HSC Maths Extension 1 exam 2026Exam: Fri 23 Oct · NESA timetable

If $x = f(t)$ and $y = g(t)$, the chain rule gives $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}$, so

The behaviour at a root $x = a$ is fixed by the multiplicity $m$ of the factor $(x - a)^m$, because near $a$ the curve looks like $y = x^m$ shifted to $a$:

A parametric curve is the path of a moving point $(x(t), y(t))$ as the parameter $t$ runs over its domain. Two facts follow that examiners test directly:

The division algorithm writes any division uniquely as $P(x) = D(x)\,Q(x) + R(x)$ with $\deg R < \deg D$. Two corollaries do most of the work in the HSC:

For $a_n x^n + a_{n-1}x^{n-1} + \dots + a_0$ with roots counted with multiplicity, the elementary symmetric functions of the roots are the coefficients (divided by $a_n$) with an alternating sign:

For a recurrence proof, the recurrence drives the step and the closed form is the target. Always start the step from $a_{k + 1} = f(a_k)$ (the rule), then substitute $a_k = g(k)$ (the hypothesis), and simplify to $g(k + 1)$. Never start from the closed form for $a_{k + 1}$, that would assume what you are proving.

For random samples of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$:

Translate the geometry into algebra:

For each whole number $n \ge 1$,

and by convention $0! = 1$. The factorial is defined only on the whole numbers $0, 1, 2, \dots$, and $n!$ counts the number of ways to arrange $n$ distinct objects in a row.

Arrangements around a circle. The number of ways to arrange $n$ distinct objects around a circle, where arrangements that are rotations of one another are regarded as the same, is

The multiplication principle. If a selection is made in $r$ stages, with $n_1$ choices at the first stage, $n_2$ at the second, and so on to $n_r$ at the last, and the choices are independent, then the number of ways to complete the whole selection is

The identical-elements formula. If $n$ objects consist of $r_1$ alike of one type, $r_2$ alike of another, and so on up to $r_k$ alike of the last type (so $r_1 + r_2 + \cdots + r_k = n$), the number of distinct arrangements of all of them is

$|x - a|$ is the distance from $x$ to $a$. So for $k > 0$:

A function can only change sign at a zero (where $f(x) = 0$) or at a discontinuity (where $f(x)$ is undefined, which in this course means a zero of a denominator). Between consecutive such points the sign is constant, so it can be read off from any single test value in that interval.

For non-zero polynomials $P$ and $Q$ of degrees $n$ and $m$:

The sum and product relations. Reading the elementary symmetric functions of the zeroes off the coefficients, the sign alternates at every step (minus, plus, minus, plus).

At a zero $x = \alpha$ of multiplicity $m$, the graph of $y = P(x)$ behaves like this:

The remainder theorem. When a polynomial $P(x)$ is divided by $x - a$, the remainder is