10 examples · updated 25 August 2026
WORKED EXAMPLES, DONE THE WAY THE BOOKS DO THEM.
Every one carries a route line before the working, numbered steps rather than an answer letter, and the named trap that each wrong option was built to catch. All written from scratch to the published ESAT content specification.
Mathematics 1
MATHEMATICS 1
4 EXAMPLES
- How do you find the total surface area of a square-based pyramid? Find the slant height first, from the right triangle whose legs are the perpendicular height and half the base. The perpendicular height is not the slant height, and using it as one is the single most common way to lose this question. M1.7 · MENSURATION
- What is the coefficient of x³ in the expansion of (2 − x/2)⁶? The coefficient is −20. The powers of 2 and of ½ cancel exactly, so the whole term collapses to −C(6,3), and the minus sign survives because the x term is being raised to an odd power. M1.3 · BINOMIAL EXPANSION
- How do you solve 3^(2x) − 10 × 3^x + 9 = 0? Substitute u = 3^x. The equation becomes u² − 10u + 9 = 0, which factorises to (u − 1)(u − 9) = 0, giving 3^x = 1 and 3^x = 9, so x = 0 and x = 2 and the two solutions sum to 2. M1.2 · INDICES AND LOGARITHMS
- How many solutions does sin 2x = ½ have between 0° and 360°? There are four solutions. Doubling the angle doubles the interval the angle travels through, so 2x runs from 0° to 720° — two complete turns — and sin θ = ½ has two solutions in every turn. M1.5 · TRIGONOMETRY
Mathematics 2
MATHEMATICS 2
3 EXAMPLES
- How do you find the coefficient of x⁶ in the derivative of (1 + 2x)⁷? Differentiate first, then pick the term. Doing it the other way round means expanding seven terms you will throw away, and in an ESAT Mathematics 2 module you have about 89 seconds. M2.3 · SERIES
- Why does plotting log y against log x give a straight line? A log-log plot is straight exactly when y and x obey a power law y = kxⁿ. Taking logs of both sides gives log y = n log x + log k, which is the equation of a line whose gradient is the power n and whose vertical intercept is log k, not k. M2.2 · LOGARITHMS
- How many distinct real roots does x⁴ + 3x² − 4 = 0 have? There are two distinct real roots, x = 1 and x = −1. Substituting u = x² gives u = 1 and u = −4, and only the non-negative value of u can come from a real x, so the pair of roots from u = −4 does not exist. M2.1 · ALGEBRA
Physics
PHYSICS
3 EXAMPLES
- How fast is a block moving at the bottom of a smooth slope? On a smooth slope the speed at the bottom depends only on the vertical height dropped, not on the length of the slope or its angle. Use v = √(2gh) with h the vertical drop, which for a 10 m slope at 30° is 5 m, giving 10 m/s. P2.1 · MECHANICS AND ENERGY
- How do you find the primary current in an ideal transformer? Work out the secondary side completely first, then cross to the primary using conservation of power, because an ideal transformer transfers all of it. Voltage scales with the turns ratio and current scales with its inverse, so a transformer that steps voltage down steps current up. P3.2 · ELECTRICITY
- What does a voltmeter read across a potential divider? A voltmeter of finite resistance sits in parallel with the resistor it measures and lowers the resistance of that branch, so it always reads less than the unloaded divider would suggest. The effect is negligible only when the meter resistance is much larger than the resistor it is across, which is exactly what this question removes. P3.2 · ELECTRICITY
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