HSC Maths Ext 2 case studies
HSC Maths Ext 2 · Put your knowledge to work in context.
Mathematics Extension 2 is the most demanding mathematics course in the NSW Higher School Certificate, taken alongside Mathematics Extension 1 and typically chosen by students aiming at mathematics-heavy university degrees or who simply enjoy hard problems. It is assessed by a written examination in which full working must be shown, and marks are awarded for reasoning and proof, not just final answers. The HSC Maths Ext 2 case studies are 5 cases and about 2,928 words.
1A vanishing derivative at an endpoint of the interval
A student is analysing f(x) = x^3 - 3x on the closed interval [0, 2]. They compute f'(x) = 3x^2 - 3, solve f'(x) = 0 to get x = 1 (rejecting x = -1 as outside the interval), and then evaluate f(1) = -2.
573 words, about 3 min2Deciding whether an induction argument is valid
+ (2n - 1) = n^2 + 1. Their argument: 'Base case n = 1: LHS = 1, RHS = 1 + 1 = 2.
605 words, about 3 min3Choosing coordinates for a projectile with a sloped landing
A ball is launched from the origin with speed 20 m/s at angle θ above the horizontal, over horizontal ground, and lands on a plane inclined at 30° to the horizontal that rises away from the launch point. A student sets up axes along and perpendicular to the inclined plane, resolves gravity into components -g sin30° along the plane and -g cos30° perpendicular to it, and writes the perpendicular displacement as zero at landing.
566 words, about 3 min4A locus problem where squaring hides a branch
A student is asked to describe the locus of points z in the complex plane satisfying z - 2 = z + 2i . They square both sides, write (z - 2)(z̄ - 2) = (z + 2i)(z̄ - 2i), expand, cancel z z̄, and obtain -2z - 2z̄ + 4 = 2i z̄ - 2i z + 4.
588 words, about 3 min5When a series test gives no information
A student must determine whether Σ (from n = 1 to ∞) of a n converges, where a n = (n^2 + 1)/(n^3 + n). They apply the ratio test: a {n+1}/a n → 1, so they conclude the ratio test is inconclusive, then apply the comparison test against 1/n, noting a n ≈ n^2/n^3 = 1/n, and conclude the series diverges.
596 words, about 3 min