Mathematics — UK A-Level
Comprehensive UK A-Level Mathematics flashcards (AQA/Edexcel/OCR common core) covering pure mathematics (proof, algebra, functions, coordinate geometry, sequences and series, trigonometry, exponentials and logarithms, differentiation, integration, numerical methods, vectors), statistics (sampling, data, probability, distributions, hypothesis testing), and mechanics (kinematics, forces, moments, projectiles).
Ämne: Matematik · Nivå: Gymnasium (16–19) · 428 kort
Innehåll
- A mathematical proof is a logical argument that establishes the truth of a statement beyond doubt, using axioms, definitions and previously proven results.
- Proof by deduction starts from known facts or axioms and uses logical steps to reach the required conclusion.
- Proof by exhaustion breaks a statement into a finite number of cases and verifies the statement holds in every case.
- Disproof by counter-example: a single example for which a statement is false is enough to prove that the statement is not always true.
- Proof by contradiction assumes the negation of the statement is true, then derives a logical contradiction, showing the original statement must be true.
- The symbol ⇒ means 'implies'. The symbol ⇔ means 'if and only if' and indicates that the implication works in both directions.
- A necessary condition must hold for a statement to be true; a sufficient condition guarantees the statement is true. A ⇔ B means A is both necessary and sufficient for B.
- The classic proof that √2 is irrational uses proof by contradiction, assuming √2 = a/b in lowest terms and deriving that both a and b are even.
- Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ.
- Negative and fractional indices: a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a, and a⁰ = 1 for any a ≠ 0.
- To rationalise a denominator of the form a + √b, multiply numerator and denominator by the conjugate a − √b.
- The quadratic formula gives the roots of ax² + bx + c = 0 as x = (−b ± √(b² − 4ac)) / (2a).
- The discriminant of ax² + bx + c is b² − 4ac. If it is positive there are two real roots, if zero one repeated root, if negative no real roots.
- Completing the square writes ax² + bx + c in the form a(x + p)² + q, revealing the vertex of the parabola at (−p, q).
- The factor theorem states that (x − a) is a factor of polynomial f(x) if and only if f(a) = 0.
- The remainder theorem states that when polynomial f(x) is divided by (x − a), the remainder is f(a).
- A function is a relation in which each input from the domain maps to exactly one output in the range.
- A composite function fg(x) means apply g first, then f to the result: fg(x) = f(g(x)).
- A function has an inverse f⁻¹ only if it is one-to-one. The graph of f⁻¹(x) is the reflection of f(x) in the line y = x.
- The modulus function |x| gives the non-negative magnitude of x: |x| = x if x ≥ 0, and |x| = −x if x < 0.
- Graph transformation y = f(x) + a translates the curve up by a; y = f(x + a) translates it left by a.
- Graph transformation y = af(x) stretches the curve vertically by scale factor a; y = f(ax) stretches it horizontally by scale factor 1/a.
- Partial fractions decompose a rational expression into a sum of simpler fractions, used to simplify integration and binomial expansions.
- The gradient of the line joining (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁).
- The equation of a straight line through (x₁, y₁) with gradient m is y − y₁ = m(x − x₁).
- Two lines are parallel if their gradients are equal, and perpendicular if the product of their gradients is −1 (m₁m₂ = −1).
- The distance between points (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²), from Pythagoras' theorem.
- The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r².
- The angle in a semicircle is a right angle: if a triangle is inscribed in a circle with one side as the diameter, the opposite angle is 90°.
- A tangent to a circle is perpendicular to the radius at the point of contact.
- The perpendicular from the centre of a circle to a chord bisects the chord.
- Parametric equations express x and y separately in terms of a third variable (parameter), e.g. x = t², y = 2t. Eliminating the parameter gives the Cartesian equation.
- An arithmetic sequence has a common difference d between consecutive terms. The nth term is uₙ = a + (n − 1)d.
- The sum of the first n terms of an arithmetic series is Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.
- A geometric sequence has a common ratio r between consecutive terms. The nth term is uₙ = arⁿ⁻¹.
- The sum of the first n terms of a geometric series is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.
- A geometric series converges to a sum to infinity S∞ = a/(1 − r) only when |r| < 1.
- Sigma notation Σ indicates a sum. For example, Σ (from r=1 to n) of r = 1 + 2 + 3 + ... + n.
- An increasing sequence has each term greater than the previous; a decreasing sequence has each term smaller. A periodic sequence repeats its values at fixed intervals.
- The binomial expansion of (a + b)ⁿ for positive integer n is Σ (from r=0 to n) of ⁿCᵣ aⁿ⁻ʳ bʳ, where ⁿCᵣ = n!/(r!(n−r)!).
- The binomial coefficients ⁿCᵣ form the rows of Pascal's triangle, where each entry is the sum of the two entries above it.
- For the binomial expansion of (1 + x)ⁿ with non-integer or negative n, the series is valid only when |x| < 1.
- A recurrence relation defines each term using previous terms, e.g. uₙ₊₁ = 2uₙ + 1 with a given first term u₁.
- The Pythagorean identity is sin²θ + cos²θ = 1.
- The tangent identity is tanθ = sinθ / cosθ.
- The sine rule states a/sinA = b/sinB = c/sinC for any triangle with sides a, b, c opposite angles A, B, C.
- The cosine rule states a² = b² + c² − 2bc·cosA, used when given two sides and the included angle, or three sides.
- The area of a triangle using two sides and the included angle is (1/2)ab·sinC.
- Radians measure angles using arc length: 2π radians = 360°, so π radians = 180° and 1 radian ≈ 57.3°.
- The arc length of a sector is s = rθ and its area is A = (1/2)r²θ, where θ is measured in radians.