Topic 1 · Algebra
Sequences, Series & the Binomial Theorem
A geometric series with first term \(u_1\) and ratio \(r\) converges if and only if \(|r|<1\), with sum to infinity \(S_\infty = \frac{u_1}{1-r}\).
Key Formulae to Memorise
u_n = u_1 · r^(n−1) [nth geometric term]
S_n = u_1(1−r^n)/(1−r), r ≠ 1
Binomial: (a+b)^n = Σ C(n,k) a^(n−k) b^k, k=0..n
Worked Example
Find the sum to infinity of the series \(6 + 4 + \frac{8}{3} + \cdots\)
Here \(u_1=6\), \(r=\frac{2}{3}\). Since \(|r|<1\): \(S_\infty = \frac{6}{1-\frac{2}{3}} = \frac{6}{\frac{1}{3}} = 18\).
01
The first three terms of a geometric sequence are \(k+2,\; 3k,\; 5k+4\) where \(k \in \mathbb{R}\).
- Show that \(3k^2 - 2k - 8 = 0\) and hence find the two possible values of \(k\).
- Given that the series converges, find the sum to infinity.
Enter the sum to infinity (part b):
02
Find the coefficient of \(x^3\) in the expansion of \(\left(2x - \dfrac{1}{x}\right)^7\).
Enter the coefficient (integer):
Topic 2 · Functions
Inverse Functions & Transformations
A function \(f\) has an inverse \(f^{-1}\) if and only if \(f\) is bijective (one-to-one and onto). The graph of \(f^{-1}\) is the reflection of \(f\) in the line \(y=x\).
Key Results
Domain of f⁻¹ = Range of f
f(f⁻¹(x)) = x and f⁻¹(f(x)) = x
Quadratic: complete the square → restrict domain → find inverse
Worked Example
Let \(f(x)=\ln(2x-1)\) for \(x>\frac{1}{2}\). Find \(f^{-1}(x)\).
Let \(y=\ln(2x-1)\Rightarrow e^y=2x-1\Rightarrow x=\frac{e^y+1}{2}\). Hence \(f^{-1}(x)=\frac{e^x+1}{2}\), domain \(\mathbb{R}\).
03
Let \(f(x) = e^{2x+1}\) and \(g(x) = \dfrac{x-3}{2}\), for \(x \in \mathbb{R}\).
- Find \((f \circ g)(x)\).
- Find the value of \(x\) for which \(f^{-1}(x) = g(x)\).
Enter the x-value for part (b), to 3 significant figures:
Topic 3 · Trigonometry & Geometry
Compound Angles & General Solutions
Essential Identities
sin(A±B) = sinAcosB ± cosAsinB
cos(2A) = 1 − 2sin²A = 2cos²A − 1 = cos²A − sin²A
R·sin(θ+α): R=√(a²+b²), tanα = b/a
General: sinθ=k ⟹ θ = arcsin k + 2nπ or π − arcsin k + 2nπ
Worked Example
Write \(3\sin\theta + 4\cos\theta\) in the form \(R\sin(\theta+\alpha)\).
\(R=\sqrt{9+16}=5\), \(\tan\alpha=\frac{4}{3}\Rightarrow\alpha\approx0.927\text{ rad}\). Answer: \(5\sin(\theta+0.927)\).
04
Solve the equation \(2\cos^2\theta - \cos\theta - 1 = 0\) for \(\theta \in [0,\, 2\pi]\). Give your answers in exact form.
- Factor the equation and find all solutions.
- State the number of solutions in the given interval.
Enter the number of solutions (part b):
05
Given that \(\sin\alpha = \dfrac{3}{5}\) and \(0 < \alpha < \dfrac{\pi}{2}\), and \(\cos\beta = -\dfrac{5}{13}\) with \(\dfrac{\pi}{2} < \beta < \pi\),
- Find the exact value of \(\cos(\alpha + \beta)\).
Enter exact decimal or fraction (e.g. -0.923 or type -12/65 ≈ value):
Topic 4 · Complex Numbers
Modulus-Argument Form & De Moivre's Theorem
Polar Form
z = r(cosθ + i sinθ) = re^(iθ), r=|z|, θ=arg(z)
De Moivre: z^n = r^n(cos nθ + i sin nθ)
nth roots: z_k = r^(1/n) · e^(i(θ+2kπ)/n), k=0,1,...,n−1
Worked Example
Write \(z = 1 + i\sqrt{3}\) in polar form and find \(z^6\).
\(r=2\), \(\theta=\frac{\pi}{3}\). So \(z=2e^{i\pi/3}\). Then \(z^6=2^6 e^{i\cdot 2\pi}=64(\cos 2\pi+i\sin 2\pi)=64\).
06
Let \(z = \sqrt{3} - i\).
- Write \(z\) in the form \(r e^{i\theta}\) where \(-\pi < \theta \leq \pi\).
- Find the real part of \(z^{10}\).
Enter the real part of z¹⁰ (integer):
07
The equation \(z^3 = -8\) has three roots in \(\mathbb{C}\).
- Find all three roots in the form \(a+bi\) where \(a,b\in\mathbb{R}\).
- Show that the roots form the vertices of an equilateral triangle in the Argand plane and state the side length.
Enter the side length of the triangle (exact decimal):
Topic 5 · Vectors
Vector Lines, Planes & Intersections
Essential Formulae
Line: r = a + λb
Plane: r·n = a·n (normal vector form)
Angle between lines: cosθ = |b₁·b₂| / (|b₁||b₂|)
Distance point to plane: d = |a·n − c| / |n|
Worked Example
Find the angle between lines \(\mathbf{r}_1 = \begin{pmatrix}1\\0\\0\end{pmatrix}+\lambda\begin{pmatrix}1\\1\\0\end{pmatrix}\) and \(\mathbf{r}_2=\mu\begin{pmatrix}0\\1\\1\end{pmatrix}\).
\(\cos\theta=\frac{|(1)(0)+(1)(1)+(0)(1)|}{\sqrt{2}\cdot\sqrt{2}}=\frac{1}{2}\). So \(\theta=60°\).
08
Two planes \(\Pi_1\) and \(\Pi_2\) are given by:
\[\Pi_1:\; 2x - y + z = 4 \quad \text{and} \quad \Pi_2:\; x + y - 2z = 1\]
- Find the acute angle between the planes, giving your answer in degrees to one decimal place.
- Find the vector equation of their line of intersection.
Enter the acute angle in degrees (1 d.p.):
09
Line \(L\) passes through \(A(1,2,-1)\) with direction vector \(\mathbf{d} = \begin{pmatrix}2\\-1\\3\end{pmatrix}\). Point \(P\) has coordinates \((3, 0, 5)\).
- Find the shortest distance from \(P\) to line \(L\), giving your answer to 3 significant figures.
Enter the shortest distance (3 s.f.):
Topic 6 · Statistics & Probability
Conditional Probability & Distributions
Key Formulae
P(A|B) = P(A∩B) / P(B)
Bayes: P(A|B) = P(B|A)·P(A) / P(B)
Normal: X~N(μ,σ²), standardise Z=(X−μ)/σ
Binomial: P(X=k) = C(n,k)·p^k·(1−p)^(n−k)
Worked Example
If \(P(A)=0.4\), \(P(B)=0.5\), \(P(A\cap B)=0.2\), find \(P(A|B)\).
\(P(A|B)=\frac{0.2}{0.5}=0.4\). Note \(A\) and \(B\) are independent since \(P(A|B)=P(A)\).
10
A factory has two machines. Machine A produces 60% of items; machine B produces 40%. The defect rate for A is 3% and for B is 5%. An item is selected at random and found to be defective.
- Find the probability that it came from machine A, to 3 decimal places.
Enter P(A | defective) to 3 d.p.:
11
The heights of adult males in a country are normally distributed with mean \(\mu = 175\text{ cm}\) and standard deviation \(\sigma = 8\text{ cm}\).
- Find the probability that a randomly selected male is taller than 185 cm.
- 10% of males are shorter than height \(h\text{ cm}\). Find \(h\).
Enter h to 3 s.f. (part b):
Topic 7 · Calculus — Differentiation
Advanced Differentiation Techniques
Rules to Master
Chain: d/dx[f(g(x))] = f'(g(x))·g'(x)
Product: d/dx[uv] = u'v + uv'
Quotient: d/dx[u/v] = (u'v − uv')/v²
Implicit: differentiate both sides w.r.t. x, collect dy/dx
d/dx[arctan x] = 1/(1+x²); d/dx[arcsin x] = 1/√(1−x²)
Worked Example
If \(y = x^2 e^{3x}\), find \(\frac{dy}{dx}\).
Product rule: \(\frac{dy}{dx} = 2x e^{3x} + x^2 \cdot 3e^{3x} = e^{3x}(2x+3x^2) = xe^{3x}(2+3x)\).
12
The curve \(C\) is defined by \(x^3 + y^3 - 3xy = 5\).
- Find \(\dfrac{dy}{dx}\) in terms of \(x\) and \(y\).
- Find the gradient of \(C\) at the point \((2, 1)\).
Enter the gradient at (2,1) as a decimal:
13
A closed cylindrical tin has a total surface area of \(300\pi\) cm². Find the radius of the cylinder that maximises its volume, giving your answer in exact form.
Enter the optimal radius in cm (exact value as decimal, e.g. 10):
Topic 8 · Calculus — Integration
Integration Techniques
Standard Results & Methods
∫u dv = uv − ∫v du [integration by parts]
∫f'(x)/f(x) dx = ln|f(x)| + C
∫sin²x dx = x/2 − sin(2x)/4 + C (use double angle)
Partial fractions: decompose then integrate term by term
Volume of revolution: V = π∫[a→b] y² dx
Worked Example
Evaluate \(\int_0^1 x e^x \,dx\).
Integration by parts, \(u=x\), \(dv=e^x dx\): \([xe^x]_0^1 - \int_0^1 e^x dx = e - [e^x]_0^1 = e - (e-1) = 1\).
14
Find the exact value of \(\displaystyle\int_1^e x^2 \ln x \,dx\).
Enter the exact value to 4 significant figures:
15
The region \(R\) is enclosed by the curve \(y = \sqrt{x+1}\), the \(x\)-axis, and the lines \(x=0\) and \(x=3\).
- Find the area of \(R\).
- Find the volume of the solid formed when \(R\) is rotated \(2\pi\) radians about the \(x\)-axis. Give your answer as an exact multiple of \(\pi\).
Enter the volume divided by π (i.e. the coefficient):
Topic 9 · Differential Equations
Separable & Homogeneous ODEs
Solution Methods
Separable: dy/dx = f(x)g(y) → ∫1/g(y)dy = ∫f(x)dx
Integrating factor: for dy/dx + P(x)y = Q(x), IF = e^(∫P dx)
Logistic: dP/dt = kP(M−P) → P = M/(1+Ae^(−kMt))
Worked Example
Solve \(\dfrac{dy}{dx} = 2xy\), given \(y(0)=3\).
Separate: \(\frac{1}{y}dy = 2x\,dx\). Integrate: \(\ln|y|=x^2+C\). Apply IC: \(C=\ln 3\). Solution: \(y = 3e^{x^2}\).
16
Consider the differential equation \(\dfrac{dy}{dx} = \dfrac{x^2}{y}\), with initial condition \(y = 2\) when \(x = 0\).
- Solve the differential equation to find \(y\) as a function of \(x\).
- Hence find the value of \(y\) when \(x = 3\), to 3 significant figures.
Enter y when x = 3 (3 s.f.):
17
Solve the differential equation \(\dfrac{dy}{dx} + \dfrac{2}{x}y = x^2\), given that \(y = 1\) when \(x = 1\).
- Find the integrating factor.
- Find the particular solution.
- Find the value of \(y\) when \(x = 2\).
Enter y when x = 2 (exact decimal):
Topic 10 · Maclaurin Series & Limits
Power Series Expansions
Standard Maclaurin Series
e^x = 1 + x + x²/2! + x³/3! + ··· (all x)
sin x = x − x³/3! + x⁵/5! − ··· (all x)
cos x = 1 − x²/2! + x⁴/4! − ··· (all x)
ln(1+x) = x − x²/2 + x³/3 − ··· (−1
(1+x)^n = 1 + nx + n(n−1)x²/2! + ··· (|x|<1 if n∉ℤ⁺)
Worked Example
Find the Maclaurin series of \(e^x \cos x\) up to and including the \(x^3\) term.
Multiply series: \((1+x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots)(1-\frac{x^2}{2}+\cdots)\). Collect: \(1 + x + 0 \cdot x^2 - \frac{x^3}{3} + \cdots\).
18
Find the Maclaurin series expansion of \(f(x) = \ln(1 + \sin x)\) up to and including the term in \(x^3\).
- State the series clearly.
- Use your series to find an approximation for \(\ln(1 + \sin 0.1)\), giving your answer to 5 significant figures.
Enter the approximation to 5 s.f. (part b):
19
Evaluate the following limit using either L'Hôpital's rule or Maclaurin series:
\[\lim_{x \to 0} \frac{e^{3x} - 1 - 3x}{x^2}\]
Enter the exact value of the limit:
Topic Mixed · Proof & Further Algebra
20
Use the principle of mathematical induction to prove that, for all positive integers \(n\):
\[\sum_{r=1}^{n} r(r+1) = \frac{n(n+1)(n+2)}{3}\]
- Verify the base case \(n=1\).
- Assume true for \(n=k\) and prove for \(n=k+1\).
- State the value of \(\displaystyle\sum_{r=1}^{10} r(r+1)\).
Enter the sum for n = 10 (part c):
Examination Complete
—
Time taken: —