📐Concept Review & Key Formulae

Arithmetic Sequences & Series

aₙ = a₁ + (n−1)d
Sₙ = n/2 · (2a₁ + (n−1)d)

d = common difference. S₁₀ with a₁=3, d=4: S₁₀ = 5·(6+36) = 210

Geometric Sequences & Series

aₙ = a₁ · rⁿ⁻¹
S∞ = a₁/(1−r), |r| < 1

r = common ratio. Converges only when |r| < 1.

Exponents & Logarithms

logₐ(xy) = logₐx + logₐy
logₐ(xⁿ) = n·logₐx

Key identity: a^(log_a x) = x. Change of base: log_b(x) = ln(x)/ln(b)

Quadratic Functions

f(x) = ax² + bx + c
vertex: x = −b/(2a)

Discriminant Δ = b²−4ac. Two real roots ↔ Δ > 0.

Binomial Theorem

(a+b)ⁿ = Σ C(n,r)·aⁿ⁻ʳ·bʳ
C(n,r) = n! / (r!(n−r)!)

The coefficient of xʳ term: C(n,r)·(coeff of x)ʳ·(const)ⁿ⁻ʳ

Differentiation

d/dx[xⁿ] = nxⁿ⁻¹
Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x)

Product rule: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v−uv')/v²

Integration

∫xⁿ dx = xⁿ⁺¹/(n+1) + C
∫ₐᵇ f(x)dx = F(b) − F(a)

Area between curves: ∫|f(x)−g(x)|dx. Always find antiderivative first.

Trigonometry

Sine rule: a/sinA = b/sinB
Cosine rule: c²=a²+b²−2ab·cosC

Exact values: sin30°=½, sin45°=√2/2, sin60°=√3/2, cos60°=½

Probability

P(A∪B) = P(A)+P(B)−P(A∩B)
P(A|B) = P(A∩B)/P(B)

Independent events: P(A∩B) = P(A)·P(B). Mutually exclusive: P(A∩B)=0.

Normal Distribution & Statistics

Z = (X − μ) / σ
X ~ N(μ, σ²)

68% of data within 1σ, 95% within 2σ, 99.7% within 3σ of mean.

Vectors

a · b = |a||b|cosθ
a · b = a₁b₁ + a₂b₂ (+ a₃b₃)

Perpendicular ↔ a·b = 0. |a| = √(a₁²+a₂²). Parallel ↔ a = kb.

Exponential Models

N(t) = N₀ · e^(kt)
Half-life: t = ln(2)/k

k > 0: growth. k < 0: decay. To find t, isolate eˢᵒᵐᵉᵗʰⁱⁿᵍ, then take ln.

📋 Worked Example — Exam Technique

Find the sum of the first 8 terms of the arithmetic sequence 5, 9, 13, …
a₁ = 5, d = 4, n = 8
S₈ = 8/2 · (2·5 + 7·4) = 4 · (10 + 28) = 4 · 38 = 152

Find the coefficient of x² in the expansion of (3x + 2)⁴.
r = 2: C(4,2)·(3x)²·2² = 6 · 9x² · 4 = 216x² → coefficient = 216
Examination Questions
01
Arithmetic Series
The arithmetic sequence has first term \(a_1 = 3\) and common difference \(d = 4\). Find \(S_{10}\), the sum of the first 10 terms.
✦ Worked Solution
Use \(S_n = \frac{n}{2}(2a_1 + (n-1)d)\)
\(S_{10} = \frac{10}{2}(2 \cdot 3 + 9 \cdot 4) = 5(6 + 36) = 5 \times 42\)
Answer: C — \(S_{10} = 210\)
02
Geometric Sequences
A geometric sequence has first term \(a_1 = 2\) and common ratio \(r = 3\). Find the 5th term \(a_5\).
✦ Worked Solution
Use \(a_n = a_1 \cdot r^{n-1}\)
\(a_5 = 2 \cdot 3^{4} = 2 \cdot 81 = 162\)
Answer: B — \(a_5 = 162\)
03
Logarithms
Evaluate \(\log_2 32\).
✦ Worked Solution
We need \(x\) such that \(2^x = 32\)
\(32 = 2^5\), so \(\log_2 32 = 5\)
Answer: B — \(\log_2 32 = 5\)
04
Quadratic Functions
Find the coordinates of the vertex of \(f(x) = x^2 - 6x + 8\).
✦ Worked Solution
Vertex x-coordinate: \(x = \dfrac{-b}{2a} = \dfrac{6}{2} = 3\)
\(f(3) = 9 - 18 + 8 = -1\)
Answer: A — Vertex at \((3,\ -1)\)
05
Binomial Theorem
Find the coefficient of \(x^3\) in the expansion of \((2x + 1)^5\).
✦ Worked Solution
The \(x^3\) term uses \(r = 3\) (power of \(2x\) is 3):
\(\binom{5}{3}(2x)^3(1)^2 = 10 \cdot 8x^3 \cdot 1 = 80x^3\)
Answer: B — coefficient of \(x^3\) is 80
06
Differentiation
Given \(f(x) = 3x^4 - 2x^2 + 5\), find \(f'(1)\).
✦ Worked Solution
\(f'(x) = 12x^3 - 4x\)
\(f'(1) = 12(1)^3 - 4(1) = 12 - 4 = 8\)
Answer: C — \(f'(1) = 8\)
07
Definite Integration
Evaluate \(\displaystyle\int_0^3 (2x + 1)\,dx\).
✦ Worked Solution
Antiderivative: \(F(x) = x^2 + x\)
\(F(3) - F(0) = (9 + 3) - (0 + 0) = 12\)
Answer: D — the integral equals 12
08
Trigonometry — Exact Values
Find the exact value of \(\sin 60°\).
✦ Worked Solution
From the 30-60-90 triangle: sides are 1, √3, 2
\(\sin 60° = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{\sqrt{3}}{2}\)
Answer: C — \(\sin 60° = \dfrac{\sqrt{3}}{2}\)
09
Sine Rule
In triangle \(ABC\), side \(a = 8\), angle \(A = 30°\), and angle \(B = 45°\). Find the length of side \(b\).
✦ Worked Solution
Sine rule: \(\dfrac{b}{\sin B} = \dfrac{a}{\sin A}\)
\(b = \dfrac{8 \sin 45°}{\sin 30°} = \dfrac{8 \cdot \tfrac{\sqrt{2}}{2}}{\tfrac{1}{2}} = 8\sqrt{2}\)
Answer: B — \(b = 8\sqrt{2}\)
10
Probability — Addition Rule
Events \(A\) and \(B\) have \(P(A) = 0.4\), \(P(B) = 0.5\), and \(P(A \cap B) = 0.2\). Find \(P(A \cup B)\).
✦ Worked Solution
Addition rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
\(= 0.4 + 0.5 - 0.2 = 0.7\)
Answer: C — \(P(A \cup B) = 0.7\)
11
Normal Distribution
The marks in an exam are normally distributed with mean \(\mu = 70\) and standard deviation \(\sigma = 5\). A student scores 75. Find the standardised score (\(z\)-score).
✦ Worked Solution
\(z = \dfrac{X - \mu}{\sigma} = \dfrac{75 - 70}{5} = \dfrac{5}{5} = 1\)
Answer: B — \(z = 1\)
12
Vectors — Dot Product
Find the scalar (dot) product of vectors \(\mathbf{a} = \begin{pmatrix}3\\4\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}1\\-2\end{pmatrix}\).
✦ Worked Solution
\(\mathbf{a} \cdot \mathbf{b} = (3)(1) + (4)(-2) = 3 - 8 = -5\)
Answer: A — dot product is \(-5\)
13
Composite Functions
Let \(f(x) = 2x + 1\) and \(g(x) = x^2\). Find \(f(g(3))\).
✦ Worked Solution
First compute \(g(3) = 3^2 = 9\)
Then \(f(g(3)) = f(9) = 2(9) + 1 = 19\)
Answer: C — \(f(g(3)) = 19\)
14
Logarithm Laws
Simplify \(\log_6 4 + \log_6 9\).
✦ Worked Solution
\(\log_6 4 + \log_6 9 = \log_6(4 \times 9) = \log_6 36\)
\(36 = 6^2\), so \(\log_6 36 = 2\)
Answer: B — the expression equals 2
15
Discriminant
For what values of \(k\) does \(f(x) = x^2 - 5x + k\) have two distinct real roots?
✦ Worked Solution
Two distinct real roots require discriminant \(\Delta > 0\)
\(\Delta = b^2 - 4ac = 25 - 4k > 0 \implies k < \dfrac{25}{4}\)
Answer: C — \(k < \dfrac{25}{4}\)
16
Chain Rule
Given \(y = (2x + 1)^3\), find \(\dfrac{dy}{dx}\) at \(x = 1\).
✦ Worked Solution
Chain rule: \(\dfrac{dy}{dx} = 3(2x+1)^2 \cdot 2 = 6(2x+1)^2\)
At \(x=1\): \(6(2+1)^2 = 6 \times 9 = 54\)
Answer: B — \(\dfrac{dy}{dx}\bigg|_{x=1} = 54\)
17
Definite Integration — Area
Evaluate \(\displaystyle\int_1^4 (3x^2 - 2x)\,dx\).
✦ Worked Solution
Antiderivative: \(F(x) = x^3 - x^2\)
\(F(4) - F(1) = (64 - 16) - (1 - 1) = 48 - 0 = 48\)
Answer: C — the integral equals 48
18
Cosine Rule
In triangle \(ABC\), \(a = 5\), \(b = 7\), and angle \(C = 60°\). Find the length of side \(c\).
✦ Worked Solution
\(c^2 = a^2 + b^2 - 2ab\cos C = 25 + 49 - 2(5)(7)\cos 60°\)
\(= 74 - 70 \times \dfrac{1}{2} = 74 - 35 = 39\)
Answer: B — \(c = \sqrt{39}\)
19
Binomial Distribution
\(X \sim B(5,\ 0.3)\). Find \(P(X = 2)\), giving your answer to 4 significant figures.
✦ Worked Solution
\(P(X=2) = \binom{5}{2}(0.3)^2(0.7)^3\)
\(= 10 \times 0.09 \times 0.343 = 0.3087\)
Answer: C — \(P(X=2) = 0.3087\)
20
Exponential Decay
A radioactive substance decays according to \(N(t) = 200\,e^{-0.05t}\). Find the time \(t\) (in years) when \(N = 100\). Give your answer to 2 decimal places.
✦ Worked Solution
\(100 = 200\,e^{-0.05t} \implies e^{-0.05t} = 0.5\)
\(-0.05t = \ln(0.5) \implies t = \dfrac{-\ln 0.5}{0.05} = \dfrac{\ln 2}{0.05} \approx 13.86\)
Answer: C — \(t \approx 13.86\) years
FINAL RESULT
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📋 Answer Key & Worked Solutions