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# 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.

- Euclid’s Lemma: given an integer
- 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(A*U*B) + n(A*U*B).*

- 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 x
^{2}≥ 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.