CSEC Mathematics Paper 02 January 2022 Study Guide

Examination Overview and Instructions

The Caribbean Examinations Council (CXC) Caribbean Secondary Education Certificate (CSEC) Mathematics Paper 02 General Proficiency for January 2022 consists of ten structured questions divided into two sections. Section I contains seven questions, while Section II contains three questions. Candidates are required to answer all questions and clearly show all working in the provided spaces. The permitted examination materials include an electronic calculator and a geometry set. A dedicated list of formulae is provided on page 4 of the booklet to assist with calculations. The examination duration is set at 2 hours and 40 minutes.

Section I: Computational Arithmetic and Financial Mathematics

Question 1 focuses on numerical operations and real-world financial applications. The first part requires finding the exact value of the expression $\frac{8.9+31.6}{0.75 \times 5.4}$. Additionally, students must compute the value of $3.9 \tan(18^{\circ})$, rounding the result to one decimal place. The financial mathematics portion involves Ria, who earns an hourly rate of $\$13.50$. For a week where she worked $40\,hours$, her earnings must be calculated. In August, she worked for 4 weeks with gross earnings totaling $\$2463.75$. Her regular work week consisted of $40\,hours$, and overtime was remunerated at $1.5$ times the standard hourly rate. Students must demonstrate that Ria worked exactly $15\,hours$ of overtime during that month. Furthermore, a $20\%\text{ tax}$ deduction is applied to her gross earnings, and the remaining balance must be determined. Finally, the question explores simple interest where Ria invests $\$219$ for $3\,years$ at an annual rate of $4.5\%$. The total interest received after the 3-year period must be calculated.

Section I: Algebra, Linear Inequalities, and Geometric Perimeter

Question 2 addresses algebraic factorization, geometric expressions, and inequalities. The first task is to factorize the expression $3n^2 + 15mp$ completely. Next, the question presents a quadrilateral with side lengths defined in terms of $x$ as follows: $(2x-1)\,cm$, $(15-2x)\,cm$, $(3x-7)\,cm$, and $(2x+5)\,cm$. Students are required to write a simplified expression for the perimeter of this quadrilateral in terms of $x$. Given that the perimeter is exactly $32\,cm$, the length of the longest side must be identified. The final part of this question involves solving a compound inequality: -1 < \frac{2-4x}{3} < 5. Candidates must determine all integer values of $x$ that satisfy this range.

Section I: Geometric Properties and Transformations

Question 3 evaluates knowledge of quadrilateral properties, angle geometry, and transformations. Part (a) asks students to match specific quadrilaterals (Trapezium, Rhombus, Kite, Square, Rectangle) to their properties. Specifically, they must identify which quadrilateral has no lines of symmetry and rotational symmetry of order one, which has exactly two lines of symmetry and four right angles, and which has exactly one line of symmetry but no rotational symmetry. Part (b) provides a diagram of four straight lines, including two parallel lines, with given angles of $71^{\circ}$ and $54^{\circ}$. From this, the values of angles $q$ and $r$ must be determined, and a geometrical reason must be provided to explain why $2p = 71^{\circ}$. Part (c) involves transformations on a square grid. Triangle $X'Y'Z'$ is given as the image of Triangle $XYZ$ after an enlargement with a scale factor of $2$ and a center of enlargement at $(5, 1)$. Students must draw the original object Triangle $XYZ$. Additionally, Triangle $X'Y'Z'$ is mapped onto Triangle $X''Y''Z''$ via a reflection in line $P$, and the equation of this mirror line $P$ must be stated.

Section I: Functions and Inverse Mappings

Question 4 defines three distinct functions: $f(x) = 2x - 1$, $g(x) = 3x + 2$, and $h(x) = 5^{x}$. The student is tasked with evaluating the outputs for specific inputs, including $f(-3)$, $h(0)$, and the composite function value $g^{2}(-3)$. Further algebraic work is required to find the simplified expression for the composite function $gf(x)$. The question also requires finding the inverse function $g^{-1}(x)$ and then using that inverse to determine the value of $x$ when $g^{-1}(x) = 4$.

Section I: Statistical Data Analysis and Probability

Question 5 utilizes a cumulative frequency diagram representing the walking distances of $120\,students$ to school. Students must extract information from the graph, such as the number of students who walked at most $1\,km$. Additionally, the diagram is used to estimate the median distance, the lower quartile, and the interquartile range. The probability part of the question asks for the likelihood that a randomly chosen student walked more than $1.5\,km$. Finally, candidates must complete a frequency table containing distance intervals, midpoints ($x$), and numbers of students ($f$). The intervals included are 0 < d \le 0.5 with midpoint $0.25$ and $f=12$, 0.5 < d \le 1.0 with midpoint $0.75$ and $f=24$, and 1.0 < d \le 1.5 with midpoint $1.25$ and $f=46$. Values for the remaining intervals (1.5 < d \le 2.0, 2.0 < d \le 2.5, and 2.5 < d \le 3.0) must be used to calculate an estimate for the mean distance walked by the total group of students.

Section I: Solid Geometry and Volumetric Rates

Question 6 involves a three-dimensional scenario where water is served from a cylindrical container into cone-shaped cups. The cylinder has a radius of $12\,cm$ and a height of $20\,cm$, and is initially full. The cups have a radius of $3\,cm$ and a height of $8\,cm$. Students must calculate the volume of water in the cylinder in litres, using the conversion $1000\,cm^{3} = 1\,litre$, and providing the answer to two decimal places. Water flows through a pipe at a rate of $7.8\,ml/s$. Using the formula for the volume of a cone, $V = \frac{1}{3} \pi r^{2} h$, students must find the time in seconds required to fill one empty cup. The question concludes by asking for the total number of cups that can be completely filled from the full cylindrical container.

Section I: Number Patterns and Sequences

Question 7 explores a sequence of figures constructed from lines of unit length and dots placed at vertices. The figures consist of octagons and squares. In Figure 1, there are $8\,dots$ and a perimeter of $8$. Figure 2 has $16\,dots$ and a perimeter of $14$. Figure 3 has $24\,dots$ and a perimeter of $20$. Students are required to draw Figure 4 and complete a table relating the Figure Number ($n$) to the Number of Dots ($D$) and the Perimeter ($P$). They must determine these values for Figure 4 and for a figure where the perimeter is $86$. Finally, the relationship between dots and perimeter is analyzed; for a figure where the difference $D - P = 36$, the specific figure number $n$ must be calculated.

Section II: Advanced Algebra and Graphical Analysis

Question 8 centers on the quadratic function $f(x) = 3 + 5x - x^{2}$. Candidates must complete a table of values for $x$ ranging from $-1$ to $6$. Using these values, the graph of the function must be plotted on a grid. From the graph, students identify the equation of the axis of symmetry and the maximum value of the function. A linear function, $y = 3 - \frac{1}{2}x$, is then introduced. Candidates must find the coordinates where this line crosses the x-axis and the y-axis, and subsequently draw the line on the same grid as the quadratic function. The intersection points between the quadratic curve and the linear line provide the solutions to the simultaneous equations $y = 3 + 5x - x^{2}$ and $y = 3 - \frac{1}{2}x$, which must be determined from the graph.

Section II: Circle Theorems and Trigonometric Bearings

Question 9 is divided into circle geometry and trigonometry. Part (a) depicts a circle with center $O$, diameter $RP$, and tangent $AB$ at point $P$. Points $P, Q, R, S$ lie on the circumference. Given angles include $\angle APQ = 3x^{\circ}$, $\angle QPR = 2x^{\circ}$, $\angle RPS = x^{\circ}$, and $\angle QSP = 54^{\circ}$. Students must determine the values of $x$, $y$, and $z$, providing detailed reasons based on circle theorems (e.g., angle between tangent and chord, angles in a semicircle). Part (b) describes the spatial relationship between four towns: $L, M, N$, and $R$. Distance measurements are $LR = 18\,km$, $LN = 12\,km$, and $MN = 10\,km$. Known angles include $\angle RLN = 25^{\circ}$ and $\angle LMN = 88^{\circ}$. The bearing of town $N$ from town $M$ is specified as $50^{\circ}$. Students must calculate the angle $\angle MLN$, the distance $NR$, and the bearing of Town $R$ from Town $L$.

Section II: Vectors, Matrices, and Linear Transformations

Question 10 covers vector geometry and matrix algebra. Part (a) utilizes a coordinate grid with points $O$, $P$, and $R$. Students must write the position vector $\mathbf{OR}$ in the form $\begin{pmatrix} x \ y \end{pmatrix}$. A point $Q$ is defined such that the vector $\mathbf{QR} = \begin{pmatrix} 2 \ -4 \end{pmatrix}$. Students must plot $Q$ and calculate the magnitude of the vector $\mathbf{QR}$, denoted $|QR|$. Furthermore, they must prove via calculation that the shape $OPQR$ is a parallelogram. Part (b) requires solving a matrix equation for unknowns $x$ and $y$: $\begin{pmatrix} 2 &amp; -1 \ 5 &amp; -4 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 6 \ 6 \end{pmatrix}$. Part (c) involves a transformation matrix $T = \begin{pmatrix} 0 &amp; 1 \ 1 &amp; 0 \end{pmatrix}$. This matrix maps the point $S(2, 5)$ onto $S'(5, 2)$. Candidates must provide a full description of the single transformation represented by matrix $T$.