Anglican High School Secondary 4 Preliminary Examination 2025 Mathematics Study Guide

General Examination Information and Mathematical Formulae

• Examination Details: Anglican High School Secondary Four Preliminary Examination 2025. Subject: Mathematics (4052). Paper 1 and Paper 2 duration: 2 hours 15 minutes each. Total marks: 90 per paper.

• General Instructions:     * Show all working; omission of essential working results in loss of marks.     * Accuracy: If not specified/exact, give answers to 3 significant figures. Degrees to 1 decimal place.     * Constants: For $\pi$, use calculator value or $3.142$.

• Compound Interest Formula: Total amount $= P(1 + \frac{r}{100})^n$.

• Mensuration Formulae:     * Curved surface area of a cone: $\pi r l$     * Surface area of a sphere: $4 \pi r^2$     * Volume of a cone: $\frac{1}{3} \pi r^2 h$     * Volume of a sphere: $\frac{4}{3} \pi r^3$     * Area of triangle ABC: $\frac{1}{2} ab \sin(C)$     * Arc length: $r \theta$ ($\theta$ in radians)     * Sector area: $\frac{1}{2} r^2 \theta$ ($\theta$ in radians)

• Trigonometry Formulae:     * Sine Rule: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$     * Cosine Rule: $a^2 = b^2 + c^2 - 2bc \cos(A)$

• Statistics Formulae:     * Mean: $\bar{x} = \frac{\sum fx}{\sum f}$     * Standard Deviation: $\sqrt{\frac{\sum fx^2}{\sum f} - (\frac{\sum fx}{\sum f})^2}$

Algebra and Equations

• Indices and Simplification:     * Simplifying $\frac{2}{27 \sqrt{a^3}} \times a^{-2}$ results in $\frac{2}{9ab^2}$ (positive index form).     * Expanding and simplifying $(x-y)^2 - (x+1)(x-2y)$ : The result is $y^2 - x + 2y$.

• Factorisation (Grouping):     * Factorize completely: $ax - 2cy - cx + 2ay = (x + 2y)(a - c)$.

• Linear and Fractional Equations:     * Solving $\frac{3}{4} - \frac{9-x}{4} = 3 - 2(4 - x)$: Evaluates to $x = 16$.     * Solving $\frac{2x}{3-2x} = \frac{7}{x-3}$: Solutions are $x = 1.93$ or $x = -0.260$.

• Rearranging Subject of Formula:     * Given $D = (a-c)^2 - a^2$, to make $c$ the subject: $C = a \pm \sqrt{D + a^2}$.

• Algebraic Fractions:     * Simplifying $\frac{2}{9x^2-1} + \frac{1}{1-3x}$: The result in simplest form is $\frac{-3x}{9x^2-1}$.

• Completing the Square:     * Expressing $-x^2 + 4x - 3$ in the form $a(x+b)^2 + c$: $-(x-2)^2 + 1$.     * Turning point coordinates: $(2, 1)$.

• Inequalities:     * Solving \frac{x-3}{2} < \frac{9+x}{4} \leq 7: Result is -3 < x \leq 19.

Number Theory and Patterns

Prime Factorization and Properties:     * $2250 = 2 \times 3^2 \times 5^3$.     * To make $\frac{2250h}{k}$ a perfect cube: Smallest positive integers are $h = 3, k = 2$.     * Integers with $LCM = 2250$ and $HCF = 45$: The two smallest possible integers are $90$ (even) and $1125$ (odd).

• Real-World Scenario: Simultaneous Flashing:     * Green light flashes every $6\,s$, red light every $8\,s$. They next flash simultaneously at the Least Common Multiple (LCM) of 6 and 8, which is $24\,s$.

• Patterns and Energy Harvesting:     * A device captures energy where the fraction of energy in the $n^{th}$ second follows a pattern: $\frac{1}{2} \times \frac{1}{1}$ for 1st, $\frac{1}{2^2} \times \frac{3}{2}$ for 2nd, $\frac{1}{2^3} \times \frac{4}{3}$ for 3rd.     * General formula for the $n^{th}$ second: $\frac{1}{2^n} \times \frac{n+1}{n}$.     * Success threshold: At least $49.8\%$ of maximum capacity is harvested when $K \geq 249$ seconds.

Geometry and Proportionality

• Direct Proportionality:     * Pendulum period $P$ is directly proportional to $\sqrt{L}$. If $P$ increases by $25\%$, the new $P' = 1.25P$: $\frac{P'}{P} = 1.25 = \frac{\sqrt{L'}}{\sqrt{L}}$. Thus $\frac{L'}{L} = 1.25^2 = 1.5625$, representing a $56.25\%$ increase in length.

• Similarity and Scale:     * Pyramids: Great Pyramid surface area $= 8.30 \times 10^4\,m^2$, volume $= 2.45 \times 10^4\,m^3$. Replica volume $= 9.22 \times 10^{-6}\,m^3$.     * Using volume ratio $(\frac{l_1}{l_2})^3 = \frac{V_1}{V_2}$ and area ratio $(\frac{l_1}{l_2})^2 = \frac{A_1}{A_2}$, the surface area of the replica is $4.33 \times 10^{-2}\,m^2$.     * Maps: Map A scale is $1 : 200,000$. Lake area on A is $4\,cm^2$, on B is $2.56\,cm^2$. Resulting scale $n = 250,000$.

• Polygons and Angles:     * Irregular polygon: One angle is $148^{\circ}$, others are $139^{\circ}$. Calculate sides $n$ using $(n-2) \times 180 = 148 + (n-1)139$. Result: $n = 7$ sides.

• Congruency and Geometric Proof:     * In a trapezium $ABCD$ ($DC \parallel AB$) where $DO=CO$:         * Similar isosceles triangles: $\triangle AOB$ and $\triangle COD$.         * Congruency proof for $\triangle ABC \cong \triangle BAD$: $AB$ is common; $\triangle ODC$ is isosceles ($DO=CO$), thus $\angle ODC = \angle OCD$; alternate angles $\angle OBA = \angle ODC$ and $\angle OAB = \angle OCD$; Side $\text{AC = AO + OC}$, $\text{BD = BO + OD}$, hence $AC=BD$. Property used: SAS.

• Circle Geometry:     * Properties utilized: Tangent-radius theorem ($90^{\circ}$), alternate segment theorem, diameter subtends right angle ($90^{\circ}$).     * Calculated angles: $\angle ACG = 70^{\circ}$, $\angle AEX = 30^{\circ}$, $\angle FOA = 60^{\circ}$, $\angle GCE = 98^{\circ}$.

• Radians and Sectors:     * A metal sector with radius $20\,cm$, thickness $50\,cm$, and volume $2000\,cm^3$     * Find $\angle POQ$: $\text{Volume = Area of cross-section } \times \text{ thickness}$. $2000 = (\frac{1}{2} \times 20^2 \times \theta) \times 50 \rightarrow \theta = 0.2\,\text{radians}$.     * Recasting: If melted into a sphere, the radius is $r = 7.82\,cm$.

Statistics and Probability

• Data Analysis (Stem-and-Leaf):     * Sample size $n = 24$ students.     * Interquartile range (IQR): $22.5\,marks$.     * Median: $46\,marks$.     * Mode: $37\,marks$.

• Standard Deviation Analysis:     * Calculation for orange masses: Estimate is $\sigma = 12.0$.     * Effect of error: If each orange is overestimated by $5\,g$, the standard deviation remains unchanged because it measures the spread, which is unaffected by a constant shift.

• Cumulative Frequency and Box Plots:     * Comparing School A and B commute times: School B has a larger range and interquartile range compared to School A, indicating more varied commute times. Median commute for B ($22\,min$) is higher than A ($17\,min$).

• Probability Calculations:     * Probability of two girls in white T-shirts: Part of a set with $N$ children. Solving quadratic gives $N=10$.     * Boy in white vs girls in red: Utilizing given combined probability $\frac{108}{665}$ to find specific counts $a, b, c$.

Trigonometry and Kinematics

• Advanced Trigonometry:     * Obtuse/Acute possibilities: $\angle BAC$ can be $53.9^{\circ}$ or $126.1^{\circ}$ based on sine rule solutions.     * Shortest distance from a point to a line segment: Calculated as the perpendicular height using $\text{Area} = \frac{1}{2} \times \text{base} \times \text{perpendicular height}$.

• Graphing and Kinematics:     * Acceleration: $a = \frac{v - u}{t}$. Note conversion from minutes to hours ($\div 60$).     * Distance: Defined as the area under the speed-time graph.     * Average Speed: For a race total distance $62 \times 4.940\,km = 306.28\,km$. To finish in $\leq 2\,hours$, remaining laps must be at minimum average speed.

Financial Mathematics and Matrices

• Banking and Loans:     * Compound Interest (Monthly): If $6\%$ per annum compounded monthly, $r = 0.5\%$ and $n = 60$ for 5 years.     * Hire Purchase: Total cost $= \text{Deposit } + (\text{Monthly instalment } \times \text{ Duration})$. Rate of interest calculated on the original loan amount.

• Matrices:     * Purchase Matrix $Y$: Columns represent different types of pies (Chicken, Apple, Taro); rows represent individuals (Peter, Jane).     * Matrix $W = YX$: Represents total expenditure and savings relative to different bakeries.

Real-World Case Study: Cookie Business Optimization

• Operational Costs:     * Depreciation of 3 ovens: Total cost $\$10,950$. Lifespan: 2 years. Calculation per day: $\frac{10950}{2 \times 365} = \$15$     * Fixed costs: Electricity/water ($\$6$), delivery per location ($\$3.50$).     * Variable costs: Materials per size (Large: $20\,c$, Medium: $15\,c$, Small: $11\,c$), packaging containers ($\$0.30$ to $\$0.60$).

• Profit Maximization:     * Identify container efficiency (fitting exactly 20 cookies based on diameter and thickness).     * Suggesting strategy: Reducing material waste or bulk purchasing ingredients to lower cost price without affecting the final selling price ($\$1.00$, $\$0.80$, $\$0.50$).