Study Reference
Key Concepts & Formulae
Review before attempting the practice questions below
IB Mathematics AI HL — Core Concepts Master Quiz
Name: _________________________ Date: _____________ Score: _____ / 20
Part I — Concept Review
1
Number & Algebra — Sequences & Series
- Arithmetic sequence: \(u_n = u_1 + (n-1)d\)
- Arithmetic sum: \(S_n = \dfrac{n}{2}(2u_1 + (n-1)d)\)
- Geometric sequence: \(u_n = u_1 \cdot r^{n-1}\)
- Geometric sum: \(S_n = \dfrac{u_1(r^n - 1)}{r - 1}\), \(r \neq 1\)
- Infinite geometric sum (|r|<1): \(S_\infty = \dfrac{u_1}{1-r}\)
Logarithm laws: \(\log(ab)=\log a+\log b\), \(\log\tfrac{a}{b}=\log a - \log b\), \(\log a^n = n\log a\)
Worked Example
Find the 20th term of an arithmetic sequence where \(u_1=3\) and \(d=5\).
→ \(u_{20} = 3 + (20-1)(5) = 3 + 95 = \mathbf{98}\)
2
Functions
- Vertex form of quadratic: \(f(x)=a(x-h)^2+k\), vertex at \((h,k)\)
- To complete the square: \(x^2+bx = (x+\tfrac{b}{2})^2 - \tfrac{b^2}{4}\)
- Inverse function \(f^{-1}\): swap \(x\) and \(y\), solve for \(y\)
- Exponential: \(a^x = b \Rightarrow x = \log_a b\)
Worked Example
Find the vertex of \(f(x)=x^2-6x+13\).
→ \(f(x)=(x-3)^2+4\), vertex = \(\mathbf{(3,4)}\)
3
Trigonometry
- Exact values: \(\sin30°=\tfrac{1}{2}\), \(\cos30°=\tfrac{\sqrt3}{2}\), \(\tan30°=\tfrac{1}{\sqrt3}\)
- \(\sin60°=\tfrac{\sqrt3}{2}\), \(\cos60°=\tfrac{1}{2}\), \(\tan60°=\sqrt3\)
- Area of triangle: \(A=\tfrac{1}{2}ab\sin C\)
- Cosine rule: \(c^2=a^2+b^2-2ab\cos C\)
- Sine rule: \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)
Worked Example
Triangle with \(a=6\), \(b=8\), \(C=30°\). Find the area.
→ \(A=\tfrac{1}{2}(6)(8)\sin30°=24\times0.5=\mathbf{12}\)
4
Differential & Integral Calculus
- Power rule: \(\dfrac{d}{dx}(x^n)=nx^{n-1}\)
- Maximum/minimum: set \(f'(x)=0\), check \(f''(x)\)
- Integral power rule: \(\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1}+C\)
- Chain rule: \(\dfrac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)\)
- Chain rule for rates: \(\dfrac{dV}{dt}=\dfrac{dV}{dr}\cdot\dfrac{dr}{dt}\)
Worked Example
Differentiate \(f(x)=3x^4-2x^3+5x-7\).
→ \(f'(x)=12x^3-6x^2+5\)
5
Statistics & Probability
- Population mean: \(\bar{x}=\dfrac{\sum x_i}{n}\); population variance: \(\sigma^2=\dfrac{\sum(x_i-\bar{x})^2}{n}\)
- Addition rule: \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
- Binomial: \(X\sim B(n,p)\), \(P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}\)
- Normal: \(Z=\dfrac{X-\mu}{\sigma}\), use standard normal table
Worked Example
\(X\sim B(10,0.3)\). Find \(P(X=3)\).
→ \(\binom{10}{3}(0.3)^3(0.7)^7 = 120(0.027)(0.0824) \approx \mathbf{0.2668}\)
6
Matrices & Vectors
- 2×2 determinant: \(\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\)
- Dot product: \(\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2+a_3b_3\)
- If \(\mathbf{a}\cdot\mathbf{b}=0\), vectors are perpendicular
- Magnitude: \(|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}\)
Worked Example
Find \(\det\begin{pmatrix}3&1\\2&4\end{pmatrix}\).
→ \(3(4)-1(2)=12-2=\mathbf{10}\)
Part II — Practice Questions
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