TMUA Maths Mistakes: 20 Errors the Examiners Flag

Most "common TMUA mistakes" lists talk about study habits: ration the past papers, do not neglect Paper 2, practise against the clock. Those matter, and we cover them in common TMUA mistakes. This page is the other half, and almost nobody writes it: the specific pieces of maths that go wrong.

Every error below is one the exam board flags in its own free notes, the Notes on Mathematics for TMUA Paper 1 and ESAT Mathematics 2 and the Notes on Logic and Proof, both written by the team that develops and sets the test. That matters more than it sounds. When the people who write the questions stop mid-page to say "be very careful not to" do a particular thing, they are quietly telling you where the wrong options come from. Each entry below gives the mistake, a concrete instance, why it is tempting, and the check that catches it.

Powers, roots and surds

1. $(a^3)^2$ is $a^6$, not $a^9$

A bracketed power and a stacked power look almost identical on paper and mean different things. The board sets $(a^m)^n$ and $a^{m^n}$ side by side: $(a^3)^2 = a^3 \times a^3 = a^6$, whereas $a^{3^2} = a^{3 \times 3} = a^9$ (p.8). It is tempting because handwriting flattens the tower, and because the index law $(a^m)^n = a^{mn}$ is so familiar that you apply it to the wrong shape.

The check: find the outermost operation before you touch any rule. A bracket means the power acts on everything inside it. A stacked index means the top power is evaluated first.

2. Writing $\sqrt5 \times \sqrt5 = 25$

The board calls this out by name in the middle of a rationalising example: it is very easy to be careless and write $\sqrt{5} \times \sqrt{5} = 25$, and you should make sure you do not (p.14). The correct value is $5$, because $\sqrt{5}$ squared undoes the root. The temptation is pure pattern-matching: you see two fives, and $5 \times 5 = 25$ fires before the root does.

The check: whenever a root multiplies itself, the root vanishes and you are left with the number underneath. $\sqrt{a} \times \sqrt{a} = a$, always. More on this in our surds and indices guide.

3. Saying $\sqrt{49 - 12\sqrt5} = 2 - 3\sqrt5$

The board sets this as an exercise and warns you off the obvious answer in a footnote: it is not $2 - 3\sqrt{5}$, and you should work out why (p.12). The algebra tempts you because $(2 - 3\sqrt{5})^2$ really does equal $49 - 12\sqrt{5}$. But $3\sqrt{5} \approx 6.7$, so $2 - 3\sqrt{5}$ is negative, and the square-root sign always means the positive root. The answer is $3\sqrt{5} - 2$.

The check: after any factorise-the-surd move, glance at the sign of your answer. A square root can never come out negative, so if it does, flip the bracket.

4. Dropping the $\pm$, or inventing one

Two mistakes that are mirror images. The board's convention, stated explicitly, is that $\sqrt{a}$ is always positive, so $\sqrt{64}$ is $8$ and not $-8$; if you want both you must write $\pm\sqrt{64}$ (p.11). But $x^2 = 25$ is a different statement, and it has two solutions, $x = \pm 5$. The Notes on Logic and Proof build a whole worked example out of the confusion: start from $x = \sqrt{25}$, square both sides, and you have manufactured a solution $x = -5$ that was never there (p.72).

The check: ask which direction you are travelling. Taking a square root of a number gives one positive value. Solving an equation in $x^2$ gives two. The trap on the paper is usually the option that keeps only the positive root of a quadratic.

The "everything is linear" family

A large share of the board's warnings share a single root cause: assuming an operation distributes over a sum, or over a quotient, when it does not.

5. $(x+y)^2 = x^2 + y^2$

The board's phrasing here is memorable. It notes that it might seem "obvious" to write this, but it is "generally mathematical bunkum" (p.110). The correct expansion is $(x + y)^2 = x^2 + 2xy + y^2$, and it is the missing $2xy$ that the distractors are built from.

The check: squaring is not a distributive operation. If you ever find yourself writing $\text{op}(x + y) = \text{op}(x) + \text{op}(y)$, stop and expand properly.

6. $f(x+3) = f(x) + f(3)$, and $f(x^2) = [f(x)]^2$

Both appear in the same board footnote, with the flat verdict that they "are not generally true" (p.126). Take $f(x) = x^2$: then $f(x + 3) = (x + 3)^2 = x^2 + 6x + 9$, while $f(x) + f(3) = x^2 + 9$. The two differ by $6x$. For the second, take $f(x) = x + 1$: $f(x^2) = x^2 + 1$, but $[f(x)]^2 = x^2 + 2x + 1$.

The check: $f$ is a rule, not a multiplier. Substitute the whole input into the rule, then simplify. Never split the input first.

7. $f(3x) = 3x^2$ when $f(x) = x^2$

The same error wearing a transformation costume. The board is blunt: to find $f(3x)$ you replace $x$ by $3x$, giving $f(3x) = (3x)^2 = 9x^2$, and "we do NOT write $f(3x) = 3x^2$" (p.150). The pull is that the $3$ looks like a coefficient sitting outside, when in fact it is part of the input.

The check: put the substitution in a bracket every single time. $f(3x)$ means "wherever you see $x$, write $(3x)$", brackets included. This is the same discipline that keeps graph transformations straight.

8. Splitting the integral of a fraction into two integrals

Writing $\int \frac{f(x)}{g(x)}\, dx = \frac{\int f(x)\, dx}{\int g(x)\, dx}$ earns the board's bluntest note in the whole document: it is "very wrong, so don't do it" (p.126). Integration is linear over sums, which is why you can integrate term by term, and that success tempts you into assuming it is linear over quotients too. It is not. Try $\int_1^2 \frac{x^2}{x}\, dx$: the left-hand side is $\int_1^2 x\, dx = \frac{3}{2}$, while $\frac{\int_1^2 x^2 dx}{\int_1^2 x\, dx} = \frac{7/3}{3/2} = \frac{14}{9}$.

The check: simplify the fraction into powers of $x$ before you integrate. That is the only move the specification asks for anyway, as our integration guide sets out.

Binomial expansions

9. Writing $2x^7$ where you mean $(2x)^7$

Finding the $x^7$ term in $(3 + 2x)^8$, the board stresses that both the $2$ and the $x$ are raised to the seventh power, and adds in a footnote that a common error is to write $2x^7$ without the bracket and forget that the $2$ must be raised too (p.56). The correct term is $\binom{8}{7} \times 3^1 \times (2x)^7 = 3072x^7$. Drop the bracket and you get $48x^7$, which is out by a factor of $64$.

The check: write the bracket before you write the power, every time. If the answer options are miles apart, a missing bracket is the first thing to suspect.

10. Losing the minus inside $(-3x)^5$

The board's second worked example is the coefficient of $x^5$ in $(2 - 3x)^7$, and it says outright that this illustrates a mistake students often make (p.57). The tempting expression treats the term as $\binom{7}{5} 2^2 \cdot 3x^5$, which gets you $252$. The correct expression raises the whole of $-3x$ to the fifth power: $\binom{7}{5} 2^2 (-3x)^5 = 21 \times 4 \times (-243) x^5 = -20412x^5$.

The check: an odd power keeps a negative sign, an even power kills it. When a binomial has a minus in it, write the bracketed term $(-3x)$ down first and only then decide the power.

Sequences and series

11. The fence-post error

How many terms are there in $ar^m + ar^{m+1} + \cdots + ar^n$? The board asks exactly this, then answers in a footnote with a shout: "NO! it is $n - m + 1$ terms", and names it as the fence-post error, adding that it is very easy to make (p.53). The instinct to subtract is strong, because subtraction is what measures distance. But counting posts is not the same as counting gaps.

The check: test your formula on a tiny case you can count on your fingers. From $ar^2$ to $ar^4$ there are three terms, and $4 - 2 + 1 = 3$. The board admits to counting on its own fingers when a fence-post error is a risk.

12. Using $S_n - S_m$ for part of a geometric progression

Straight after the fence-post warning, the board flags the natural companion slip: it is tempting to write $ar^m + \cdots + ar^n$ as $S_n - S_m$, "but that would be wrong" (p.53). With the usual convention that $S_k = a + ar + \cdots + ar^{k-1}$ covers $k$ terms, the correct difference is $S_{n+1} - S_m$, and the board draws attention to the $n + 1$ in its own working.

The check: before subtracting two sums, write down the first and last term each one actually contains. The difference is a fence-post question in disguise, so it fails in exactly the same way. Our sequences and series guide has the full toolkit.

Coordinate geometry

13. Calling the radius 25 when $r^2 = 25$

The circle $(x - 2)^2 + (y - 3)^2 = 25$ has centre $(2, 3)$ and radius $5$. The board spells this out and adds the warning in-line: be very careful not to say its radius is $25$ (p.71). It is tempting because the number is sitting right there, and because under time pressure you read the equation rather than parse it.

The check: the right-hand side is always $r^2$. Say "$r$ squared equals" out loud as you read it, and root the number before you use it. The related trap is that not every equation of the form $x^2 + y^2 + ax + by + c = 0$ is a circle at all: complete the square on $x^2 + y^2 - 4x - 6y + 20 = 0$ and you reach $(x - 2)^2 + (y - 3)^2 = -7$, which no real point satisfies (p.72).

14. Assuming $c$ is the $y$-intercept in $ax + by + c = 0$

In $y = mx + c$ the constant $c$ is the $y$-intercept. In $ax + by + c = 0$ it is not, and the board says so directly: do not assume that the $c$ in $ax + by + c = 0$ is the $y$-intercept (p.68). It calls the shared letter unfortunate, which is exactly why the mistake happens. For $2x + 4y - 8 = 0$, rearranging gives $y = 2 - \frac{x}{2}$, so the intercept is $2$, not $-8$.

The check: never read a feature off a line until it is in $y = mx + c$ form. Rearranging takes five seconds and removes the whole class of error. More in our coordinate geometry guide.

15. Getting the gradient upside down or the wrong sign

Computing the gradient from two points, the board gives two separate warnings in one sentence: keep the order of the $x$ and the $y$ on the top and bottom the same way round, otherwise the sign comes out wrong, and make sure $y$ goes on top and $x$ on the bottom (p.68). So $\frac{y_1 - y_2}{x_2 - x_1}$ has the right size and the wrong sign, while $\frac{x_1 - x_2}{y_1 - y_2}$ is the reciprocal of what you wanted.

The check: always subtract in the same direction top and bottom, and sanity-check against the picture. If the line visibly falls left to right, the gradient must be negative.

Trigonometry

16. Rearranging before you have listed every solution

This is the single most expensive trig habit on the paper. Solving $\sin(2x + 60°) = \frac{1}{2}$ for $-360° < x < 360°$, the board lists every value of $2x + 60$ first, going deliberately beyond the range because dividing by $2$ will pull them back in, and only then rearranges. That gives eight solutions: $x = -315, -195, -135, -15, 45, 165, 225, 345$.

Rearrange first to the basic solution $x = -15$, then generate the rest, and you get only $x = -165, -15, 195, 345$. The board walks through this incorrect route in full and concludes that things have gone wrong, so it is better to find all your solutions first and then rearrange at the end (p.96 to p.97). Note that the bad method does not merely lose solutions, it invents two that are not solutions at all.

The check: treat the whole bracket as the angle. Solve for $2x + 60$ across a range wide enough to survive the rearrangement, then divide and shift as the last step. Our trigonometry guide drills the pattern.

Graphs and transformations

17. Getting the order of two horizontal transformations wrong

A horizontal squash and a horizontal translation do not commute, and the board devotes a page to it. Going from $\cos x$ to $\cos\left(2x + \frac{\pi}{6}\right)$, if you squash horizontally by a factor of $2$ first and then translate by $\frac{\pi}{6}$, you land on $\cos\left(2x + \frac{\pi}{3}\right)$, which is wrong; the board prints "OH NO !!!" next to it (p.154). Translate first and then squash and you get the right answer. Its footnote explains why: replacing $x$ by $2x$ first means the later translation is scaled by that $2$ as well.

The check: factorise the bracket before you describe anything. $\cos\left(2x + \frac{\pi}{6}\right) = \cos\left(2\left(x + \frac{\pi}{12}\right)\right)$ tells you the shift is $\frac{\pi}{12}$, not $\frac{\pi}{6}$, if you insist on squashing first. Advice to "apply them outside-in" or "inside-out" consistently is not enough here, which is why the board tests it.

Inequalities and invalid deductions

18. Writing $1 \lt x \lt -5$

The board calls this a howler (p.30). A chained inequality reads as one sentence, so $1 < x < -5$ claims $1 < -5$ along the way. What the writer meant was two separate statements: $x < -5$ or $1 < x$. The board also flags the related "$-2 < x$ and $x < -4$", which no $x$ satisfies, and says its own preference is to use "or".

The check: a chained inequality is only legal if the two outer numbers are themselves in the right order. If the solution set is two disjoint pieces, you need the word "or", not a chain.

19. Doing the same thing to both sides of an inequality

Inequalities are not equations, and the Notes on Logic and Proof devote a section of the error taxonomy to this (p.73). Three ways it breaks:

• Squaring both sides. $-5 < 4$ is true, but squaring gives $25 < 16$, which is false.
• Multiplying by a negative. $1 < 2$ is true, but multiplying by $-1$ gives $-1 < -2$, which is false.
• Applying a function that is not increasing. $\frac{\pi}{4} < \frac{\pi}{3}$ is true, but taking cosines gives $\cos\frac{\pi}{4} < \cos\frac{\pi}{3}$, which is false because cosine is decreasing there.

The commonest live version is multiplying by an expression whose sign you do not know. Faced with $\frac{2x + 5}{x + 3} > 1$, the board declines to multiply by $x + 3$, because $x + 3$ is negative for some $x$ and the inequality sign would then be wrong for those values. It multiplies by $(x + 3)^2$ instead, which is never negative, reaching $(x + 3)(x + 2) > 0$ (p.29). Multiply by $x + 3$ carelessly and you get $x > -2$, silently losing the entire branch $x < -3$.

The check: before multiplying an inequality by anything with a letter in it, ask whether that thing could be negative. If it could, square it, or move everything to one side and study the sign of the quotient. Our inequalities guide works through both routes.

20. Cancelling a factor that might be zero, and un-doing a trig function

The specification's own list of proof errors (Err2) names two deductions as invalid, and the notes repeat them: claiming that if $ab = ac$ then $b = c$, and assuming that if $\sin A = \sin B$ then $A = B$. Neither follows. The first fails when $a = 0$, which makes $ab = ac$ true for any $b$ and $c$ at all. The second fails at $A = 30°$ and $B = 150°$, where both sines are $\frac{1}{2}$.

This is not a hypothetical. Here is the real Paper 2 question built on the first of them. Work out which statements are actually forced before you reveal the answer:

The check: cancelling is division, and division by zero is not allowed. Every time you cancel a factor, ask what happens if that factor is zero. Every time you strip a function off both sides, ask whether the function is one-to-one on the range you were given.

The mistakes at a glance

How to actually stop making these

Reading a list of errors changes nothing on its own. Two habits do.

Name the slip when you review. When you get a question wrong, do not write "careless". Write which of the twenty it was. The board's own advice is to collect examples of where working goes wrong and to look out for those specific patterns in your own work, and a named error is one you can search for next time. This is the same labelling habit that makes Paper 2 proof techniques click.

Build the checks into the working, not the review. Most of the twenty above are prevented by a bracket, a sign check or a rearrangement that costs a few seconds. Bracket every substitution. Root the right-hand side of a circle equation on sight. Factorise the bracket before describing a transformation. Ask "could this be negative?" before multiplying an inequality. Under a four-minute-per-question clock the temptation is to skip these, which is precisely when they earn their keep.

The bank at CrackTMUA is built around this: every worked solution names the trap rather than only listing steps, and the spaced-repetition engine brings back the questions where you fell for one. If you want the study-process half of this page rather than the mathematical half, common TMUA mistakes is the companion.

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