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A-Level Further Maths Flashcards: Pure Mathematics

A-Level Further Maths — Pure Mathematics Flashcards

20 flashcards with spaced repetition. Press Space to flip, then rate your recall (1—4).

What These Flashcards Cover

These flashcards cover the advanced pure mathematics content unique to Further Maths. You will work with complex numbers, matrices, hyperbolic functions, and more sophisticated techniques for solving differential equations.

Key areas:

  • Complex Numbers: i2=−1i^{2} = -1. A complex number z=x+iyz = x + iy has modulus ∣z∣=x2+y2|z| = \sqrt{x^{2} + y^{2}} and argument arg⁡(z)=tan⁡−1(yx)\arg(z) = \tan^{-1}\left(\frac{y}{x}\right). Polar form: z=r(cos⁡θ+isin⁡θ)=reiθz = r(\cos\theta + i\sin\theta) = re^{i\theta}. De Moivre’s theorem: zn=rn(cos⁡nθ+isin⁡nθ)z^{n} = r^{n}(\cos n\theta + i\sin n\theta).
  • Argand Diagrams: Represent complex numbers as points on a plane with real axis (horizontal) and imaginary axis (vertical). The modulus is the distance from the origin; the argument is the angle from the positive real axis.
  • Roots of Unity: The nnth roots of unity are e2πik/ne^{2\pi ik/n} for k=0,1,…,n−1k = 0, 1, \ldots, n-1. They lie on a circle of radius 1 in the Argand diagram, equally spaced at angles of 2πn\frac{2\pi}{n}.
  • Matrices: A matrix is a rectangular array of numbers. Matrix addition and multiplication follow specific rules. Determinant of a 2×22 \times 2: det⁡(abcd)=ad−bc\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc. Inverse: 1ad−bc(d−b−ca)\frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}. A matrix is singular if its determinant is zero.
  • Proof by Induction: To prove P(n)P(n) for all n≥1n \geq 1: (1) Show P(1)P(1) is true (base case). (2) Assume P(k)P(k) is true (inductive hypothesis). (3) Show P(k+1)P(k+1) is true using the hypothesis. Used for series summation, divisibility, and matrix results.
  • Hyperbolic Functions: cosh⁡x=ex+e−x2\cosh x = \frac{e^{x} + e^{-x}}{2}, sinh⁡x=ex−e−x2\sinh x = \frac{e^{x} - e^{-x}}{2}, tanh⁡x=sinh⁡xcosh⁡x\tanh x = \frac{\sinh x}{\cosh x}. Identity: cosh⁡2x−sinh⁡2x=1\cosh^{2}x - \sinh^{2}x = 1. Derivatives: ddxsinh⁡x=cosh⁡x\frac{d}{dx}\sinh x = \cosh x, ddxcosh⁡x=sinh⁡x\frac{d}{dx}\cosh x = \sinh x.
  • Differential Equations (Advanced): First-order linear: dydx+P(x)y=Q(x)\frac{dy}{dx} + P(x)y = Q(x), solved using integrating factor μ=e∫P dx\mu = e^{\int P\,dx}. Second-order with constant coefficients: ay′′+by′+cy=0ay'' + by' + cy = 0 solved using auxiliary equation am2+bm+c=0am^{2} + bm + c = 0.
  • Polar Coordinates: (r,θ)(r, \theta) where x=rcos⁡θx = r\cos\theta, y=rsin⁡θy = r\sin\theta. Area in polar coordinates: A=12∫r2 dθA = \frac{1}{2}\int r^{2}\,d\theta. Converting between Cartesian and polar forms.

Intuition

Complex numbers extend the number line into a plane. Where real numbers describe position along a line, complex numbers describe position on a 2D surface (the Argand diagram). Multiplication by ii is a rotation of 90 degrees.

Matrices are a shorthand for systems of linear equations. When you multiply a matrix by a vector, you are transforming that vector — rotating, scaling, shearing, or reflecting it. The determinant measures how much the matrix scales areas.

Hyperbolic functions are the “cousins” of trigonometric functions but based on exponentials rather than circles. Where cos⁡\cos and sin⁡\sin describe circular motion, cosh⁡\cosh and sinh⁡\sinh describe growth and decay.

Common Pitfalls

  1. Argument range. The argument arg⁡(z)\arg(z) should be in (−π,π](-\pi, \pi]. The calculator gives tan⁡−1(y/x)\tan^{-1}(y/x), which may need adjusting by adding or subtracting π\pi depending on the quadrant.
  2. Matrix multiplication is not commutative. AB≠BAAB \neq BA as a rule. Multiplying in the wrong order gives a different (often wrong) answer.
  3. Forgetting the base case in induction. Without showing P(1)P(1) is true, the inductive argument has no foundation. You must always verify the base case.
  4. Sign errors in the auxiliary equation. For ay′′+by′+cy=0ay'' + by' + cy = 0, the auxiliary equation is am2+bm+c=0am^{2} + bm + c = 0. A common error is using am2−bm+c=0am^{2} - bm + c = 0 or forgetting to divide through by aa.

Cross-References