International Baccalaureate · Mathematics · AA SL

Core Concepts
Master Quiz

20 exam-style questions spanning every essential topic in the AA SL syllabus

⏱ 40 Minutes ✦ 20 Questions ◆ Multiple Choice ★ IB Style
⏱ 40:00
0 / 20 answered
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Concept Review

Click each topic to expand key formulas, memory aids, and worked examples

Arithmetic Sequence

\(u_n = u_1 + (n-1)d\)    \(S_n = \frac{n}{2}(2u_1 + (n-1)d)\)
🔑 Must Memorise
  • \(d\) = common difference = \(u_2 - u_1\)
  • \(u_n\) is the nth term; \(u_1\) is the first term
  • \(S_n = \frac{n}{2}(u_1 + u_n)\) — sum using first and last

Geometric Sequence

\(u_n = u_1 \cdot r^{n-1}\)    \(S_n = \frac{u_1(r^n - 1)}{r-1}\), \(r \neq 1\)    \(S_\infty = \frac{u_1}{1-r}\), \(|r|<1\)
✎ Worked Example

AP: \(u_1 = 3,\ d = 5\). Find \(u_{10}\).

\(u_{10} = 3 + 9(5) = 3 + 45 = \mathbf{48}\)

GP: \(u_1=2,\ r=3\). Find \(S_5 = \frac{2(3^5-1)}{3-1} = \frac{2 \times 242}{2} = \mathbf{242}\)

Laws of Exponents

\(a^m \cdot a^n = a^{m+n}\)   \(\frac{a^m}{a^n} = a^{m-n}\)   \((a^m)^n = a^{mn}\)   \(a^0 = 1\)

Logarithm Rules

\(\log_a(xy)=\log_a x+\log_a y\)   \(\log_a\!\left(\frac{x}{y}\right)=\log_a x-\log_a y\)   \(\log_a x^n=n\log_a x\)

Quadratic Formula & Discriminant

\(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)    \(\Delta = b^2-4ac\)
🔑 Must Memorise
  • \(\Delta > 0\): two distinct real roots
  • \(\Delta = 0\): one repeated root
  • \(\Delta < 0\): no real roots
  • Completing the square: \(x^2+bx = (x+\tfrac{b}{2})^2 - \tfrac{b^2}{4}\)

Binomial Theorem

\((a+b)^n = \sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\)    \(\binom{n}{r}=\frac{n!}{r!(n-r)!}\)
✎ Worked Example

Coefficient of \(x^2\) in \((1+x)^5\):

\(\binom{5}{2} = \frac{5!}{2!\cdot3!} = \mathbf{10}\)

Key Definitions

Domain: all valid inputs (x-values). Range: all possible outputs (y-values).

Inverse Function

To find \(f^{-1}(x)\): swap \(x\) and \(y\), solve for \(y\).

Composite Function

\((f \circ g)(x) = f(g(x))\) — apply \(g\) first, then \(f\)
🔑 Must Memorise
  • The graph of \(f^{-1}\) is the reflection of \(f\) in \(y=x\)
  • \(f(f^{-1}(x)) = f^{-1}(f(x)) = x\)
  • Exponential growth: \(y = ka^x\) where \(a>1\)
  • Exponential decay: \(y = ka^x\) where \(0<a<1\)
✎ Worked Example

\(f(x) = 3x - 2\). Find \(f^{-1}(x)\).

Let \(y = 3x-2 \Rightarrow x = \frac{y+2}{3} \Rightarrow f^{-1}(x) = \frac{x+2}{3}\)

\(f^{-1}(7) = \frac{7+2}{3} = 3\)

Differentiation Rules

\(\frac{d}{dx}[x^n] = nx^{n-1}\)    \(\frac{d}{dx}[e^x] = e^x\)    \(\frac{d}{dx}[\ln x] = \frac{1}{x}\)

Chain, Product & Quotient Rules

\(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
\(\frac{d}{dx}[uv] = u'v + uv'\)    \(\frac{d}{dx}\!\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}\)

Integration

\(\int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\quad n \neq -1\)
🔑 Must Memorise
  • Tangent slope at \(x=a\) is \(f'(a)\)
  • Max/min: set \(f'(x)=0\) and check \(f''(x)\)
  • \(f''(a)>0\): local min; \(f''(a)<0\): local max
  • Definite integral = signed area under curve
✎ Worked Example

\(f(x) = 3x^2 + 2x\). Find \(f'(x)\) and \(f'(2)\).

\(f'(x) = 6x + 2\)  →  \(f'(2) = 6(2)+2 = \mathbf{14}\)

\(\int_0^3(2x+1)\,dx = [x^2+x]_0^3 = (9+3)-0 = \mathbf{12}\)

Key Trig Values (Radians)

\(\sin 0=0,\ \cos 0=1\)   \(\sin\frac{\pi}{6}=\frac{1}{2},\ \cos\frac{\pi}{6}=\frac{\sqrt3}{2}\)   \(\sin\frac{\pi}{4}=\frac{\sqrt2}{2},\ \cos\frac{\pi}{3}=\frac{1}{2}\)

Probability

\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)

Statistics

\(\bar{x} = \frac{\sum x}{n}\)    \(z = \frac{x - \mu}{\sigma}\)

Vectors

\(\mathbf{a}\cdot\mathbf{b} = a_1 b_1 + a_2 b_2\)    \(|\mathbf{a}| = \sqrt{a_1^2+a_2^2}\)    \(\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\)
🔑 Must Memorise
  • Normal dist: ~68% within 1σ, ~95% within 2σ
  • If \(\mathbf{a}\cdot\mathbf{b}=0\), vectors are perpendicular
  • Area of circle: \(\pi r^2\); Circumference: \(2\pi r\)
  • Exponential model: \(P = P_0 e^{kt}\)
✎ Worked Example

P(A)=0.4, P(B)=0.3, P(A∩B)=0.1 → P(A∪B) = 0.4+0.3−0.1 = 0.6

Mean of {2,4,6,8,10} = 30/5 = 6

z-score: X=75, μ=70, σ=5 → z = (75−70)/5 = 1.0

Exam Questions

Select one answer per question. Instant feedback is provided after each response.

Instructions: Each question tests a core IB Math AA SL concept. Choose the best answer from options A–D. After selecting, immediate feedback and a step-by-step solution will appear. Your timer started when you opened this page — aim to complete all 20 questions within 40 minutes.
01
Sequences & Series
An arithmetic sequence has first term \(u_1 = 3\) and common difference \(d = 5\). What is the value of \(u_{10}\)?
02
Sequences & Series
A geometric sequence has \(u_1 = 2\) and common ratio \(r = 3\). What is the sum of the first 5 terms, \(S_5\)?
03
Algebra
Simplify \(2^3 \times 2^4\).
04
Logarithms
Solve for \(x\): \(\log_2 x = 5\).
05
Quadratics
What are the roots of the equation \(x^2 - 5x + 6 = 0\)?
06
Functions
Given \(f(x) = 2x + 3\), find \(f(4)\).
07
Differentiation
Given \(f(x) = 3x^2 + 2x\), find \(f'(2)\).
08
Integration
Evaluate \(\displaystyle\int_0^3 (2x + 1)\,dx\).
09
Trigonometry
What is the exact value of \(\sin 30°\)?
10
Binomial Theorem
In the expansion of \((1+x)^5\), what is the coefficient of \(x^2\)?
11
Vectors
Find the dot product of \(\mathbf{a} = \begin{pmatrix}2\\3\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}4\\-1\end{pmatrix}\).
12
Probability
Events A and B are such that \(P(A) = 0.4\), \(P(B) = 0.3\), and \(P(A \cap B) = 0.1\). Find \(P(A \cup B)\).
13
Statistics
The data set is \(\{2, 4, 6, 8, 10\}\). What is the mean?
14
Exponential Models
A population grows according to \(P = 100\,e^{0.05t}\). What is the population when \(t = 10\), correct to the nearest whole number?
15
Completing the Square
Write \(x^2 + 6x + 5\) in the form \((x + p)^2 + q\).
16
Inverse Functions
Given \(f(x) = 3x - 2\), find \(f^{-1}(7)\).
17
Geometry
A circle has radius \(r = 5\) cm. What is its exact area?
18
Normal Distribution
A normally distributed variable has mean \(\mu = 70\) and standard deviation \(\sigma = 5\). Find the \(z\)-score for \(X = 75\).
19
Composite Functions
Let \(f(x) = x^2\) and \(g(x) = x + 1\). Find \(f(g(2))\).
20
Rate of Change
Find the gradient of the tangent to \(f(x) = x^3 - 2x\) at the point where \(x = 1\).
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✦ Complete Solutions & Explanations

Step-by-step workings for all 20 questions