Read and memorise these before tackling the exam questions.
A complex number $z = a + bi$ has modulus $|z|=\sqrt{a^2+b^2}$ and argument $\arg(z)=\arctan(b/a)$ (adjusted for quadrant).
Key formula: $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$
▸ EXAMPLE
$z = -1+\sqrt{3}i$ → $|z|=2$, $\arg(z)=\tfrac{2\pi}{3}$
⚡
De Moivre's Theorem
Topic 1
For any integer $n$:
$$(\cos\theta+i\sin\theta)^n = \cos(n\theta)+i\sin(n\theta)$$
Shorthand: $(e^{i\theta})^n = e^{in\theta}$
▸ EXAMPLE
$\left(\cos\tfrac{\pi}{6}+i\sin\tfrac{\pi}{6}\right)^6 = \cos\pi+i\sin\pi = -1$
$$(a+b)^n = \sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$$
General term: $T_{r+1}=\binom{n}{r}a^{n-r}b^r$
Find term with $x^k$: set power of $x$ equal to $k$, solve for $r$.
▸ EXAMPLE
$(2x-x^{-1})^6$: term with $x^2$ has $r=2$, coeff $=240$
∀
Proof by Induction
Topic 1
3 steps: (1) Base case ($n=1$). (2) Assume true for $n=k$. (3) Prove true for $n=k+1$ using the assumption.
Classic: $\sum_{j=1}^{n}(2j-1)=n^2$
▸ INDUCTIVE STEP
$k^2+(2k+1)=(k+1)^2$ ✓
→
Vectors & Cross Product
Topic 3
Dot product: $\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta$.
Cross product $\mathbf{a}\times\mathbf{b}$: perpendicular to both, $|\mathbf{a}\times\mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin\theta$.
Angle formula: $\cos\theta=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$
▸ EXAMPLE
$\mathbf{a}=(1,1,0)$, $\mathbf{b}=(1,0,1)$: $\cos\theta=\tfrac{1}{2}$, $\theta=60°$
▦
Matrices & Determinants
Topic 1
For $A=\bigl[\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\bigr]$: $\det(A)=ad-bc$.
$A^{-1}=\dfrac{1}{\det A}\bigl[\begin{smallmatrix}d&-b\\-c&a\end{smallmatrix}\bigr]$ (if $\det A\neq 0$)
Singular matrix: $\det A = 0$ ⇒ no inverse
▸ EXAMPLE
$A=\bigl[\begin{smallmatrix}2&1\\5&3\end{smallmatrix}\bigr]$: $\det=1$, $A^{-1}=\bigl[\begin{smallmatrix}3&-1\\-5&2\end{smallmatrix}\bigr]$
∫
Calculus: IBP & Related Rates
Topic 5
Integration by parts: $\int u\,dv = uv - \int v\,du$.
Related rates: chain rule $\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}$.
Key result: $\int xe^x\,dx = (x-1)e^x+C$
▸ EXAMPLE
$V=\tfrac{4}{3}\pi r^3$, $\tfrac{dV}{dt}=4\pi r^2\tfrac{dr}{dt}$
$$e^x=1+x+\tfrac{x^2}{2!}+\tfrac{x^3}{3!}+\cdots$$
$$\sin x=x-\tfrac{x^3}{3!}+\tfrac{x^5}{5!}-\cdots$$
Substitute: for $e^{2x}$, replace $x$ with $2x$.
▸ EXAMPLE
$e^{2x}$: coeff of $x^3$ is $\dfrac{(2)^3}{3!}=\dfrac{4}{3}$
Σ
Sequences & Series
Topic 1
Geometric series sum to infinity: $S_\infty=\dfrac{a}{1-r}$, $|r|<1$.
Condition for convergence: $|r|<1$.
Also: $S_n=\dfrac{a(1-r^n)}{1-r}$
▸ EXAMPLE
$4+\tfrac{4}{3}+\tfrac{4}{9}+\cdots$: $r=\tfrac{1}{3}$, $S_\infty=6$
🎲
Probability Distributions
Topic 4
Binomial: $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.
Normal: standardise with $Z=\dfrac{X-\mu}{\sigma}$.
68-95-99.7 rule: $P(\mu\pm\sigma)\approx0.6827$
▸ EXAMPLE
$X\sim B(8,0.3)$: $P(X=3)=\binom{8}{3}(0.3)^3(0.7)^5\approx 0.2541$