IB Math AI HL — Full Syllabus Practice Quiz

Topic Overview

Key Concepts & Formulae

T1 Number & Algebra

Arithmetic: $u_n = u_1 + (n-1)d$, $S_n = \frac{n}{2}(2u_1+(n-1)d)$
Geometric: $u_n = u_1 \cdot r^{n-1}$, $S_n = \frac{u_1(r^n-1)}{r-1}$
Compound Interest: $A = P\left(1+\frac{r}{n}\right)^{nt}$
Logarithm: $\log_a b = c \Leftrightarrow a^c = b$

★ Remember: geometric sequences require $r \neq 0, 1$. Logs: domain must be positive.

Quick Example

$u_1=3, d=5 \Rightarrow u_{10}=3+9\times5=$ 48

T2 Functions & Modelling

Exponential: $f(x) = ka^x + c$ or $N = N_0 e^{kt}$
Logarithmic: $f(x) = a\ln x + b$
Optimisation: set $f'(x) = 0$, check $f''(x)$

★ For exponential growth/decay: solve $N = N_0 e^{kt}$ by taking $\ln$ both sides.

Quick Example

$N=2000, N_0=500 \Rightarrow t = \frac{\ln 4}{0.2} \approx$ 6.93

T3 Geometry & Trigonometry

Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B}$
Cosine Rule: $a^2 = b^2+c^2-2bc\cos A$
Distance: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Vectors: $\mathbf{u}\cdot\mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta$

★ Sine rule: use when given angle–side–angle or angle–angle–side.

Quick Example

$A=40°, B=70°, b=10 \Rightarrow a = \frac{10\sin40°}{\sin70°} \approx$ 6.84

T4 Statistics & Probability

Conditional: $P(A|B) = \frac{P(A\cap B)}{P(B)}$
Binomial: $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$
Normal: $z = \frac{x-\mu}{\sigma}$
Chi-squared: $\chi^2 = \sum\frac{(O-E)^2}{E}$
Poisson: $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$

★ For hypothesis testing: if p-value < significance level, reject $H_0$.

Quick Example

$X\sim N(70,25)$: $P(X<75)=P(Z<1)\approx$ 0.8413

T5 Calculus

Differentiation: $\frac{d}{dx}[x^n] = nx^{n-1}$
Integration: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$
Area between curves: $\int_a^b[f(x)-g(x)]\,dx$
Linear regression: $\hat{y} = bx + a$, $b = \frac{S_{xy}}{S_{xx}}$

★ Area is always positive — take $|f(x)-g(x)|$ when curves cross.

Quick Example

$f(x)=x^2-6x+10$: set $f'(x)=2x-6=0 \Rightarrow$ min at $x=3$, $f(3)=$ 1

T6 Matrices & Transformations

$\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad-bc$
$A^{-1} = \frac{1}{\det A}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$ (if $\det A\neq0$)

★ If det = 0, the matrix is singular — no inverse exists.

Quick Example

$M=\begin{pmatrix}2&1\\3&4\end{pmatrix}$: $\det M=8-3=5$, $M^{-1}=\frac{1}{5}\begin{pmatrix}4&-1\\-3&2\end{pmatrix}$

Practice Exam

20 Questions — Full Syllabus

Select one answer per question. Instant feedback is shown after each attempt. Detailed solutions appear at the end.

Number & Algebra — Arithmetic Sequence

An arithmetic sequence has first term $u_1 = 3$ and common difference $d = 5$. Find the 10th term $u_{10}$.

Number & Algebra — Geometric Sequence

A geometric sequence has $u_1 = 2$ and common ratio $r = 3$. What is the 5th term $u_5$?

Number & Algebra — Compound Interest

$\$5{,}000$ is invested at an annual interest rate of $4\%$, compounded monthly. What is the amount after 3 years? (Give your answer to the nearest cent.)

Number & Algebra — Logarithms

Solve: $\log_2 x + \log_2(x-2) = 3$.

Statistics — Linear Regression

A data set has values: $x = \{1,2,3,4,5\}$ and $y = \{2.1, 3.9, 6.2, 7.8, 10.1\}$. Which of the following best describes the Pearson correlation coefficient $r$?

Statistics — Normal Distribution

The heights of students are normally distributed with mean $\mu = 70$ cm and standard deviation $\sigma = 5$ cm. Find $P(X < 75)$.

Probability — Binomial Distribution

A fair trial is repeated 10 times with probability of success $p = 0.3$. Calculate $P(X = 3)$ where $X \sim B(10, 0.3)$.

Probability — Conditional Probability

Events $A$ and $B$ are independent with $P(A) = 0.4$ and $P(B) = 0.3$. Find $P(A \mid B)$.

Calculus — Differentiation

Let $f(x) = 3x^4 - 2x^2 + 5x - 1$. Find $f'(2)$.

Calculus — Integration

Evaluate $\displaystyle\int_0^2 (2x^3 - 3x + 1)\,dx$.

Functions — Exponential Model

A population grows according to $N = 500\,e^{0.2t}$. Find the time $t$ (in years) when the population reaches $2{,}000$.

Geometry — Distance Formula

Find the exact distance between the points $A(1, 2)$ and $B(5, 6)$.

Trigonometry — Sine Rule

In triangle $ABC$, angle $A = 40°$, angle $B = 70°$, and side $b = 10$ cm. Find side $a$ to 3 significant figures.

Statistics — Chi-Squared Test

In a chi-squared goodness-of-fit test, observed frequencies are $25, 35, 40$ with equal expected frequencies (total $= 100$). Calculate the test statistic $\chi^2$.

Geometry — Vectors & Dot Product

Vectors $\mathbf{u} = \begin{pmatrix}3\\4\end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix}1\\2\end{pmatrix}$. Find the angle $\theta$ between them to 1 decimal place.

Algebra — Matrix Inverse

Find the inverse of the matrix $M = \begin{pmatrix}2 & 1\\3 & 4\end{pmatrix}$.

Probability — Poisson Distribution

A call centre receives an average of $\lambda = 3$ calls per minute. Using a Poisson model, find $P(X = 2)$.

Calculus — Optimisation

Find the coordinates of the minimum point of $f(x) = x^2 - 6x + 10$.

Statistics — Hypothesis Testing (t-test)

A sample of 6 measurements gives values: $12.1, 11.8, 12.3, 12.5, 11.9, 12.2$. A one-sample $t$-test is conducted against $H_0: \mu = 12.0$ at the $5\%$ significance level. The calculated $p$-value is $0.2617$. What is the correct conclusion?

Calculus — Area Between Curves

Find the exact area enclosed between $y = x$ and $y = x^2$ for $0 \leq x \leq 1$.

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Key Concepts & Formulae

20 Questions — Full Syllabus

Detailed Solutions

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