Edexcel AS Pure Maths, AS-Level Pure Maths, pure maths As level QUESTIONS WITH CORRECT ANSWERS

Edexcel AS Pure Maths, AS-Level Pure Maths, pure maths As level QUESTIONS WITH CORRECT ANSWERS Quadratic Formula - ANSWER Domain - ANSWER The set of possible inputs for a function. Range - ANSWER The set of possible outputs of a function. Discriminant - ANSWER b² - 4ac > 0 then two distinct real roots. b² - 4ac = 0 then one repeated real root. b² - 4ac < 0 then a quadratic function has no real roots. Types of Lines for Regions - ANSWER If y < f(x) or y > f(x) then the curve y = f(x) is not included in the region, and is represented by a dotted line. If y ≤ f(x) or y ≥ f(x) then the curve y = f(x) is included in the region, and is represented by a solid line. Graph Translations - ANSWER y = f(x) + a is a translation of the graph y = f(x) by a upwards. y = f(x + a) is a translation of the graph y = f(x) by a to the left. Graph Stretches - ANSWER y = af(x) is a stretch of the graph y = f(x) by a scale factor of a in the vertical direction. y = f(ax) is a stretch of the graph y = f(x) by a scale factor of 1/a in the horizontal direction. Graph Reflections - ANSWER y = -f(x) is a reflection of the graph of y = f(x) in the x-axis. y = f(-x) is a reflection of the graph of y = f(x) in the y-axis. Gradient of Equation - ANSWER m = (y₂ - y₁) ÷ (x₂ - x₁) Equation of a Line - ANSWER y - y₁ = m(x - x₁) with coords (x₁, y₁) Distance Formula - ANSWER √((x₂ - x₁)² + (y₂ - y₁)²) from (x₁, y₁) to (x₂, y₂) Perpendicular Bisector - ANSWER -1/m where m is original gradient Standard Equation of a Circle - ANSWER (x - a)² + (y - b)² = r² with centre (a, b) and radius r Equation of a Circle (fg) - ANSWER x² + y² + 2fx + 2gy + c = 0 with centre (-f, -g) and radius √(f² + g² - c) Circle Theorems - ANSWER • Tangent to a circle is perpendicular to the radius of the circle at the point of intersection. • Perpendicular bisector of a chord will go through the circle centre. • If triangle forms across the circle, its diameter is the hypotenuse of the right-angled triangle. • Equations of the perpendicular bisectors of two different chords will intersect at the circle centre. Factor Theorem - ANSWER If f(p) = 0, (x - p) is a factor of f(x) Mathematical Proofs - ANSWER • State any info/assumptions • Show every step clearly • Make sure every step follows logically from the previous step • Cover all possible cases • Write a statement of proof at the end of your working Truth by Exhaustion - ANSWER Break the statement into smaller cases and prove each case separately. Truth by Counter-Example - ANSWER Find one example that does not work for the statement. Pascal's Triangle - ANSWER The (n + 1)th row of Pascal's triangle gives the coefficients in the expansion of (a + b)ⁿ Factorial Formula - ANSWER n! = n x (n - 1) x (n - 2) x ... x 2 x 1 Factorials in Pascal's Triangle - ANSWER The number of ways of choosing r from a group of n items is: ⁿCᵣ = n! ÷ (r! x (n - r)!) Binomial Expansion - ANSWER (a + b)ⁿ = (ⁿCᵣ)(aⁿ⁻¹bʳ) (a + b)ⁿ = aⁿ + (ⁿC₁)(aⁿ⁻¹b) + (ⁿC₂)(aⁿ⁻²b²) + ... + (ⁿCᵣ)(aⁿ⁻ʳbʳ) + ... + bⁿ Cosine Rule (a²) - ANSWER Cosine Rule (cos(A)) - ANSWER Sine Rule - ANSWER Sine Rule Solutions - ANSWER Sometimes produces two possible solutions for a missing angle: sin(θ) = sin(180 - θ) Sine Graph - ANSWER Cosine Graph - ANSWER Tangent Graph - ANSWER CAST Diagram - ANSWER Trigonometry - ANSWER Trig Triangles (30, 60, 90) - ANSWER Trig Triangles (45, 90) - ANSWER Principal Value - ANSWER When you use the inverse trig function on your calculator, the angle you get is the principal value. Sine and Cosine Formulae - ANSWER sin²(θ) + cos²(θ) = 1 tan(θ) = sin(θ) ÷ cos(θ) Triangle Law for Vector Addition - ANSWER A→B + B→C = A→C If A→B = a, B→C = b and A→C = c, then a + b = c Vector Rules - ANSWER • P→Q = R→S, then line segments PQ and RS are equal in length and are parallel. • A→B = -(B→A) • Any vector parallel to the vector a may be written as λa Vector Magnitude - ANSWER a = xi + yj → |a| = √(x² + y²) Unit Vector - ANSWER In the direction of a, unit vector is a ÷ |a| Position Vector - ANSWER O→P = pi + qj A→B = O→B - O→A First Derivative Formula - ANSWER Differentiation Formula - ANSWER dy/dx = anxⁿ⁻¹ Function's Gradient - ANSWER Increasing on the interval [a, b] if f'(x) ≥ 0 for all values of a < x < b.