1. Arithmetic

• Numbers
  • Euclid’s Lemma: given an integer a and a positive integer b, there unique exist integers k and r such that a = bk + r, where 0 ≤ r < b
  • Fundamental theorem of Arithmetic: any positive integer n > 1 can be expressed as a product of powers of prime numbers.
  • The above properties without formal proof. As corollaries, the following to be proved:
    • If a prime divides the product of two integers, then it divides at least one of them, and 
    • If a divides bc and HCF (a,b) = 1, then a divides c. Proof of irrationality of √2, √3, and √5 using this.
• Sets
  • Revision of set operations (union, intersection, set difference, complement, symmetric difference)
  • Properties of set operations
  • Commutativity and associativity of intersection =
  • DeMorgan’s laws
  • Relation between the number of elements in two sets to the number of elements in their union and intersection (principle of inclusion-exclusion): n(A) + n(B) = n(AUB) + n(AUB).
• Progressions
  • Concept of a sequence
  • Arithmetic Progression: n-th term
  • Sum of n terms of an AP
  • Problems based on this
  • Geometric Progression: n-th term
  • Sum to n terms
  • Sum of an infinite GP, when the common ratio |r| < 1
  • Harmonic progression: n-th term
  • Arithmetic, Geometric and harmonic mean of two positive real numbers
  • Relation among them (AM ≥ GM ≥ HM)
  • A simple proof based on x² ≥ 0 for any real number x.
• Permutation, Combination, and Probability
  • Fundamental principle of counting
  • Meaning of permutation
  • Meaning of Combination
  • Notations for permutation and combination
  • Difference between permutation and combination
  • Problems based on these principles
  • Random experiment
  • Event
  • Sample Space
  • Types of events(mutually exclusive, complementary, certain, impossible)
  • Definition of probability
  • Problems on probability based on permutation and combination
• Statistics
  • Standard Deviation of grouped and ungrouped data
  • Calculation of standard deviation by the direct method
  • Coefficient of variation
  • Construction and interpretation of pie-charts.

2. Algebra

• Surds
  • Like and unlike surds
  • Addition, Subtraction and multiplication rule
  • Rationalization of simple surds
• Polynomials
  • Division of one polynomial by another
  • Concept of degree
  • Synthetic division method
  • Remainder theorem: p(x) = (x – a)h(x) + p(a)
• Quadratic Equations
  • Meaning of a quadratic expression and a quadratic equation
  • Simple problems on pure and adfected equations
  • Solutions by factorization and its limitation
  • Solution using formula
  • Relation between roots and coefficients
  • Discriminant and nature of roots
  • Graphs of quadratic expressions
  • Nature of quadratic expression in terms of associated discriminant
  • Factorizing a quadratic expression using roots
  • Graphical method of solving a quadratic equation
  • Limitation of the graphical method
  • Word problems leading to quadratic equations

3.Geometry

• Triangles
  • Similarity of triangles
  • Basic proportionality theorem (Thale’s theorem)
  • A formal proof using areas
  • Theorem: If two triangles are equiangular, their corresponding sides are proportional
  • Theorem: If two triangles are similar, then the ratio of their areas is equal to the square of the ratio of any two corresponding sides.
  • Revision of right-angled triangles leading to Pythagoras’ theorem
  • Theorem: (Pythagoras) in a right-angles triangle, the square on the hypotenuse is the sum of the squares on the other two sides.
  • Problems based on Pythagoras theorem
  • Theorem: (Converse of Pythagoras theorem) If in a triangle, the square on one side is equal to the sum of the squres on the remaining two sides, then the angle opposite to the larger side is a right-angle. Problems based on this result
• Circles
  • Meaning of tangent
  • Point of contact
  • Properties of a tangent
  • Radius drawn from the point of contact of a tangent to the circle is perpendicular to the tangent – converse of this statement (no proof, only verification)
  • Meaning of touching circles – touching externally and touching internally
  • Common tangents – direct and transverse
  • Theorem: If two circles touch each other, then the centres and the point of contact are collinear
  • Theorem: The tangents drawn from an external point to a circle are (i) equal, (ii) equally inclined to the line joining the point to the centre, and (iii) subtend equal angles at the centre
• Construction
  • Construction of chord of given length
  • Verification of the properties
    • Equal chords are equi-distant from the centre
    • Angles in the same segment are equal
    • Angles in the major-segment are acute angles; angles in the minor segment are obtuse angles; and angles in the semi-circle are right-angles

Note: This is the proposed syllabus as per the Karnataka Government.