HSC Maths Ext 1 common mistakes
HSC Maths Ext 1 · Understand the traps and how to avoid them.
When finding the inverse of a function, the candidate swaps x and y but then solves for y and labels the result f(x) instead of f⁻¹(x), or forgets to state the domain and range of the inverse, giving an answer that ignores any restriction on the original domain.
The swap-and-solve procedure is learned as a mechanical routine, so the candidate focuses on algebra and never re-checks which variable the answer should be expressed in, and domain restriction is treated as an optional extra rather than the definition of invertibility.
What to do instead
After solving, write the answer explicitly as f⁻¹(x) and test one point: if f(a)=b then f⁻¹(b) must equal a. Before starting, write the domain and range of f in a box, then state that the domain of f⁻¹ equals the range of f and vice versa; only proceed once the function is confirmed one-to-one on the stated domain.
The candidate differentiates or integrates a composite trigonometric expression such as sin²(3x) or tan(x²) by differentiating only the outer function, writing 2sin(3x) or sec²(x²), omitting the chain-rule factor.
Under time pressure the outer derivative is salient and the inner derivative is treated as a formality; the candidate has practised isolated standard derivatives far more than composites, so the inner factor is not part of the retrieved pattern.
What to do instead
Adopt a written two-line ritual: first name the inner function u, write du/dx beside it, then write the outer derivative evaluated at u, and only then multiply. Do five composite derivatives per study session out loud, saying the inner derivative aloud each time, until the verbal step becomes automatic.
When integrating to find an area or a volume of revolution, the candidate computes the antiderivative correctly but substitutes the limits in the wrong order or drops the subtraction, and when the curve crosses the axis within the interval, integrates straight across the crossing and reports a signed total as the area.
The definite integral is rehearsed as an algebraic evaluation task, so the geometric meaning (signed area) is not consulted at the moment of setting up; sketching is skipped because it feels like wasted time.
What to do instead
Always sketch the curve and shade the region before writing any integral, and split the integral at every x-intercept, taking absolute values of each piece. Write the evaluation as F(b) − F(a) with the upper limit substituted first, and sanity-check that an area answer is positive and of plausible size relative to the sketch.
In counting problems the candidate cannot decide whether order matters and cannot decide whether repetition is allowed, so they apply a permutation when a combination is required (or vice versa), and they add counts for cases that should be multiplied.
The formulas are memorised as symbols rather than as answers to two yes/no questions, so the choice is made by pattern-matching on keywords like 'arrangements' instead of by analysing the structure of the selection.
What to do instead
Before computing, answer in writing: does swapping two chosen items give a different outcome, and can an item be chosen more than once? Then apply the multiplication principle for successive stages and addition only for mutually exclusive cases. Verify small cases by listing all outcomes by hand and checking the count matches the formula.
In an induction proof the candidate writes the statement for n = k, then writes the statement for n = k + 1, and asserts that the second follows from the first without any algebraic link; the crucial step from assumption to target is replaced by 'therefore it is true'.
The structure of the argument is copied from a model proof without understanding that the middle step is the whole proof; the candidate mistakes restating the goal for proving it.
What to do instead
Write the target for n = k + 1 first, then start from the assumption for n = k and manipulate it explicitly towards the target, showing every algebraic line. Finish with a sentence naming both the base case and the inductive step before concluding. Practise by writing the linking line as a separate numbered step so it cannot be skipped.
The candidate computes a scalar product or magnitude correctly but then confuses the scalar projection with the vector projection, or uses the dot product formula with an unnormalised direction vector when finding the perpendicular distance from a point to a line.
Both formulas look similar and are recalled as a single blurred template; the distinction between a scalar and a vector output is not checked because the candidate is focused on getting a number.
What to do instead
Label the expected type of every answer before computing: a projection length is a scalar, a projection vector must be written in component form. For distance problems, always normalise the direction vector explicitly and write the formula with the unit vector substituted, then check the answer is positive and no larger than the distance to the line's nearest endpoint.
When asked for the probability of a union or a conditional probability, the candidate adds probabilities of overlapping events without subtracting the intersection, or computes P(A|B) using P(A) alone, ignoring the conditioning information.
The addition rule is remembered in its mutually exclusive form, and conditional probability is read as ordinary probability because the vertical bar is not interpreted as a restriction of the sample space.
What to do instead
Draw a two-way table or Venn diagram for every probability question and shade the region asked for before writing a formula. State the conditional formula explicitly with the conditioning event as the denominator, and check the answer lies between 0 and 1 and is consistent with the shaded region.
When solving a trigonometric equation over a stated interval, the candidate finds only the principal solution from the inverse function and stops, missing the second or further solutions generated by the symmetry of the circle.
The inverse trigonometric function returns a single value, and the candidate treats that output as the complete solution set rather than as one representative of a family.
What to do instead
After finding the reference angle, immediately write the general solution using the appropriate symmetry (for example, supplement for sine, period for tangent), then substitute successive integer values and test each against the stated interval. Mark all solutions on a sketched unit circle to confirm none are missed.
When finding stationary points or classifying them, the candidate solves f′(x) = 0 but never checks the sign of f′ on either side, and when a parameter appears in the function, differentiates treating the parameter as a variable or as a constant inconsistently.
The derivative is seen as the goal rather than as a tool, so the second half of the procedure is dropped; parameters are unfamiliar symbols and their role is decided arbitrarily mid-calculation.
What to do instead
Treat every stationary-point question as a three-part task: solve, build a sign table of f′ on intervals either side, then state the nature and coordinates. Decide at the start that the parameter is a constant, write 'a is constant' at the top of the working, and differentiate accordingly throughout.
The candidate substitutes a value into a rational or radical expression to evaluate a limit and, on obtaining an indeterminate form, concludes the limit does not exist rather than factoring or rationalising to remove the source of the indeterminacy.
Substitution works for most limits encountered in early practice, so the indeterminate result is read as a final answer instead of as a signal that algebraic simplification is required.
What to do instead
Adopt the rule that an indeterminate result is never an answer: it triggers one of three actions, factorise and cancel, multiply by the conjugate, or divide by the highest power. Write the chosen action beside the limit before executing it, and confirm the final value by checking the simplified expression at the limit point.