** CBSE Sample Papers**

# (Syllabus) CBSE Class 9th & 10th : Mathematics (Code No. 041)

##

CBSE Board Syllabus (2011)

Class : 9th & 10th

**Mathematics (Code No. 041)**

CBSE Board Syllabus (2011)

Class : 9th & 10th

**About Us: **The Syllabus in the subject of Mathematics has undergone
changes from time to time in accordance with growth of the subject and emerging
needs of the society. The present revised syllabus has been designed in
accordance with National Curriculum Framework 2005 and as per guidelines given
in Focus Group on Teaching of Mathematics which is to meet the emerging needs of
all categories of students. Motivating the topics from real life problems and
other subject areas, greater emphasis has been laid on applications of various
concepts.

The curriculum at Secondary stage primarily aims at enhancing the capacity of students to employ Mathematics in solving day-to-day life problems and studying the subject as a separate discipline. It is expected that students should acquire the ability to solve problems using algebraic methods and apply the knowledge of simple trigonometry to solve problems of heights and distances. Carrying out experiments with numbers and forms of geometry, framing hypothesis and verifying these with further observations form inherent part of Mathematics learning at this stage. The proposed curriculum includes the study of number system, algebra, geometry, trigonometry, menstruation, statistics, graphs and coordinate geometry etc. The teaching of Mathematics should be imparted through activities which may involve the use of concrete materials, models, patterns, charts, pictures posters, games, puzzles and experiments.

**OBJECTIVES
**The broad objectives of teaching of Mathematics at secondary stage are
to help the learners to: consolidate the Mathematical knowledge and skills
acquired at the upper primary stage; acquire knowledge and understanding,
particularly by way of motivation and visualization, of basic concepts, terms,
principles and symbols and underlying processes and skills. develop mastery of
basic algebraic skills; develop drawing skills; feel the flow of reasons while
proving a result or solving a problem. apply the knowledge and skills acquired
to solve problems and wherever possible, by more than one method. to develop
positive ability to think, analyze and articulate logically; to develop
awareness of the need for national integration, protection of environment,
observance of small family norms, removal of social barriers, elimination of sex
biases;

to develop necessary skills to work with modern technological devices such as calculators, computers etc;

to develop interest in Mathematics as a problem-solving tool in various fields for its beautiful structures and patterns, etc;

to develop reverence and respect towards great Mathematicians for their contributions to the field of Mathematics.

to develop interest in the subject by participating in related competitions.

to acquaint students with different aspects of mathematics used in daily life.

to develop an interest in students to study mathematics as a discipline.

**Course Structure **

Class IX

Class IX

**UNIT I : NUMBER SYSTEMS**

**1. REAL NUMBERS (20) Periods
**Review of representation of natural numbers, integers, rational numbers
on the number line. Representation of terminating / non-terminating recurring
decimals, on the number line through successive magnification.

Rational numbers as recurring/terminating decimals.

Examples of nonrecurring / non terminating decimals such as √2, √3, √5 etc. Existence of non-rational numbers (irrational numbers) such as √2, √3 and their representation on the number line. Explaining that every real number is represented by a unique point on the number line and conversely, every point on the number line represents a unique real number.

Existence of √x for a given positive real number x (visual proof to be emphasized).

Definition of nth root of a real number.

Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws.)

Rationalization (with precise meaning) of real numbers of the type (& their combinations).

1/(a + b√x) & 1/(√x + √y) where x and y are natural number and a, b are integers.

**UNIT II : ALGEBRA**

**1. POLYNOMIALS (25) Periods
**Definition of a polynomial in one variable, its coefficients, with
examples and counter examples, its terms, ero polynomial. Degree of a
polynomial. Constant, linear, quadratic, cubic polynomials; monomials,
binomials, trinomials. Factors and multiples. Zeros/roots of a polynomial /
equation. State and motivate the Remainder Theorem with examples and analogy to
integers. Statement and proof of the Factor Theorem. Factorization of ax

^{2}+ bx + c, a ≠ 0 where a, b, c are real numbers, and of cubic polynomials using the Factor Theorem.

Recall of algebraic expressions and identities. Further identities of the type (x + y + z)

^{2}= x

^{2}+ y

^{2}+ z

^{2}+ 2xy + 2yz + 2zx, (x ± y)

^{3}= x

^{3}± y

^{3}± 3xy (x ± y).

x

^{3}+ y

^{3}+ z

^{3}— 3xyz = (x + y + z) (x

^{2}+ y

^{2}+ z

^{2}— xy — yz — zx) and their use in factorization of polymonials. Simple expressions reducible to these polynomials.

**2. LINEAR EQUATIONS IN TWO VARIABLES (12) Periods**

Recall of linear equations in one variable. Introduction to the equation in two
variables. Prove that a linear equation in two variables has infinitely many
solutions and justify their being written as ordered pairs of real numbers,
plotting them and showing that they seem to lie on a line. Examples, problems
from real life, including problems on Ratio and Proportion and with algebraic
and graphical solutions being done simultaneously.

**UNIT III : COORDINATE GEOMETRY**

**1. COORDINATE GEOMETRY (9) Periods**

The Cartesian plane, coordinates of a point, names and terms associated with the
coordinate plane, notations, plotting points in the plane, graph of linear
equations as examples; focus on linear equations of the type ax + by + c = 0 by
writing it as y = mx + c and linking with the chapter on linear equations in two
variables.

**UNIT IV : GEOMETRY**

**1. INTRODUCTION TO EUCLID’S GEOMETRY (6) Periods**

History – Euclid and geometry in India. Euclid’s method of formalizing observed
phenomenon into rigorous mathematics with definitions, common/obvious notions,
axioms/postulates and theorems. The five postulates of Euclid. Equivalent
versions of the fifth postulate. Showing the relationship between axiom and
theorem.

1. Given two distinct points, there exists one and only one line through
them.

2. (Prove) two distinct lines cannot have more than one point in common.

**2. LINES AND ANGLES (10) Periods**

1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles
so formed is 180o and the converse.

2. (Prove) If two lines intersect, the vertically opposite angles are equal.

3. (Motivate) Results on corresponding angles, alternate angles, interior angles
when a transversal intersects two parallel lines.

4. (Motivate) Lines, which are parallel to a given line, are parallel.

5. (Prove) The sum of the angles of a triangle is 180o.

6. (Motivate) If a side of a triangle is produced, the exterior angle so formed
is equal to the sum of the two interiors opposite angles.

**3. TRIANGLES (20) Periods**

1. (Motivate) Two triangles are congruent if any two sides and the included
angle of one triangle is equal to any two sides and the included angle of the
other triangle (SAS Congruence).

2. (Prove) Two triangles are congruent if any two angles and the included side
of one triangle is equal to any two angles and the included side of the other
triangle (ASA Congruence).

3. (Motivate) Two triangles are congruent if the three sides of one triangle are
equal to three sides of the other triangle (SSS Congruene).

4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of
one triangle are equal (respectively) to the hypotenuse and a side of the other
triangle.

5. (Prove) The angles opposite to equal sides of a triangle are equal.

6. (Motivate) The sides opposite to equal angles of a triangle are equal.

7. (Motivate) Triangle inequalities and relation between ‘angle and facing side’
inequalities in triangles.

**4. QUADRILATERALS (10) Periods**

1. (Prove) The diagonal divides a parallelogram into two congruent triangles.

2. (Motivate) In a parallelogram opposite sides are equal, and conversely.

3. (Motivate) In a parallelogram opposite angles are equal, and conversely.

4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides
is parallel and equal.

5. (Motivate) In a parallelogram, the diagonals bisect each other and
conversely.

6. (Motivate) In a triangle, the line segment joining the mid points of any two
sides is parallel to the third side and (motivate) its converse.

**5. AREA (4) Periods**

Review concept of area, recall area of a rectangle.

1. (Prove) Parallelograms on the same base and between the same parallels have
the same area.

2. (Motivate) Triangles on the same base and between the same parallels are
equal in area and its converse.

**6. CIRCLES (15) Periods**

Through examples, arrive at definitions of circle related concepts, radius,
circumference, diameter, chord, arc, subtended angle.

1. (Prove) Equal chords of a circle subtend equal angles at the center and
(motivate) its converse.

2. (Motivate) The perpendicular from the center of a circle to a chord bisects
the chord and conversely, the line drawn through the center of a circle to
bisect a chord is perpendicular to the chord.

3. (Motivate) There is one and only one circle passing through three given
non-collinear points.

4. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant
from the center(s) and conversely.

5. (Prove) The angle subtended by an arc at the center is double the angle
subtended by it at any point on the remaining part of the circle.

6. (Motivate) Angles in the same segment of a circle are equal.

7. (Motivate) If a line segment joining two points subtendes equal angle at two
other points lying on the same side of the line containing the segment, the four
points lie on a circle.

8. (Motivate) The sum of the either pair of the opposite angles of a cyclic
quadrilateral is 180o and its converse

**7. CONSTRUCTIONS (10) Periods**

1. Construction of bisectors of line segments & angles, 60o, 90o, 45o angles
etc., equilateral triangles.

2. Construction of a triangle given its base, sum/difference of the other two
sides and one base angle.

3. Construction of a triangle of given perimeter and base angles.

**UNIT V : MENSURATION**

**1. AREAS (4) Periods
**Area of a triangle using Hero’s formula (without proof) and its
application in finding the area of a quadrilateral.

**2. SURFACE AREAS AND VOLUMES (10) Periods**

Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and
right circular cylinders/cones.

**UNIT VI : STATISTICS AND PROBABILITY
1. STATISTICS (13) Periods**

Introduction to Statistics : Collection of data, presentation of data — tabular form, ungrouped / grouped, bar graphs, histograms (with varying base lengths), frequency polygons, qualitative analysis of data to choose the correct form of presentation for the collected data. Mean, median, mode of ungrouped data.

**2. PROBABILITY (12) Periods**

History, Repeated experiments and observed frequency approach to probability.
Focus is on empirical probability. (A large amount of time to be devoted to
group and to individual activities to motivate the concept; the experiments to
be drawn from real – life situations, and from examples used in the chapter on
statistics).

INTERNAL ASSESSMENT 20 Marks

Evaluation of activities 10 Marks

Project Work 05 Marks

Continuous Evaluation 05 Marks

**UNIT I : NUMBER SYSTEMS**

**1. REAL NUMBERS (15) Periods**

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.

**UNIT II : ALGEBRA**

**1. POLYNOMIALS (6) Periods**

Zeros of a polynomial. Relationship between zeros and coefficients of quadratic
polynomials. Statement and simple problems on division algorithm for polynomials
with real coefficients.

**2. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES (15) Periods**

Pair of linear equations in two variables and their graphical solution.
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. Simple
problems on equations reducible to linear equations may be included.

**3. QUADRATIC EQUATIONS (15) Periods**

Standard form of a quadratic equation *ax*^{2} + *bx *+ *
c *= 0, *(a *≠ 0). Solution of the quadratic equations (only real roots)
by factorization, by completing the square and by using quadratic formula.
Relationship between discriminate and nature of roots.Problems related to day to
day activities to be incorporated.

**4. ARITHMETIC PROGRESSIONS (8) Periods**

Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms.

**UNIT III : TRIGONOMETRY**

**1. INTRODUCTION TO TRIGONOMETRY (12) Periods**

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of
their existence (well defined); motivate the ratios, whichever are defined at 0^{o}
& 90^{o}. Values (with proofs) of the trigonometric ratios of 30^{o},
45^{o }& 60^{o}. Relationships between the ratios.

**2. TRIGONOMETRIC IDENTITIES (16) Periods**

Proof and applications of the identity sin^{2} *A *+ cos^{2}*A
*= 1. Only simple identities to be given. Trigonometric ratios of
complementary angles.

**3. HEIGHTS AND DISTANCES (8) Periods**

Simple and believable problems on heights and distances. Problems should not
involve more than two right triangles. Angles of elevation / depression should
be only 30^{o}, 45^{o}, 60^{o}.

**UNIT IV : COORDINATE GEOMETRY**

**1. LINES (In two-dimensions) (15) Periods**

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.

**UNIT V : GEOMETRY**

**1. TRIANGLES (15) Periods**

Definitions, examples, counter examples of similar triangles.

1. (Prove) If a line is drawn parallel to one side of a triangle to intersect
the other two sides in distinct points, the other two sides are divided in the
same ratio.

2. (Motivate) If a line divides two sides of a triangle in the same ratio, the
line is parallel to the third side.

3. (Motivate) If in two triangles, the corresponding angles are equal, their
corresponding sides are proportional and the triangles are similar.

4. (Motivate) If the corresponding sides of two triangles are proportional,
their corresponding angles are equal and the two triangles are similar.

5. (Motivate) If one angle of a triangle is equal to one angle of another
triangle and the sides including these angles are proportional, the two
triangles are similar.

6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of
a right triangle to the hypotenuse, the triangles on each side of the
perpendicular are similar to the whole triangle and to each other.

7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio
of the squares on their corresponding sides.

8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum
of the squares on the other two sides.

9. (Prove) In a triangle, if the square on one side is equal to sum of the
squares on the other two sides, the angles opposite to the first side is a right
traingle.

**2. CIRCLES (8) Periods
**Tangents to a circle motivated by chords drawn from points coming closer and
closer to the point.

1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.

2. (Prove) The lengths of tangents drawn from an external point to circle are equal.

**3. CONSTRUCTIONS (8) Periods
**1. Division of a line segment in a given ratio (internally)

2. Tangent to a circle from a point outside it.

3. Construction of a triangle similar to a given triangle.

**UNIT VI : MENSURATION**

**1. AREAS RELATED TO CIRCLES (12) Periods**

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 60^{o}, 90^{o} & 120^{o}
only. Plane figures involving triangles, simple quadrilaterals and circle should
be taken.)

**2. SURFACE AREAS AND VOLUMES (12) Periods**

(i) Problems on finding surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of a cone.

(ii) Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with combination of not more than two different solids be taken.)

**UNIT VII : STATISTICS AND PROBABILITY**

**1. STATISTICS (15) Periods**

Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph.

**2. PROBABILITY (10) Periods**

Classical definition of probability. Connection with probability as given in Class IX. Simple problems on single events, not using set notation.

**INTERNAL ASSESSMENT 20 Marks**

Evaluation of activities 10 Marks

Project Work 05 Marks

Continuous Evaluation 05 Marks

**RECOMMENDED BOOKS
**1. Mathematics – Textbook for class IX – NCERT Publication

2. Mathematics – Textbook for class X – NCERT Publication

3. Guidelines for Mathematics Laboratory in Schools, class IX- CBSE Publication

4. Guidelines for Mathematics Laboratory in Schools, class X – CBSE Publication