Total marks: 100 · Duration: 2 hours 40 minutes
Instructions to candidates
- This paper consists of TWO sections: Section I and Section II.
- Section I has SEVEN questions worth a total of 64 marks. Answer ALL questions in Section I.
- Section II has THREE questions worth 12 marks each. Answer ALL questions in Section II.
- Write your answers in the spaces provided. All working must be shown clearly.
- A silent, non-programmable electronic calculator, mathematical instruments and graph paper may be used.
- Diagrams are NOT drawn to scale. Where a question describes a shape in words, you may find it helpful to sketch it first.
- Take π to be 22/7 or 3.14 unless the question says otherwise.
- In this paper a 2 × 2 matrix is written row by row, with a semicolon between the rows. For example, (3 2 ; 1 4) has first row 3, 2 and second row 1, 4. A column vector is written (x, y).
Paper
Section I (64 marks)
Question 1
(a) Using a calculator, or otherwise, evaluate
(3.25² − 1.8) ÷ (0.4 × 2.5)
(i) giving your answer exactly (2 marks)
(ii) giving your answer correct to 2 significant figures. (1 mark)
(b) The cash price of a television is $4800. It can also be bought on hire purchase by paying a deposit of 15% of the cash price, followed by 18 monthly instalments of $260.
(i) Calculate the deposit. (1 mark)
(ii) Calculate the total hire purchase price. (2 marks)
(iii) Express the extra amount paid on hire purchase as a percentage of the cash price. (2 marks)
(c) Mr Ali changes US$250 into Trinidad and Tobago dollars. The exchange rate is US$1 = TT$6.78. How many TT dollars does he receive? (1 mark)
Total: 9 marks
Question 2
In a class of 40 students:
- 22 students play cricket (C)
- 18 students play football (F)
- x students play both cricket and football
- 5 students play neither game.
(a) Draw a Venn diagram to represent this information. Show the number of students in each region, in terms of x where necessary. (2 marks)
(b) Write an equation in x and solve it. (3 marks)
(c) How many students play only one of the two games? (2 marks)
(d) A student is chosen at random from the class. Find the probability that the student plays football only. (2 marks)
Total: 9 marks
Question 3
(a) Write as a single fraction in its simplest form
2x/3 − (x − 1)/4 (2 marks)
(b) Factorise completely 2y² − 8. (2 marks)
(c) Solve the simultaneous equations
2x + 3y = 12 5x − y = 13 (3 marks)
(d) Given that p = 2q² − r, calculate the values of q when p = 74 and r = 2. (2 marks)
Total: 9 marks
Question 4
(a) A circular pond has a radius of 7 m. It is surrounded by a concrete path 3.5 m wide, so that the outer edge of the path is also a circle with the same centre. Use π = 22/7.
(i) Calculate the area of the path. (3 marks)
(ii) Concrete costs $40 per square metre. Calculate the cost of concreting the path. (1 mark)
(b) A water tank in the shape of a cuboid measures 2.4 m long, 1.5 m wide and 0.8 m high.
(i) Calculate the volume of the tank in cubic metres. (1 mark)
(ii) Express this volume in litres. [1 m³ = 1000 litres] (1 mark)
(iii) The empty tank is filled by a pump that delivers 12 litres per minute. How many hours does it take to fill the tank? (2 marks)
(c) Convert a speed of 72 km/h to metres per second. (1 mark)
Total: 9 marks
Question 5
The marks scored by 50 students in a test are shown in the table.
| Marks | Frequency | Cumulative frequency |
|---|---|---|
| 1 – 10 | 4 | |
| 11 – 20 | 7 | |
| 21 – 30 | 12 | |
| 31 – 40 | 15 | |
| 41 – 50 | 8 | |
| 51 – 60 | 4 |
(a) State the modal class. (1 mark)
(b) Copy and complete the cumulative frequency column. (2 marks)
(c) On graph paper, using upper class boundaries (10.5, 20.5, …), draw the cumulative frequency curve for the data. Use a scale of 2 cm to represent 10 marks on the horizontal axis and 2 cm to represent 10 students on the vertical axis. (3 marks)
(d) Use your graph to estimate the median mark. (1 mark)
(e) A student is chosen at random. Find the probability that the student scored more than 40 marks. (2 marks)
Total: 9 marks
Question 6
(a) Triangle ABC has vertices A(1, 2), B(4, 2) and C(1, 6).
(i) Calculate the length of BC. (1 mark)
(ii) Triangle ABC is reflected in the x-axis to give triangle A′B′C′. State the coordinates of A′, B′ and C′. (2 marks)
(iii) Triangle ABC is enlarged, centre the origin, with scale factor 2, to give triangle A″B″C″. State the coordinates of A″, B″ and C″. (2 marks)
(iv) Calculate the area of triangle ABC, and hence state the area of triangle A″B″C″. (2 marks)
(b) Using a ruler, a pencil and a pair of compasses only, construct triangle PQR in which PQ = 8 cm, angle QPR = 60° and PR = 6 cm. Measure and state the length of QR. (2 marks)
Total: 9 marks
Question 7
Study the pattern in the table below.
| Row (n) | Sum of odd numbers | Result |
|---|---|---|
| 1 | 1 | 1 = 1² |
| 2 | 1 + 3 | 4 = 2² |
| 3 | 1 + 3 + 5 | 9 = 3² |
| 4 | 1 + 3 + 5 + 7 | 16 = 4² |
| 5 |
(a) Complete Row 5 of the table. (2 marks)
(b) For Row 10, state
(i) the last odd number in the sum (1 mark)
(ii) the result. (1 mark)
(c) Write, in terms of n,
(i) the last odd number in Row n (1 mark)
(ii) the result for Row n. (1 mark)
(d) The result in one row is 625. Find the number of the row, and the last odd number in that row. (2 marks)
(e) Use your answers above to find the value of 21 + 23 + 25 + … + 49. (2 marks)
Total: 10 marks
Section II (36 marks)
Question 8 — Algebra, Relations, Functions and Graphs
(a) The functions f and g are defined by f(x) = 2x − 3 and g(x) = x² + 1.
(i) Evaluate fg(2). (1 mark)
(ii) Find an expression for f⁻¹(x). (2 marks)
(b) Copy and complete the table for y = x² − 2x − 3.
| x | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| y | 5 | −3 | −3 | 5 |
(2 marks)
(c) On graph paper, using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis, draw the graph of y = x² − 2x − 3 for −2 ≤ x ≤ 4. (2 marks)
(d) Use your graph to
(i) state the roots of x² − 2x − 3 = 0 (1 mark)
(ii) state the coordinates of the minimum point (1 mark)
(iii) write down the equation of the axis of symmetry (1 mark)
(iv) solve the equation x² − 2x − 3 = 2. (2 marks)
Total: 12 marks
Question 9 — Measurement, Geometry and Trigonometry
(a) P, Q, R and S are points on a circle with centre O. PR is a diameter of the circle. Angle QPR = 34°. S lies on the circle on the same side of the chord QR as P. TQ is the tangent to the circle at Q, where T lies on the opposite side of the chord QR from P.
Calculate, giving a reason for each answer, the size of
(i) angle PQR (1 mark)
(ii) angle QRP (1 mark)
(iii) angle QSR (1 mark)
(iv) angle TQR. (2 marks)
(b) A ship leaves a harbour, H, and sails 12 km on a bearing of 060° to a point A. It then sails 9 km on a bearing of 150° to a point B.
(i) Show that angle HAB = 90°. (2 marks)
(ii) Calculate the distance HB. (2 marks)
(iii) Calculate the bearing of B from H, correct to the nearest degree. (3 marks)
Total: 12 marks
Question 10 — Vectors and Matrices
(a) A = (3 2 ; 1 4) and B = (2 −1 ; 0 3).
(i) Find the product AB. (2 marks)
(ii) Find the determinant of A, and hence write down A⁻¹. (3 marks)
(iii) Use A⁻¹ to solve the simultaneous equations
3x + 2y = 8 x + 4y = 6 (2 marks)
(b) Relative to the origin O, the position vectors of P and Q are OP = (3, 1) and OQ = (1, 4).
(i) Find the vector PQ. (1 mark)
(ii) Calculate |PQ|, correct to 3 significant figures. (1 mark)
(iii) M is the midpoint of PQ. Find the position vector OM. (1 mark)
(iv) The point R has position vector OR = (−1, 7). Show that P, Q and R lie on a straight line. (2 marks)
Total: 12 marks
END OF TEST — TOTAL: 100 MARKS