SACE Specialist Mathematics exam 2026Exam: Tue 10 Nov · SACE Board timetable

The base case is the first domino. The inductive step guarantees that whenever one domino falls, the next falls too. Together they knock down every domino from $n_0$ onward, so no individual case needs checking.

If $w = r(\cos\theta + i\sin\theta)$, the $n$ distinct $n$th roots of $w$ are

Each root has modulus $r^{1/n}$, and successive arguments differ by $\dfrac{2\pi}{n}$.

It returns the distance of $x$ from $0$, so $|x| \ge 0$ always. Useful facts: $|x|^2 = x^2$, $|ab| = |a||b|$, and $|x| = c$ (for $c > 0$) means $x = c$ or $x = -c$. Also $|x - a|$ is the distance between $x$ and $a$ on the number line.

$(f \circ g)(x)$ is defined only when $x$ is in the domain of $g$ and $g(x)$ lies in the domain of $f$. Always check the inner output is allowed as an input to the outer function - the natural domain of the composite can be smaller than the domain of $g$ alone.

Two nonzero vectors are perpendicular if and only if $\mathbf{a} \cdot \mathbf{b} = 0$. This is because $\cos 90^{\circ} = 0$. It is the fastest way to check perpendicularity and to find an unknown component that makes two vectors orthogonal.

The direction of the line through points $P$ and $Q$ is $\vec{PQ} = Q - P$. Any nonzero scalar multiple of a direction vector describes the same line, so directions are only defined up to scale.

You split the integrand into a part to differentiate ($u$) and a part to integrate ($dv$). The goal is that the new integral $\int v\,du$ is simpler than the original.

A good $u$ is usually the "inside" of a composite function, or whatever sits under a root, inside a power, or in an exponent - provided its derivative appears (possibly times a constant) elsewhere in the integrand. After substituting, the integral must contain only $u$ and $du$.

You only need one arbitrary constant. Combine the constants from both integrals into a single $C$ on the right-hand side. Then apply the initial condition to solve for $C$ before, or after, rearranging for $y$ - whichever is cleaner.