Mathematics (CODE  041)
Class X
One Paper

Time : Three Hours

Marks : 100

Unit


Marks

Algebra 

26

Commercial Mathematics 

12

Geometry 

22

Trigonometry 

10

Mensuration 

10

Statistics 

12

Coordinate Geometry 

08


Total :

100

Unit  I : ALGEBRA
Linear
Equations in Two Variables :
System of linear
equations in two variables. Solution of the system of linear equations (i) Graphically.
(ii) By algebraic methods : (a) Elimination by substitution (b) Elimination
by equating the coefficients. (c) Cross multiplication. Applications of Linear
equations in two variables in solving simple problems from different areas. (Restricted
upto two equations with integral values as a point of solution. Problems related
to life to be incorporated).
Polynomials : HCF and LCM of polynomials
by factorisation.
Rational
Expressions :
Meaning of rational expressions.
Reduction of rational expressions to lowest terms using factorisation. Four fundamental
operations on rational expressions. (Properties like commutativity, associativity,
distributive law etc. not to be discussed. Cases involving Factor theorem may also
be given).
Quadratic
Equations :
Standard form of a quadratic equation
ax2+bx+c=0, (a . 0). Solution of ax2+bx+c=0 by (i) factorisation (ii) quadratic
formula. Application of quadratic equations in solving wordproblems from different
areas. (Roots should be real) (Problems related to daytoday activities to be incorporated).
Arithmetic
Progressions (AP) :
Introduction to AP by pattern of number. General term of an AP, Sum to nterms of
an AP. Simple problems. (Common difference should not be irrational number).
Unit  II : COMMERCIAL
MATHEMATICS
Instalments :
Installment payments and
installment buying (Number of installments should not be more than 12 in case of
buying). (Only equal installments should be taken. In case of payments through equal
installments, not more than three installments should be taken.
Income Tax :
Calculation of Income Tax for salaried
class. (In case of income tax problems, annual salary should be exclusive of HRA).
Unit  III : GEOMETRY
A number of propositions
in Geometry are listed below. Some of them have already been learnt at the upper
primary stage through activities/ experiments. At this stage, the purpose is to
acquaint the pupil with the nature and method of a geometrical proof. In order to
ensure that the burden on the pupil is not much, only proofs of some selected propositions
may be asked in the examination. These propositions may be selected in such a way
that they reflect different types of proofs such as direct proof, proof by contradiction,
proof by exhaustion, proof by various applications of previous propositions, Keeping
this in view, some propositions have been marked with "*". (For the verification
of geometrical results, teacher may use different types of activities such as models,
paper cutting, paper folding, measurement etc. The students should also be encouraged
to perform these activities themselves). In view of this:

The truth of the unstarred
propositions should be brought home to the pupils by either recalling them from
earlier classes or by verifying them experimentally in the present class.

The proofs of only ‘*’ marked
propositions may be asked in the examination.

The riders on ‘*’ propositions
only may be asked in the examination. However, they may involve the use of other
results (unstarred ones).

The unstarred propositions
should not be asked as riders/exercises in the examination.
Similar
Triangles :

If a line is drawn parallel
to one side of a triangle, the other two sides are divided in the same ratio.

If a line divides any two
sides of a triangle in the same ratio, the line is parallel to the third side.

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

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

If one angle of a triangle
is equal to one angle of the other and the sides including these angles are proportional,
the triangles are similar.

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.

The ratio of the areas of
similar triangles is equal to the ratio of the squares on their corresponding sides.

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

In a triangle, if the square
on one side is equal to the sum of the squares on the remaining two, the angle opposite
the first side is a right angle.

The internal bisector of
an angle of a triangle divides the opposite side in the ratio of the sides containing
the angle and its converse.
Circles
:

Two circles are congruent
if and only if they have equal radii.

Equal chords of a circle
subtend equal angles at the centre and conversely, if the angles subtended by the
chords at the centre (of a circle) are equal, then the chords are equal.

Two arcs of a circle are
congruent if the angles subtended by them at the centre are equal and its converse.

If two arcs of a circle
are congruent, their corresponding chords are equal and its converse.

The perpendicular from the
centre of a circle to a chord bisects the chord and conversely, the line drawn through
the centre of a circle to bisect a chord is perpendicular to the chord.

There is one and only one
circle passing through three given noncollinear points.

Equal chords of a circle
(or congruent circles) are equidistant from the centre(s) and conversely, chords
of a circle (or of congruent circles) that are equidistant from the centre(s) are
equal.

The angle subtended by an
arc at the centre is double the angle subtended by it at any point on the remaining
part of the circle.

The angle in a semicircle
is a right angle and its converse.

Angles in the same segment
of a circle are equal.

If a line segment joining
two points subtends equal angles at two other points lying on the same side of the
line containing the segment, the four points lie on a circle.

The sum of the either pair
of the opposite angles of a cyclic quadrilateral is 180°.

If a pair of opposite angles
of a quadrilateral is supplementary, then the quadrilateral is cyclic.

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

The lengths of tangents
drawn from an external point to a circle are equal.

If two chords of a circle
intersect inside or outside the circle, then the rectangle formed by the two parts
of one chord is equal in area to the rectangle formed by the two parts of the other.

Converse of proposition
16.

If PAB is a secant to a
circle intersecting it at A and B and PT is a tangent, then PA × PB=PT2.

If a line touches a circle
and from the point of contact a chord is drawn, the angles which this chord makes
with the given line are equal respectively to the angles formed in the corresponding
alternate segments and the converse.

If two circles touch each
other internally or externally, the point of contact lies on the line joining their
centres. (Concept of common tangents to two circles should be given)
Constructions
:

Construction of tangents
to a circle (i) At a point on it without using the centre. (ii) At a point onit
using the centre. (iii) From a point outside it. [(i) Proofs of constructions not
required. (ii) Constructions using ruler and compasses only].

Construction of incircle
and circumcircle of a triangle with given sides.

Construction of a triangle,
given base, vertical angle and either altitude or median through the vertex.

Construction of figures
(triangles, quadrilaterals) similar to the given figure as per the given scale factor.
Unit  IV : TRIGONOMETRY
Trigonometric
Identities :

Sin2A + Cos2A = 1

Sec2A = 1 + tan2A

Cosec2A = 1 + Cot2A

Proving simple identities
based on the above.
Trigonometric
ratios of complementary angles :
Heights
and Distances :
Simple Problems on
heights and distances.
(i) Problems should not involve more than two right triangles.
(ii) Angles of elevation/depression should be only 30°, 45°, 60°.
Unit  V : MENSURATION
Volumes
and Surface Areas :

Problems on finding volumes
and surface areas of combinations of right circular cone,rightcircular cylinder,
hemisphere and sphere. Frustum of a cone.

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  VI : STATISTICS
Mean :
Mean of grouped data. (Calculation by assuming
assumed mean should also be discussed).
Probability
: Elementary idea of probability as a measure
of uncertainty (for single event only).
Pictorial
Representation of Data :
Reading and construction
of pie chart. [(i) Sub parts of a pie chart should not exceed five). (ii) Central
angles should be in multiples of 5 degrees.]
Unit  VII
: COORDINATE GEOMETRY
Coordinate
Geometry :
Distance between two points. Section
formula. (internal division only.)
