Revision Notes: Real Numbers Class 10

Learning Objectives

Master real numbers, polynomials, and coordinate geometry
Solve linear equations in two variables and quadratic equations
Understand trigonometry and its applications
Study circles, constructions, and mensuration
Apply statistics and probability concepts

Key Concepts

Real Numbers

The Fundamental Theorem of Arithmetic states every composite number can be uniquely expressed as a product of primes. Euclid's Division Lemma: a = bq + r, where 0 ≤ r < b. Used to find HCF.

Rational numbers: Terminating or repeating decimal expansion
Irrational numbers: √2, √3, π — non-terminating, non-repeating
HCF × LCM = Product of two numbers

Polynomials & Quadratic Equations

A polynomial p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀. Zeroes of a polynomial are values of x where p(x) = 0.

Quadratic formula: x = (-b ± √(b²-4ac)) / 2a
Discriminant: D = b²-4ac. D>0: two real roots, D=0: equal roots, D<0: no real roots
Sum of roots: -b/a, Product of roots: c/a

Trigonometry

For a right triangle with angle θ: sin θ = Opposite/Hypotenuse, cos θ = Adjacent/Hypotenuse, tan θ = Opposite/Adjacent.

Identity: sin²θ + cos²θ = 1
Values: sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1
Applications: Height and distance problems

Coordinate Geometry

Distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]. Section formula: Point dividing line in ratio m:n = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)). Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2).