📖 Concept Review & Key Formulae
T01
Arithmetic Sequences & Series
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\( u_n = u_1 + (n-1)d \) | \( S_n = \dfrac{n}{2}(2u_1 + (n-1)d) = \dfrac{n}{2}(u_1 + u_n) \)
Worked Example
Find the 20th term of the sequence \(3, 8, 13, 18, \ldots\)
\( u_1 = 3,\; d = 5 \implies u_{20} = 3 + 19(5) = \mathbf{98} \)
T02
Geometric Sequences & Series
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\( u_n = u_1 \cdot r^{n-1} \) | \( S_n = \dfrac{u_1(r^n - 1)}{r-1},\; r \neq 1 \) | \( S_\infty = \dfrac{u_1}{1-r},\; |r|<1 \)
Worked Example
Find the 5th term of \(2, 6, 18, 54, \ldots\)
\( u_1 = 2,\; r = 3 \implies u_5 = 2 \times 3^4 = \mathbf{162} \)
T03
Exponents & Logarithms
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\( a^m \cdot a^n = a^{m+n} \) | \( \log_a(xy) = \log_a x + \log_a y \) | \( a^x = b \Leftrightarrow x = \log_a b \)
Worked Example
Evaluate \(\log_2 32\).
\( 2^x = 32 = 2^5 \implies x = \mathbf{5} \)
T04
Quadratic Functions
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\( x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \) | \( f(x)=a(x-h)^2+k \), vertex \((h,k)\) | \(\Delta = b^2-4ac\)
Worked Example
Solve \(x^2 - 5x + 6 = 0\). Discriminant: \(\Delta = 25 - 24 = 1 > 0\). Roots: \(x = 3\) or \(x = 2\).
T05
Trigonometry
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Exact values: \(\sin 30°= \tfrac{1}{2}\), \(\cos 60°= \tfrac{1}{2}\), \(\tan 45°= 1\), \(\sin 45°= \tfrac{\sqrt{2}}{2}\)
Sine rule: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\) | Cosine rule: \(c^2 = a^2 + b^2 - 2ab\cos C\)
Sine rule: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\) | Cosine rule: \(c^2 = a^2 + b^2 - 2ab\cos C\)
Worked Example
Find the exact value of \(\tan 45°\).
\(\tan 45° = \dfrac{\sin 45°}{\cos 45°} = \dfrac{\sqrt{2}/2}{\sqrt{2}/2} = \mathbf{1}\)
T06
Binomial Theorem
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\( (a+b)^n = \displaystyle\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \) | \(\binom{n}{k} = \dfrac{n!}{k!(n-k)!}\)
Worked Example
Find the coefficient of \(x^2\) in \((1+x)^5\).
\(\binom{5}{2} = \dfrac{5!}{2!\,3!} = \mathbf{10}\)
T07
Functions & Composite Functions
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Domain: all valid inputs. Range: all outputs produced.
\( (f \circ g)(x) = f(g(x)) \) — apply \(g\) first, then \(f\).
\( (f \circ g)(x) = f(g(x)) \) — apply \(g\) first, then \(f\).
Worked Example
If \(f(x)=2x+1\) and \(g(x)=x^2\), find \(g(f(2))\).
\(f(2) = 5\), then \(g(5) = 25\). Answer: \(\mathbf{25}\).
T08
Differentiation
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Power rule: \(\dfrac{d}{dx}[x^n] = nx^{n-1}\)
Chain rule: \(\dfrac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
Chain rule: \(\dfrac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
Worked Example
If \(f(x) = x^3 - 4x\), find \(f'(2)\).
\(f'(x) = 3x^2 - 4 \implies f'(2) = 12 - 4 = \mathbf{8}\)
T09
Integration
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\(\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C \quad (n \neq -1)\)
\(\displaystyle\int_a^b f(x)\,dx = [F(x)]_a^b = F(b) - F(a)\)
\(\displaystyle\int_a^b f(x)\,dx = [F(x)]_a^b = F(b) - F(a)\)
Worked Example
Evaluate \(\displaystyle\int_0^3 2x\,dx\).
\([x^2]_0^3 = 9 - 0 = \mathbf{9}\)
T10
Probability, Statistics & Vectors
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\(P(A \cup B) = P(A)+P(B)-P(A \cap B)\)
Normal: \(Z = \dfrac{X-\mu}{\sigma}\) | \(|\mathbf{v}| = \sqrt{v_1^2+v_2^2}\) | \(\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2\)
Normal: \(Z = \dfrac{X-\mu}{\sigma}\) | \(|\mathbf{v}| = \sqrt{v_1^2+v_2^2}\) | \(\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2\)
Worked Example
Find the magnitude of vector \(\mathbf{v} = \begin{pmatrix}3\\4\end{pmatrix}\).
\(|\mathbf{v}| = \sqrt{9+16} = \sqrt{25} = \mathbf{5}\)
✏️ Practice Questions
1
The first term of an arithmetic sequence is \(3\) and the common difference is \(5\). What is the 20th term?
Sequences
2
The first term of a geometric sequence is \(2\) and the common ratio is \(3\). Find the 5th term.
Sequences
3
Find the sum of the first 10 terms of the arithmetic series with first term \(1\) and common difference \(2\).
Series
4
Simplify \(2^3 \times 2^5\).
Exponents
5
Evaluate \(\log_2 32\).
Logarithms
6
What are the solutions to \(x^2 - 5x + 6 = 0\)?
Quadratics
7
The function \(f(x) = (x-3)^2 + 4\). What is the minimum value of \(f(x)\)?
Quadratics
8
What is the exact value of \(\sin 30°\)?
Trigonometry
9
What is the exact value of \(\tan 45°\)?
Trigonometry
10
Find the coefficient of \(x^2\) in the expansion of \((1+x)^5\).
Binomial Theorem
11
What is the domain of \(f(x) = \sqrt{x - 2}\)?
Functions
12
Let \(f(x) = 2x + 1\) and \(g(x) = x^2\). Find \(g(f(2))\).
Functions
13
If \(f(x) = x^3 - 4x\), find \(f'(2)\).
Differentiation
14
Evaluate \(\displaystyle\int_0^3 2x\,dx\).
Integration
15
For events \(A\) and \(B\), \(P(A) = 0.4\), \(P(B) = 0.5\), and \(P(A \cap B) = 0.2\). Find \(P(A \cup B)\).
Probability
16
\(X \sim N(50, 10^2)\). Which \(z\)-score corresponds to \(X = 60\)?
Statistics
17
Find the magnitude of the vector \(\mathbf{v} = \begin{pmatrix}3\\4\end{pmatrix}\).
Vectors
18
Calculate the dot product \(\begin{pmatrix}2\\3\end{pmatrix} \cdot \begin{pmatrix}4\\-1\end{pmatrix}\).
Vectors
19
Find the mean of the data set \(\{2, 4, 6, 8, 10\}\).
Statistics
20
An investment of \(\$1000\) earns \(5\%\) interest per year, compounded annually. What is the value after 3 years (to the nearest dollar)?
Financial Maths
FINAL SCORE
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Answer Key & Full Solutions
Detailed step-by-step explanations for all 20 questions
1
C — 98
\(u_{20} = u_1 + (n-1)d = 3 + 19 \times 5 = 3 + 95 = \mathbf{98}\)
2
B — 162
\(u_5 = 2 \times 3^{5-1} = 2 \times 81 = \mathbf{162}\)
3
B — 100
\(S_{10} = \dfrac{10}{2}(2(1) + 9(2)) = 5 \times 20 = \mathbf{100}\)
4
C — \(2^8\)
\(2^3 \times 2^5 = 2^{3+5} = 2^8 = 256\). When multiplying with the same base, add the exponents.
5
B — 5
\(\log_2 32 = x \Leftrightarrow 2^x = 32 = 2^5 \implies x = \mathbf{5}\)
6
B — \(x = 2\) and \(x = 3\)
Factor: \(x^2-5x+6 = (x-2)(x-3) = 0 \implies x = 2\) or \(x = 3\). Verify: \(2+3=5\) ✓, \(2 \times 3=6\) ✓.
7
B — 4
In vertex form \(f(x)=(x-3)^2+4\), the vertex is \((3,4)\). Since \((x-3)^2 \geq 0\), the minimum is \(f(3) = 0 + 4 = \mathbf{4}\).
8
B — \(\frac{1}{2}\)
From the unit circle or 30-60-90 triangle: \(\sin 30° = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{1}{2}\).
9
D — 1
\(\tan 45° = \dfrac{\sin 45°}{\cos 45°} = \dfrac{\sqrt{2}/2}{\sqrt{2}/2} = \mathbf{1}\). In the 45-45-90 triangle, opposite = adjacent.
10
B — 10
The \(x^2\) term in \((1+x)^5\) is \(\binom{5}{2}x^2\). \(\binom{5}{2} = \dfrac{5!}{2!\,3!} = \dfrac{20}{2} = \mathbf{10}\).
11
B — \(x \geq 2\)
The expression under the square root must be non-negative: \(x - 2 \geq 0 \implies x \geq 2\). Domain: \([2, \infty)\).
12
B — 25
Step 1: \(f(2) = 2(2)+1 = 5\). Step 2: \(g(f(2)) = g(5) = 5^2 = \mathbf{25}\).
13
B — 8
Differentiate: \(f'(x) = 3x^2 - 4\). Substitute \(x=2\): \(f'(2) = 3(4) - 4 = 12 - 4 = \mathbf{8}\).
14
B — 9
\(\displaystyle\int_0^3 2x\,dx = \left[x^2\right]_0^3 = 3^2 - 0^2 = 9 - 0 = \mathbf{9}\).
15
B — 0.7
\(P(A \cup B) = P(A)+P(B)-P(A \cap B) = 0.4 + 0.5 - 0.2 = \mathbf{0.7}\).
16
B — \(z = 1.0\)
\(z = \dfrac{X - \mu}{\sigma} = \dfrac{60 - 50}{10} = \dfrac{10}{10} = \mathbf{1.0}\).
17
B — 5
\(|\mathbf{v}| = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = \mathbf{5}\). This is the 3-4-5 Pythagorean triple.
18
B — 5
\(\begin{pmatrix}2\\3\end{pmatrix} \cdot \begin{pmatrix}4\\-1\end{pmatrix} = (2)(4) + (3)(-1) = 8 - 3 = \mathbf{5}\).
19
B — 6
\(\bar{x} = \dfrac{2+4+6+8+10}{5} = \dfrac{30}{5} = \mathbf{6}\).
20
B — \(\$1158\)
\(A = P(1+r)^t = 1000(1.05)^3 = 1000 \times 1.157625 \approx \mathbf{\$1158}\).