QCAA Specialist Mathematics common mistakes
QCAA Specialist Mathematics · Understand the traps and how to avoid them.
When negating a statement that contains quantifiers, the candidate swaps 'for all' to 'there exists' but leaves the inner predicate unchanged, producing a false negation such as negating 'for all x, P(x)' as 'there exists x, P(x)'.
The candidate treats negation as a mechanical token swap and does not re-evaluate the truth of the resulting statement, so the required change to the predicate (adding 'not') is forgotten.
What to do instead
Write the negation in two stages: first flip each quantifier in order and reverse the inequality or predicate, then test the result on a single counterexample value to confirm the truth values are opposite. Practise with nested quantifiers where the order of the swap matters.
In an induction proof the candidate writes the algebra for the inductive step but never states the inductive hypothesis as an explicit assumption, or never verifies the base case.
Candidates imitate the look of an induction argument from worked examples without internalising that the hypothesis is a premise to be used and the base case anchors the chain.
What to do instead
Use a fixed four-line scaffold: state the proposition P(n); prove P(1) (or the stated base); write 'Assume P(k) is true for some integer k' and display it; then derive P(k+1) explicitly citing where the hypothesis was substituted. Mark each line as base, assumption, or step before writing prose.
The candidate performs arithmetic on complex numbers in Cartesian form but forgets that division requires multiplying numerator and denominator by the conjugate, instead dividing real and imaginary parts separately.
Real-number intuition suggests componentwise operations are valid, and the conjugate method is a procedure the candidate has memorised without understanding it enforces a real denominator.
What to do instead
Always rewrite division as a fraction and show the conjugate multiplication explicitly, then verify the result by multiplying back: (quotient) times (divisor) should recover the dividend. Do three verification checks per practice set until the reflex is automatic.
When finding roots of a complex equation, the candidate gives only the principal root and omits the other roots, or states arguments without restricting them to a consistent principal range such as (-pi, pi].
The candidate applies real-number square-root habits and forgets that the nth root of a complex number produces n distinct values spaced around a circle.
What to do instead
Before solving, note the degree of the equation and count how many roots are expected. Use modulus-argument form, add 2k pi to the argument, and list every k from 0 to n-1, then convert back to Cartesian form and plot the roots to confirm they are evenly spaced.
The candidate computes a dot product or cross product correctly but then misinterprets what it means, for example concluding two vectors are parallel from a zero dot product or perpendicular from a zero cross product.
The two products are learned close together and their geometric meanings are stored as similar-sounding facts, so they get interchanged under time pressure.
What to do instead
Attach each product to its geometric test in a written summary: zero dot product means perpendicular, zero cross product means parallel. Before answering any geometry question, state which test is needed and why, then compute the matching product only.
When solving a matrix equation or finding an inverse, the candidate multiplies matrices in the wrong order, writing AB when BA is required, or assumes matrix multiplication commutes.
Scalar algebra habits carry over, and the non-commutativity of matrix multiplication is a rule the candidate knows in theory but does not apply when rearranging equations.
What to do instead
Treat every matrix equation as left-multiplication or right-multiplication explicitly: to remove a factor on the left, multiply both sides on the left. Write the order of every product above the expression before computing, and check dimensions to catch impossible orderings.
The candidate differentiates or integrates a composite function without applying the chain rule, for example writing the derivative of sin(3x) as cos(3x) or the integral of e^(2x) as e^(2x).
Pattern recognition fires on the outer function alone, and the inner derivative is treated as a detail rather than a required factor.
What to do instead
Adopt a spoken routine: name the outer function, name the inner function, differentiate the outer, then multiply by the derivative of the inner. For integrals, substitute u for the inner function and convert dx before integrating. Say the routine aloud during practice until it becomes silent habit.
In a separable differential equation the candidate separates variables but omits the constant of integration, or applies an initial condition before integrating and therefore cannot determine the constant.
The candidate treats the constant as optional decoration and rushes to substitute the initial condition as soon as a formula appears.
What to do instead
Impose a fixed order: separate, integrate both sides, add one constant, then substitute the initial condition to solve for it. Write 'plus C' as a mandatory line in every solution, and check the final particular solution satisfies both the differential equation and the initial condition.
The candidate computes a probability for a compound event by adding probabilities of overlapping events without subtracting the intersection, or treats dependent events as independent.
The addition rule is remembered in its simplest disjoint form, and the phrase 'and' is loosely associated with multiplication regardless of whether the events influence each other.
What to do instead
Before computing, draw a two-way table or Venn diagram and mark whether the events overlap and whether one changes the chance of the other. State the applicable rule in words first (add then subtract the overlap, or condition the second probability on the first), then substitute numbers.
When solving a trigonometric equation the candidate gives only the solution in the principal range and misses the second solution in the given interval, or loses solutions when dividing both sides by a trigonometric factor.
Inverse trigonometric functions return a single principal value, and dividing by a factor silently discards the case where that factor is zero.
What to do instead
Find the general solution first using the symmetry of the relevant quadrant, then filter to the stated interval by listing candidate angles systematically. Never divide by a trigonometric expression: factorise and set each factor to zero, then check each case against the interval.