CSEC Mathematics Paper 02 Study Notes

Caribbean Secondary Education Certificate (CSEC) Examination Mathematics Paper 02 - General Proficiency 2019

Examination Information

The Caribbean Examination Council administers this examination, which includes the following critical details:

• Test Code: 01234020
• Subject: Mathematics
• Proficiency Level: General
• Date: 04 January 2019 (a.m.)

General Instructions (Page 3)

Students must follow these instructions carefully to ensure their exam responses are clear and structured:

1. The examination consists of TWO sections: Section I contains SEVEN questions, and Section II contains THREE questions.
2. Candidates are required to answer ALL questions.
3. Answers must be written in the spaces provided within the booklet; candidates must not write in the margins.
4. All calculations and workings must be clearly shown for evaluation purposes.
5. A list of important formulae will be provided on page 4 of this booklet.
6. If additional space is required for answers, candidates should use the extra pages at the end of the booklet, ensuring to draw a line through any original answers and mark the question number on the extra pages.
7. Required materials include an electronic calculator and a geometry set.
8. The candidates are instructed not to turn the page until instructed to do so.

List of Mathematical Formulae (Page 4)

The following formulae are vital for completing examination questions:

• Volume of a prism: $V = Ah$ where A is the area of the cross-section and h is the perpendicular length.
• Volume of a cylinder: $V = \frac{22}{7} r^2 h$ where r is the radius and h is the height.
• Volume of a right pyramid: $V = \frac{1}{3} Ah$ where A is the area of the base and h is the height.
• Circumference of a circle: $C = 2\frac{22}{7} r$.
• Arc length: $S = \frac{ heta}{360} imes 2\frac{22}{7} r$.
• Area of a circle: $A = \frac{22}{7} r^2$.
• Area of a sector: $A = \frac{ heta}{360} imes \frac{22}{7} r^2$.
• Area of a trapezium: $A = \frac{1}{2}(a + b)h$ where a and b are lengths of the parallel sides and h is the distance between them.
• Quadratic equation roots: For $ax^2 + bx + c = 0$, the solutions are x = rac{-b ext{±} rac{ ext{√}(b^2 - 4ac)}{2a}.

Trigonometric Ratios:

• $ext{sin} heta = \frac{ ext{opposite}}{ ext{hypotenuse}}$
• $ext{cos} heta = \frac{ ext{adjacent}}{ ext{hypotenuse}}$
• $ext{tan} heta = \frac{ ext{opposite}}{ ext{adjacent}}$

Area of a Triangle:

• For base b and height h: $ext{Area} = \frac{1}{2} bh$.
• For sides a, b, c: $ext{Area} = \frac{1}{4} ext{√} s(s-a)(s-b)(s-c)$ where $s = \frac{a + b + c}{2}$.
• Sine Rule: $\frac{a}{ ext{sin} A} = \frac{b}{ ext{sin} B} = \frac{c}{ ext{sin} C}$.
• Cosine Rule: $a^2 = b^2 + c^2 - 2bc ext{cos} A$.

Section I Questions (Pages 5-12)

All questions require clear workings:

Question 1 (9 marks)

1. (a) Compute the following:
   1. (i) Evaluate $3.8 imes 10^2 + 1.7 imes 10^3$, providing the answer in standard form (2 marks).
   2. (ii) Evaluate $\frac{1}{2} imes \frac{3}{5} - \frac{3}{12}$, presenting the answer as a fraction in lowest terms (2 marks).
   3. (b) Convert the decimal number 6 to a binary number (1 mark).
   4. (c) Calculate the value of a car originally bought for $65,000 that depreciates by 8% each year after 2 years (2 marks).
   5. (d) A student's exam performance is assessed with the following results:
      | Paper | Percentage Obtained | Maximum Mark |
      |-------|---------------------|---------------|
      | 01 | 55% | 30 |
      | 02 | 60% | 50 |
      | 03 | 80% | 20 |
      Determine, as a percentage, the student's final mark for the Mathematics examination (2 marks).

Question 2 (9 marks)

2. (a)
   1. (i) Manipulate $y = \frac{x}{5} + 3p$ to make x the subject (2 marks).
   2. (ii) Factorize and solve the equation $2x^2 - 9x = 0$ (2 marks).
   3. (b) A rectangular plot's width is defined as 3 meters shorter than its length, denoted by $l$, enclosing an area of 378 square meters. Confirm the expression $l^2 - 3l - 378 = 0$ (2 marks).
   4. (c) The force $F$ on an object is directly proportional to the extension $e$, expressed as $F = ke$. Identify the values of $x$ and $y$ in the incomplete proportionality table as shown (3 marks).

Question 3 (9 marks)

3. (a) Construct triangle ABC following these specifications: AB = 5cm, with angle ABC = 90° and angle BAC = 60° (4 marks).
   1. (b) Referencing a right triangle with side lengths a, b, c:
      • (i) Derive an expression for c in terms of a and b (1 mark).
      • (ii) Provide an expression for $ext{sin} heta + ext{cos} heta$ in terms of a, b, c (2 marks).
      • (iii) Confirm that $( ext{sin} heta)^2 + ( ext{cos} heta)^2 = 1$ using the results from (i) and (ii) (2 marks).

Question 4 (9 marks)

4. (a) Given the function $h(x) = \frac{2x + 3}{5 - x}$:
   1. (i) Identify the value of x for which the function is undefined (1 mark).
   2. (ii) Determine an expression for the inverse function h^-1(x) (3 marks).
   3. (b) An L-shaped graph intersects the x-axis and y-axis. Determine:
      • (i) The gradient of the line (2 marks).
      • (ii) The equation of the line (1 mark).
      • (iii) The equation of a perpendicular line through point P (2 marks).

Section II Questions (Pages 25-36)

Algebra, Relations, Functions, and Graphs

8. (a) Complete the table for the function $f(x) = 3 + 2x - x^2$ and subsequently draw the function's graph for $-2 ext{ to } 4$ (3 marks).
   • Determine:
     • (i) Maximum point coordinates (1 mark).
     • (ii) Range for which f(x) > 0 (2 marks).
     • (iii) Gradient of $f(x)$ at $x=1$ (1 mark).

Geometry and Trigonometry

9. (a)
   • Determine angles given the circle's construction with known angles (situation not drawn to scale). Specifically calculate:
     • (i) Angle PTR (2 marks).
     • (ii) Angle TPQ (2 marks).
     • (iii) The obtuse angle POR (2 marks).
   • (b) Use the angles of depression from a lighthouse to two approaching ships (12° and 20° respectively), and their distance of 110 m apart:
     • (i) Illustrate the diagram and calculate distance TS2 to Ship 2 (3 marks).
     • (ii) Calculate the height of the lighthouse TB (2 marks).

Vectors and Matrices

10. (a) Discuss matrix P, showing whether it is singular or not by calculation (2 marks).
    • Compute values a and b from PQ = R (3 marks).
    • Clarify the impossibility of the matrix product QP (1 mark).
    • (b) For parallelogram OABC, where X and Y are midpoints of AB and BC respectively, show how $OX + OY = k(r + s)$ using vector methods (3 marks).

Additional Space

Any extra responses can be written on the designated extra space pages. Ensure to mark the question number on these pages if utilized.

Receipt Instructions for Candidates

Candidates should ensure to fill out all requested information accurately in block capitals and keep their receipt safe after submission of the examination booklet. This slip is signed by the supervisor/invigilator for authentication after the exam is completed.