Download A-Level Core Formula Sheet and more Cheat Sheet Mathematical Statistics in PDF only on Docsity! MATHEMATICS & STATISTICS/ A-LEVEL CORE FORMULA SHEET Quadratic Equation ax2 + bx+ c = 0 ⇒ x = −b± √ b2 − 4ac 2a Logs and Exponentials y = bx ⇔ x = logb(y), for b, y > 0 logb(p) + logb(q) = logb(pq) logb(p)− logb(q) = logb(p/q) logb(p k) = k logb(p) blogb(x) = x, logb(b x) = x ln(x) = loge(x), eln(x) = ln(ex) = x Odd and Even Functions f(−x) = −f(x) ⇔ f(x) is odd f(−x) = f(x) ⇔ f(x) is even Straight Lines Line with gradient m through (x1, y1) has equation (y − y1) = m(x− x1) Lines with gradients m1 and m2 are perpendicular if m1m2 = −1 Circles Circle with centre C(a, b) and radius r has equation (x− a)2 + (y − b)2 = r2 Algebra, Geometry and Functions Arithmetic Sequences For an arithmetic sequence with first term a, last term l, and common difference d: nth term = un = a+ (n− 1)d Sum to n terms = Sn = 1 2n(2a+ (n− 1)d) = 1 2n(a+ l) Geometric Sequences For a geometric sequence with first term a and common ratio r: nth term = un = arn−1 Sum to n terms = Sn = a(1− rn) 1− r , for r 6= 1 Sum to infinity = S∞ = a 1− r , for |r| < 1 Binomial Series Binomial coefficient is nCr = ( n r ) = n! r!(n− r)! For n ∈ N, (a+ b)n = an + nC1a n−1b+ nC2a n−2b2 + . . . + nCra n−rbr + . . .+ bn For n ∈ R and |b| < |a|, (a+ b)n = an + nan−1b+ n(n− 1) 2! an−2b2 + . . . + n(n− 1) · · · (n− r + 1) r! an−rbr + . . . Sequences and Series Trapezium Rule∫ b a y dx ≈ h 2 (y0 + yn) + h(y1 + y2 + . . .+ yn−1), where h = b− a n , xk = a+ kh, xn = b, and yk = f(xk) Newton-Raphson Iteration To solve f(x) = 0, use xn+1 = xn − f(xn) f ′(xn) Numerical Methods Radians 2π radians = 360◦ For a sector of angle θ radians in a circle of radius r: Arc length = s = θr Sector area = A = 1 2θr 2 Triangles Sine rule: a sin(A) = b sin(B) = c sin(C) Cosine rule: a2 = b2 + c2 − 2bc cos(A) Area = 1 2ab sin(C) Trig Identities Pythagorean Identities: sin2(θ) + cos2(θ) ≡ 1 tan2(θ) + 1 ≡ sec2(θ) 1 + cot2(θ) ≡ cosec2(θ) Sum/Difference Identities sin(a± b) = sin(a) cos(b)± sin(b) cos(a) cos(a± b) = cos(a) cos(b)∓ sin(a) sin(b) tan(a± b) = tan(a)± tan(b) 1∓ tan(a) tan(b) Double Angle Formulae sin(2θ) = 2 sin(θ) cos(θ) cos(2θ) = cos2(θ)− sin2(θ) tan(2θ) = 2 tan(θ) 1− 2 tan(θ) Small Angle Approximations When θ (in radians) is small: sin(θ) ≈ θ, cos(θ) ≈ 1− 1 2θ 2, tan(θ) ≈ θ Trigonometry 1
