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MATHEMATICS
& STATISTICS/
A-LEVEL CORE
FORMULA SHEET
Quadratic Equation
ax^2 + bx + c = 0 ⇒ x = −b ±
b^2 − 4 ac 2 a 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(pk) = k logb(p) blogb(x)^ = x, logb(bx) = 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 (x 1 , y 1 ) has equation (y − y 1 ) = m(x − x 1 ) Lines with gradients m 1 and m 2 are perpendicular if m 1 m 2 = − 1 Circles Circle with centre C(a, b) and radius r has equation (x − a)^2 + (y − b)^2 = r^2
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 = 12 n(2a + (n − 1)d) = 12 n(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^ + nC 1 an−^1 b + nC 2 an−^2 b^2 +...
- nCr an−r^ br^ +... + bn For n ∈ R and |b| < |a|,
(a + b)n^ = an^ + nan−^1 b + n(n − 1) 2! an−^2 b^2 +...
n(n − 1) · · · (n − r + 1) r! an−r^ br^ +...
Sequences and Series
Trapezium Rule ∫ (^) b
a
y dx ≈ h 2 (y 0 + yn) + h(y 1 + y 2 +... + 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 2 π radians = 360◦ For a sector of angle θ radians in a circle of radius r: Arc length = s = θr Sector area = A = 12 θr^2
Triangles
Sine rule: a sin(A)
b sin(B)
c sin(C) Cosine rule: a^2 = b^2 + c^2 − 2 bc cos(A) Area = 12 ab sin(C)
Trig Identities Pythagorean Identities: sin^2 (θ) + cos^2 (θ) ≡ 1 tan^2 (θ) + 1 ≡ sec^2 (θ) 1 + cot^2 (θ) ≡ cosec^2 (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) = cos^2 (θ) − sin^2 (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 − 12 θ^2 , tan(θ) ≈ θ
Trigonometry
Table of Derivatives
Function Derivative xn^ nxn−^1 akx^ k ln(a)akx ekx^ kekx ln(kx) (^1) x sin (kx) k cos (kx) cos (kx) −k sin (kx) tan(kx) k sec^2 (kx) sec(kx) k sec(kx) tan(kx) cosec(kx) −k cosec(kx) cot(kx) cot(kx) −k cosec^2 (kx)
Rules
Differentiation from first principles:
f ′(x) = lim h→ 0
f (x + h) − f (x) h
Chain rule:
y = f (g(x)) ⇒ dy dx = f ′(g(x)) × g′(x)
or y = f (g(x)) = f (u) ⇒ dy dx
dy du
×
du dx Product rule:
y = uv ⇒ dy dx
du dx v + u dv dx
Calculus Quotient rule:
y = u v
dy dx
du dx v − u dv dx v^2 Inverse rule: 1
dy dx = dx dy Parametric rule: For a curve given by x = f 1 (t) and y = f 2 (t), dy dx
dy/dt dx/dt Table of Integrals Function Integral (remember to add +c!) xn^ x n+ n+1 (n^6 =^ −1) x−^1 = (^1) x ln(x) ekx^1 k ekx sin (kx) − (^1) k cos (kx) cos (kx) (^1) k sin (kx) tan(kx) (^1) k ln | sec (kx)| sec^2 (kx) (^1) k tan(kx) cosec^2 (kx) − (^1) k cot(kx) cot(kx) (^1) k ln | sin(kx)| f ′(x) f (x) ln^ |f^ (x)| Rules Definite integration:
∫ (^) b
a
f (x) dx = F (b) − F (a), where F (x) is the integral of f (x)
Integration by parts:
u dv dx dx = uv −
du dx v dx
Forces Weight = mg Friction: F ≤ μR Newton’s 2nd law: F = ma
Mechanics
Kinematics For 1D motion with constant acceleration: v = u + at s = ut + 12 at^2 s = vt − 12 at^2 s = 12 (u + v)t v^2 = u^2 + 2as For 1D motion with variable acceleration: r (position), v, and a are all functions of t; the above suvat equations no longer apply, so use
v = dr dt , a = dv dt , r =
v dt, v =
a dt
Probability P (A ∪ B) = P (A) + P (B) − P (A ∩ B) P (A|B) =
P (A ∩ B)
P (B)
, for P (B) 6 = 0
Summary Statistics
Mean = ¯x = Σxi n =
Σfixi Σfi^ ,^ Variance =^ σ
2
Standard dev. = σ =
Σx^2 i n −^ ¯x
Σfix^2 i Σfi^ −^ x¯
2
Outliers are any data outside of the interval ¯x ± 2 σ or Q 1 − 1 .5IQR and Q 3 + 1.5IQR Binomial Distribution If X ∼ B(n, p), then P (X = r) = nCr pr^ (1 − p)n−r Mean of X = np, variance of X = np(1 − p) Normal Distribution If X ∼ N (μ, σ^2 ), then Z = X^ −^ μ σ with Z ∼ N (0, 1) Hypothesis test for the mean: if X ∼ N (μ, σ^2 ), then
X¯ ∼ N
μ, σ^2 n
and X¯ − μ σ/
n ∼^ N^ (0,^ 1)
Statistics
