HSC Maths Advanced exam 2026Exam: Mon 19 Oct · NESA timetable

The average rate of change of $y$ over the interval $x_1 < x < x_2$ is the gradient of the chord joining the points $(x_1, y_1)$ and $(x_2, y_2)$ on the curve:

It is the constant rate that would give the same total change across the interval.

The logarithm is the index. For a base $a > 0$ with $a \neq 1$, $y = \log_a x$ means $x = a^y$. The base of the log is the base of the power; the value of the log is the index. So $\log_2 8 = 3$ because $8 = 2^3$. The number inside a log must be positive, since every power of a positive base is positive.

To convert degrees to radians multiply by $\dfrac{\pi}{180}$; to convert radians to degrees multiply by $\dfrac{180}{\pi}$. Cancel the resulting fraction to keep the exact form (for example $150^\circ = \dfrac{5\pi}{6}$). One radian is $\dfrac{180}{\pi} \approx 57.30^\circ$.

To evaluate a power with a complicated index, work in this order:

The two line tests classify every graph into one of four types. Run the vertical test first, then the horizontal test:

A quadratic $y = ax^2 + bx + c$ (with $a \ne 0$) graphs as a parabola.

The shape of $y = x^n$ depends on whether $n$ is even or odd.

The three special expansions are $(A + B)^2 = A^2 + 2AB + B^2$, $(A - B)^2 = A^2 - 2AB + B^2$ and $(A + B)(A - B) = A^2 - B^2$. Read forwards they expand; read backwards they factor. The single most common error is forgetting the $2AB$ cross term in a perfect square, so write it every time.

For $y = \sin x$ and $y = \cos x$ (in degrees):

For $y = a\sin x$ or $y = a\cos x$ the amplitude is $|a|$ and the period stays $360\degree$. The tangent curve has no amplitude (it climbs without bound) and a period of $180\degree$.

The sine rule holds in every triangle:

For an acute angle $\theta$ in a right-angled triangle, with sides named relative to $\theta$:

The Pythagorean identity holds for every angle $\theta$:

Single application (one strip, from $x = a$ to $x = b$):

Write the curve as $x = g(y)$. For the region between the curve and the $y$-axis from $y = a$ to $y = b$:

A rate of change proportional to the current value, $\frac{dN}{dt} = kN$, always solves to $N(t) = N_0 e^{kt}$, where $N_0$ is the initial value. $k > 0$ gives growth and $k < 0$ gives decay. To establish this in an exam, differentiate $N_0 e^{kt}$ and show it equals $kN$; do not just quote the solution.

For $y = a \sin(b(x - h)) + d$ (cosine is identical; tangent drops the amplitude):

The identities worth knowing cold:

The three rules for any triangle $ABC$, side $a$ opposite angle $A$ and so on: