JCE Mathematics common mistakes
JCE Mathematics · Understand the traps and how to avoid them.
When solving an equation by squaring both sides (e.g. √(x+3) = x − 3), the candidate squares, solves the resulting quadratic, and reports both roots without checking either in the original equation, so an extraneous root survives into the final answer.
Squaring is not a reversible operation: it maps both x and −x to the same value, so it can create solutions that were never true. Candidates treat each algebraic step as automatically equivalence-preserving because most earlier equation-solving (adding, multiplying, factorising) was, and they have often memorised 'square both sides' as a procedure rather than an implication.
What to do instead
After any non-reversible step (squaring, multiplying by an expression containing the unknown, raising to an even power), substitute every candidate root back into the equation as originally written and reject any that fail. Train the habit of writing '⇒' for one-directional steps and only claiming '⇔' when the step is reversible; explicitly mark the check as part of the solution, not optional.
The candidate differentiates or integrates a composite function by acting on the outer function only - writing d/dx[sin(3x²)] = cos(3x²), or ∫e^(x²)dx = e^(x²) - and never introduces the inner derivative or the substitution factor.
The power rule and the standard derivatives are learned first and rehearsed heavily, so the pattern 'derivative of sin is cos' fires before the chain rule is considered. Inner functions are visually buried inside brackets, and candidates read the expression as a single named function rather than as a composition of two.
What to do instead
Before differentiating, explicitly identify the inner function and write it down (here u = 3x², du/dx = 6x), then assemble the answer as outer′ × inner′. For integration, write the substitution u = ..., du = ... dx before touching the integral. Drill a mixed set where some items need the chain rule and some do not, so the decision to check for a composition becomes automatic.
The candidate computes a probability of a union as P(A) + P(B) even when the events can occur together, or computes P(A and B) as P(A) × P(B) for events that are not independent - for example adding the probabilities of drawing a king or a heart and getting 13/52 + 4/52 = 17/52.
The addition and multiplication rules are first met in simple, mutually exclusive or independent settings (dice, coins), and the simplified formulas stick. The overlap term is invisible unless the candidate consciously tests for it, and 'or means add, and means multiply' is a memorable but incomplete heuristic.
What to do instead
Force a set-based check before every calculation: write the general rules P(A∪B) = P(A) + P(B) − P(A∩B) and P(A∩B) = P(A)P(B|A), then decide whether the correction term is zero. For overlapping categories, sketch a two-way table or Venn diagram and count the cells; the arithmetic then confirms the formula rather than replacing it.
When a question asks for the area of a region bounded by a curve and a line, the candidate integrates from one x-intercept to the other without checking whether the curves cross inside that interval, so a region that lies partly below the axis is added with the wrong sign and the area comes out too small.
The definite integral is taught as 'the area under the curve', and in the majority of introductory examples the curve stays above the axis, so the sign issue never surfaces. Finding intersection points feels like the hard part, and candidates stop once they have limits rather than inspecting the geometry between them.
What to do instead
Always sketch the two graphs and mark every intersection point in the interval. Split the integral at each crossing and integrate |upper − lower| on each subinterval, or integrate (curve − line) rather than the curve alone. As a sanity check, confirm the computed area is positive and roughly matches the sketch's proportions.
The candidate treats a logarithm as a multiplier of its argument - writing log(2x) = 2 log x, or log(a + b) = log a + log b - and then 'simplifies' an expression using these false identities.
Logarithm laws look structurally like distribution and factor-pulling, and candidates pattern-match to the algebra they know from linear expressions. The valid laws (log(ab) = log a + log b, log(xⁿ) = n log x) are similar enough in shape that the invalid versions feel like harmless rearrangements.
What to do instead
Verify any log manipulation with a numerical spot-check on simple values (e.g. log(2·3) = log 6 ≈ 0.778 versus log 2 + log 3 ≈ 0.778, but log(2+3) = log 5 ≈ 0.699 ≠ log 2 + log 3). Write the three valid laws side by side with the two common invalid ones and label them, so the contrast is memorised rather than the laws in isolation.
In a related-rates or optimisation problem, the candidate differentiates correctly but substitutes the given numerical value of the variable before differentiating - for example setting r = 5 in V = (4/3)πr³ and then reporting dV/dt = 0, or substituting x before finding dy/dx.
Substitution is the standard first move in almost every other algebra question, so it is applied reflexively. Candidates do not distinguish between 'evaluate a quantity at a point' and 'differentiate a function, then evaluate the derivative at a point', and the two operations look interchangeable in notation.
What to do instead
Adopt a fixed written order: (1) write the relationship in symbols, (2) differentiate with respect to the independent variable, (3) only then substitute the instantaneous values. Keep the variable symbolic until the derivative line is complete, and state explicitly 'this derivative is a function, evaluated at t = ...' before plugging in.
The candidate answers a 'hence' or 'show that' question by starting from the result given in the question and manipulating it until something true appears, treating the printed statement as an assumption rather than a target.
'Show that' items appear to offer a free starting point, and working backwards from a known answer feels efficient. Candidates also confuse verification (checking a stated result) with derivation (producing it), and many have never been penalised explicitly for the logical direction of the argument.
What to do instead
For 'show that', begin from the given information and derive forward to the printed form, keeping each line justified. If working backwards as a scratch method, rewrite the final solution forwards. For 'hence', make the link to the previous part explicit (e.g. 'using the factorisation from part (a)'), since an independent method usually loses the 'hence' mark.
When finding the inverse of a function or the image of a point under a transformation, the candidate applies the operations in the same order as the original - inverting f(x) = 3x + 2 as (x − 2)/3 written as x/3 − 2, or rotating a point about the wrong centre - because the order of inverse operations is reversed.
Inverse operations must undo the original in reverse sequence, but candidates read the expression left to right and invert each operation in place. The same left-to-right reading error appears in composite functions, where fg(x) is evaluated as g(f(x)).
What to do instead
Write the original as a chain of operations (multiply by 3, then add 2) and undo it by reversing the chain (subtract 2, then divide by 3), showing each step. For composites, state the convention explicitly - fg(x) means apply g first - and test with a numerical value to confirm the order.
The candidate computes a trigonometric ratio for a non-right triangle using SOHCAHTOA, or uses the sine rule to find an angle without considering the ambiguous case, returning only the acute solution when an obtuse one also fits the given data.
Right-triangle trigonometry is over-practised relative to the sine and cosine rules, and diagrams in textbooks are usually drawn to scale with acute angles, so the obtuse possibility is never visible. The ambiguous case arises from sin θ = sin(180° − θ), a fact easily forgotten under time pressure.
What to do instead
First classify the triangle: right-angled (use SOHCAHTOA), or not (use sine/cosine rule). When using the sine rule for an angle, always compute 180° minus the found angle and test both against the given sides and the angle sum. Sketching the triangle roughly, and checking whether the largest angle faces the longest side, catches most wrong answers.
The candidate reports a confidence interval or hypothesis-test conclusion without checking the conditions for the procedure - using a normal approximation for a small sample from a heavily skewed population, or interpreting a p-value as the probability that the null hypothesis is true.
The mechanics of the test (formula, table value, comparison with α) are what get drilled, while the assumptions are stated once in the textbook and rarely assessed. The p-value's definition is counterintuitive, and the phrase 'probability of the null hypothesis' is repeated loosely in everyday speech.
What to do instead
Before computing, write the conditions in a checklist (random sampling, sample size, expected counts, independence) and state whether each holds. Interpret a p-value in a fixed sentence: 'If the null hypothesis were true, the probability of data at least this extreme is p' - never 'the probability the null is true'. Practise writing the conclusion in context, including the decision and its scope.