FY13CE Pure Mathematics common mistakes
FY13CE Pure Mathematics · Understand the traps and how to avoid them.
When asked to prove a statement by contradiction or by counterexample, the candidate restates the claim, or tests three numerical cases and declares it 'always true', instead of setting up the negation of the statement and deriving a contradiction, or producing one fully specified counterexample with the values substituted.
Proof is the least practised skill in the course, so candidates fall back on the verification habit built up in algebra: they check examples rather than argue generally. Testing cases feels productive and confirms belief in the result, so the candidate stops as soon as a few examples agree, without noticing that the general case is what is being marked.
What to do instead
Before writing, classify the demand: 'show that', 'prove', 'disprove', 'find a counterexample'. For disproof, give one explicit counterexample with the arithmetic shown. For proof by contradiction, write the negation of the conclusion as your first line ('assume sqrt(2) = p/q in lowest terms') and mark with an arrow where the contradiction appears. For proof by induction, label the base case, the inductive hypothesis, and the inductive step separately. Practise writing the negation of quantified statements ('for all', 'there exists') until it is automatic.
The candidate applies a function to both sides of an equation or inequality without checking whether the function is injective or order-preserving: squaring both sides of an equation and keeping only the positive root, or dropping the reversal of an inequality when multiplying by a negative quantity or a variable of unknown sign.
Early algebra taught 'do the same to both sides' as a universal licence, and the fine print about one-to-one functions and sign was never made explicit. Squaring feels like a clean simplification, and the spurious roots or lost negative solutions are not visible in the final line, so the error goes undetected.
What to do instead
Adopt a two-line habit: after squaring, write 'check for extraneous roots' and substitute each answer back. For inequalities, never multiply by an expression containing the variable; instead move everything to one side, factor, and use a sign table or sketch. When solving trigonometric equations, state the general solution with the period (for example x = 30 degrees + 360k or x = 150 degrees + 360k) rather than a single value.
The candidate treats a sequence as if it were a series: uses the arithmetic mean formula for the sum of n terms, confuses the nth term with the sum to n terms, or answers a 'sum to infinity' question for a geometric series whose common ratio has modulus greater than or equal to one.
The notation a_n, S_n and the summation symbol look similar under time pressure, and both the nth-term and sum formulas contain the same parameters a and d or r. The convergence condition |r| < 1 is memorised as a phrase but not checked against the actual value, so a divergent series is summed anyway.
What to do instead
Annotate every question with 'term' or 'sum' before selecting a formula, and write the formula from memory each time rather than recalling a shape. For geometric problems, compute r explicitly and write '|r| = ... < 1, so the sum to infinity exists' as a visible step; if |r| >= 1, write 'diverges' and stop. Keep a one-line card of the four formulas (arithmetic term, arithmetic sum, geometric term, geometric sum) and rehearse them until they are distinct.
The candidate differentiates or integrates a function without using the chain, product or quotient rule where required, for example writing the derivative of sin(3x) as cos(3x), the derivative of e^(x^2) as e^(x^2), or the integral of cos(2x) as sin(2x).
Pattern matching on familiar functions dominates: the outer function is recognised and its standard derivative recalled, while the inner linear or composite argument is treated as part of the name rather than as a function to be differentiated. The missing factor is invisible because the answer has the right form.
What to do instead
Before differentiating, underline every composite argument and write 'u = ...' explicitly. Then apply the rule and multiply by du/dx, cancelling the constant factor at the end. For integration, always check by differentiating the answer: if d/dx[sin(2x)] gives 2cos(2x), the required antiderivative must carry the factor one half. Do five composite examples of each rule daily until the inner factor is written automatically.
The candidate uses degrees while the calculus requires radians, or mixes them within one question: differentiating sin(x) as cos(x) after working in degrees, or evaluating an arc length or sector area using a degree measure in a formula derived for radians.
School mathematics up to this point is dominated by degrees, and the calculator is usually left in degree mode. The derivative formulas for trigonometric functions and the arc length formula r times theta are only valid in radians, but nothing in the notation signals which unit is in force.
What to do instead
Put the calculator in radian mode at the start of every calculus or trigonometry session and check the mode indicator before each question. Write 'radians' next to any angle used in a derivative, integral, arc length or sector area. When a question gives degrees, convert with a visible multiplication by pi/180 before differentiating.
The candidate solves a logarithmic or exponential equation by 'cancelling' the log or the base: writing log(a + b) = log a + log b, or taking logs of a sum term by term, or solving e^x = 5 by writing x = ln 5 - ln e.
The logarithm laws for products and quotients are over-generalised to sums because the notation looks distributive. The inverse relationship between exp and ln is understood, but the candidate applies it to a sum instead of isolating the exponential term first.
What to do instead
Write the three valid laws (product, quotient, power) on the top of the page and treat any sum inside a log as a warning sign. For equations, isolate the exponential or logarithmic term completely before applying the inverse, then solve the resulting linear equation. Verify by substitution: log 2 + log 3 = log 6, whereas log(2 + 3) is not log 2 + log 3.
The candidate substitutes into a formula for the equation of a line, tangent or normal without checking the sign of the gradient or the position of the point: using the negative reciprocal for a parallel line, or writing the tangent at a stationary point with a finite gradient.
The perpendicular gradient rule is memorised as 'flip and change sign' but is applied indiscriminately, and the candidate does not pause to consider whether the gradient is zero or undefined. The point-gradient form is then used mechanically with the wrong m.
What to do instead
Before writing the line, state the gradient explicitly and justify it: 'gradient of tangent = f'(a) = ...; gradient of normal = -1/that'. If f'(a) = 0, write 'tangent is horizontal, y = ...' and 'normal is vertical, x = ...' rather than forcing the formula. For parallel and perpendicular questions, write the two gradients side by side and check the relationship.
The candidate performs integration by parts or substitution without changing the limits, or forgets the constant of integration, or omits the modulus in ln|x| and the plus-or-minus in inverse trigonometric and square-root results.
Definite integrals are treated as an extension of indefinite ones, so the limits are carried along mechanically while the variable of integration has changed. The constant of integration is dropped because it 'does not matter', and the modulus is dropped because it is rarely visible in the final answer.
What to do instead
For substitution, write the new limits as a separate line ('when x = 0, u = 1; when x = 2, u = 5') before integrating, and never return to x. For indefinite integrals, write '+ C' as part of the answer line, not as an afterthought. For ln and inverse trig, write the modulus or the domain restriction explicitly and check the sign of the integrand over the interval.
The candidate computes a vector product or line intersection by treating vectors as scalars: adding magnitudes instead of components, using the dot product to find a perpendicular vector, or declaring two lines to intersect without solving the simultaneous equations in the parameter.
Vector notation is new and visually similar to coordinate notation, so component-wise operations are mixed with magnitude operations. The condition for perpendicularity (dot product zero) and for parallelism (scalar multiple) are memorised as phrases without the associated computation.
What to do instead
Write vectors in column form and perform every operation component by component. For perpendicularity, compute the dot product and set it to zero; for a perpendicular vector in two dimensions, swap components and negate one. For line intersection, set the two parametric forms equal, solve two of the three equations, then substitute into the third to confirm consistency; if the third fails, state that the lines are skew.
The candidate applies a matrix method without checking the conditions: inverting a matrix whose determinant is zero, using Cramer's rule or an inverse for a system with no unique solution, or assuming that matrix multiplication is commutative when forming a transformation composition.
The inverse formula is learned as a procedure, and the determinant is computed only as an intermediate step rather than as a test. Composition order is confused because function notation f(g(x)) and matrix product AB are not linked explicitly in the candidate's mind.
What to do instead
Compute the determinant first and write 'det = ...; since det is non-zero, the inverse exists' (or 'det = 0, so no inverse'). For systems, check the determinant before choosing a method, and interpret a zero determinant geometrically (parallel lines or coincident lines). For compositions, write the order in words ('apply B first, then A, so the matrix is AB') and test with a simple vector.