Cameroon GCE Ordinary Level Mathematics Paper 2 (June 2020) Comprehensive Study Guide

Cameroon General Certificate of Education (GCE) Board Ordinary Level Examination - June 2020

Subject Title: Mathematics
Subject Code: 0570
Paper Number: 2
Examination Date: June 2020
Time Duration: Two and a Half hours
Registration Requirements: - Registration Centre Number - Centre Name - Candidate’s Full Names - Candidate Identification Number
Examination Structure and General Instructions: - The paper consists of two sections: Section A and Section B. - Candidates must answer ALL questions in both sections. - Responses for Section A must be written in the spaces provided. - Section B questions carry equal marks. - Good English and orderly presentation of answers are required. - All steps in calculations must be shown, providing answers at each stage. - Calculators are permitted for use.

Section A: Mathematics Proficiency and Analytical Problems (15 Questions)

1. Arithmetic Simplification: - Task: Simplify the expression involving the following numerical components: $3$ , $6$ , $11$ , $4$ , $7$ , and $12$ . Based on the layout, the expression structure relates to: $\frac{3}{4} \times \frac{6}{7} \div \frac{11}{12}$ .
2. Set Theory and Representation: - Given Sets: - A = { x \mid 0 < x < 4, x \in \mathbb{Z} } - B = { x \mid -1 \leq x < 2, x \in \mathbb{Z} } - Required Tasks: - (a) List the elements of set $A$ . - (b) List the elements of set $B$ . - (c) Draw a Venn diagram to represent these sets.
3. Formal Logic Table: - Task: Complete the logic table provided in Figure 1 for variables $p$ , $q$ , and the operations $\neg q$ (NOT $q$ ) and $p \land \neg q$ (p AND NOT $q$ ). - Given Truth Values: - $p=T, q=T$ - $p=T, q=F$ - $p=F, q=T$ - $p=F, q=F$
4. Inequalities and Number Lines: - (a) Solve the combined inequality: -7 < 5x - 2 < 3 - (b) Represent the solution set on a real number line.
5. Mathematical Functions: - Functions defined on $\mathbb{R}$ (the set of real numbers): - $g : x \mapsto x^2 + 2$ - $h : x \mapsto x + 2$ - Required Calculations: - (a) $g(2)$ - (b) $h^{-1}(x)$ - (c) $h(g(x))$
6. Direct Variation: - Conditions: $y \propto x$ where $y = 10$ when $x = \frac{1}{3}$ . - Required Tasks: - (a) Find the relation between $x$ and $y$ . - (b) Determine the value of $x$ when $y = 6$ .
7. Trigonometry and Surveying: - Scenario: The angle of elevation of the top of a tower from point $M$ ( $20\,m$ from the base) is $050^\circ$ . - Required Tasks: - (a) Calculate the height of the tower to one decimal place. - (b) State the angle of depression of point $M$ from the top of the tower.
8. Geometry of Composite Shapes (Figure 2): - Components: Semicircle $AEB$ and rectangle $ABCD$ . - Dimensions: $AC = 20\,m$ and $AB = 14\,m$ . - Task: Find the area of the shaded region in the figure.
9. Sequences and Summations: - Sum of the first $n$ terms formula: $S_n = 2n^2 - n$ , where $n \in \mathbb{Z}^{+}$ . - Required Tasks: - (a) Find the first three terms of the sequence. - (b) State the specific type of sequence described.
10. Matrix Algebra: - Given Matrices: - $A = \begin{pmatrix} 5 & 2 \ 3 & -1 \end{pmatrix}$ - $B = \begin{pmatrix} -3 & 0 \ 1 & 4 \end{pmatrix}$ - Operations To Perform: - (a) State the transpose of matrix $A$ . - (b) Find the matrix resulting from $A - B$ .
11. Circle Geometry (Figure 3): - Context: $APQ$ is a circum-circle. Chord $AP$ subtends an angle of $120^\circ$ at the center $O$ of the circle. - Task: Calculate the values of the angles labeled $x$ and $y$ .
12. Coordinate Geometry and Line Segments: - Points: $A(1, 3)$ , $B(2, y)$ , and $C(4, 6)$ . - Condition: Points $A$ , $B$ , and $C$ lie on a straight line. - Required Tasks: - (a) Determine the value of $y$ . - (b) Calculate the length of line segment $AB$ , leaving the answer in surd form. - (c) Determine the ratio $AB : BC$ .
13. Probability and Random Sampling: - Box contents: $6$ red balls, $4$ white balls, and $5$ blue balls. - Task: Determine the probability of drawing: - (a) A white ball. - (b) A ball that is NOT red. - (c) A ball that is either red or white.
14. Network Graphs (Figure 4): - Requirements: Identify and state individual counts for: - (a) The number of nodes. - (b) The number of edges. - (c) The number of regions.
15. Statistics via Bar Charts (Figure 5): - Context: Bar chart showing goals per match for a team across $10$ matches. - Required Findings: - (a) The number of goals scored in two matches by the team. - (b) The total number of goals scored by the team during the last football season.

Section B: Multi-Part Advanced Problems (4 Questions)

1. Financial Mathematics and Matrix Operations: - (i) Property Purchase: - Michael, Peter, and John buy a plot for $2,800,000\,FCFA$ . - 65% of the total value is paid as an initial deposit. - (a) Calculate the initial deposit amount. - (b) The deposit is split in the ratio $5:3:2$ . Calculate Peter's specific contribution. - (c) A processing fee of $91,000\,FCFA$ is required for documents. Find the percentage of the initial deposit that this fee represents. - (d) Calculate the remaining balance to be paid to the seller. - (ii) Matrix Analysis: - Given Matrix $M = \begin{pmatrix} 1 & 4 \ 2 & 3 \end{pmatrix}$ . - Required Tasks: (a) Find the determinant of $M$ , (b) Find the adjugate of $M$ , (c) Determine the inverse of matrix $M$ .
2. Data Interpretation and Analytical Geometry: - (i) Physics Test Statistics: - Student scores: $70, 80, 78, 98, 84, 67, 98, 70, 80, 100, 87, 83, 70, 70, 88, 91, 70, 78, 88, 88$ . - Required Tasks: - (a) Represent scores in a frequency distribution table. - (b) Determine the mode. - (c) Determine the median score. - (d) Determine the mean score. - (ii) Linear Equations: - Line $l_1$ passes through $P(3, 4)$ and $Q(1, -2)$ and intersects the y-axis at point $R$ . - Required Tasks: (a) Determine the equation of line $l_1$ , (b) State the coordinates of point $R$ .
3. Geometric Transformations and Graphing: - Triangle Vertices: $A(1, 4)$ , $B(1, 1)$ , and $C(3, 1)$ . - Required Tasks: - (a) Find coordinates of $A'B'C'$ after rotating $ABC$ through $90^\circ$ anticlockwise about the origin. - (b) Plot $ABC$ and $A'B'C'$ on a graph ( $1\,cm = 1\,unit$ scale for $-4 \leq x \leq 4$ and $-2 \leq y \leq 10$ ). - (c) Draw the mirror line defined by the equation $x + y = 6$ . - (d) Reflect triangle $ABC$ across the mirror line $x + y = 6$ . - (e) Determine the resulting coordinates for triangle $A''B''C''$ .
4. Calculus, Function Analysis, and Vectors: - (i) Quadratic Function Graphing: - Function: $f(x) = 5 + 3x - x^2$ for values of $x$ from $-2$ to $+5$ . - Tasks: (a) Draw the graph using a scale of $1\,cm$ to $1\,unit$ on both axes. (b) Use the graph to solve $5 + 3x - x^2 = 0$ . (c) Determine the gradient of the curve at the y-axis intercept. - (ii) Vector Calculations: - Given position vectors: $\vec{P} = 3\mathbf{i} + 2\mathbf{j}$ and $\vec{Q} = -\mathbf{i} + 2\mathbf{j}$ . - Equation provided: $\vec{OP} = 3\vec{OQ} + 2\vec{OR}$ . - Tasks: (a) Find $2\vec{OR}$ expressed in terms of $\mathbf{i}$ and $\mathbf{j}$ . (b) Determine the position vector of point $R$ .