IB Mathematics — Analysis & Approaches

Essential 20 Questions

Higher Level · All Topics · Exam-Style Practice

T1 Sequences & Series
Key Formulas
  • Arithmetic: \(u_n = a + (n-1)d\),   \(S_n = \dfrac{n}{2}(2a + (n-1)d)\)
  • Geometric: \(u_n = ar^{n-1}\),   \(S_n = \dfrac{a(r^n-1)}{r-1}\)
  • Infinite GP (|r|<1): \(S_\infty = \dfrac{a}{1-r}\)
Quick Example
Find the sum of the AP: 2, 5, 8, …, 50
a=2, d=3, n=17 → S = 17/2 × (4+48) = 442
T2 Functions & Inverses
Key Facts
  • To find \(f^{-1}(x)\): swap \(x \leftrightarrow y\), solve for \(y\)
  • Domain of \(f^{-1}\) = Range of \(f\)
  • Composite: \((f \circ g)(x) = f(g(x))\)
Quick Example
If \(f(x) = \dfrac{3x+1}{x-2}\), find \(f^{-1}(x)\).
Swap, solve: \(f^{-1}(x) = \dfrac{2x+1}{x-3}\)
T3 Trigonometry
Key Formulas
  • Unit circle exact values: sin30°=½, sin45°=√2/2, sin60°=√3/2
  • Double angle: \(\sin 2\theta = 2\sin\theta\cos\theta\)
  • Pythagorean: \(\sin^2\theta + \cos^2\theta = 1\)
Quick Example
\(\sin\!\left(\dfrac{5\pi}{6}\right) = \sin\!\left(\pi - \dfrac{\pi}{6}\right) = \sin\dfrac{\pi}{6}\)
= 1/2
T4 Exponentials & Logarithms
Key Laws
  • \(\log_a(xy) = \log_a x + \log_a y\)
  • \(\log_a(x^n) = n\log_a x\)
  • Change of base: \(\log_a b = \dfrac{\ln b}{\ln a}\)
Quick Example
\(\log_2 27 \cdot \log_3 4\) using change of base chain rule
= \(\dfrac{3\ln 3}{\ln 2} \cdot \dfrac{2\ln 2}{\ln 3}\) = 6
T5 Differentiation & Integration
Core Rules
  • Chain rule: \(\dfrac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)\)
  • Product rule: \((uv)' = u'v + uv'\)
  • \(\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C\), \(n \ne -1\)
Quick Example
Differentiate \(y = \sin(3x^2)\)
\(\dfrac{dy}{dx} = 6x\cos(3x^2)\)
T6 Vectors
Key Formulas
  • Dot product: \(\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta\)
  • Perpendicular iff \(\mathbf{a}\cdot\mathbf{b} = 0\)
  • Cross product magnitude: \(|\mathbf{a}\times\mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta\)
T7 Complex Numbers
Key Formulas
  • Polar form: \(z = r(\cos\theta + i\sin\theta) = re^{i\theta}\)
  • De Moivre: \(z^n = r^n(\cos n\theta + i\sin n\theta)\)
  • Modulus: \(|z| = \sqrt{a^2+b^2}\)
Quick Example
\((1+i\sqrt{3})^6\): r=2, θ=π/3
\(r^6 = 64\), \(6\theta = 2\pi\) → result = 64
T8 Probability & Statistics
Key Formulas
  • Conditional: \(P(A|B) = \dfrac{P(A\cap B)}{P(B)}\)
  • Normal: standardise \(Z = \dfrac{X - \mu}{\sigma}\)
  • Binomial: \(P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}\)
40:00
0 / 20 answered
0 ✓