Mathematics (CODE - 041)

Class X

One Paper
Time : Three Hours
Marks : 100
Commercial Mathematics  
Coordinate Geometry  


Total :

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 co-efficients. (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 word-problems from different areas. (Roots should be real) (Problems related to day-to-day activities to be incorporated).

Arithmetic Progressions (AP) : Introduction to AP by pattern of number. General term of an AP, Sum to n-terms of an AP. Simple problems. (Common difference should not be irrational number).


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).


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 :

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

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

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

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

  5. 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.

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

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

  9. 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.

  10. 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 :

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

  2. 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.

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

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

  5. 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.

  6. There is one and only one circle passing through three given non-collinear points.

  7. 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.

  8. 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.

  9. The angle in a semi-circle is a right angle and its converse.

  10. Angles in the same segment of a circle are equal.

  11. 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.

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

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

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

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

  16. 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.

  17. Converse of proposition 16.

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

  19. 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.

  20. 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 :

  1. 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].

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

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

  4. Construction of figures (triangles, quadrilaterals) similar to the given figure as per the given scale factor.


Trigonometric Identities :

  1. Sin2A + Cos2A = 1

  2. Sec2A = 1 + tan2A

  3. Cosec2A = 1 + Cot2A

  4. Proving simple identities based on the above.

Trigonometric ratios of complementary angles :

  • Sin (90°–A) = Cos A

  • Cos (90°–A) = Sin A

  • Tan (90°–A) = Cot A

  • Cosec (90°–A) = Sec A

  • Sec (90°–A) = Cosec A

  • Cot (90°–A) = Tan A
    Problems based on above.

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°.


Volumes and Surface Areas :

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

  2. 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).


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.]


Co-ordinate Geometry : Distance between two points. Section formula. (internal division only.)

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