Title: InkMaths / Further Maths: core results
On every note of this deck, unless a note says otherwise:
Footer: Continue on InkMaths · © 2026 InkMaths. All rights reserved.

Front: RECALL
State where the nth roots of a non-zero complex number sit.
Back: All n of them lie on a circle of radius the nth root of the modulus, equally spaced 2π/n apart. Find one, then rotate. Joining them gives a regular n-sided polygon centred on the origin, and for n greater than 1 the roots always sum to zero. Zero is the exception. zⁿ = 0 has the single root z = 0, of multiplicity n, and no polygon.

Front: RECALL
State which matrix acts first in a product.
Back: In AB the right-hand matrix acts first, so AB means do B then A. Matrix multiplication does not commute, and a rotation followed by a reflection is not the same map as the reverse. Write the composite in the order the question describes the transformations, then reverse it.

Front: STANDARD RESULT · LEARN IT
Cubic root relations:
az^3 + bz^2 + cz + d = 0: Σα = ?, Σαβ = ?, αβγ = ?\(az^{3} + bz^{2} + cz + d = 0: Σα = ?, Σαβ = ?, αβγ = ?\)az^3 + bz^2 + cz + d = 0: Σα = ?, Σαβ = ?, αβγ = ?
Back: -ba, ca, -da\(-\frac{b}{a}, \frac{c}{a}, -\frac{d}{a}\)-(b/a), (c/a), - d/a

Front: STANDARD RESULT · LEARN IT
Hyperbolic identity:
cosh^2 x - sinh^2 x = ?\(\text{cosh}^{2} x - \text{sinh}^{2} x = ?\)cosh^2 x - sinh^2 x = ?
Back: cosh^2 x - sinh^2 x = 1\(\text{cosh}^{2} x - \text{sinh}^{2} x = 1\)cosh^2 x - sinh^2 x = 1

Front: STANDARD RESULT · LEARN IT
Multiplying complex numbers:
|zw| = ?, arg(zw) = ?\(|zw| = ?, \text{arg}(zw) = ?\)|zw| = ?, arg(zw) = ?
Back: |z||w|, arg z + arg w, then add or take 2π to land in (-π, π]\(|z||w|, \text{arg } z + \text{arg } w, \text{ then add or take 2π to land in } (-π, π]\)|z||w|, arg z + arg w, then add or take 2π to land in (-π, π]

Front: RECALL
State the conjugate root theorem and what it is for.
Back: A polynomial with real coefficients has its non-real roots in conjugate pairs. Given one root you get a second free, and the two multiply to a real quadratic factor you can divide out. A question that hands you 2 + 3i is handing you z² − 4z + 13.

Front: RECALL
State how to sum a series that does not start at r = 1.
Back: The booklet gives Σr, Σr² and Σr³ from 1 to n and nothing else. For a sum from n + 1 to 2n, take the sum to 2n and subtract the sum to n. Split the terms first, and remember Σ of a constant k over n terms is nk.

Front: STANDARD RESULT · LEARN IT
Volume of revolution:
V = ? (about the x-axis)\(V = ? \text{(about the x-axis)}\)V = ? (about the x-axis)
Back: V = π ∫ y^2 dx\(V = π \text{∫} y^{2} \, dx\)V = π ∫ y^2 dx

Front: STANDARD RESULT · LEARN IT
Second order equations:
ay'' + by' + cy = 0: auxiliary equation?\(ay'' + by' + cy = 0: \text{auxiliary equation?}\)ay′′ + by′ + cy = 0: auxiliary equation?
Back: am^2 + bm + c = 0\(am^{2} + bm + c = 0\)am^2 + bm + c = 0

Front: STANDARD RESULT · LEARN IT
cosh in terms of e:
cosh x = ?\(\text{cosh } x = ?\)cosh x = ?
Back: cosh x = e^x + e^-x2\(\text{cosh } x = \frac{e^{x} + e^{-x}}{2}\)cosh x = (e^x + e^(-x))/2

Front: RECALL
State which particular integral to try.
Back: Match the shape of the right-hand side. A constant wants λ, a linear term λx + μ, a quadratic λx² + μx + ν, ke^px\(ke^{px}\)ke^px wants λe^px\(λe^{px}\)λe^px, and a sine or cosine wants λ cos ωx + μ sin ωx with both terms there. If your trial already solves the homogeneous equation, multiply by x and try again.

Front: STANDARD RESULT · LEARN IT
2 × 2 determinant and inverse:
rows (a, b), (c, d): det = ?, inverse = ?\(\text{rows } (a, b), (c, d): \text{det} = ?, \text{inverse} = ?\)rows (a, b), (c, d): det = ?, inverse = ?
Back: ad − bc, and for det ≠ 0 1ad − bc times rows (d, −b), (−c, a)\(ad − bc, \text{ and for det ≠ 0 } \frac{1}{ad − bc} \text{ times rows } (d, −b), (−c, a)\)ad − bc, and for det ≠ 0 (1/(ad − bc)) times rows (d, − b), (−c, a)

Front: STANDARD RESULT · LEARN IT
Logarithmic form:
arsinh x = ?\(\text{arsinh } x = ?\)arsinh x = ?
Back: arsinh x = ln(x + √x^2 + 1)\(\text{arsinh } x = \text{ln}(x + \sqrt{x^{2} + 1})\)arsinh x = ln(x + √(x^2 + 1))

Front: STANDARD RESULT · LEARN IT
De Moivre's theorem:
(cos θ + i sin θ)^n = ?\((\cos θ + \text{i} \sin θ)^{n} = ?\)(cos θ + i sin θ)^n = ?
Back: (cos θ + i sin θ)^n = cos nθ + i sin nθ\((\cos θ + \text{i} \sin θ)^{n} = \cos nθ + \text{i} \sin nθ\)(cos θ + i sin θ)^n = cos nθ + i sin nθ

Front: STANDARD RESULT · LEARN IT
Polar area:
A = ?\(A = ?\)A = ?
Back: A = 12 ∫ r^2 dθ\(A = \frac{1}{2} \text{∫} r^{2} \, dθ\)A = (1/2) ∫ r^2 dθ

Front: STANDARD RESULT · LEARN IT
Integrating factor:
dydx + P(x)y = Q(x): IF = ?\(\frac{dy}{dx} + P(x)y = Q(x): \text{IF} = ?\)dy/dx + P(x)y = Q(x): IF = ?
Back: IF = e^∫ P dx\(\text{IF} = e^{\text{∫} P \, dx}\)IF = e^(∫ P dx)

Front: RECALL
State what the determinant of a transformation matrix tells you.
Back: Its size is the area scale factor in two dimensions and the volume scale factor in three. A negative determinant means the image has been reflected. A determinant of zero collapses the plane onto a line, so the matrix is singular and no inverse exists.

Front: RECALL
State how to divide by a complex number.
Back: Multiply top and bottom by the conjugate of the denominator. (c + di)(c − di) = c² + d², a real number, so the bottom clears and you are left with a real part and an imaginary part you can read off. Never split a fraction with a complex denominator term by term.

Front: RECALL
State the two ways to get a Maclaurin series.
Back: Either differentiate repeatedly and evaluate every derivative at x = 0, or take a standard series from the booklet and substitute. Substituting is faster and safer, but the validity condition moves with the substitution, so ln(1 + 2x) inherits the window of ln(1 + x) with 2x in place of x: −½ < x ≤ ½, right-hand end still included.