Mathematics · Analysis & Approaches

IB AA SL Core Concepts

20 Exam-Style Questions — All Core Topics

20 Questions Multiple Choice 40 Minutes SL Level
Concept Review
Memorise These
Topic 1 · Sequences

Arithmetic & Geometric

\(u_n = u_1 + (n-1)d\)
\(S_n = \frac{n}{2}(2u_1+(n-1)d)\)
\(u_n = u_1 r^{n-1}\)
\(S_n = \frac{u_1(r^n-1)}{r-1}\), \(S_\infty = \frac{u_1}{1-r}\,\,(|r|<1)\)
Example: \(u_1=3,\,d=4\) → \(u_{15}=3+14(4)=59\)
Topic 1 · Algebra

Exponents & Logarithms

\(\log_a(xy)=\log_a x+\log_a y\)
\(\log_a(x^n)=n\log_a x\)
\(a^x=b \Leftrightarrow x=\log_a b\)
Example: \(\log_2 x+\log_2(x-2)=3\)
→ \(x(x-2)=8\) → \(x=4\)
Topic 1 · Algebra

Binomial Theorem

\((a+b)^n=\displaystyle\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\)
\(\binom{n}{r}=\frac{n!}{r!(n-r)!}\)
Example: Coeff of \(x^3\) in \((2+x)^5\):
\(\binom{5}{3}\cdot2^2=10\cdot4=40\)
Topic 2 · Functions

Quadratics & Discriminant

\(\Delta = b^2-4ac\)
\(\Delta>0\): 2 roots · \(\Delta=0\): 1 · \(\Delta<0\): none
Vertex: \(x=-\dfrac{b}{2a}\)
Example: \(x^2+kx+4=0\) one solution → \(k^2-16=0\) → \(k=\pm4\)
Topic 3 · Trigonometry

Identities & Special Angles

\(\sin^2\theta+\cos^2\theta=1\)
\(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\)
\(\sin\tfrac{\pi}{6}=\tfrac{1}{2},\;\sin\tfrac{\pi}{4}=\tfrac{\sqrt2}{2},\;\sin\tfrac{\pi}{3}=\tfrac{\sqrt3}{2}\)
\(\cos\tfrac{\pi}{4}=\tfrac{\sqrt2}{2},\;\tan\tfrac{\pi}{4}=1\)
Example: \(\sin\theta=\frac{3}{5}\) → \(\cos\theta=\frac{4}{5}\) (Q1)
Topic 5 · Calculus

Differentiation

\(\frac{d}{dx}[x^n]=nx^{n-1}\)
Tangent at \(x=a\): \(y-f(a)=f'(a)(x-a)\)
Example: \(f(x)=x^3-3x^2+2\)
\(f'(x)=3x^2-6x\), \(f'(2)=0\)
Topic 5 · Calculus

Integration

\(\displaystyle\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\)
Area \(=\displaystyle\int_a^b f(x)\,dx\)
Example: \(\displaystyle\int_0^2(2x+3)\,dx=[x^2+3x]_0^2=10\)
Topic 4 · Statistics

Probability & Normal Dist.

\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
\(X\sim N(\mu,\sigma^2),\; Z=\dfrac{X-\mu}{\sigma}\)
Example: \(X\sim N(50,\,64)\),\, \(P(X<58)=P(Z<1)\approx0.8413\)
Topic 3 · Geometry

Vectors & Dot Product

\(\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2\)
\(\cos\theta=\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}\)
Example: \((2,3)\cdot(4,-1)=8-3=5\)
Topic 1 · Counting

Permutations & Combinations

\(P(n,r)=\dfrac{n!}{(n-r)!}\)
\(C(n,r)=\dfrac{n!}{r!(n-r)!}\)
Example: Arrange 4 from 6: \(P(6,4)=\frac{6!}{2!}=360\)
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Time Remaining
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0 of 20 answered 0%
01
Arithmetic Sequences
The first term of an arithmetic sequence is 3 and the common difference is 4. What is the 15th term?
02
Geometric Series
Find the sum of the first 6 terms of the geometric series with first term 2 and common ratio 3.
03
Quadratic Equations · Discriminant
The equation \(x^2 + kx + 4 = 0\) has exactly one real solution. What are the possible values of \(k\)?
04
Exponential Equations
Solve for \(x\): \(\quad 3^{2x} = 81\)
05
Logarithms
Solve: \(\;\log_2 x + \log_2(x - 2) = 3\)
06
Differentiation
Let \(f(x) = x^3 - 3x^2 + 2\). Find \(f'(2)\).
07
Integration
Evaluate: \(\displaystyle\int_0^2 (2x + 3)\,dx\)
08
Trigonometry · Pythagorean Identity
Given \(\sin\theta = \dfrac{3}{5}\) and \(\theta\) is in the first quadrant, find \(\cos\theta\).
09
Vectors · Dot Product
Find the scalar (dot) product of vectors \(\vec{a} = \begin{pmatrix}2\\3\end{pmatrix}\) and \(\vec{b} = \begin{pmatrix}4\\-1\end{pmatrix}\).
10
Binomial Theorem
Find the coefficient of \(x^3\) in the expansion of \((2 + x)^5\).
11
Probability
Events \(A\) and \(B\) satisfy \(P(A)=0.4\), \(P(B)=0.3\), and \(P(A\cap B)=0.1\).
Find \(P(A\cup B)\).
12
Normal Distribution
\(X \sim N(50,\, 8^2)\). Find \(P(X < 58)\). Give your answer to 4 significant figures.
13
Composite Functions
Let \(f(x) = 2x + 1\) and \(g(x) = x^2\). Find \(g(f(2))\).
14
Calculus · Optimisation
Let \(f(x) = x^3 - 6x^2 + 9x\). The local maximum value of \(f\) is:
15
Infinite Geometric Series
An infinite geometric series has first term 12 and common ratio \(\dfrac{1}{2}\). Find its sum to infinity.
16
Quadratic Equations
Solve \(x^2 - 5x + 6 = 0\). The solutions are:
17
Trigonometry · Exact Values
What is the exact value of \(\tan\!\left(\dfrac{\pi}{4}\right)\)?
18
Calculus · Tangent Lines
Find the equation of the tangent to \(f(x) = x^2 + 3x\) at the point where \(x = 1\).
19
Statistics · Mean
For the data set \(\{2,\, 5,\, 7,\, 8,\, 8\}\), find the mean.
20
Counting Principles · Permutations
In how many ways can 4 different books be selected and arranged in order from a shelf of 6 different books?
Final Score
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Full Solutions & Explanations

Complete worked solutions for all 20 questions

01Arithmetic SequencesAnswer: C · 59
Use \(u_n = u_1 + (n-1)d\) with \(u_1=3\), \(d=4\), \(n=15\):
\(u_{15} = 3 + (15-1)\times 4 = 3 + 56 = \mathbf{59}\)
02Geometric SeriesAnswer: B · 728
\(S_n = \dfrac{u_1(r^n-1)}{r-1}\) with \(u_1=2\), \(r=3\), \(n=6\):
\(S_6 = \dfrac{2(3^6-1)}{3-1} = \dfrac{2\times728}{2} = \mathbf{728}\)
03DiscriminantAnswer: C · k = ±4
Exactly one real root ⟺ discriminant \(\Delta = 0\).
\(\Delta = k^2 - 4(1)(4) = k^2 - 16 = 0 \Rightarrow k = \pm 4\)
04Exponential EquationsAnswer: B · x = 2
Express both sides as powers of 3:
\(3^{2x} = 81 = 3^4 \Rightarrow 2x = 4 \Rightarrow x = \mathbf{2}\)
05LogarithmsAnswer: C · x = 4
Apply the product law: \(\log_2[x(x-2)] = 3 \Rightarrow x(x-2) = 2^3 = 8\)
\(x^2 - 2x - 8 = 0 \Rightarrow (x-4)(x+2)=0 \Rightarrow x=4\) or \(x=-2\).
Reject \(x=-2\) (log undefined for negative argument). So \(x = \mathbf{4}\).
06DifferentiationAnswer: B · 0
\(f'(x) = 3x^2 - 6x\)
\(f'(2) = 3(4) - 6(2) = 12 - 12 = \mathbf{0}\)
This means \(x=2\) is a stationary point of \(f\).
07IntegrationAnswer: C · 10
\(\displaystyle\int_0^2(2x+3)\,dx = \bigl[x^2+3x\bigr]_0^2 = (4+6)-(0) = \mathbf{10}\)
08Pythagorean IdentityAnswer: B · 4/5
\(\sin^2\theta + \cos^2\theta = 1 \Rightarrow \cos^2\theta = 1 - \tfrac{9}{25} = \tfrac{16}{25}\)
Since \(\theta\) is in Q1, \(\cos\theta = +\dfrac{4}{5}\).
09Dot ProductAnswer: B · 5
\(\vec{a}\cdot\vec{b} = (2)(4) + (3)(-1) = 8 - 3 = \mathbf{5}\)
10Binomial TheoremAnswer: C · 40
General term: \(\dbinom{5}{r}(2)^{5-r}(x)^r\). For \(x^3\), set \(r=3\):
\(\dbinom{5}{3}\cdot 2^2 = 10 \times 4 = \mathbf{40}\)
11ProbabilityAnswer: B · 0.6
Addition rule: \(P(A\cup B) = P(A)+P(B)-P(A\cap B) = 0.4+0.3-0.1 = \mathbf{0.6}\)
12Normal DistributionAnswer: B · 0.8413
Standardise: \(Z = \dfrac{58-50}{8} = 1\)
\(P(X < 58) = P(Z < 1) = \mathbf{0.8413}\) (standard normal table).
13Composite FunctionsAnswer: C · 25
Step 1: \(f(2) = 2(2)+1 = 5\)
Step 2: \(g(f(2)) = g(5) = 5^2 = \mathbf{25}\)
14OptimisationAnswer: C · 4
\(f'(x) = 3x^2-12x+9 = 3(x-1)(x-3)\). Critical points at \(x=1\) and \(x=3\).
\(f''(x) = 6x-12\). At \(x=1\): \(f''(1)=-6 < 0\) → local maximum.
\(f(1) = 1-6+9 = \mathbf{4}\).
15Infinite Geometric SeriesAnswer: B · 24
Since \(|r|=\frac{1}{2} < 1\), the series converges:
\(S_\infty = \dfrac{u_1}{1-r} = \dfrac{12}{1-\frac{1}{2}} = \dfrac{12}{\frac{1}{2}} = \mathbf{24}\)
16Quadratic EquationsAnswer: B · x = 2 and x = 3
Factorise: \(x^2-5x+6 = (x-2)(x-3) = 0\)
Solutions: \(x = \mathbf{2}\) and \(x = \mathbf{3}\).
17Exact Trigonometric ValuesAnswer: C · 1
In a 45–45–90 triangle: \(\sin\!\tfrac{\pi}{4}=\cos\!\tfrac{\pi}{4}=\dfrac{\sqrt{2}}{2}\)
\(\tan\!\dfrac{\pi}{4} = \dfrac{\sin\frac{\pi}{4}}{\cos\frac{\pi}{4}} = \dfrac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = \mathbf{1}\)
18Tangent LinesAnswer: A · y = 5x − 1
\(f'(x) = 2x+3\), so the gradient at \(x=1\) is \(m = f'(1)=5\).
Point: \(f(1) = 1+3 = 4\). Tangent: \(y-4=5(x-1) \Rightarrow y = 5x-1\).
19Statistics · MeanAnswer: B · 6
\(\bar{x} = \dfrac{2+5+7+8+8}{5} = \dfrac{30}{5} = \mathbf{6}\)
20PermutationsAnswer: C · 360
Order matters → use permutations:
\(P(6,4) = \dfrac{6!}{(6-4)!} = \dfrac{720}{2} = \mathbf{360}\)