QCE General Mathematics exam 2026Exams: Thu 5 Nov to Fri 6 Nov · QCAA timetable

If the residuals are randomly scattered above and below zero with no pattern, a linear model is appropriate. If they show a clear pattern (especially a curve, such as positive, then negative, then positive), the association is non-linear and a linear model is not appropriate, however high $R^2$ is.

Trend: long-term direction. Seasonality: systematic, calendar-related movements that repeat each cycle. Irregular fluctuations: unsystematic, short-term movements that follow no pattern.

Same meridian (or the equator): $D = 111.2 \times \text{angular distance}$

Same parallel of latitude $\theta$: $D = 111.2\cos\theta \times \text{angular distance}$

$D$ is in kilometres and the angular distance is in degrees (convert minutes to decimal degrees before multiplying).

Degree $=$ number of edge ends at a vertex (a loop counts 2). Sum of degrees $= 2e$, so there is always an even number of odd-degree vertices. A simple graph has no loops and no multiple edges; a complete graph joins every pair of vertices exactly once and has $\frac{n(n-1)}{2}$ edges on $n$ vertices.

A cut separates the source from the sink. Its capacity is the sum of the capacities of the edges crossing it from the source side to the sink side only. Every cut capacity is an upper bound on the flow, and by the maximum-flow minimum-cut theorem the maximum flow equals the capacity of the minimum cut.

Euler's formula for a connected planar graph: $v + f - e = 2$ ($v$ vertices, $f$ faces including the outer face, $e$ edges). Rearranged: $f = 2 - v + e$, $e = v + f - 2$, $v = 2 - f + e$. Combine with the sum of degrees, which equals $2e$, to find $e$ from vertex degrees.

Perpetuity: $A = \dfrac{d}{i}$, so $d = Ai$ and $i = \dfrac{d}{A}$. The payment equals the interest each period, so the balance never changes and the payments continue forever. $i$ must be the rate for the same period as the payment: monthly payments need the monthly rate.