Class 10 Mathematics Karnataka Board Syllabus

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)
  • 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.

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