A Level Further Maths formulas

Further Pure Mathematics 1

Roots of polynomial equations

• Quadratic ax² + bx + c = 0: α + β = −b/a and αβ = c/a
• Cubic ax³ + bx² + cx + d = 0: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a
• Quartic ax⁴ + bx³ + cx² + dx + e = 0: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a
• Sum of squares: Σα² = (Σα)² − 2Σαβ
• Sum of reciprocals for a cubic: 1/α + 1/β + 1/γ = (αβ + βγ + γα)/(αβγ)
• New roots 1/α: put x = 1/y. New roots α²: put x = √y. New roots α + k: put x = y − k. New roots kα: put x = y/k
• Each root satisfies the equation, so for x³ + px² + qx + r = 0: Σα³ = −pΣα² − qΣα − 3r

Rational functions and graphs

• Vertical asymptote: x = a where the denominator is zero at x = a (and the numerator is not)
• Horizontal asymptote: y = 0 if the top has lower degree; y = (ratio of leading coefficients) if the degrees are equal
• Oblique asymptote: if the top has degree one more than the bottom, divide; y = quotient is the asymptote
• Set of values of y: rearrange to a quadratic in x and solve b² − 4ac ≥ 0 as an inequality in y
• y = 1/f(x): zeros of f become vertical asymptotes, a maximum of f becomes a minimum, and the sign of y is unchanged
• y² = f(x): exists only where f(x) ≥ 0, is y = ±√f(x), and is symmetrical about the x-axis
• y = |f(x)|: reflect the part below the x-axis in the x-axis. y = f(|x|): keep the part for x ≥ 0 and reflect it in the y-axis

Summation of series

• Σ_r=1ⁿ r = n(n + 1)/2
• Σ_r=1ⁿ r² = n(n + 1)(2n + 1)/6
• Σ_r=1ⁿ r³ = n²(n + 1)²/4
• Σ_r=1ⁿ 1 = n, so a constant c sums to cn, not c
• Sum from r = m + 1 to n = (sum from 1 to n) − (sum from 1 to m)
• Method of differences: if u_r = f(r) − f(r + 1), then Σ_r=1ⁿ u_r = f(1) − f(n + 1)
• Sum to infinity = limit of S_n as n → ∞, if this limit is finite; terms such as 1/n and 1/(n + 2) tend to 0

Matrices

• AB exists only if (number of columns of A) = (number of rows of B). In general AB ≠ BA, but AI = IA = A
• For M with rows (a, b) and (c, d): det M = ad − bc, and M⁻¹ = [1/(ad − bc)] × the matrix with rows (d, −b) and (−c, a)
• Singular means det M = 0 (no inverse). Also det(AB) = det A × det B
• (AB)⁻¹ = B⁻¹A⁻¹ and (ABC)⁻¹ = C⁻¹B⁻¹A⁻¹
• Rotation through θ anticlockwise about the origin: rows (cos θ, −sin θ) and (sin θ, cos θ). A⁻¹ is the transformation that undoes A
• AB means do B first, then A
• Area of image = |det M| × area of object. A negative determinant means the shape has also been turned over

Polar coordinates

• x = r cos θ and y = r sin θ
• r² = x² + y² and tan θ = y/x (use a sketch to choose the correct quadrant for θ)
• Area of a sector = ½ ∫ r² dθ between θ = α and θ = β
• To integrate squares: cos² θ = ½(1 + cos 2θ) and sin² θ = ½(1 − cos 2θ)
• If r depends on cos θ only, the curve is symmetrical about the initial line; if on sin θ only, about the line θ = π/2
• The curve passes through the pole when r = 0; those values of θ give the tangents at the pole
• r = a is a circle with centre the pole; θ = α is a half-line from the pole; r = 2a cos θ is a circle of diameter 2a through the pole

Vectors

• Plane: r·n = p is the same as ax + by + cz = d with n = (a, b, c) and d = a·n for any point a on the plane. For r = a + λb + μc the normal is b × c
• a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁), and |a × b| = |a||b| sin θ
• Line with direction d and plane with normal n: if d·n ≠ 0 they meet at one point; if d·n = 0 the line is parallel to the plane, or lies in it if one of its points is on the plane
• Angle between a line and a plane: sin φ = |d·n|/(|d||n|). Angle between two planes: cos φ = |n₁·n₂|/(|n₁||n₂|)
• Distance from (x₁, y₁, z₁) to ax + by + cz = d is |ax₁ + by₁ + cz₁ − d|/√(a² + b² + c²)
• The line of intersection of two planes has direction n₁ × n₂
• Skew lines r = a + sb and r = c + td: common perpendicular has direction b × d, and shortest distance = |(c − a)·(b × d)|/|b × d|

Proof by induction

• Four parts: (1) show true for n = 1; (2) assume true for n = k; (3) show true for n = k + 1 using the assumption; (4) write the conclusion
• Sums: (sum to k + 1 terms) = (assumed sum to k terms) + (the (k + 1)th term), then factorise to the required form
• Divisibility: write f(k + 1) in terms of f(k), or show that f(k + 1) − m × f(k) is a multiple of the divisor for a suitable number m
• Matrix powers: M^k+1 = M^k × M, with the assumed form put in for M^k
• Sequences: put the assumed formula for u_k into the recurrence relation to get u_k+1
• If the result starts at n = N, the base case is n = N, not n = 1
• Conjecture questions: calculate cases n = 1, 2, 3, guess the formula, then prove it

Further Pure Mathematics 2

Hyperbolic functions

• sinh x = (e^x − e^−x)/2, cosh x = (e^x + e^−x)/2, tanh x = sinh x/cosh x = (e^2x − 1)/(e^2x + 1)
• sech x = 1/cosh x, cosech x = 1/sinh x, coth x = 1/tanh x
• cosh² x − sinh² x = 1, 1 − tanh² x = sech² x, coth² x − 1 = cosech² x
• sinh 2x = 2 sinh x cosh x; cosh 2x = cosh² x + sinh² x = 2cosh² x − 1 = 1 + 2sinh² x
• sinh⁻¹ x = ln(x + √(x² + 1)) for all x; cosh⁻¹ x = ln(x + √(x² − 1)) for x ≥ 1
• tanh⁻¹ x = ½ ln((1 + x)/(1 − x)) for −1 < x < 1
• cosh x ≥ 1 and is even; sinh x is odd; y = tanh x has asymptotes y = 1 and y = −1

Matrices

• det A ≠ 0: unique solution x = A⁻¹b, three planes meet at one point
• det A = 0: no unique solution. Consistent means infinitely many solutions (planes meet in a line, or are the same plane); inconsistent means no common point (parallel planes or a triangular prism)
• Eigenvalues: solve the characteristic equation det(A − λI) = 0. Eigenvectors: solve (A − λI)e = 0 for each λ
• For a 2 × 2 matrix: λ² − (trace)λ + det A = 0, so sum of eigenvalues = trace (a + d) and product = det A
• If Ae = λe then Aⁿe = λⁿe, and A⁻¹e = (1/λ)e when A is non-singular
• A = QDQ⁻¹ and Aⁿ = QDⁿQ⁻¹; the columns of Q are eigenvectors in the same order as the eigenvalues on the diagonal of D
• A satisfies its own characteristic equation: for 2 × 2, A² − (trace)A + (det A)I = 0

Differentiation

• d/dx: sinh x → cosh x; cosh x → sinh x; tanh x → sech² x; sech x → −sech x tanh x
• d/dx: sin⁻¹ x → 1/√(1 − x²); cos⁻¹ x → −1/√(1 − x²)
• d/dx: sinh⁻¹ x → 1/√(x² + 1); cosh⁻¹ x → 1/√(x² − 1); tanh⁻¹ x → 1/(1 − x²)
• With ax in place of x use the chain rule, e.g. d/dx sinh⁻¹(ax) = a/√(a²x² + 1)
• Parametric: dy/dx = (dy/dt) ÷ (dx/dt), and d²y/dx² = [d/dt (dy/dx)] ÷ (dx/dt)
• Implicit: differentiate the whole equation twice with respect to x, using d/dx(y²) = 2y(dy/dx), then substitute the point and the value of dy/dx
• Maclaurin: f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + …

Integration

• ∫ sinh x dx = cosh x + c; ∫ cosh x dx = sinh x + c; ∫ sech² x dx = tanh x + c
• ∫ 1/√(a² − x²) dx = sin⁻¹(x/a) + c; ∫ 1/(x² + a²) dx = (1/a) tan⁻¹(x/a) + c
• ∫ 1/√(x² + a²) dx = sinh⁻¹(x/a) + c; ∫ 1/√(x² − a²) dx = cosh⁻¹(x/a) + c for x > a
• Arc length: ∫ √(1 + (dy/dx)²) dx, or ∫ √((dx/dt)² + (dy/dt)²) dt, or in polars ∫ √(r² + (dr/dθ)²) dθ
• Surface area about the x-axis: 2π ∫ y √(1 + (dy/dx)²) dx; about the y-axis: 2π ∫ x √(1 + (dx/dy)²) dy
• Rectangles of unit width, f decreasing: f(2) + f(3) + … + f(n) < ∫₁ⁿ f(x) dx < f(1) + f(2) + … + f(n − 1)
• Limit of a sum: as n → ∞, (1/n) Σ_r=1ⁿ f(r/n) → ∫₀¹ f(x) dx

Complex numbers

• de Moivre: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)
• Multiply: multiply the moduli and add the arguments. Divide: divide the moduli and subtract the arguments
• Multiple angles: expand (cos θ + i sin θ)ⁿ by the binomial theorem; the real part is cos nθ and the imaginary part is sin nθ
• If z = cos θ + i sin θ: zⁿ + z⁻ⁿ = 2 cos nθ and zⁿ − z⁻ⁿ = 2i sin nθ; so (2 cos θ)ⁿ = (z + 1/z)ⁿ and (2i sin θ)ⁿ = (z − 1/z)ⁿ
• nth roots of unity: z = e^2πki/n for k = 0, 1, …, n − 1; their sum is 0 (for n ≥ 2)
• nth roots of re^iα: z = r^1/n e^{i(α + 2πk)/n} for k = 0, 1, …, n − 1
• C + iS method: C + iS is a geometric or binomial series in e^iθ; sum it, then take the real part for C and the imaginary part for S

Differential equations

• For dy/dx + Py = Q (coefficient of dy/dx must be 1): integrating factor I = e^{∫P dx}, then y × I = ∫(Q × I) dx + C
• General solution = complementary function + particular integral; a second order equation has two arbitrary constants, a first order equation has one
• First order dy/dx + ay = f(x): the CF is Ae^−ax
• Second order ay′′ + by′ + cy = f(x): solve the auxiliary equation am² + bm + c = 0. Distinct real roots m₁, m₂: Ae^m₁x + Be^m₂x. Repeated root m: (A + Bx)e^mx. Complex roots p ± qi: e^px(A cos qx + B sin qx)
• PI trial forms: polynomial of the same degree for a polynomial (include every lower power); ke^bx for ae^bx; λ cos px + μ sin px for a cos px + b sin px (always both terms)
• If the trial form is already part of the CF, multiply it by x (by x² if it matches a repeated root)
• x = e^t gives x dy/dx = dy/dt and x² d²y/dx² = d²y/dt² − dy/dt; y = ux gives dy/dx = u + x du/dx

Further Mechanics

Motion of a projectile

• For initial speed V at angle θ above the horizontal: horizontal velocity = V cos θ (constant); vertical velocity = V sin θ − gt
• Position: x = (V cos θ)t and y = (V sin θ)t − ½gt², with y measured upwards from the point of projection
• Speed at any instant = √(v_x² + v_y²); direction to the horizontal from tan(angle) = v_y ÷ v_x
• Greatest height = (V sin θ)²/(2g), reached when the vertical velocity is zero; the speed there is V cos θ, not zero
• On level ground only: time of flight = (2V sin θ)/g and range = (V² sin 2θ)/g
• Trajectory: y = x tan θ − gx²/(2V² cos² θ), which can be written y = x tan θ − gx²(1 + tan² θ)/(2V²)
• Use g = 10 m s⁻² unless the question gives another value

Equilibrium of a rigid body

• Moment = force × perpendicular distance from the point to the line of action = Fd sin θ, where θ is the angle between the force and the line of length d (unit N m)
• Uniform triangular lamina: centre of mass is 2/3 of the way along each median from the vertex, so 1/3 of the height from each side; its coordinates are the means of the three vertices
• From the list of formulae: solid hemisphere 3r/8 from the flat face; solid cone h/4 from the base; sector of radius r and angle 2α at (2r sin α)/(3α) from the centre; semicircular lamina 4r/(3π) from the straight edge
• Composite body: x̄ = Σ(mx) ÷ Σm and ȳ = Σ(my) ÷ Σm; for a uniform lamina use areas in place of masses, and give a removed part a negative area
• Equilibrium: resultant force = 0 and sum of moments about any point = 0; take moments about a point where an unknown force acts to remove it
• A body hanging freely from a point rests with its centre of mass vertically below that point
• On a plane at angle θ: a body topples when the vertical line through the centre of mass passes outside the base (for a block of base width b and height h, when tan θ > b/h); it slides when tan θ > μ

Circular motion

• Angular speed ω = 2π ÷ (time for one revolution); speed = radius × angular speed (v = rω)
• Acceleration towards the centre = v²/r = rω²; resultant force towards the centre = mv²/r = mrω²
• Conical pendulum, string of length L at angle θ to the vertical: T cos θ = mg, T sin θ = mrω², with r = L sin θ
• Vertical circle, energy: v² = u² − 2gh, where h is the height gained since the speed was u
• String of length r: at the lowest point T − mg = mv²/r; at the highest point T + mg = mv²/r
• Complete circles on a string (or on the inside of a surface): speed at the top must satisfy v² ≥ gr, so speed at the bottom must satisfy u² ≥ 5gr
• Complete circles on a light rod, or a bead on a wire or in a tube: the particle only has to reach the top, so u² > 4gr at the bottom

Hooke's law

• Hooke's law: tension = modulus × extension ÷ natural length (T = λx/l)
• Elastic potential energy = modulus × extension² ÷ (2 × natural length) (EPE = λx²/(2l)), in joules; this also equals ½Tx
• Extension = stretched length − natural length; never put the full length into the formulae
• Particle hanging at rest on a vertical string: λe/l = mg, where e is the extension at equilibrium
• Energy: kinetic energy + gravitational potential energy + elastic potential energy is constant when no other force does work; subtract the work done against friction if the surface is rough
• Speed is greatest at the equilibrium position (acceleration zero); the particle is at rest for an instant at the greatest extension
• A slack string has zero tension and zero elastic potential energy

Linear motion under a variable force

• Newton's second law: resultant force = mass × acceleration (F = ma)
• Acceleration a = dv/dt = v dv/dx, and v = dx/dt
• Force given in terms of t: use m dv/dt = F(t) and integrate with respect to t
• Force given in terms of x: use mv dv/dx = F(x), which gives ½mv² = ∫F dx + C
• Force given in terms of v: use m dv/dt = F(v) for time, or mv dv/dx = F(v) for distance, and separate the variables
• Always add the constant of integration and find it from the initial conditions, or use limits on both integrals
• Terminal (steady) speed is the speed at which the acceleration is zero

Momentum

• Momentum = mass × velocity; total momentum before = total momentum after (m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂), with one direction taken as positive
• Newton's experimental law: speed of separation = e × speed of approach (v₂ − v₁ = e(u₁ − u₂)), with 0 ≤ e ≤ 1
• e = 1: perfectly elastic, kinetic energy conserved. e = 0: inelastic, the spheres have the same velocity after impact
• Direct impact with a fixed surface: rebound speed = e × impact speed; a ball dropped from height h rebounds to height e²h
• Oblique impact with a smooth fixed wall: the component parallel to the wall is unchanged; the component perpendicular to the wall is reversed and multiplied by e
• If the path makes angle α with the wall before impact and β after, tan β = e tan α
• Oblique impact of two smooth spheres: components perpendicular to the line of centres are unchanged; use momentum and Newton's law along the line of centres

Further Probability & Statistics

Continuous random variables

• f(x) ≥ 0 for all x, and the integral of f(x) over the whole range = 1 (use this to find an unknown constant k)
• P(a < X < b) = ∫ f(x) dx from a to b = F(b) − F(a)
• F(x) = P(X ≤ x) = ∫ f(t) dt from the lower end of the range up to x; f(x) = F′(x). F rises from 0 to 1 and never decreases
• Median m: F(m) = 0.5. Lower quartile: F = 0.25. The p% percentile: F = p/100
• E(X) = ∫ x f(x) dx, and in general E(g(X)) = ∫ g(x) f(x) dx; Var(X) = E(X²) − [E(X)]²
• Piecewise PDF: integrate each piece over its own interval, and when building F add the area from all earlier pieces
• For Y = g(X) with g increasing: F_Y(y) = P(X ≤ g⁻¹(y)) = F_X(g⁻¹(y)); differentiate for the PDF and state the new range of y

Inference using normal and t-distributions

• Unbiased variance estimate: s² = Σ(x − x̄)²/(n − 1) = [Σx² − (Σx)²/n]/(n − 1)
• One-sample t-test: t = (x̄ − μ₀)/(s/√n), with ν = n − 1
• Confidence interval for μ: x̄ ± t × s/√n, where t is the table value for ν = n − 1 (for 95% use p = 0.975)
• Pooled estimate: s_p² = [(n₁ − 1)s₁² + (n₂ − 1)s₂²]/(n₁ + n₂ − 2) = [Σ(x − x̄)² + Σ(y − ȳ)²]/(n₁ + n₂ − 2)
• 2-sample t-test: t = (x̄ − ȳ)/[s_p√(1/n₁ + 1/n₂)], with ν = n₁ + n₂ − 2; needs independent samples from normal populations with a common variance
• Paired t-test: find the n differences d, then t = d̄/(s_d/√n) with ν = n − 1; needs the differences to be normally distributed
• Known variances (or large samples): z = (x̄ − ȳ)/√(σ₁²/n₁ + σ₂²/n₂); a confidence interval for a difference is (x̄ − ȳ) ± (table value) × (the same standard error)

χ2-tests

• Test statistic: χ² = Σ(O − E)²/E, summed over all classes or cells; the totals of O and E must be equal
• Goodness of fit: E = total frequency × probability of the class under H₀; the last class is usually 'this value or more', so the probabilities add to 1
• Goodness of fit: ν = (number of classes after combining) − 1 − (number of parameters estimated from the data)
• Contingency table: E = (row total × column total) ÷ grand total, and ν = (rows − 1)(columns − 1)
• If any E is less than 5, combine neighbouring classes (or rows or columns) before working out χ² and ν
• H₀: the distribution fits (or the factors are independent). Reject H₀ if χ² is greater than the critical value

Non-parametric tests

• Sign test: under H₀ the number of values above the median is B(n, ½); find the tail probability and compare it with the significance level. For large n use N(n/2, n/4) with a continuity correction
• Wilcoxon signed-rank: find the differences from the median, rank them by size ignoring sign (1 = smallest), then P = sum of ranks of positive differences and Q = sum for negative ones
• Signed-rank test statistic: T = the smaller of P and Q; check with P + Q = n(n + 1)/2. Reject H₀ if T ≤ the critical value
• Signed-rank, large n: T is approximately N(n(n + 1)/4, n(n + 1)(2n + 1)/24)
• Wilcoxon rank-sum (sample sizes m ≤ n): rank all m + n values together; R_m = sum of the ranks in the smaller sample; W = the smaller of R_m and m(n + m + 1) − R_m. Reject H₀ if W ≤ the critical value
• Rank-sum, large samples: R_m is approximately N(m(n + m + 1)/2, mn(n + m + 1)/12)
• Paired data: sign test or Wilcoxon matched-pairs signed-rank test on the differences. Two independent samples: Wilcoxon rank-sum test. The signed-rank tests need a symmetrical distribution (of the values or of the differences); the rank-sum test needs two distributions of the same shape

Probability generating functions

• G(t) = E(t^X) = Σ P(X = r)t^r; P(X = r) is the coefficient of t^r, and G(1) = 1
• Mean: E(X) = G′(1). Variance: Var(X) = G″(1) + G′(1) − [G′(1)]²
• Binomial B(n, p) with q = 1 − p: G(t) = (q + pt)ⁿ
• Poisson with mean λ: G(t) = e^{λ(t − 1)}
• Geometric (number of trials up to and including the first success): G(t) = pt/(1 − qt)
• Discrete uniform on 1, 2, ..., n: G(t) = (t + t² + ... + tⁿ)/n = t(1 − tⁿ)/[n(1 − t)]
• If X and Y are independent: G_X+Y(t) = G_X(t) × G_Y(t)