Edexcel A-Level Maths Pure Year 2|2023 LATEST UPDATE|GUARANTEED SUCCESS

Contradiction A contradiction is a disagreement between two statements, which means that both cannot be true. Proof by contradiction is a powerful technique. Proof by contradiction To prove a statement by contradiction you start by assuming it is not true. You then use logical steps to show that this assumption leads to something impossible (either a contradiction of the assumption, or a contradiction of a fact you know to be true). You can conclude that your assumption was incorrect and the original statement was true. A rational number can be written as a/b, where a and b are integers. An irrational number cannot be expressed in the form a/b, where a and b are integers. To multiply fractions cancel any common factors, then multiply the numerators and multiply the denominators. To divide two fractions multiply the first fraction by the reciprocal of the second fraction. To add or subtract two fractions, find a common denominator. Splitting a fraction into partial fractions A single fraction with two distinct linear factors in the denominator can be split into two separate fractions with linear denominators. Two methods to find the constants of partial fractions substitution and equating coefficients. The method of partial fractions ca also be used when there are more than two distinct linear factors in the denominator. A single fraction with a repeated linear factor in the denominator can be split into two or more separate fractions. Improper algebraic fraction One whose numerator has a degree equal to or larger than the denominator. An improper fraction must ... before you can express it in partial fractions be converted to a mixed fraction. To connect an improper fraction into a mixed fraction you can either use: - algebraic division - or the relationship F(x) = Q(x) x divisor + remainder Degree of a polynomial the largest exponent in the expression You can measure angles in units called radians 2⫪ radians = 360⁰ ⫪ radians = 180⁰ 1 radian = 180⁰/⫪ 30⁰ = ⫪/6 radians 45⁰ = ⫪/4 radians 60⁰ = ⫪/3 radians 90⁰ = ⫪/2 radians 180⁰ = ⫪ radians 360⁰ = 2⫪ radians sin(⫪/6) = 1/2 sin(⫪/3) = √3/2 sin(⫪/4) = 1/√2 = √2/2 cos(⫪/6) = √3/2 cos(⫪/3) = 1/2 cos(⫪/4) = 1/√2 = √2/2 tan(⫪/6) = 1/√3 = √3/3 tan(⫪/3) = √3 tan(⫪/4) = 1 sin(⫪ - θ) = sinθ sin(⫪ + θ) = -sinθ sin(2⫪ - θ) = -sinθ cos(⫪ - θ) = -cosθ cos(⫪ + θ) = -cosθ cos(2⫪ - θ) = cosθ tan(⫪ - θ) = -tanθ tan(⫪ + θ) = tanθ tan(2⫪ - θ) = -tanθ Using radians greatly simplifies the formula for arc length. To find the arc length l of a sector of circle use the formula l = rθ, where r is the radius of the circle and θ is the angle, in radians, contained by the sector. Using radians also greatly simplifies the formula for the area of a sector To find the area A of a sector of a circle use the formula A=(1/2)r²θ, where r is the radius of the circle and θ is the angle, in radians, contained by the sector. Minor sector The smaller area enclosed by a radii and an arc Major sector The larger area enclosed by a radii and an arc