IB Mathematics · Analysis & Approaches SL
Core Concepts
Practice Quiz
All major topics · Exam-style · 20 questions
20 Questions
40 Min
All Topics
SL Level
Topic 1 · Number & Algebra
Key Concepts to Memorise
- Arithmetic sequence: \(u_n = u_1 + (n-1)d\), Sum: \(S_n = \tfrac{n}{2}(2u_1+(n-1)d)\)
- Geometric sequence: \(u_n = u_1 r^{n-1}\), Sum: \(S_n = \dfrac{u_1(r^n-1)}{r-1}\)
- Laws of logarithms: \(\log(ab)=\log a+\log b\), \(\log a^n = n\log a\)
- Change of base: \(\log_b a = \dfrac{\ln a}{\ln b}\)
\(u_n = u_1 + (n-1)d \qquad S_n = \frac{n}{2}(u_1+u_n)\)
Worked Example
Find the 10th term of the arithmetic sequence \(5, 8, 11, \ldots\)
Solution: \(u_1=5,\ d=3\). So \(u_{10}=5+(10-1)\times3=5+27=\mathbf{32}\)
The first term of an arithmetic sequence is 3 and the common difference is 4. What is the 15th term?
\(u_{15} = 3 + (15-1)\times 4 = 3 + 56 = \mathbf{59}\)
A) 55 B) 59 C) 63 D) 51
Find the sum of the first 5 terms of the geometric series with \(u_1 = 2\) and common ratio \(r = 3\).
\(S_5 = \dfrac{2(3^5-1)}{3-1} = \dfrac{2 \times 242}{2} = \mathbf{242}\)
A) 120 B) 160 C) 242 D) 486
Evaluate \(\log_2 32\).
\(2^x = 32 = 2^5 \Rightarrow x = \mathbf{5}\)
A) 4 B) 5 C) 6 D) 3
Key Concepts to Memorise
- Inverse: if \(f(x)=y\), then \(f^{-1}(y)=x\). Reflect in the line \(y=x\).
- Vertex of quadratic: \(x = -\dfrac{b}{2a}\), vertex \(= \left(-\dfrac{b}{2a},\ f\!\left(-\dfrac{b}{2a}\right)\right)\)
- Exponential: \(a^x = b \Rightarrow x = \log_a b\)
Vertex form: \(f(x) = a(x-h)^2 + k\), vertex at \((h, k)\)
Worked Example
Find the vertex of \(f(x)=x^2-6x+5\).
Solution: \(x = -\tfrac{-6}{2}=3\). \(y = 9-18+5 = -4\). Vertex: \(\mathbf{(3,-4)}\).
Let \(f(x) = 2x + 3\). Find \(f^{-1}(7)\).
\(f^{-1}(y) = \dfrac{y-3}{2}\). So \(f^{-1}(7) = \dfrac{7-3}{2} = \mathbf{2}\)
A) 2 B) 4 C) 5 D) 3
Find the vertex of \(f(x) = x^2 - 4x + 3\).
\(x = -\dfrac{-4}{2}=2\). \(y = 4-8+3 = -1\). Vertex: \(\mathbf{(2,-1)}\).
A) (2,3) B) (2,1) C) (2,−1) D) (−2,−1)
Solve for \(x\): \(2^x = 16\).
\(2^x = 2^4 \Rightarrow x = \mathbf{4}\)
A) 3 B) 4 C) 5 D) 8
Key Concepts to Memorise
- Exact values: \(\sin30°=\tfrac{1}{2},\ \cos60°=\tfrac{1}{2},\ \tan45°=1,\ \sin45°=\tfrac{\sqrt{2}}{2}\)
- Sine rule: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\)
- Cosine rule: \(a^2 = b^2+c^2-2bc\cos A\)
- Area of triangle: \(\text{Area} = \tfrac{1}{2}ab\sin C\)
\(\sin^2\theta + \cos^2\theta = 1\)
Worked Example
In triangle \(ABC\), \(a=10,\ A=30°,\ B=60°\). Find \(b\).
Solution: \(\dfrac{b}{\sin 60°}=\dfrac{10}{\sin 30°}\Rightarrow b = \dfrac{10\times\frac{\sqrt{3}}{2}}{\frac{1}{2}} = 10\sqrt{3}\approx17.3\)
What is the exact value of \(\sin 30°\)?
From the unit circle, \(\sin 30° = \mathbf{\dfrac{1}{2}}\). Note: \(\sin45°=\tfrac{\sqrt{2}}{2}\) and \(\sin60°=\tfrac{\sqrt{3}}{2}\).
A) 1/2 B) √2/2 C) √3/2 D) 1
In triangle \(ABC\), \(a = 8\), \(\hat{A} = 30°\) and \(\hat{B} = 45°\). Find the exact value of side \(b\).
\(\dfrac{b}{\sin 45°} = \dfrac{8}{\sin 30°} \Rightarrow b = \dfrac{8 \times \frac{\sqrt{2}}{2}}{\frac{1}{2}} = \dfrac{4\sqrt{2}}{\frac{1}{2}} = \mathbf{8\sqrt{2}}\)
A) 4√2 B) 4√6 C) 8√2 D) 8√3
In triangle \(ABC\), \(b = 5\), \(c = 7\) and \(\hat{A} = 60°\). Find the exact length of side \(a\).
\(a^2 = 5^2+7^2-2(5)(7)\cos60° = 25+49-70\times\tfrac{1}{2} = 74-35 = 39\). So \(a = \mathbf{\sqrt{39}}\).
A) √29 B) √39 C) √49 D) √74
Topic 5 · Calculus — Differentiation
Key Concepts to Memorise
- Power rule: \(\dfrac{d}{dx}x^n = nx^{n-1}\)
- Chain rule: \(\dfrac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)\)
- Gradient at a point: substitute \(x\)-value into \(f'(x)\)
- Stationary points: set \(f'(x) = 0\) and solve
\(\frac{d}{dx}(ax^n) = anx^{n-1}\)
Worked Example
Differentiate \(f(x) = 5x^3 - 2x + 1\).
Solution: \(f'(x) = 15x^2 - 2\)
Find \(\dfrac{d}{dx}\!\left(3x^4 - 2x^2 + 5\right)\).
\(\dfrac{d}{dx}(3x^4) = 12x^3\), \(\dfrac{d}{dx}(-2x^2) = -4x\), \(\dfrac{d}{dx}(5)=0\). Answer: \(\mathbf{12x^3-4x}\).
A) 12x³+4x B) 3x³−2x C) 12x⁴−4x D) 12x³−4x
Find the gradient of \(f(x) = (2x+1)^3\) at \(x = 0\).
\(f'(x) = 3(2x+1)^2 \cdot 2 = 6(2x+1)^2\). At \(x=0\): \(f'(0)=6(1)^2 = \mathbf{6}\).
A) 3 B) 6 C) 12 D) 24
Topic 5 · Calculus — Integration
Key Concepts to Memorise
- Power rule: \(\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C\quad (n\neq -1)\)
- Definite integral: \(\displaystyle\int_a^b f(x)\,dx = [F(x)]_a^b = F(b)-F(a)\)
- Area between curve and \(x\)-axis: \(\displaystyle A = \int_a^b |f(x)|\,dx\)
\(\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\)
Worked Example
Evaluate \(\displaystyle\int_0^1 4x^3\,dx\).
Solution: \([x^4]_0^1 = 1 - 0 = \mathbf{1}\)
Evaluate \(\displaystyle\int_0^2 (3x^2 + 2x)\,dx\).
\(\displaystyle\int_0^2(3x^2+2x)\,dx = \bigl[x^3+x^2\bigr]_0^2 = (8+4)-(0) = \mathbf{12}\)
A) 8 B) 10 C) 12 D) 14
Topic 4 · Statistics & Probability
Key Concepts to Memorise
- Standard deviation (population): \(\sigma = \sqrt{\dfrac{\sum(x-\bar{x})^2}{n}}\)
- Independent events: \(P(A\cap B) = P(A)\cdot P(B)\)
- Addition rule: \(P(A\cup B) = P(A)+P(B)-P(A\cap B)\)
- Binomial: \(P(X=k) = \dbinom{n}{k}p^k(1-p)^{n-k}\)
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
Worked Example
Data: \(1, 3, 5\). Find the population standard deviation.
Solution: \(\bar{x}=3\). Variance \(=\tfrac{(1-3)^2+(3-3)^2+(5-3)^2}{3}=\tfrac{8}{3}\). \(\sigma = \sqrt{\tfrac{8}{3}} \approx 1.63\)
The data set is \(\{2, 4, 6, 8, 10\}\). Find the population standard deviation.
\(\bar{x}=6\). Variance \(= \dfrac{16+4+0+4+16}{5} = \dfrac{40}{5}=8\). \(\sigma = \sqrt{8} = 2\sqrt{2} \approx 2.83\).
A) √6 B) 2√2 C) 3 D) 4
Events \(A\) and \(B\) are independent with \(P(A)=0.3\) and \(P(B)=0.5\). Find \(P(A\cup B)\).
Independent: \(P(A\cap B)=0.3\times0.5=0.15\). \(P(A\cup B)=0.3+0.5-0.15=\mathbf{0.65}\).
A) 0.15 B) 0.80 C) 0.65 D) 0.50
If \(X\sim B(10,\,0.3)\), find \(P(X=3)\) correct to 3 significant figures.
\(P(X=3)=\dbinom{10}{3}(0.3)^3(0.7)^7 = 120\times0.027\times0.0824 \approx \mathbf{0.267}\)
A) 0.267 B) 0.300 C) 0.233 D) 0.147
If \(X\sim N(50,\,10^2)\), find \(P(X>60)\) correct to 3 significant figures.
\(Z = \dfrac{60-50}{10} = 1\). \(P(Z>1)=1-\Phi(1)\approx 1-0.841 = \mathbf{0.159}\)
A) 0.841 B) 0.159 C) 0.500 D) 0.023
Key Concepts to Memorise
- Magnitude: \(|\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2}\)
- Dot product: \(\mathbf{a}\cdot\mathbf{b} = a_1 b_1+a_2 b_2+a_3 b_3\)
- Angle between vectors: \(\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\)
- Parallel: \(\mathbf{a}=k\mathbf{b}\). Perpendicular: \(\mathbf{a}\cdot\mathbf{b}=0\)
\(\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta\)
Worked Example
Find \(\mathbf{a}\cdot\mathbf{b}\) where \(\mathbf{a}=(2,1,-1)\) and \(\mathbf{b}=(3,-2,4)\).
Solution: \(2\times3+1\times(-2)+(-1)\times4=6-2-4=\mathbf{0}\). The vectors are perpendicular.
Find \(\mathbf{a}\cdot\mathbf{b}\) where \(\mathbf{a}=(1,2,3)\) and \(\mathbf{b}=(2,-1,1)\).
\(\mathbf{a}\cdot\mathbf{b} = (1)(2)+(2)(-1)+(3)(1) = 2-2+3 = \mathbf{3}\)
A) 3 B) 0 C) 7 D) −3
Find the magnitude of vector \(\mathbf{v} = (3,\,4)\).
\(|\mathbf{v}| = \sqrt{3^2+4^2} = \sqrt{9+16} = \sqrt{25} = \mathbf{5}\)
A) 7 B) √7 C) 5 D) 25
Mixed Topics · Final Questions
Evaluate the definite integral \(\displaystyle\int_1^4 (x^2-1)\,dx\).
\(\displaystyle\int_1^4(x^2-1)\,dx = \left[\frac{x^3}{3}-x\right]_1^4\)
\(= \left(\frac{64}{3}-4\right)-\left(\frac{1}{3}-1\right) = \frac{64}{3}-4-\frac{1}{3}+1 = \frac{63}{3}-3 = 21-3 = \mathbf{18}\)
A) 15 B) 20 C) 21 D) 18
An investment of \$1000 earns 5% annual interest compounded annually. What is the value after 3 years? Give your answer to the nearest cent.
\(A = 1000\times(1.05)^3 = 1000\times1.157625 = \mathbf{\$1157.63}\)
A) $1150.00 B) $1157.63 C) $1200.00 D) $1050.00
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