Cambridge IGCSE International Mathematics Paper 4 (Extended) Study Guide

General Examination Guidelines and Instructions

The Cambridge IGCSE International Mathematics Paper 4 (Extended) for the October/November 2023 session (0607/43) is a two-hour and 15-minute examination totalling $120$ marks. Candidates are instructed to use a black or dark blue pen, while HB pencils are permitted for diagrams and graphs. Necessary materials include geometrical instruments, tracing paper, and a graphic display calculator where appropriate. All working must be shown clearly to receive marks for correct methods, even if final answers are incorrect. Numerical answers should be given to $3$ significant figures, while angles in degrees should be given to $1$ decimal place unless otherwise specified. For calculations involving $ext{r}$, the calculator value should be used.

Reference Mathematical Formula List

The examination provides a comprehensive list of formulas for reference. For the quadratic equation $ax^2 + bx + c = 0$, the solutions are found using $x = \frac{-b ext{!} ext{ (likely } ext{±)} ext{ } ext{√}(b^2 - 4ac)}{2a}$. For cylinders with radius $r$ and height $h$, the curved surface area is $A = 2 ext{π}rh$ and the volume is $V = ext{π}r^2h$. For cones with radius $r$, height $h$, and sloping edge $l$, the curved surface area is $A = ext{π}rl$ and the volume is $V = \frac{1}{3} ext{π}r^2h$. Spheres of radius $r$ have a curved surface area of $A = 4 ext{π}r^2$ and a volume of $V = \frac{4}{3} ext{π}r^3$. The volume of a pyramid with base area $A$ and height $h$ is $V = \frac{1}{3}Ah$. Trigonometric rules include the Sine Rule: $\frac{a}{ ext{sin}(A)} = \frac{b}{ ext{sin}(B)} = \frac{c}{ ext{sin}(C)}$; the Cosine Rule: $a^2 = b^2 + c^2 - 2bc ext{cos}(A)$; and the Area of a Triangle: $ext{Area} = \frac{1}{2}bc ext{sin}(A)$.

Arithmetic and Speed Calculations (Question 1)

Calculations involve comparing world records and determining average speeds. In the men's triple jump, the record was 15.52 ext{ m} in 1911 and increased to 18.29 ext{ m} by 2021. Candidates must calculate 15.52 ext{ m} as a percentage of 18.29 ext{ m}. For running records, the 2021 women's 800 ext{ m} world record was set at 1 ext{ minute} ext{ }53 ext{ seconds}; the task is to find the average speed in $ext{m/s}$. Additionally, the 2021 men's world record speed for the 100 ext{ m} was given as 37.58 ext{ km/h}, requiring the calculation of the total time taken for the run in seconds.

Statistical Analysis and Measures of Central Tendency (Question 2)

A data set of 12 scores from Sunni's games includes: $7, 17, 4, 20, 15, 12, 11, 16, 6, 18, 9,$ and $20$. Analysis targets five statistical measures: (i) the mode, representing the most frequent score; (ii) the median, the middle value of the ordered set; (iii) the mean, defined as the total sum divided by the number of scores; (iv) the range, or the difference between the highest and lowest scores; and (v) the upper quartile ($Q_3$). An evaluative component requires explaining why the mode may not be the optimal measure of average to represent this specific data set.

Geometric Transformations (Question 3)

Geometric problems focus on transformations of triangle A on a coordinate grid. Part (a) requires a translation of triangle A by the vector egin{pmatrix} -5 \ 2 – ext{e} ext{ (likely } egin{pmatrix} -5 \ 2 ext{ or } -2 ext{ depending on OCR interpretation)} ext{)} ext{0} ext{ (given as a vector box)} ext{)} ext{… transcript says } egin{pmatrix} -5 \ 2 ext{ (interpreted as } egin{pmatrix} -5 \ 2 ext{)} ext{)} ext{/ } egin{pmatrix} 5 \ -2 ext{)} . Part (b) involves describing a single transformation mapping triangle A onto triangle C. Part (c)(i) defines triangle D through a sequence: first, a reflection in the line $y = -1$, followed by a $90^°$ clockwise rotation about the point $(-1, 1)$. Part (c)(ii) asks for the description of the single transformation that could replace this sequence to map A directly to D.

Functional Sketching and Analysis (Question 4)

The function $f(x) = \frac{x^2}{3} - 1 - \frac{x}{…} - \frac{7}{x^2} - \frac{3}{2} ext{ (transcript formatting: } x^2/3 - 1/x - x^2/7 - 3/2 ext{)}$ is studied for the interval $-3 ext{≤} x ext{≤} 4$. Tasks include sketching the graph of $y = f(x)$, solving the equation $f(x) = 0$, and determining the values of a constant $k$ for which the horizontal line $f(x) = k$ has exactly two solutions. Furthermore, candidates must identify the range of $x$ values for which the gradient (derivative) of $f(x)$ is negative.

Compound Growth and Value Projection (Question 5)

A picture's value increases by 60 ext{%} every 5 ext{ years}. Given the 2020 valuation of $20,000$, the value recorded in 2015 must be calculated. Students must show that the value in 2040 (four 5-year intervals later) will be exactly $131,072$. Finally, the year in which the picture's value first exceeds $1,000,000$ must be determined using exponential growth calculation methods.

Pyramid Mensuration and Sectioning (Question 6)

Pyramid VABCD has a square base with side AD = 10 ext{ cm} and a vertical height VO = 12 ext{ cm}. Part (a)(i) requests the volume ($V = \frac{1}{3} imes ext{base area} imes ext{height}$). Part (a)(ii) requires showing that the slant height from vertex V to the midpoint M of side CD ($VM$) is 13 ext{ cm}. In part (b), a smaller pyramid VPQRS is removed from the top, where face PQRS is parallel to base ABCD and QR = 8 ext{ cm}. Candidates must calculate the volume of the remaining solid (the frustum ABCDPQRS) and its total surface area, including the base, top, and four trapezoidal faces.

Cumulative Frequency and Grouped Data (Question 7)

Data for $240$ marathon runners is presented via a cumulative frequency curve over the time interval 150 ext{ min} to 220 ext{ min}. Using the curve, students estimate the median time (at $120^{ ext{th}}$ runner) and the interquartile range ($Q_3 - Q_1$). The fastest 20 ext{%} of runners (the top $48$) receive medals; the longest time among these medalists must be estimated. The curve is also used to complete a frequency table for intervals: 150 < t ext{≤} 160, 160 < t ext{≤} 170, etc., with provided frequencies including $16$ and $32$. From this table, an estimate for the mean marathon time is calculated using mid-interval values.

Algebraic Solving and Rearrangement (Question 8)

Algebraic tasks include: (a) evaluating $v = u + at$ for $u = 60, a = -32,$ and $t = 3$; (b)(i) solving the linear equation $\frac{x}{6} + \frac{x}{2} = -9$; (b)(ii) solving the absolute value or fractional equation $\frac{2}{3} (x - 7) = ext{… (transcript: } 2 / (3x-x) ext{ etc.)}$; (c) solving the quadratic $3x^2 - 11x + 6 = 0$ via factorisation. Part (d) requires rearranging the formula $y = \frac{x + ax}{5} + \frac{b}{3} ext{ (reconstructed)}$ to make $x$ the subject. Part (e) involves simplifying the rational algebraic expression $\frac{(x - y)}{(ax - bx + ay - by)} ext{ (interpreted from textual cues: } 9x^2 ext{ etc.)}$.

Set Theory and Probability (Question 9)

Venn diagram skills are assessed through shading regions ($A ext{∩} B'$ and $P' ext{∪} Q$). A survey of $120$ students regarding History ($H$), Geography ($G$), and Economics ($E$) provides numerical data: $x$ students in $H$ only, $25$ in $E$ only, $32$ in $G$ only, $9$ in $H ext{∩} G ext{∩} E'$, $4$ in all three, $17$ in $G ext{∩} E ext{∩} H'$, $7$ in $H ext{∩} E ext{∩} G'$, and $5$ students studying none. Students must find the value of $x$ and solve various probabilities: (ii) a random student studying both History and Geography; (iii) from Economics students, finding one who studies $G$ but not $H$ and another who studies $H$ but not $G$; (iv) selecting three students where two study exactly two subjects and one studies all three.

Functions, Asymptotes, and Composite Solutions (Question 10)

Three functions are defined: $f(x) = \frac{4}{x-1}$, $g(x) = x-3$, and $h(x) = 4 - 2(x+2)$. Tasks include evaluating $g(-3)$, finding the composite value $f(h(4))$, and finding the general expression for $g(f(x))$ in simplest form. Part (a)(iv) requires finding the inverse function $h^{-1}(x)$. Part (b) focuses on sketching the graph of $y = \frac{g(x)}{f(x)}$ for $-2 ext{≤} x ext{≤} 4$, identifying its vertical asymptote, and solving graphically for $h(x) = \frac{g(x)}{f(x)}$. This equation is finally rearranged into the quadratic form $ax^2 + bx + c = 0$ to find constants $a, b,$ and $c$.

Trigonometry and Shortest Distance (Question 11)

Part (a) involves a right-angled triangle with sides 12 ext{ cm} and 18 ext{ cm} to find angle $p$. Part (b) presents a quadrilateral $ABCD$ with sides AB = 230 ext{ m}, BC = 150 ext{ m}, CD = 190 ext{ m}, and DA = 180 ext{ m} (based on transcript figures). Students must use the Cosine Rule to show angle $ABC = 67.0^°$. Additionally, the shortest distance from vertex A to the side BC (the perpendicular height of triangle ABC) must be calculated.

Coordinate Geometry and Bisectors (Question 12)

The line defined by $y = \frac{x}{3} + 7$ is analyzed to find its $y$-intercept and its intersection with the horizontal line $y = 2$. Part (b) utilizes points $A(-5, 8)$ and $B(1, -2)$ to find the equation of the perpendicular bisector of the segment $AB$. This requires finding the midpoint of $AB$, the gradient of $AB$, the negative reciprocal gradient for the perpendicular line, and finally the linear equation in the form $y = mx + c$.