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 facorization method, meaning of a rational expression, addition, subtraction, multiplication of rational expressions, factorization of expressions using remainder theorem ratio and proportion, their properties and applications.

2. Commercial Mathematics ( Marks : 10)

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 Saving 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.

3. Mensuration (Marks : 10)

Area and Volume

Area of four walls of a room, area of a circle, sector and segment of a circle; surface area and volume of cube, cuboid, cone, cylinder, sphere;

4. Trigonometry ( Marks : 14)

Trigonometrical Identities

Sin² A + Cos² A = 1; Sec² A = 1 + tan² A; cosec² A = 1 + cos² A
Proving simple identities based upon the above;
Trigonometrical ratio of complementary angles
Sin (90⁰ – A) = cos A, cosec (90⁰ – A) = sec A
Cos (90⁰-A) = sin A, sec (90⁰-A) = consec A
Tan (90⁰-A) = cot A, cot (90⁰-A) = tan A
Simple problems based upon the above

Heights and Distances
Solution of simple problems of height and distance using trigonometrical tables and logarithmic tables.

5. Geometry (Marks : 26)

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. The proofs of only the star – marked prepositions may be asked in the Board Examination. In order to achieve the objectives of teaching geometry, the solving of riders (Exercises) covering all the propositions should be taught and tested.

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, he line must be parallel to the third side.
3. If in two triangles, the corresponding angles are equal (i.e. the two triangles are equiangular) their corresponding sides are proportional.
4. If corresponding sides of two triangles are proportional then the triangles are similar.
5. If the corresponding sides of two triangles are proportional, the triangles are equiangular.
6. If in two triangles, one pair of corresponding sides are proportional and the included angles are equal than the two triangles are similar.
7. If a perpendicular is drawn from the vertex of the right angle of a right angled triangle to the hypotenus, the triangle on each side of the perpendicular are similar to the whole triangle and to each other.
8. The ratio of the areas of similar triangles is equal to the ratio of the squares on the corresponding sides.
9. In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
10. In a triangle, if the square on one side is equal to the sum of the squares on the remaining two sides, the angle opposite the first side is a right angle.

Circle

1. If two arcs of a circle are congruent, their corresponding chords are equal.
2. If two chords of a circle are equal, then their corresponding arcs are congruent.
3. The perpendicular from the centre of a circle to a chord bisects the chord.
4. The line joining the centre of a circle to the mid – point of a chord is perpendicular to the chord.
5. There is one and only one circle passing through three given non – collinear points.
6. Equal chords of a circle (or of congruent circles) are equidistant from the centres.
7. Chords of a circle (or of congruent circles) that are equidistant from the centres are equal.
*8. The degree measure of an arc of a circle is twice the angle subtended by it at any point of the alternate segment of circle with respect to the arc.
*9. The angle in a semi – circle is a right angle.
*10. The arc of a circle subtending a right angle at any point of the circle in its alternate segment is a semi – circle.
*11. Angles in the same segment of a circle are equal.
12. 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 the same circle.
13. Equal chords of a circle subtend equal angles at the centre.
14. If the angles subtend by two chords of a circle at the centre are equal, the chords are equal.
*15. The sum of either pair of opposite angles of a cycle quadrilateral is 180.
16. If the sum of any pair of opposite angles of a quadrilateral is 180⁰.
17. A tangent at any point of a circle is perpendicular to the radius through the point of contact.
18. The lengths of the two tangents drawn from an external point to a circle are equal.
19. If two chords of a circle intersect inside or outside the circle then the rectangle formed by the two parts of a chord is equal in area to the rectangle formed by the two parts of the other.
*20. IF PAB is a recent to a circle intersecting the circle at A and B PT is a tangent at T, than PA x PB = PT².
*21. If a chord is drawn through the point of contact of a tangent to a circle then the angles which this chord makes with the given tangent are equal respectively to the angles formed in the corresponding alternate segments.
22. If a line is drawn through an end point of a chord of circle so that the angle formed with the chord is equal to the angle subtended by the chord in the alternate segment, then the line is a tangent to the circle.
23. If two circles touch each other (internally or externally) the point of contact lies on the line through the centres.

Constructions :

1. Construction of a tangent to a circle at a given point on the circle when the centre is (I) known (ii) unknown.

2. Construction of a triangle, given base, vertical angle and either altitude or median through vertex.
3. Construction of figures (triangles, quadrilaterals, etc.) similar to the given figures as per the given scale factor.
4. Division of a given line segment, internally/externally in a given ratio.

6. Statistics (Marks : 10)

Mean of grouped data, median and mode of ungrouped data, descriptive explanation of mortality tables (CDR, SDR. IMR), cost of living index and price index.