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Unit 1

Financial Mathematics

Simple interest is linear (same interest each year); compound interest grows exponentially. If the question says just 'interest', assume compound.
Compounding period matters: for k periods per year at nominal rate r%, use $F V = P V (1 + r / (100 k))^{k n}$ .
Effective annual rate is what actually gets paid over one year: $(1 + r / (100 k))^{k} - 1$ , expressed as a %.

Q: €2000 at 6% p.a. compounded monthly for 5 years — value at end?

A: FV = 2000(1 + 0.06/12)^60 = 2000 × 1.005^60 ≈ €2697.70.

Q: Effective annual rate for 8% p.a. compounded quarterly?

A: (1.02)^4 − 1 = 0.0824 → 8.24%.

Unit 3

Voronoi Diagrams

A Voronoi diagram partitions the plane so every point belongs to the region of its nearest 'site' (marked point).
The boundary between two adjacent regions is the perpendicular bisector of the segment joining those two sites.
Toxic waste dump problem: the optimal location is a vertex of the Voronoi diagram — the point *furthest* from all sites. Check every vertex; the answer is the one whose distance to its 3 nearest sites is largest.

Q: Given sites at (0,0), (6,0), (3,4) — which point is equidistant from all three (circumcentre)?

A: Circumcentre is the Voronoi vertex — solve the perpendicular-bisector equations pairwise. Here: (3, 5/6) ≈ (3, 0.833).

Unit 4

Chi-squared test

Use χ² for categorical data — testing whether observed frequencies match expected.
Test statistic: $χ^{2} = \sum \frac{(f_{o} - f_{e})^{2}}{f_{e}}$ . Degrees of freedom = (rows − 1) × (cols − 1) for contingency tables.
If your GDC gives p < significance level → reject H₀ (there is an association). If χ²_calc > χ²_crit → same conclusion.

Q: 2×3 table, χ²_calc = 8.2, 5% significance. Critical value χ²_crit(df=2, 0.05) = 5.99.

A: 8.2 > 5.99, so reject H₀ at the 5% level: there is significant association between the variables.

Unit 1

Approximation and error bounds

When determining upper and lower bounds for a calculated value, ensure you consider the maximum and minimum possible values for each input based on its stated precision. Think carefully about how these extremes combine in your calculations.
A common pitfall is forgetting the rules for setting bounds. If a value is given as $x$ to the nearest $0.1$ , its range is $[x - 0.05, x + 0.05]$ . If it's given to 2 significant figures, say $2.3$ , its range is $[2.25, 2.35]$ . Do not round intermediate calculations; only apply appropriate rounding for the final upper and lower bounds if specifically instructed.
Be mindful of how operations affect bounds. For addition, $LB (a + b) = LB (a) + LB (b)$ . For subtraction, $LB (a - b) = LB (a) - UB (b)$ . For multiplication (with positive values), $LB (a \times b) = LB (a) \times LB (b)$ . Similar rules apply for upper bounds and division.

Q: A rectangle has length $L = 8.5$ cm (1 d.p.) and width $W = 3.2$ cm (1 d.p.). Find the lower bound for its perimeter.

A: Lower bound for $L = 8.45$ cm. Lower bound for $W = 3.15$ cm. Perimeter LB $= 2 (LB (L) + LB (W)) = 2 (8.45 + 3.15) = 2 (11.60) = 23.20$ cm.

Unit 4

Linear regression on the GDC

Always ensure your GDC is set to the correct mode (e.g., 'Diagnostics On' for $r$ and $r^{2}$ ) and that all data points are entered accurately into lists. Even a single data entry error can lead to a completely incorrect regression equation and subsequent analysis.
A common pitfall is incorrectly identifying the independent (x) and dependent (y) variables in context. Swapping them will give an inverse relationship, drastically altering your regression equation and the meaning of any predictions.
When asked for the regression equation, ensure you state it in the exact form requested (e.g., $y = a x + b$ or $y = m x + c$ ) and use the original variable names from the problem (e.g., $C = a T + b$ if 'Cost' and 'Temperature' are given). Avoid premature rounding; use values from your GDC to at least 3 significant figures.

Q: The GDC output for a linear regression is $y = a x + b$ with $a = 0.7891$ and $b = 12.345$ . Write down the regression equation, rounding coefficients to 3 significant figures.

A: $y = 0.789 x + 12.3$ .

Q: For data showing a strong negative linear correlation, what would be a likely value for the correlation coefficient, $r$ ?

A: $r = - 0.91$ (or any value close to $- 1$ , e.g., $- 0.85$ to $- 0.99$ ).

Unit 1

Compound interest with sinking funds

Crucial Calculator Settings: Always verify your financial calculator (or GDC's TVM solver) has P/Y (payments per year) and C/Y (compounding periods per year) correctly set based on the question. A frequent error is defaulting both to 12 when they should be different (e.g., quarterly payments, monthly compounding).
Sinking Funds = Future Value: Sinking fund questions are fundamentally about accumulating a specific future lump sum through periodic payments. You will typically be solving for the payment (PMT) given a known Future Value (FV), rather than a Present Value (PV).
Identify All Variables: Before jumping into calculations, clearly list your N, I%, PV, PMT, FV, P/Y, and C/Y values. Ensure consistent signs for PV/PMT/FV; for a sinking fund, the FV is usually positive (target amount), and PMT will be negative (outflow).

Q: A company needs to accumulate $$500,000 $i n 4 y e a r s . W h a t m o n t h l y p a y m e n t m u s t b e m a d e i n t o a f u n d e a r n i n g$ 5.4\%$ p.a. compounded monthly to reach this goal?

A: Using a TVM solver with $N = 4 \times 12 = 48$ , $I % = 5.4$ , $P V = 0$ , $F V = 500000$ , $P / Y = 12$ , $C / Y = 12$ , the required monthly payment is $P M T = $$ -9477.72.