Concept Review & Key Formulas
Study these before attempting the questions. All formulas below appear in the IB Formula Booklet unless marked ★ (must memorise).
Topic 1 — Sequences & Series
ALGEBRA
Arithmetic Sequence
$u_n = u_1 + (n-1)d$ | $S_n = \dfrac{n}{2}(2u_1 + (n-1)d)$
Geometric Sequence
$u_n = u_1 \cdot r^{n-1}$ | $S_n = \dfrac{u_1(r^n - 1)}{r-1},\; r \neq 1$
- $d$ = common difference (AP); $r$ = common ratio (GP)
- ★ Infinite GP sum: $S_\infty = \dfrac{u_1}{1-r}$, valid only when $|r| < 1$
Worked Example
AP: $u_1=3,\; d=4$. Find $S_{10}$.$S_{10} = \dfrac{10}{2}(2 \cdot 3 + 9 \cdot 4) = 5 \times 42 = \mathbf{210}$
Topic 2 — Exponents, Logarithms & Binomial Theorem
ALGEBRA
Log Laws ★ (memorise)
$\log(ab)=\log a+\log b$ | $\log\!\left(\dfrac{a}{b}\right)=\log a-\log b$ | $\log a^n = n\log a$
Binomial Theorem
$(a+b)^n = \displaystyle\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$
Worked Example
Coefficient of $x^3$ in $(2x+3)^5$: $\binom{5}{3}(2x)^3(3)^2 = 10 \cdot 8 \cdot 9 = \mathbf{720}$
Topic 3 — Functions
FUNCTIONS
- ★ Inverse: if $f(x)=y$ then $f^{-1}(y)=x$. Swap $x$ and $y$, solve for $y$.
- Composite: $(f \circ g)(x) = f(g(x))$
- Domain of $f^{-1}$ = range of $f$
Worked Example
$f(x)=2x+3$. Find $f^{-1}(7)$. Let $y=2x+3 \Rightarrow x=\dfrac{y-3}{2}$. So $f^{-1}(7)=\dfrac{7-3}{2}=\mathbf{2}$
Topic 4 — Trigonometry
GEOMETRY & TRIG
Pythagorean Identity ★
$\sin^2\theta + \cos^2\theta = 1$
Exact Values ★
$\sin 30°=\frac{1}{2}$, $\cos 60°=\frac{1}{2}$, $\tan 45°=1$, $\sin 45°=\frac{\sqrt{2}}{2}$, $\tan 60°=\sqrt{3}$
- For $\sin\theta = k$: solutions in $[0°, 360°]$ are $\theta$ and $180°-\theta$
- For $\cos\theta = k$: solutions are $\theta$ and $360°-\theta$
Worked Example
Solve $2\sin x = 1$, $0 \le x \le 2\pi$. $\sin x = \frac{1}{2}$ → $x = \dfrac{\pi}{6}$ or $x = \dfrac{5\pi}{6}$
Topic 5 — Calculus
CALCULUS
Differentiation (Power Rule)
$\dfrac{d}{dx}[x^n] = nx^{n-1}$
Integration (Power Rule)
$\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C, \quad n \neq -1$
- ★ Max/Min: set $f'(x)=0$; check $f''(x)$ — negative = max, positive = min
- ★ Area = $\displaystyle\int_a^b f(x)\,dx$ (must be positive; split if curve crosses $x$-axis)
Worked Example
$f(x)=x^3-3x^2+2$. $f'(x)=3x^2-6x$. $f'(2)=12-12=\mathbf{0}$
Topic 6 — Probability & Statistics
STATISTICS
Addition Rule
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Normal Distribution
$X \sim N(\mu, \sigma^2)$ | Standardise: $Z = \dfrac{X - \mu}{\sigma}$
- ★ Population variance: $\sigma^2 = \dfrac{\sum(x_i - \bar{x})^2}{n}$
- Normal distribution is symmetric about $\mu$
Topic 7 — Vectors & Complex Numbers
ALGEBRA
Dot Product & Angle
$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$ | $\mathbf{a} \cdot \mathbf{b} = a_1b_1+a_2b_2+a_3b_3$
Complex Numbers ★
$i^2 = -1$ | $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$
Worked Example
$(2+3i)(1-2i)=2-4i+3i-6i^2=2-i+6=\mathbf{8-i}$
Practice Questions
Quiz Complete
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📋 Full Answer Key & Worked Solutions