons, drawing of an ogive.
Unit-6: Computing (I) 10 pds.
Popular introduction to computers : what they are, what they can perform and what they cannot perform, role and use of computers in modern society etc. Meaning of a problem-algorithm, a detailed and precise step by step method of solution of the problem, illustrated by means of simple day-to-day problems (like buying an article, multiplication of numbers etc.), simple flow charting (decision boxes included but not loops), easy exercises.
Prescribed Book 1. Mathematics -A Textbook for Secondary Schools Class IX Published by NCERT
2.
47
CLASS X
One Paper 3 hours 100 Marks
Unit weightage
UNITS Marks
1. Algebra 30
2. Commercial Mathematics and Mensuration 14
3. Trigonometry 12
4. Geometry 26
5. Statistics 08
6. Computing 10
Unit 1 : Algebra 63 pds.
Linear equations in two variables
Linear equations in two variables and its graph, system of two linear equations in two
variables, solution of the system of equations by graphical and algebraic methods - consistency/
inconsistency of the equations, applications involving the system of equations from different areas.
Rational Expressions
G.C.D. & L.C.M. of polynomials by factorization method, meaning of a rational expression,
addition, subtraction, multiplication of rational expressions, factorization of expressions involving
cyclic factors, ratio and proportion. properties and their applications.
Quadratic Equation
Meaning and standard form of a quadratic equation ax2 + bx + c = 0; (a # 0).
Solution of ax2 + bx + c = 0; a # 0 (i) by factorization (ii) by quadratic formula; discriminant
of the quadratic equation and nature of the roots, formation of the quadratic equations with given
roots, applications involving quadratic equation from several areas, solution of equations reducible
to quadratic form, factorization of quadratic polynomials by using quadratic formula (when other
methods are not easily applicable).
Unit 2 : Commercial Mathematics and Mensuration 32 pds.
Banking Working of Banks and different types of Accounts (Saving Bank account, Recurring Deposit account), problems.
The teacher is expected to devote some time in telling the students as to how banking system evolved to come to its present form. More emphasis should be laid on problem solving in Savings Bank Account.
Taxes The main objective of this unit, is to acquaint the students with the concepts of national economy with special reference to different forms of taxes:
1. Direct taxes and Indirect taxes.
2. Computation of Income Tax.
3. Sales Tax.
The teacher is expected to give sufficient practice in solving problems involving Income Tax and Sales Tax only.
48
Mensuration
Area and Volume
Area of a circle, sector and segment of a circle; surface area and volume of cube, cuboid, cone, cylinder, sphere; area of four walls of a room using logarithmic tables for computational work.
Unit 3 : Trigonometry 25 pds.
Trigonometrical Identities
sin2 A + cos2 A = 1; sec2 A = 1+tan2 A; cosec2 A= 1 + cot2 A
Proving simple identities based upon the above;
Trigonometrical ratio of complementary angles
sin (90o-A) = cos A, cosec (90o-A) = sec A
cos (90o-A) = sin A, sec (90o-A) = cosec A
tan (90o-A) = cot A, cot (90o-A) = tan A
Simple problems based upon the above
Heights and Distances
Reading of trigonometrical tables, solution of simple problems on height and distance, using
trigonometrical tables and logarithmic tables.
Unit 4 : Geometry 52 pds.