Maths

Course Structure

First Term Course Syllabus

Unit I: Number Systems

1. Real Numbers

• 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

• 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

• 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 a pair of linear equations in two variables algebraically – by substitution, by elimination and by cross multiplication method
• Simple situational problems must be included
• Simple problems on equations reducible to linear equations

Unit III: Geometry

1. Triangles

• Definitions, examples, counter examples of similar triangles
• (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
• (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side
• (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar
• (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar
• (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
• (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
• (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides
• (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides
• (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 triangle

Unit IV: Trigonometry

1. Introduction to Trigonometry

• 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 and 90^o
• Values (with proofs) of the trigonometric ratios of 30^o, 45^o and 60^o
• Relationships between the ratios

2. Trigonometric Identities

• Proof and applications of the identity sin2A + cos2A = 1
• Only simple identities to be given
• Trigonometric ratios of complementary angles

Unit V: Statistics and Probability

1. Statistics

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

Second Term Course Syllabus

Unit II: Algebra

3. Quadratic Equations

• Standard form of a quadratic equation ax² + 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 discriminant and nature of roots
• Situational problems based on quadratic equations related to day to day activities to be incorporated

4. Arithmetic Progressions

• Motivation for studying Arithmetic Progression Derivation of the 9^th term and sum of the first ‘n’ terms of A.P. and their application in solving daily life problems.

Unit III: Geometry

2. Circles

• Tangents to a circle motivated by chords drawn from points coming closer and closer to the point
• (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact
• (Prove) The lengths of tangents drawn from an external point to circle are equal

3. Constructions

• Division of a line segment in a given ratio (internally)
• Tangent to a circle from a point outside it
• Construction of a triangle similar to a given triangle

Unit IV: Trigonometry

3. Heights and Distances

• 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 V: Statistics and Probability

2. Probability

• Classical definition of probability
• Simple problems on single events (not using set notation)

Unit VI: Coordinate Geometry

1. Lines (In two-dimensions)

• Concepts of coordinate geometry, graphs of linear equations
• Distance formula
• Section formula (internal division)
• Area of a triangle

Unit VII: Mensuration

1. Areas Related to Circles

• 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 and 120^o only
• Plane figures involving triangles, simple quadrilaterals and circle should be taken

2. Surface Areas and Volumes

• Problems on finding surface areas and volumes of combinations of any two of the following −
  • Cubes
  • Cuboids
  • Spheres
  • Hemispheres
  • Right circular cylinders/cones
  • Frustum of a cone