HSC Maths Ext 2 common mistakes
HSC Maths Ext 2 · Understand the traps and how to avoid them.
When proving a statement by induction, the candidate writes the algebra for the inductive step but never states or uses the assumption that the result holds for n = k, jumping straight from the base case to 'hence true for n = k+1'.
Students treat the inductive step as ordinary algebra and forget that the hypothesis 'assume true for n = k' is the logical bridge; without it the working only re-proves the base case and the argument collapses.
What to do instead
Adopt a rigid three-line scaffold before any algebra: (1) base case n = 1 verified explicitly, (2) 'Assume the statement is true for n = k, that is [write the exact assumed equation]', (3) 'Now prove for n = k+1' and explicitly substitute the assumed equation at the point where the k-case expression appears. Mark the substitution in the working so it is visible to the marker.
The candidate differentiates or integrates a function without first restricting attention to a valid interval or checking the conditions of the theorem, e.g. applying the second derivative test at a point where f''(x) = 0 and declaring it a point of inflection, or using a series expansion outside its radius of convergence.
Students memorise test procedures as decision rules and skip the hypothesis-checking step, so a zero second derivative is read as 'inflection' and a divergent expansion is summed anyway.
What to do instead
Before applying any test, write down its hypotheses as a checklist and verify each one: for the second derivative test, if f''(x) = 0 the test is inconclusive, so examine sign changes of f'' on either side; for series, state the interval of convergence and confirm the value being substituted lies strictly inside it. Practise classifying stationary points by a sign table of f' rather than by f'' alone.
When computing with complex numbers, the candidate writes modulus and argument results without accounting for the quadrant, e.g. taking arg(z) = arctan(b/a) for a point in the second or third quadrant, or writing Arg(z1 z2) = Arg(z1) + Arg(z2) when the sum leaves (-pi, pi].
The arctan formula is learned as a universal recipe and the multi-valued nature of argument is glossed over; students also conflate the principal argument with an arbitrary argument.
What to do instead
Always sketch the point in the complex plane first and determine the quadrant before writing the argument, adjusting by pi where needed. For products and quotients, use mod-arg form and then explicitly reduce the resulting angle into (-pi, pi] by adding or subtracting 2pi, showing that reduction as a line of working.
In vector geometry, the candidate uses the dot product where a projection or perpendicular distance is needed, or writes the scalar projection of b onto a as a·b rather than (a·b)/|a|, and confuses the vector equation of a line with that of a plane.
Dot and cross product formulas are memorised as bare expressions without their geometric meaning, so students cannot tell which quantity answers the question being asked.
What to do instead
For every vector problem, first name the geometric object required (angle, length, area, distance, projection) and write its formula from the definition before substituting. Remember a·b = |a||b|cos(theta) gives the angle and hence the projection, while |a x b| gives area and perpendicular distance. Check the answer's type: a scalar projection must be a number, not a vector.
The candidate treats 'proof by contradiction' and 'proof by contraposition' as interchangeable, or attempts to disprove a universal statement with a single example while trying to prove an existential statement by algebra on a general variable.
The quantifier structure of the statement is never analysed, so students pick a proof strategy by habit rather than by the logical form of what must be shown.
What to do instead
Before writing, rewrite the statement in quantifier form ('for all x, P(x) implies Q(x)' versus 'there exists x such that P(x)'). Match strategy to form: universal implication suits direct proof or contraposition, 'no such object exists' suits contradiction, existence suits construction with an explicit witness, and disproof of a universal needs one counterexample. State the strategy explicitly in the first sentence of the solution.
In mechanics, the candidate resolves forces or applies Newton's second law without a clear sign convention or free-body diagram, producing equations in which weight, normal reaction and friction appear with inconsistent signs, or applies energy conservation when non-conservative friction does work.
Students jump to equations from memory of similar problems; without a diagram the direction of each force is guessed, and the condition for conservation of mechanical energy is not checked.
What to do instead
Draw a free-body diagram for each body, mark an axis with a stated positive direction, and resolve every force along and perpendicular to that axis before writing equations. Explicitly check whether friction or an applied non-conservative force acts; if so, use the work-energy theorem with the work done against friction included rather than plain energy conservation.
When handling binomial coefficients, the candidate writes the general term of (a + b)^n incorrectly, e.g. omitting the powers of a and b or using the wrong index, and then solves for the term independent of x from a faulty expression, or treats C(n, r) as defined for r > n.
The general term is recalled loosely and the constraint 0 <= r <= n is forgotten, so the algebra is built on a wrong starting expression and impossible values of r are accepted.
What to do instead
Write the general term in full every time: T_(r+1) = C(n, r) a^(n-r) b^r, with a and b copied exactly from the bracket including signs. When a term is specified (independent of x, or a given power), equate exponents to find r and then verify 0 <= r <= n is an integer; if not, state that no such term exists.
The candidate uses a standard limit or series expansion in a context where it does not apply, e.g. applying the small-angle approximation sin(x) ~ x to a term that is not small, or substituting a limit into a quotient whose denominator tends to zero without further analysis.
Approximations are learned as formulas divorced from their validity conditions, and limit laws requiring finite limits of both numerator and denominator are applied mechanically.
What to do instead
For each approximation, record its validity condition (for instance x small in radians) and check the actual size of the argument before use. For limits, verify the quotient rule's hypotheses hold; if the denominator tends to zero, factor, rationalise, or use L'Hopital's rule only after confirming the indeterminate form, and justify each step.
When finding a recurrence relation or closed form for a sequence, the candidate guesses a formula from the first few terms and asserts it without proof, or manipulates the recurrence by treating it as an equation in n rather than in the index.
Pattern-spotting feels sufficient because the first terms fit, and the need for a rigorous link between recurrence and closed form is underestimated.
What to do instead
After conjecturing a closed form, always prove it by induction: check the base case, assume the formula at n = k, and substitute the recurrence to derive the n = k+1 case. Keep the index explicit throughout and never replace n by a specific value mid-derivation.
In further differentiation, the candidate differentiates an implicitly defined or parametrically defined function by treating y as a constant, or forgets the chain rule factor when differentiating a composite such as a function of y with respect to x.
Students default to differentiating 'x-terms normally' and lose track of which variable is the function of which, especially when y appears inside a power or trigonometric function.
What to do instead
Before differentiating, state the dependency explicitly ('y is a function of x'). For implicit differentiation, differentiate both sides term by term and attach dy/dx to every derivative of a y-expression, showing that factor. For parametric curves, write dy/dx = (dy/dt)/(dx/dt) and compute each derivative separately before dividing.