Formulae and Cheat Sheets
The sheets below collect the statements we expect students to carry in working memory by the end of each course. We publish them exactly as students may use them in the exam room: no derivations, just the result, its domain, and a one line reminder of when it applies.
The Quadratic Formula
For a quadratic equation ax^2 + bx + c = 0 with a not equal to 0, the two complex roots are given by the formula below. The expression under the radical is the discriminant; when it is positive the roots are real and distinct, when it is zero they coincide, and when it is negative they are a conjugate pair.
Euler's Identity
Euler's identity connects the five most important constants in mathematics. Read as e raised to the imaginary power i times pi, it equals negative one, so that adding one returns zero. It follows from the power series for the exponential function and is the reason complex exponentials are used to represent waves and oscillations.
The Definition of a Limit
Precisely, the limit of f(x) as x approaches a is L when, for every positive tolerance epsilon, there exists a positive delta such that inputs within delta of a (but not equal to a) produce outputs within epsilon of L. This epsilon-delta statement is a promise, not a recipe: it says f(x) can be forced as close to L as we wish by taking x close enough to a.
Everyday applications. These formulae are not classroom ornaments. Compound interest grows by repeated multiplication, so continuous compounding lands exactly on Euler's number e. The quadratic formula finds break-even points in cost models. The definition of a limit is what makes any algorithm that claims to approach an answer, such as square root or matrix iteration, trustworthy under finite computation.
Worked Examples
Each example below is written the way we ask students to write: every step justified, every line traceable to an earlier definition or theorem.
Example A: Solving a Quadratic by Factoring
Solve x^2 - 6x + 5 = 0. We look for two numbers that multiply to 5 and add to -6, namely -5 and -1, so the left side factors as (x - 5)(x - 1). A product is zero only when one factor is zero, which gives x = 5 or x = 1. Checking: 25 - 30 + 5 = 0 and 1 - 6 + 5 = 0.
Example B: A Limit That Factors
Find the limit of (x^2 - 4)/(x - 2) as x tends to 2. Substitution fails because the denominator vanishes. Factor the numerator as (x - 2)(x + 2); the cancellation of the common factor (x - 2) is legitimate because x is never equal to 2 inside a limit. What remains is x + 2, which tends to 4, so the limit equals 4.
Example C: The Chain Rule
Differentiate h(x) = sin(3x^2). Treat 3x^2 as the inner function u, whose derivative is 6x. The chain rule states h'(x) = cos(u) times u', so h'(x) = 6x cos(3x^2). The outer function is evaluated at the unmodified inner input, while the multiplier comes from the derivative of the inner function.
Past Papers
Past papers are the closest thing we have to a preview of the exam. Each paper is released with its full marking scheme so students can practice under timed conditions and then grade their own work honestly.
| Paper | Level | Year | Notes |
|---|---|---|---|
| Calculus I | beginner | 2025 | Eight questions, two hours, full marking scheme. |
| Linear Algebra | beginner | 2025 | Computational and conceptual sections. |
| Discrete Mathematics | beginner | 2024 | Combinatorics and graph theory emphasis. |
| Probability and Statistics | intermediate | 2025 | Formula sheet provided with the paper. |
| Differential Equations | intermediate | 2024 | Laplace transform tables attached. |
| Real Analysis | advanced | 2025 | Proof based, no calculators permitted. |
Access. Enrolled students download papers through their course page. Visitors may request the archive by writing to us through the contact page; we honour those requests within a week.