IB Math AA SL — Core Concepts Master Quiz

📐 IB Math AA SL — 20 Core Questions

Covers all major topics: Algebra, Functions, Trigonometry, Calculus, Statistics & Probability, and Vectors. Each question is exam-style with full worked explanations.

📋 20 Questions

⏱ 40 Minutes

🎯 MC Format

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Study the concept review below, then scroll down to attempt all 20 questions. Answers and explanations are revealed immediately after each attempt.

CONCEPT REVIEW

Key Formulae & Memory Points

Topic 1 — Algebra

Sequences & Series

An arithmetic sequence has a common difference $d$. A geometric sequence has a common ratio $r$.

$u_n = u_1 + (n-1)d$ | $S_n = \dfrac{n}{2}(u_1+u_n)$
$u_n = u_1 \cdot r^{n-1}$ | $S_n = \dfrac{u_1(r^n-1)}{r-1}$

d = u₂ − u₁ r = u₂/u₁ |r|<1 for convergent

Topic 1 — Algebra

Binomial Theorem & Logarithms

Binomial coefficient: $\binom{n}{r} = \dfrac{n!}{r!(n-r)!}$. General term: $T_{r+1} = \binom{n}{r}a^{n-r}b^r$.

$\log_b(xy)=\log_b x + \log_b y$ | $\log_b\!\left(\tfrac{x}{y}\right)=\log_b x - \log_b y$
$\log_b(x^n)=n\log_b x$ | Change of base: $\log_b x = \dfrac{\ln x}{\ln b}$

log₁₀ = lg loge = ln log 1 = 0

Topic 2 — Functions

Quadratics, Inverse & Composition

Discriminant $\Delta = b^2 - 4ac$ determines nature of roots. Vertex form: $f(x)=a(x-h)^2+k$.

$\Delta > 0$: two distinct real roots | $\Delta = 0$: one repeated root
$\Delta < 0$: no real roots
$(f \circ g)(x) = f(g(x))$

ff⁻¹(x)=x swap x & y for inverse

Topic 3 — Geometry & Trigonometry

Radian Measure & Trig Identities

$\pi$ radians $= 180°$. Arc length $l=r\theta$. Area of sector $A=\tfrac{1}{2}r^2\theta$.

$\sin^2\theta + \cos^2\theta = 1$
$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
Sine rule: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}$ | Cosine rule: $a^2=b^2+c^2-2bc\cos A$
Area: $\frac{1}{2}ab\sin C$

sin30°=½ cos60°=½ tan45°=1

Topic 5 — Calculus

Differentiation & Integration

Power rule: $\dfrac{d}{dx}[x^n] = nx^{n-1}$. Chain rule: $\dfrac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)$.

$\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ $(n \neq -1)$
$\int_a^b f(x)\,dx = F(b)-F(a)$
$\int e^x\,dx = e^x+C$ | $\int \frac{1}{x}\,dx = \ln|x|+C$

max/min when f'=0 inflection when f''=0

Topic 4 — Statistics & Topic 3 — Vectors

Probability, Normal Distribution & Vectors

$P(A \cup B) = P(A)+P(B)-P(A\cap B)$. For independent events: $P(A\cap B)=P(A)\cdot P(B)$.

$X \sim N(\mu, \sigma^2)$: z-score $= \dfrac{x - \mu}{\sigma}$
Dot product: $\mathbf{a}\cdot\mathbf{b} = a_1b_1+a_2b_2+a_3b_3$
$\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$

P(A')=1−P(A) P(Z<0)=0.5 perp: a·b=0

Worked Example

Binomial Expansion — Finding a Coefficient

Find the coefficient of $x^2$ in the expansion of $(2x+3)^4$.

Use the general term: $T_{r+1} = \binom{4}{r}(2x)^{4-r}(3)^r$

For $x^2$, we need $4-r=2$, so $r=2$.

$T_3 = \binom{4}{2}(2x)^2(3)^2 = 6 \cdot 4x^2 \cdot 9 = 216x^2$

✓ Coefficient of $x^2$ = 216

Worked Example

Definite Integration — Area Calculation

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

Integrate: $\displaystyle\int (2x^3+3x)\,dx = \frac{x^4}{2}+\frac{3x^2}{2}+C$

Apply limits: $\left[\frac{x^4}{2}+\frac{3x^2}{2}\right]_0^2 = \left(\frac{16}{2}+\frac{12}{2}\right)-(0) = 8+6$

✓ Answer = 14

— Practice Questions —

Arithmetic Sequences

The first term of an arithmetic sequence is $3$ and the common difference is $4$. What is the $10$th term?

📘 Full Solution

Use $u_n = u_1 + (n-1)d$ with $u_1=3$, $d=4$, $n=10$:

$u_{10} = 3 + (10-1)(4) = 3 + 36 = \mathbf{39}$

⚠ Common error: using $n$ instead of $(n-1)$: $3+10(4)=43$ — this is wrong.

Geometric Series

A geometric series has first term $u_1 = 2$ and common ratio $r = 3$. What is the sum of the first $5$ terms?

📘 Full Solution

$S_n = \dfrac{u_1(r^n - 1)}{r-1}$, so $S_5 = \dfrac{2(3^5-1)}{3-1} = \dfrac{2(243-1)}{2} = 242$

Verify: $2+6+18+54+162 = 242$ ✓

Quadratic Equations

Solve $x^2 - 5x + 6 = 0$. The solution set is:

📘 Full Solution

Factor: $x^2-5x+6 = (x-2)(x-3) = 0$

Find two numbers that multiply to $+6$ and add to $-5$: that is $-2$ and $-3$.

Therefore $x=2$ or $x=3$. Check: $4-10+6=0$ ✓ and $9-15+6=0$ ✓

Exponential Equations

Solve for $x$: $\;2^{2x+1} = 32$

📘 Full Solution

Write both sides as powers of $2$: $32 = 2^5$

So $2^{2x+1} = 2^5 \Rightarrow 2x+1 = 5 \Rightarrow 2x = 4 \Rightarrow x = 2$

Check: $2^{2(2)+1} = 2^5 = 32$ ✓

Logarithms

Solve $\log_2 x + \log_2(x-2) = 3$, where $x > 2$.

📘 Full Solution

Combine logs: $\log_2[x(x-2)] = 3$

So $x(x-2) = 2^3 = 8 \Rightarrow x^2 - 2x - 8 = 0 \Rightarrow (x-4)(x+2) = 0$

$x = 4$ or $x = -2$. Since $x > 2$, reject $x=-2$. Answer: $x = 4$.

Check: $\log_2 4 + \log_2 2 = 2+1 = 3$ ✓

Binomial Theorem

In the expansion of $(2x + 3)^4$, the coefficient of $x^2$ is:

📘 Full Solution

General term: $T_{r+1} = \binom{4}{r}(2x)^{4-r}(3)^r$

For $x^2$: $4-r=2 \Rightarrow r=2$

$T_3 = \binom{4}{2}(2)^2(3)^2 = 6 \times 4 \times 9 = 216$

Differentiation

Given $f(x) = 3x^4 - 2x^2 + 5$, find $f'(2)$.

📘 Full Solution

Differentiate term by term: $f'(x) = 12x^3 - 4x$

Substitute $x=2$: $f'(2) = 12(8) - 4(2) = 96 - 8 = 88$

Definite Integration

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

📘 Full Solution

$\displaystyle\int(2x^3+3x)\,dx = \frac{2x^4}{4}+\frac{3x^2}{2} = \frac{x^4}{2}+\frac{3x^2}{2}$

$\left[\frac{x^4}{2}+\frac{3x^2}{2}\right]_0^2 = \left(\frac{16}{2}+\frac{12}{2}\right) - 0 = 8+6 = 14$

Probability

Events $A$ and $B$ satisfy $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$. Find $P(A\cup B)$.

📘 Full Solution

Inclusion-exclusion: $P(A\cup B) = P(A)+P(B)-P(A\cap B)$

$= 0.4 + 0.5 - 0.2 = 0.7$

Normal Distribution

$X \sim N(70, 10^2)$. Using the standard normal table, $P(X < 80)$ is approximately:

📘 Full Solution

Standardise: $z = \dfrac{80-70}{10} = 1$

$P(X < 80) = P(Z < 1) \approx 0.8413$ (from standard normal table)

Note: $P(Z<2)=0.9772$, $P(Z<0)=0.5$

Trigonometric Equations

Solve $\sin(2x) = \dfrac{\sqrt{3}}{2}$ for $0 \leq x \leq \pi$. The solutions are:

📘 Full Solution

Let $u = 2x$, so $0 \leq u \leq 2\pi$. Solve $\sin u = \frac{\sqrt{3}}{2}$.

$\sin u = \frac{\sqrt{3}}{2}$ gives $u = \frac{\pi}{3}$ or $u = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$

Back-substitute: $x = \frac{u}{2}$, so $x = \frac{\pi}{6}$ or $x = \frac{\pi}{3}$

Vectors — Dot Product

Given $\mathbf{a} = \begin{pmatrix}2\\3\\-1\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}1\\-2\\4\end{pmatrix}$, find $\mathbf{a}\cdot\mathbf{b}$.

📘 Full Solution

$\mathbf{a}\cdot\mathbf{b} = (2)(1) + (3)(-2) + (-1)(4) = 2 - 6 - 4 = -8$

Since $\mathbf{a}\cdot\mathbf{b} \neq 0$, the vectors are NOT perpendicular.

Composite Functions

Let $f(x) = 2x+1$ and $g(x) = x^2$. Find $(g \circ f)(3)$.

📘 Full Solution

$(g \circ f)(3) = g(f(3))$. First compute $f(3) = 2(3)+1 = 7$.

Then $g(7) = 7^2 = 49$

⚠ Do NOT compute $f(g(3))$ — order matters! $f(g(3)) = f(9) = 19 \neq 49$.

Inverse Functions

Given $f(x) = \dfrac{3x-2}{x+1}$, find $f^{-1}(2)$.

📘 Full Solution

Find $f^{-1}$: let $y = \frac{3x-2}{x+1}$. Swap and solve for $x$:

$x(y-3) = -2-y \Rightarrow x = \dfrac{2+y}{3-y}$

So $f^{-1}(y) = \dfrac{2+y}{3-y}$. Then $f^{-1}(2) = \dfrac{2+2}{3-2} = \dfrac{4}{1} = 4$

Verify: $f(4) = \frac{3(4)-2}{4+1} = \frac{10}{5} = 2$ ✓

Inequalities

Solve $|x - 3| < 5$.

📘 Full Solution

$|x-3| < 5 \Rightarrow -5 < x-3 < 5$

Add $3$ throughout: $-5+3 < x < 5+3 \Rightarrow -2 < x < 8$

Kinematics / Calculus

A particle moves along a straight line with velocity $v(t) = 3t^2 - 12t + 9$ m/s. The particle is at rest when:

📘 Full Solution

Set $v(t)=0$: $3t^2-12t+9 = 0 \Rightarrow t^2-4t+3=0$

Factor: $(t-1)(t-3)=0 \Rightarrow t=1$ or $t=3$

Check: $v(1)=3-12+9=0$ ✓, $v(3)=27-36+9=0$ ✓

Radian Measure

Convert $135°$ to radians. Give your answer in exact form.

📘 Full Solution

Multiply degrees by $\dfrac{\pi}{180}$: $135 \times \dfrac{\pi}{180} = \dfrac{135\pi}{180} = \dfrac{3\pi}{4}$

Simplify: $\gcd(135,180)=45$, so $\frac{135}{180}=\frac{3}{4}$

Area of Triangle

A triangle has sides $a = 5$ and $b = 8$, with included angle $C = 60°$. Its area is:

📘 Full Solution

Area $= \frac{1}{2}ab\sin C = \frac{1}{2}(5)(8)\sin 60°$

$= 20 \times \dfrac{\sqrt{3}}{2} = 10\sqrt{3} \approx 17.32$ sq. units

Statistics — Standard Deviation

The dataset $\{2, 4, 6, 8, 10\}$ has mean $\bar{x} = 6$. The population standard deviation is:

📘 Full Solution

Variance $\sigma^2 = \frac{1}{n}\sum(x_i-\bar{x})^2$

$= \frac{1}{5}[(2-6)^2+(4-6)^2+(6-6)^2+(8-6)^2+(10-6)^2]$

$= \frac{1}{5}[16+4+0+4+16] = \frac{40}{5} = 8$

$\sigma = \sqrt{8} = 2\sqrt{2} \approx 2.828$

Financial Mathematics

\$1{,}000 is invested at an annual interest rate of $6\%$, compounded monthly, for $2$ years. The value at the end is closest to:

📘 Full Solution

Use $A = P\!\left(1+\dfrac{r}{n}\right)^{nt}$ with $P=1000$, $r=0.06$, $n=12$, $t=2$

$A = 1000\!\left(1+\dfrac{0.06}{12}\right)^{24} = 1000(1.005)^{24}$

$(1.005)^{24} \approx 1.12716$, so $A \approx \$1{,}127.16$

Quiz Complete — Your Result

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CORRECT

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INCORRECT

Answers & Explanations

Q1. Arithmetic Sequences

Correct Answer: B — 39

$u_{10} = 3 + (10-1)(4) = 3 + 36 = 39$. Formula: $u_n = u_1 + (n-1)d$.

Q2. Geometric Series

Correct Answer: C — 242

$S_5 = \frac{2(3^5-1)}{3-1} = \frac{2 \times 242}{2} = 242$. Verify: $2+6+18+54+162=242$.

Q3. Quadratic Equations

Correct Answer: B — {2, 3}

$x^2-5x+6=(x-2)(x-3)=0 \Rightarrow x=2$ or $x=3$.

Q4. Exponential Equations

Correct Answer: B — x = 2

$2^{2x+1}=2^5 \Rightarrow 2x+1=5 \Rightarrow x=2$.

Q5. Logarithms

Correct Answer: C — x = 4

$\log_2[x(x-2)]=3 \Rightarrow x(x-2)=8 \Rightarrow x^2-2x-8=0 \Rightarrow x=4$ (reject $x=-2$).

Q6. Binomial Theorem

Correct Answer: C — 216

$T_3 = \binom{4}{2}(2)^2(3)^2 = 6 \times 4 \times 9 = 216$.

Q7. Differentiation

Correct Answer: B — 88

$f'(x)=12x^3-4x$; $f'(2)=12(8)-4(2)=96-8=88$.

Q8. Definite Integration

Correct Answer: C — 14

$\left[\frac{x^4}{2}+\frac{3x^2}{2}\right]_0^2 = 8+6 = 14$.

Q9. Probability

Correct Answer: B — 0.7

$P(A \cup B) = 0.4+0.5-0.2=0.7$.

Q10. Normal Distribution

Correct Answer: C — 0.8413

$z=(80-70)/10=1$; $P(Z<1) \approx 0.8413$.

Q11. Trigonometric Equations

Correct Answer: B — x = pi/6 and x = pi/3

$2x=\pi/3$ or $2x=2\pi/3 \Rightarrow x=\pi/6$ or $x=\pi/3$.

Q12. Vectors — Dot Product

Correct Answer: C — -8

$\mathbf{a}\cdot\mathbf{b} = 2(1)+3(-2)+(-1)(4)=2-6-4=-8$.

Q13. Composite Functions

Correct Answer: C — 49

$f(3)=7$; $g(f(3))=g(7)=49$.

Q14. Inverse Functions

Correct Answer: B — 4

$f^{-1}(y)=\frac{2+y}{3-y}$; $f^{-1}(2)=\frac{4}{1}=4$. Verify: $f(4)=\frac{10}{5}=2$.

Q15. Inequalities

Correct Answer: C — -2 < x < 8

$|x-3|<5 \Rightarrow -5<x-3<5 \Rightarrow -2<x<8$.

Q16. Kinematics

Correct Answer: C — t = 1 and t = 3

$3t^2-12t+9=0 \Rightarrow (t-1)(t-3)=0$.

Q17. Radian Measure

Correct Answer: B — 3pi/4

$135 \times \frac{\pi}{180} = \frac{3\pi}{4}$.

Q18. Area of Triangle

Correct Answer: B — 10 root 3

Area $= \frac{1}{2}(5)(8)\sin 60° = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3}$.

Q19. Standard Deviation

Correct Answer: C — 2 root 2

$\sigma^2=\frac{40}{5}=8$; $\sigma=2\sqrt{2}\approx2.828$.

Q20. Financial Mathematics

Correct Answer: C — $1,127.16

$A=1000(1.005)^{24}\approx\$1{,}127.16$.