FY12CE Mathematics common mistakes
FY12CE Mathematics · Understand the traps and how to avoid them.
When differentiating a composite function, the candidate differentiates only the outer function and drops the inner derivative - writing d/dx sin(3x²) = cos(3x²) or d/dx e^(x²) = e^(x²).
Chain rule is learned as a pattern ('derivative of sin is cos') and under timed conditions the candidate pattern-matches the outer function and stops, treating the argument as a single variable. Nested functions with polynomial or exponential inner parts are where the omission is most likely, because the inner derivative is not visually salient.
What to do instead
Before writing any answer, explicitly name the inner function u and write the chain as outer′(u)·u′ on the page, even for one-line answers. Drill a mixed set of 20 composites (trig, exponential, log, power, inverse trig) and mark each inner derivative in a different colour until the step becomes automatic. On the exam, do a final pass over every derivative checking that each composite has a visible inner-derivative factor.
The candidate computes a definite integral or area but substitutes the limits in the wrong order or forgets to change limits after substitution, producing a sign-flipped or wrong-valued answer, e.g. evaluating ∫₀² with F(0) − F(2).
The upper-minus-lower convention competes in memory with the substitution rule's 'evaluate at old limits', and under pressure the candidate writes the antiderivative values in the order they appear on the page. When using u-substitution the candidate often back-substitutes inconsistently or plugs original limits into the u-antiderivative.
What to do instead
Adopt a fixed written template: bracket the antiderivative, write '= [F(x)]_a^b = F(b) − F(a)', and physically copy b above a before evaluating. For substitution, always either convert the limits to u-values in the same line or write 'back-substitute to x, then apply original limits' - never mix. Practise five substitution integrals where the limits are converted, checking that a positive integrand gives a positive result.
When finding eigenvectors, the candidate solves det(A − λI) = 0 for λ correctly but then solves (A − λI)x = 0 by row-reducing A itself rather than A − λI, or picks a single eigenvector for a repeated eigenvalue and misses the second independent one.
Row-reduction of A is a well-practised reflex, and substituting the numeric λ into every diagonal entry is error-prone mental arithmetic. For repeated eigenvalues the candidate stops once one free variable yields a vector, not checking the nullity of A − λI.
What to do instead
Write A − λI explicitly with λ substituted before any row operation, and compute its nullity to know in advance how many independent eigenvectors to expect (nullity = multiplicity for diagonalisable cases). For each eigenvalue, state the dimension of the eigenspace before choosing vectors. Verify each candidate v by computing Av and confirming it equals λv - a 30-second check that catches both errors.
The candidate computes a cross product or scalar triple product and reports a magnitude or a signed value when a vector (or the opposite orientation) was required, or takes the dot product where the cross product is needed for a normal vector.
Area, volume and normal-vector problems all invoke 'cross product' in the candidate's notes, and the operations get conflated. Sign conventions in the triple product (a·(b×c) versus (a×b)·c) and the direction of a×b are also easy to reverse under time pressure.
What to do instead
Write the geometric quantity required first - 'normal vector =', 'area = ½|a×b|', 'volume = |a·(b×c)|' - then perform the operation that matches it. Compute cross products with the determinant expansion written out in full (i, j, k rows) rather than mentally, and check orientation with the right-hand rule or by confirming a·(a×b) = 0 and b·(a×b) = 0.
The candidate solves a first-order linear ODE without dividing through by the leading coefficient, so the integrating factor uses P(x) from the un-normalised equation, or omits the constant of integration when computing the integrating factor.
The standard form y′ + P(x)y = Q(x) is memorised, but equations are often presented as x y′ + 2y = … and the candidate reads off the coefficient of y directly. The integrating factor e^∫P is treated as a formula to plug into, and constants feel irrelevant because they cancel in the multiplication step - except when they do not.
What to do instead
Always rewrite the ODE into y′ + P(x)y = Q(x) as an explicit first line, dividing every term by the coefficient of y′. Compute μ = e^∫P dx with the integral shown, and note that the arbitrary constant may be taken as zero only because it cancels - state this. Finish by differentiating the solution and substituting into the original (un-normalised) equation as a check.
When classifying the equilibrium of a 2×2 linear system, the candidate uses the trace and determinant but misassigns the cases - calling a saddle a node because det < 0 is confused with trace < 0, or declaring stability from the sign of the determinant alone.
The classification table (trace τ, determinant Δ, discriminant τ² − 4Δ) is memorised as a list rather than derived from eigenvalues, and the conditions for saddle (Δ < 0), stable node/spiral (τ < 0, Δ > 0) and centre (τ = 0, Δ > 0) blur together.
What to do instead
Compute the eigenvalues explicitly first, then classify: real parts both negative = stable, both positive = unstable, opposite signs = saddle, purely imaginary = centre. Use τ and Δ only as a cross-check, writing the rule beside the working ('Δ < 0 ⇒ saddle regardless of τ'). Practise ten systems where you state eigenvalues, then classification, then sketch the phase portrait.
The candidate computes P(A|B) as P(A)P(B), or uses P(A|B) = P(A∩B)/P(B) but substitutes P(A) for P(A∩B) when events are not independent, and applies the multiplication rule for independent events to dependent sampling without replacement.
The formula P(A∩B) = P(A)P(B) is over-learned from independence examples and gets applied as a default. Condition notation is also read backwards: candidates compute P(B|A) when P(A|B) is asked, especially in Bayes-type word problems where the given and target events are described in prose.
What to do instead
Before computing, write the two events in words and label which is given ('given B') and which is the target. Use the general rule P(A∩B) = P(A|B)P(B) unless independence is explicitly justified, and for without-replacement problems draw the tree with changing denominators. For Bayes problems, write the partition and total probability denominator explicitly before dividing.
The candidate carries out a hypothesis test and states the conclusion as 'the null hypothesis is true' or 'proven', or interprets a p-value as the probability that H₀ is true, rather than reporting a decision about evidence at the stated significance level.
Everyday language treats 'not rejected' as 'confirmed', and the p-value's definition (probability of data at least as extreme, given H₀) is easily inverted in memory to 'probability H₀ is true'. The candidate also forgets to state the test statistic, degrees of freedom and the decision rule in context.
What to do instead
Use a fixed conclusion sentence: 'At the 5% significance level there is (insufficient/sufficient) evidence to reject H₀; we cannot conclude H₀ is true, only that it is not contradicted.' Write the p-value definition once in the margin as a reminder. Always report the test statistic, its distribution and degrees of freedom, and phrase the conclusion in terms of the original variables.
In a proof by induction, the candidate verifies the base case, assumes P(k), and then manipulates the statement P(k+1) directly without ever using the inductive hypothesis - or proves the implication P(k) ⇒ P(k+1) for a specific k rather than in general.
Algebraic manipulation of the target expression feels productive, and the step 'assume P(k) is true' is written as a formality that is never invoked. The quantifier structure ('for all k ≥ n₀') is not made explicit, so a single-instance verification passes for a proof.
What to do instead
Structure the proof in four labelled lines: (1) base case n = n₀ verified numerically; (2) inductive hypothesis: assume P(k) for some arbitrary k ≥ n₀; (3) inductive step: state explicitly 'using the inductive hypothesis, P(k) gives …' and show where it is substituted; (4) conclusion: 'hence by induction P(n) holds for all n ≥ n₀'. If the hypothesis is never used, the proof is wrong - rewrite.
When applying a numerical method, the candidate reports the computed iterate as the answer without checking convergence or the error estimate, or uses a fixed number of iterations without confirming the tolerance was met; in root-finding, Newton's method is applied without checking the derivative is non-zero or the initial guess is near a root.
Numerical methods are treated as procedures to execute rather than algorithms with convergence conditions. The tolerance criterion (|x_{n+1} − x_n| < ε or |f(x_n)| < ε) is often confused, and divergence or oscillation is not recognised because the iterates are not tabulated.
What to do instead
Tabulate iterations with columns for x_n, f(x_n), and the step difference, and stop only when the stated tolerance is met, reporting the achieved error. Before starting Newton's method, check f′(x₀) ≠ 0 and that f changes sign near x₀. Compare against the known convergence order (quadratic for Newton, linear for bisection) to sanity-check the iteration count, and state the final answer to the precision the tolerance justifies.
The candidate sets up a modelling problem by substituting numbers immediately, producing an equation in the wrong variables or with mismatched units, and never states the assumptions or checks the solution against the physical constraints (non-negativity, boundary values, limiting behaviour).
Exam pressure pushes toward computation, and word problems invite premature substitution before the relationship between quantities is defined. Units (per hour versus per minute, percentage versus proportion) are silently mixed, and the resulting answer is accepted without a plausibility test.
What to do instead
Write three lines before any algebra: define every variable with its units, state the governing relationship or balance law (rate in − rate out, conservation, proportionality), and list the assumptions. Keep symbols until the final step, then substitute with units carried through. Finish by checking the solution against boundary conditions and limits (e.g. t → 0 and t → ∞) and confirming it lies in the physically admissible range.