**CARIBBEAN EXAMINATIONS COUNCIL**

**SECONDARY EDUCATION CERTIFICATE**

**EXAMINATION**

**MATHEMATICS**

**Paper 02—General Proficiency**

*2 hours 40 minutes*

**25 May 2006 (am)**

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** **

** INSTRUCTION TO CANDIDATES**

- Answer ALL questions in Section I and ANY TWO in Section II.
- Write your answers in the booklet provided.
- All working must be shown clearly.
- A list of formulae is provided on page 2 of this booklet.

**Required Examination Materials**

Electronic Calculator

Geometry Set

Graph Paper (Provided)

**DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO**

*LIST OF FORMULAE*

Volume of a prisim

*V= Ah* where A is the area of a cross-section and h is the perpendicular length.

Volume of cylinder

*V=∏r ^{2}h* where

*r*is the radius of the base and

*h*is the perpendicular height.

Circumference

*C=2∏r* where *r* is the radius of the circle.

Area of a circle

*A=∏r ^{2}* where

*r*is the radius of the circle.

Area of a trapezium

*A=1/2 (a+b)h* where *a* and *b* are the lengths of the parallel sides and *h* is the perpendicular distance between the parallel sides.

Roots of quadratic equations

If ax^{2} + bx + c = 0; then x = -b + √b2 – 4ac/ 2a

Trigonometric Ratios

Sin Ѳ = Opposite side/Hypoteneuse

Cos Ѳ = Adjacent side/Hypoteneuse

Tan Ѳ = Opposite side/Adjacent Side

Area of triangle

Area of triangle= 1/2 bh where b is the length of the base and h is the perpendicular height.

Area of triangle ABC= 1/2 ab sinC

Area of triangle ABC=√s(s-a) (s-b) (s-c) where s = (a+b+c)/2

Sine rule

**Section I**

**Answer ALL the questions in this section.**

**ALL working must be clearly shown.**

**1.** (a) Using a calculator, or otherwise, determine the value of

(12.3)^{2} – (0.246 ÷ 3) and write the answer:

(i) exactly

(ii) correct to two significant figures

**(2 marks)**

(b) The table below gives information on the values and rates of depreciation in value of two motor vehicles.

Calculate:

(i) the values of *p* and *q*

(ii) the value of the Taxi after 2 years

**(6 marks)**

(c) GUY $1.00 = US $0.01 and EC $1.00 = US $0.37.

Calculate the value of:

(i) GUY $60,000 in US $

**(2 marks)**

(ii) US $925 in EC $

**(2 marks)**

**Total 12 marks**

**2.** (a) Simplify

**(3 marks)**

(b) (i) Factorise

(a) x^{2} – 5x

**(1 mark)**

(b) x^{2} – 81

**(1 mark)**

(ii) Simplify

**(3 marks)**

(c) Two cassettes and three CD’s cost $175, while four cassettes and one CD cost $125.

(i) Given that one cassette cost $x and one CD costs $y, write two equations in x and y to represent the information.

**(2 marks)**

(ii) Calculate the cost of one cassette.

**(2 marks)**

**Total 12 marks**

**3.** (a) In the quadrilateral KLMN, not drawn to scale, LM= LN=LK, angle KLM = 140°, and angle LKN = 40°.

Giving the reason for each step of your answer, calculate the size of:

(i) angle LNK

**(2 marks)**

(ii) angle NLM

**(2 marks)**

(iii) angle KNM

**(2 marks)**

(b) In a survey of 39 students, it was found that

18 can ride a bicycle,

15 can drive a car,

x can ride a bicycle and drive a car,

3x can do neither.

B is the set of students in the survey who can ride a bicycle, and C is the set of students who can drive a car.

(i) Copy and complete the Venn diagram to represent the information.

(ii) Write an expression in x for the number of students in the survey.

(iii) Calculate the value of x.

**(5 marks)**

4. (a) Using a ruler, a pencil and a pair of compasses, construct the triangle ABC in which

AB = 8 cm

angle BAC = 60°

AC = 5 cm

**(Credit will be given for a neat, clear diagram)**

**(4 marks)**

(b) Measure and state the length of BC

**(1 mark)**

(c) Find the perimeter of triangle ABC

**(1 mark)**

(d) Draw on your diagram, the line CD which is perpendicular to AB and meets AB at D.

**(2 marks)**

(e) Determine the length of CD.

(f) Calculate the area of triangle ABC giving your answer to 1 decimal point.

**(2 marks)**

**Total 12 marks**

**5.** The diagram below shows the graph of the function f(x) = x^{2} – 2x – 3 for a ≤ x ≤ b. The tangent to the graph at (2, -3) is also drawn. Use the graph to determine the

(a) values of a and b which define the domain of the graph.

**(2 marks)**

(b) values for x which x^{2} – 2x – 3 = 0.

**(2 marks)**

(c) coordinates of the minimum point on the graph.

**(2 marks)**

(d) whole number values of x for which x^{2} – 2x – 3 < 1.

**(2 marks)**

(e) gradient of f(x) = x^{2} – 2x – 3 at x = 2.

**(3 marks)**

**Total 11 marks**

**6.** A man walks x km, due north, from point G to point H. He then walks (x+7) km due east from H to point F. the distance along a straight line from G to F is 13 km. The diagram below, **not drawn to scale**, shows the relative positions of G, H and F The direction of north is also shown.

(a) Copy the diagram and show on the diagram, the distances x km, (x+7) km and 13 km.

**(2 marks)**

(b) From the information on your diagram, write an equation in x which satisfies Pythagoras’ Theorem. Show that the equation can be simplified to give x^{2} + 7x – 60 = 0.

**(3 marks)**

(c) Solve the equation and find the distance GH.

**(2 marks)**

(d) Determine the bearing of F from G.

**(4 marks)**

**Total 11 marks**

**7**. In an agricultural experiment, the gains in mass, of 100 cows during a certain period were recorded in kilograms as shown in the table below.

(a) Copy and complete the mid-interval values column.

**(1 mark)**

(b) (i) Calculate an estimate of the mean gain in mass of the 100 cows.

Hint: EACH of the 29 cows in the “10-14” interval is assumed to have a mass of 12 kg.

**(3 marks)**

(ii) On your answer sheet, complete the drawing of the frequency polygon for the gain in mass of the cows.

**(5 marks)**

(c) Calculate the probability that a cow chosen at random from the experimental group gained 20 kg or more.

**(2 marks)**

**Total 11 marks**

**8.** The drawings below show a sequence of squares made from toothpicks.

(a) On the answer sheet provided,

(i) Draw the next shape in the sequence

**(2 marks)**

(ii) insert appropriate values in columns 2 and 3 when

a) n = 4

b) n = 7

**(4 marks)**

(b) Complete the table by inserting appropriate values at

(i) r

**(2 marks)**

(ii) s

**(2 marks)**

**Total 10 marks**

**SECTION II**

**Answer TWO questions in this section.**

**ALGEBRA AND RELATIONS, FUNCTIONS AND GRAPHS**

**9.** (a) Solve the pair of similtaneous equations

y= x + 2

y= x^{2}

**(5 marks)**

(b) A strip of wire length 32 cm is cut into two pieces.One piece is bent to form a square of side x cm. The other piece is bent to form a rectangle of length 1 cm and width 3 cm.

The diagrams below, not drawn to scale, show the square and rectangle.

(i) Write an expression in terms of l and x, for the length of the strip of wire.

**(2 marks)**

(ii) Show that l= 13 – 2x

**(2 marks)**

The sum of the areas of the square and the rectangle is represented by S.

(iii) Show that S = x^{2} – 6x + 39

**(2 marks)**

(iv) Calculate the values of x for which S = 30.25

**(4 marks)**

**Total 15 marks**

**10**. The owner of a parking lot wishes to park x vans and y cars for persons attending a function. The lot provides parking space for no more than 60 vehicles.

(i) Write an inequality to represent this information.

**(2 marks)**

To get a good bargain, he must provide parking space for at least ten cars.

(ii) Write an inequality to represent this information.

**(1 mark)**

The number of cars parked must be fewer than or equal to twice the number of vans parked.

(iii) Write an inequality to represent this information.

**(2 marks)**

(iv) (a) Using a scale of 2 cm to represent 20 vans on the x-axis, and 2 cm to represent 10 cars on the y-axis, draw the graphs of the lines associated with the inequalities at (i), (ii) and (iii) above.

**(5 marks)**

(b) Identify, by shading, the region which satisfies all three inequalities.

**(1 mark)**

The parking fee for a van is $6 and for a car is $5.

(v) Write an expression in x and y for the total fees charged for parking x vans and y cars.

**(1 mark)**

(vi) Using your graph write down the coordinates of the vertices of the shaded region.

**(1 mark)**

(vii) Calculate the maximum fees charged.

**(2 marks)**

**Total 15 marks**

**GEOMETRY AND TRIGONOMETRY**

**11.** (a) The diagram below, not drawn to scale, shows a vertical tower, FT, and a vertical antenna, TW, mounted on the top of the tower.

A point P is on the same horizontal ground as F, such that PF= 28 m, and the angles of elevation of T and W from P are 40° and 54° respectively.

(i) Copy and label the diagram clearly showing

a) the distance 28 m

b) the angles of 40° and 54°

c) any right angles

(ii) Calculate the length of the antenna TW.

**(7 marks)**

(b) The diagram below, not drawn to scale, shows a circle, centre O. The lined BD and DCE are tangents to the circle, and angle BCD = 70°.

Calculate, giving reasons for each step of your answer.

(i) angle OCE

(ii) angle BAC

(iii) angle BOC

(iv) BDC

**(8 marks)**

**Total 15 marks**

**12.** (a) The diagram below, not drawn to scale, shows parallellogram EFGH in which EF = 6 cm, EH = 4.2 cm, and angle FEH = 70°.

Calculate

(i) the length of the diagonal HF

**(3 marks)**

(ii) the area of the parallelogram EFGH

**(2 marks)**

(b) In this question use ∏=3.14 and assume the earth to be a sphere of radius 6370 km.

The diagram below, not drawn to scale, shows a sketch of the earth with the North and South poles labelled N and S respectively. The circle of latitude 41°N is shown. Arcs representing circles of longitude 4°E and 74°W are drawn but not labelled.

(i) Copy the sketch above and draw and label two arcs to represent

(a) The Equator

(b) The Greenwhich Meridian

**(2 marks)**

(ii) Two points, Y and M, on the surface of the earth have coordinates Y(41°N, 74°W) and M(41°N 4°E).

(a) Insert the points Y and M on your diagram.

**(2 marks)**

(b) Calculate, to the nearest kilometer, the circumference of the circle of latitude 41°N.

**(3 marks)**

(c) Calculate the shortest distance between Y and M measured along the circle of latitude 41°N.

**(3 marks)**

**Total 15 marks**

** VECTORS AND MATRICES**

**13.** The diagram below shows the position vectors of two points, A and C, relative to an origin O.

(a) Copy and complete the diagram to show

(i) The point B such that OABC is a parallelogram.

**(1 mark)**

**(2 marks)**

(b) Write as a column vector, in the form the vector

(c) Given that G is the midpoint of OB, use a vector method to

(i) determine the coordinates of G

**(3 marks)**

(ii) prove, using a vector method, that A, G and C lie on a straight line.

**(5 marks)**

**Total 15 marks**

**14**. (a) The value of the determinant is 9.

(i) Calculate the value of x

**(3 marks)**

(ii) For this value of x, find M^{-1}

**(2 marks)**

(iii) Show that M^{-1}M=1

**(2 marks)**

(b) The graph below shows the line segment AC and its image A’C’ after a transformation by the matrix

(i) Write in the form of a single 2×2 matrix, the coordinates of

a) A and C

**(2 marks)**

b) A’ and C’

**(2 marks)**

(ii) Using matrices only, write an equation to represent the transformation of AC onto A’C’.

**(2 marks)**

(iii) Determine the values of p, q, r and s.

**(2 marks)**

**Total 15 marks**

**END OF TEST**