# Class 10 Mathematics Karnataka Board Syllabus

### 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)
• 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
• 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

### 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.  Study Materials for Class 10

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