syllabus of 9th Raj Board

UNIT I : NUMBER SYSTEMS
1. REAL NUMBERS
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 Ö2, Ö3, Ö5 etc. Existence
of non-rational numbers (irrational numbers) such as Ö2, Ö3 and their representation
on the number line. Explaining that every real number is represented by unique point
on the number line and conversely, every point on the number line represent a unique
real number.
Existence of Öx 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)
+ a +b x x y
1 & 1
where x and y are natural number and a, b are integers.
UNIT II : ALGEBRA chtxf.kr
1. POLYNOMIALS cgqiqin
Definition of a polynomial in one variable, its coefficients, with examples and counter
examples, its terms, zero 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 ax2+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 = x2 + y2 + z2 + 2xy + 2 yz + 2zx, (x + y)3 =
x3 + y3 + 3xy (x + y).
x3 + y3 + z3 - 3xyz = (x + y + z) (x2 + y2 + z2 - xy - yz - zx) and their use in
factorization of polynomials. Simple expressions reducible to these polynomials..
2. LINEAR EQUATIONS IN TWO VARIBALES
nks pjksa sa okys jSfSfSf[kd lehdj.k
Recall of linear equations in one variable. Introduction to the equation in two variables.
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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 : GEOMETRY T;kfefr
1. INTRODUCTION TO EUCLID’S CEOMETRY
;wfwfDyM dh T;kfefr dk ifjp;
History – Geometry in India and Eulid’s geometry. 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. (Axiom) Given two distinct points, there exists one and only one line through them.
2. (Theorem) (Prove) two distinct lines cannot have more than one point in common.
2. LINES AND ANGLES js[s[kk,a vkSjSjSj dks.s.k
1. (Motivate) if a ray stand 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 interests 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 f=Hkqtqtqt
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
tringle is equal to any two angels 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 Congruence).
4. (Motivate) Two right triangle 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 side 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.
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4. QUADRILATERALS prqHqHqHkqZtqZqtZt
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 angels 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 {ks=s=Qy
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 o`r`r
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 subtends 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 jpuk,sasasa
1. Constructions of bisectors of line segments & angels, 60o, 90o, 45o angles etc.,
equilateral triangles.
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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 angels.
UNIT IV : COORDINATE GEOMETRY funsZs’Zs’Z’kkadadad T;kfefr
1. COORDINATE GEOMETRY funsZ’kkad T;kfefr
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 .
UNIT V : MENSURATION esUsUsUlwjwjwjs’s’s’ku
1. AREAS {ks=s=Qy
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 i`"`"`"Bh; {ks=s=Qy ,oa vk;ru
Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and
right circular cylinders/cones.
UNIT VI : STATISTICS AND PROBABLILTY
lkaf[;dh ,oa izkf;drk
1. STATISTICS lkafafa [;dh
Introduction to Statistics : Collection of data, presentation of data – tabular form,
ungrouped / grouped, bar graphs, histograms (with varying base length), frequency
polygons, qualitative analysis of data to choose the correct from of presentation for
the collected data. Mean, median, mode of ungrouped data.
2. PROBABILITY izkzkzkf;drk
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.
Prescribed Books :
Mathematics -Textbook for class IX NCERT's Book Published under Copyright