TMUA Maths · Maths, Economics and Computing courses
How many real solutions does the equation |x2 − 4| = 3 − x have?
The answer is D — four. The modulus splits the problem in two, and the trap is solving only one half. Where |x| ≥ 2 the bracket is positive, so x2 − 4 = 3 − x, giving x2 + x − 7 = 0 and x = −1 ± √292 ≈ 2.19 and −3.19. Where |x| < 2 the bracket is negative, so 4 − x2 = 3 − x, giving x2 − x − 1 = 0 and x = 1 ± √52 ≈ 1.62 and −0.62. All four survive the checks: each lies in the region it came from, and each keeps the right-hand side non-negative. Sketching y = |x2 − 4| against the line y = 3 − x gets you there faster than the algebra — the line cuts the W-shape four times, and that picture is worth more than any formula.
TARA Problem solving · Oxford and UCL courses
A wooden cube is painted red on all six faces, then cut into 27 identical smaller cubes. One small cube is picked at random and rolled. What is the probability that the face showing is red?
The answer is C — one third. The instinct is to sort the small cubes into corners, edges, faces and centre, and most people lose several minutes there. They needn’t: every small face is equally likely to end up showing, so all you need is the proportion of them that are red. There are 27 × 6 = 162 small faces in total, and the red ones are exactly the surface of the original cube — six sides of nine squares, so 54. That gives 54162 = 13. The trap is answering a slightly different question: the chance that a cube has a red face somewhere is 2627, which feels like what is being asked and is not. TARA’s Problem Solving module rewards spotting the shortcut, because there is no time to take the long road.
UCAT Decision making · Medicine and Dentistry
Every applicant to Medicine at this university must sit the UCAT. Jamila is sitting the UCAT.
Which conclusion follows?
The answer is C — you cannot tell. The statement says that all Medicine applicants sit the UCAT. It does not say that only Medicine applicants sit it. Jamila might be applying for Dentistry, or somewhere else entirely, or she may not have decided. Choosing A is affirming the consequent, and it is the single most common error in UCAT Decision Making. D fails for the same reason — sitting the test does not tie her to any particular university.
LNAT Comprehension · Law
Supporters of later school start times argue that they raise exam results. But every study they cite was carried out in schools that introduced other changes at the same time — smaller classes, extra tutoring, new timetables. Until we see evidence from a school where only the start time changed, the case is unproven.
Which best states the author’s main point?
The answer is B. The author’s claim is narrow, and it is about the evidence rather than about start times. A and D go further than the passage does — “unproven” is not the same as “wrong”, and the author never advises against later starts. C is the sharpest trap: the passage names the other changes as things that muddy the picture, not as proven causes of anything. The LNAT consistently rewards the answer that stops exactly where the passage stops, which is why able readers who over-interpret tend to score below their ability.
ESAT Physics · Engineering and Science courses
A ball is thrown vertically upwards at 20 m/s from the top of a 25 m cliff, and lands at the foot of the cliff. Taking g = 10 m/s² and ignoring air resistance, how long is it in the air?
The answer is C — 5 seconds. Take upwards as positive and measure displacement from the throwing point, so the landing point is at −25 m. Then −25 = 20t − 5t2, which rearranges to t2 − 4t − 5 = 0 and factorises to (t − 5)(t + 1) = 0. Discard the negative root and t = 5 s. Answering 4 s means treating the flight as symmetrical and forgetting the ball ends up 25 m below where it started; answering 2 s means stopping at the top of the arc. The habit worth building is committing to a sign convention before writing anything down — most marks lost on ESAT mechanics go to signs rather than to physics. For the record, it hits the ground at 30 m/s.
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