HSC Mathematics Advanced: what actually gets examined

Better responses

• Applying transformations one at a time, separating the horizontal change from the vertical one and drawing a small intermediate sketch for each stage before the final graph.
• Sketching a hyperbola with both asymptotes ruled and labelled, intercepts marked, and the branches approaching the asymptotes without ever crossing them.
• Reading the horizontal and vertical translations straight off the parameters in the equation, then using one given point to fix the remaining constant instead of solving simultaneous equations.
• Using the symmetry of a parabola, via its axis of symmetry, to locate the vertex or an unknown coefficient rather than grinding through longer algebra.
• Writing a domain or range in proper notation, either interval notation or inequalities, with each end point correctly included or excluded.
• Solving an inequality from a graph by reading where one curve sits below the other and expressing the answer as a combined inequality.

Where marks were lost — and how to keep them

• Keeping dilations and translations apart: a factor inside the bracket rescales x-values, a factor outside rescales y-values, and a shift right by h means replacing x with x minus h.
• Handling a quadratic whose leading coefficient is not 1 when it is presented in transformed form: complete the square or expand and match coefficients, without slips in the perfect square.
• Rejecting solutions the context rules out after solving a quadratic, such as a negative translation or a value that makes an exponential non-positive.
• Telling inverse variation from direct variation and linking k over x to a reciprocal graph; plotting from a table on the grid's actual scale and joining with a smooth curve rather than ruled segments.
• Clearing algebraic fractions and changing the subject of a formula cleanly, especially when a numerator has two terms or brackets fix the order of operations.
• Interpreting the discriminant: two points of intersection need a positive discriminant, and any further condition in the question also restricts the answer.

Better responses

• Working entirely in radians when the domain is given in radians, and in degrees when it is in degrees, with the calculator in the matching mode and exact values used for the standard angles.
• Solving an equation in a compound angle by first rescaling the domain for the whole argument, finding every quadrant solution there, then dividing back to get each value of x.
• Reading amplitude, centre line and period from a graph or context and using period equals 2 pi over b to find b, keeping the vertical shift distinct from the amplitude.
• Choosing the right tool for a triangle: a plain trigonometric ratio when it is right-angled, otherwise the sine or cosine rule copied exactly from the reference sheet with the substitution shown.
• Using the sector and arc formulas with the angle in radians and working out what fraction of a full circle an arc really is, rather than assuming a semicircle.
• Sketching the trigonometric curve in a practical problem to see when two heights are equal or where the function is decreasing, then confirming with algebra.

Where marks were lost — and how to keep them

• Finding the obtuse solution in the ambiguous case of the sine rule and, more generally, allowing for solutions in the other quadrants of a specified domain.
• Labelling sides and angles on a 3D or bearings diagram, never assuming an angle is 90 or 45 degrees from how the figure looks, and relating a bearing to the angle inside the triangle.
• Proving an identity from one side only, using a common denominator and the Pythagorean identities; the tan and sec forms are not on the reference sheet and must be known.
• Drawing a sine or cosine graph on an axis measured in plain units such as time, recognising when the period is longer than the window shown and drawing only the required part.
• Understanding how the coefficient b, the horizontal scale factor and the period of a trigonometric function depend on one another.
• Picking the right area formula: the perpendicular sides for half base times height, the included angle for half ab sin C, and telling a segment from a sector or semicircle.

Better responses

• Interpreting the base of an exponential model: a base above 1 grows by base minus 1 per period, a base below 1 decays, and the initial value comes from t equals 0 with e to the power 0 being 1.
• Isolating the exponential term first, then taking logarithms of both sides or converting to log form to solve for the exponent, with each rearrangement written down.
• Sketching an exponential as a smooth curve approaching a labelled asymptote, with the axes labelled and time on the horizontal axis, built by transforming the basic curve.
• Using logarithm laws confidently in exact answers, such as rewriting ln of a half as minus ln 2, combining two logs into one, or changing the base.

Where marks were lost — and how to keep them

• Turning a percentage into the right multiplier: a 5.5% rise is a factor of 1.055 and a 3% fall is 0.97, so the percentage change and the factor are different numbers.
• Applying logarithm laws to equations with negative indices and to inequalities, where dividing by a negative logarithm reverses the inequality sign.
• Moving between logarithmic and index form to solve an equation, including when the equation comes from a cumulative distribution function.
• Remembering that 'initial' means t equals 0, that e to the power 0 equals 1, and that e itself is a constant, not a variable to solve for.

Better responses

• Matching the rule to the structure, product for x times a trig function, chain for a power of a bracket, quotient for a fraction, writing u, v, u' and v' and simplifying first where that avoids a harder rule.
• Establishing the nature of a stationary point with a second-derivative value or a table of first-derivative signs, and comparing end-point values whenever a global maximum or minimum is wanted.
• Setting up an area between curves as top minus bottom integrated between the intersection x-values, showing the limits substituted in brackets and treating a negative result as a prompt to recheck.
• Leaning on the reference sheet for standard derivatives and integrals, including a to the power x, and rewriting roots and fractions in index form before integrating.
• Applying the trapezoidal rule with the right count, n applications need n plus 1 function values, transcribing table values exactly, and linking concavity to whether the estimate is over or under.
• Recognising that 'hence' means using the derivative just found, whether for a reverse chain rule integral or a product-rule result that feeds a definite integral.

Where marks were lost — and how to keep them

• Finding a tangent: evaluate the derivative at the point to get a numerical gradient, then use point-gradient form; when the gradient is given, f'(x) equals m is an equation to solve.
• Including the constant of integration and evaluating it from an initial condition, and integrating with respect to the right variable when other pronumerals appear in the expression.
• Reading gradient and concavity from the signs of f' and f'', justifying a change of concavity at any point of inflection, and separating horizontal points of inflection from other kinds.
• Distinguishing area from a definite integral for regions below the x-axis, splitting a shaded region into a triangle plus an integral, and knowing some regions need geometry, not integration.
• Keeping the calculator in radians for any trigonometric calculus and writing trig derivatives in correct notation, such as sec squared x rather than sec of 2x.
• In optimisation, building the function from the constraint first, simplifying before differentiating, rejecting solutions outside the domain, and checking whether a quadratic even needs calculus.

Better responses

• Building the recurrence explicitly, A1 from the principal and then A2 in terms of A1 before substituting, showing that interest applies for the whole period before the repayment is taken off.
• Writing out at least three terms to confirm whether a sequence is arithmetic or geometric and to fix the first term, ratio and number of terms before choosing a sum formula.
• Converting the rate and the number of periods to the compounding period first, then locating the table factor and knowing whether to multiply by it or divide by it.
• Isolating the (1 plus r) to the power n term and solving for n with logarithms, then reading the non-integer answer as a count of full payments plus a smaller final one.
• Seeing the connection between an annuity table's factors and the sum of a geometric series, and between one part of a question and the next.

Where marks were lost — and how to keep them

• Using the table when one is supplied instead of rebuilding the series, and reading its factor once, since the rate and periods are already built into that factor.
• Telling an annuity from a single compound-interest sum, and a 'total' (the sum of the series) from the nth term.
• Counting the terms in a geometric series correctly by stating the first and last term explicitly; an off-by-one error in n changes every later step.
• Deriving a given result in a 'show that' question rather than working backwards from it, and not proving a formula the question says to take as given.
• Stating the first term and the common ratio before applying the limiting sum formula, remembering the size restriction on the ratio, and simplifying the fraction.
• Solving simultaneous equations from a geometric sequence by dividing to eliminate the first term, and remembering that an even power gives two possible common ratios.

Better responses

• Drawing and labelling the normal curve, placing the mean, the z-scores and the shaded region on it, before reading the table or applying the empirical rule.
• Using the z-score formula from the reference sheet and then the table for a non-integer z-score, reserving the 68-95-99.7 rule for whole-number z-values.
• Interpreting bivariate data in context: the slope as a rate in the variables' own units, the intercept as the value when x is zero, and form, direction and strength named in syllabus terms.
• For a probability density function, setting the total area to 1 to find an unknown, integrating from the lower bound to x for the CDF, and setting the CDF to 0.5 or 0.25 for a median or percentile.
• Organising multi-stage probability with a tree diagram or list, multiplying along branches and adding across, and using the complement for 'at least one'.
• Giving as many distinct, contextual observations as there are marks, each stated with the statistical term, such as median, interquartile range, skew or increasing at a decreasing rate.

Where marks were lost — and how to keep them

• Reading the table as the probability below a z-score and taking the complement or the difference of two values for 'greater than' or 'between', then finishing with the count asked for.
• Keeping z-scores, percentages and probabilities distinct, and comparing z-scores rather than raw scores when two data sets have different means and standard deviations.
• Telling a PDF from a CDF: the mode is the maximum of f (an end point if f only decreases), the median is where F reaches 0.5, and P(X greater than a) is 1 minus P(X less than a).
• Recognising 'given' as conditional probability and using the intersection over the condition's probability; independence means the conditional probability equals the unconditional one.
• Explaining why extrapolation is unreliable: a regression line used outside the data range can give impossible values, and the answer must say so in the context of the variables.
• Separating the slope from the correlation coefficient, the mean or expected value from the mode, and remembering that the standard deviation is the square root of the variance.