**CARIBBEAN EXAMINATIONS COUNCIL**

**SECONDARY EDUCATION CERTIFICATE**

**EXAMINATION**

**MATHEMATICS**

**Paper 02—General Proficiency**

*2 hours 40 minutes*

**26 May 2005**

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** INSTRUCTION TO CANDIDATES**

**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 the section**

**All working must be clearly shown**

**1** (a) Calculate the EXACT value of:

**(3 marks)**

(b) The table below shows Amanda’s shopping bill. Some numbers were removed and replaced with letters.

(i) Calculate the values of A, B, C and D.

**(5 marks)**

(ii) Amanda sold 6 of the 12 stickers which she had bought for 75 cents each, and the remaining stickers at 40 cents each. Show using calculations whether Amanda made a profit or loss on buying and selling stickers.

**(3 marks)**

**2**. Factorise:

(i) 5a^{2}b + ab^{2}

**(2 marks)**

(ii) 9k^{2}-1

**(2 marks)**

(iii) 2y^{2} – 5y + 2

**(2 marks)**

b) Expand and simplify:

(2x + 5)(3x – 4)

**(2 marks)**

(c) Adam, Imran and Shakeel were playing a card game.

Adam scored *x* points.

Imran scored 3 points fewer than Adam.

Shakeel scored twice as many points as Imran.

Together they scored 39 points.

(i) Write down, in terms of *x*, an expression for the number of points scored by Shakeel.

**(2 marks)**

(ii) Write an equation which may be used to find the value of x.

**(2 marks)**

**Total 12 marks**

**3** (a) In the diagram shown below, the Universal set (U), represents all the students in a class. The set M represents the students who take music. The set D represents the students who take Drama. If 24 students take Music, calculate:

(i) the number of students who take BOTH Music and Drama.

(ii) the number of students who take Drama only.

**(4 marks)**

(b) A straight line passes through the point P (-3, 5) and has a gradient of 2/3.

(i) Write down the equation of this line in the form of *y=mx+c.*

**(5 marks)**

(ii) Show that this line is parallel to the line 2x-3y=0.

**(2 marks)**

**Total 11 marks**

**4**. The figures shown below, not drawn to scale represent the cross sections of two circular pizzas. Both pizzas are equally thick and contain the same toppings.

(i) Is a medium pizza twice as large as a small pizza? Use calculations to support your answer.

**(5 marks)**

(ii) A medium pizza is cut into three equal parts, and each part is sold for $15.95. A small pizza is sold for $12.95. Which is the better buy? Use calculations to support your answer.

**(5 marks)**

**Total 10 marks**

**5**. On graph paper draw the x-axis and the y-axis. Using a scale of 1 cm to represent 1 unit on both axes, draw the triangle DEF with vertices D (1,1), E(3,1), and F(1,4).

**(3 marks)**

(b) Draw the images of triangle DEF under reflection in the line x=4. Name the image triangle D’E’F’.

(ii)Draw the triangle D’E’F under the translation . Name the image D”E”F”.

(iii) Name the type of transformation that maps triangle DEF unto D”E”F”.

**(5 marks)**

(c) A vertical stick of height 1.8 m casts a shadow of length 2 m on the horizontal as shown in the diagram below, not drawn to scale.

Calculate to the nearest degree, the angle of elevation of the sun.

**(4 marks)**

**Total 12 marks**

**6** (a) In the diagram below, ABCDE is a pentagon. Angle BAE=108 degrees, angle ABC=90. Angle AED= 80 degrees and angle ADC=57 degrees and AE is parallel to CD.

Calculate the size of the angle marked:

(i) x degrees

(ii) y degrees

**(4 marks)**

**Show all steps in your calculations and give reasons for your answers. **

(b) The function f and g are defined by:

Evaluate:

(i) g(3) + g(-3)

(ii) f^{-1}(6)

(iii) fg(2)

**(8 marks)**

**Total 12 marks**

**7**. The table below gives the distribution of heights of 400 female applicants for the Police Service.

(a) Using a horizontal scale of 2 cm to represent a height of 5 cm and a vertical scale of 2 cm to represent 50 applicants, draw a cumulative frequency curve of the heights. **Start your horizontal scale at 150 cm.**

**(5 marks)**

(b) Use your graph to estimate:

(i) the number of applicants whose heights are less than 170 cm.

**(1 mark)**

(ii) the median height of applicants.

**(2 marks)**

(iii) the height that 25% of the applicants are less than.

**(2 marks)**

(iv) the probability that an applicant selected at random has a height that is no more than 162 cm.

**(2 marks)**

**Credit will be given for drawing the appropriate lines on the graph to show how the estimates were obtained. **

**Total 12 marks**

**8** (a) Study the number pattern in the table below and complete lines (i), (ii) and (iii) in your answer booklet.

**(7 marks)**

(b) Show that:

(a – b)^{2} (a + b) + ab(a + b) = a^{2} + b^{2}

**(3 marks)**

** Total 10 marks**

**Section II**

**Answer TWO questions in the section**

**ALGEBRA AND RELATIONS, FUNCTIONS AND GRAPHS**

**9** (a) Write *5x ^{2} + 2x – 7* in the form

*a(x + b)*, where

^{2}+ c*a*,

*b*and

*c*are real numbers.

**(4 marks)**

(b) Hence or, otherwise, determine

(i) the minimum value y = *5x ^{2} + 2x – 7*

(ii) the value of x at which the minimum occurs

**(3 marks)**

(c) Find the values of x for which *5x ^{2} + 2x – 7=0*

**(3 marks)**

(d) Sketch the graph of y = *5x ^{2} + 2x – 7, *clearly showing

(i) the coordinates of the minimum point

(ii) the value of the y-intercept

(iii) the points where the graph cuts the x-axis.

**(5 marks)**

**10**. The speed-time graph below shows the movement of a cyclist.

Using the graph, calculate

(i) the acceleration of the cyclist during the first 15 secs.

(ii) the distance traveled by the cyclist between the period t = 15 and t = 35 seconds.

**(6 marks)**

(b) The graph below represents the 5-hour journey of an athlete.

(i) What was the average speed during the first 2 hours?

(ii) What did the athlete do between 2 and 3 hours after the start of the journey?

(iii) what was the average speed on the return journey?

**(5 marks)**

(c) The diagram below shows a triangular region bounded by the lines and the line HK.

(i) Write the equation of the line HK.

**(1 mark)**

(ii) Write the set of **three inequalities** which define the shaded region GHK.

**(3 marks)**

**Total 15 marks**

**GEOMETRY AND TRIGONOMETRY**

In the diagram above, not drawn to scale, P and Q are midpoints of the sides XY and XZ of triangle XYZ. Given than XP= 7.5 cm and XQ = 4.5 cm and the area of triangle XPQ = 13.5 cm calculate,

(i) the size of angle PXQ, expressing your answer correct to the nearest degree.

(ii) the area of triangle YXZ.

**(6 marks)**

The figure SJKM above, not drawn to scale, is a trapezium with SJ parallel to MK, angle MJK = 124˚, angle MSJ = 136˚, and SM=SJ=50 metres.

(i) Calculate the size of

(a) angle SJM

(b) angle JKM

**(3 marks)**

(ii) Calculate, expressing your answer correct to ONE decimal place, the length of

a) MJ

b) JK

**(6 marks)**

**12**. In this question, assume the earth to be a sphere of radius 6,400 km and use ∏=3.14.

The latitudes and longitudes of Antigua and Belize are given in the table below.

(a) Draw a sketch of the earth showing the location of Antigua and Belize, their associated cities of latitudes and longitude, the equator and the Greenwhich Meridian.

**(6 marks)**

(b) Calculate the shortest distance between Antigua and Belize measured along their common circle of latitude.

**(5 marks)**

(c) A town, Bahia Blanka, situated in South America, lies on a meridian 62˚W and has a latitude off 38˚S. Calculate the shortest distance between Antigua and Bahia Blanka measured along the common circle of longitude.

**(4 marks)**

**Total 15 marks**

**VECTORS AND MATRICES**

In the figure above, not drawn to scale, ABCD is a parallelogram such that and The point P is on DB such that DP:PB = 1:2.

(a) Express in terms of x and y:

**(5 marks)**

**(2 marks)**

(c) Given that E is the mid-point of DC, prove that A, D and E are collinear.

**(4 marks)**

(d) Given that , use a vector method to prove that triangle AED is isosceles.

**(4 marks)**

**Total 15 marks**

(i) Show that M is a non-singular matrix.

(ii) Write down the inverse of M.

(iii) Write down the 2×2 matrix which is equal to the product of MxM^{-1}.

(iv) Pre-multiply both sides of the following matrix equation by M^{-1}.

Hence solve for x and y.

**(7 marks)**

(b) (i) Write down the 2×2 matrix, R, which represents a reflection in the Y axis.

(ii) Write down the 2×2 matrix, N, which represents a clockwise rotation of 180˚ of the origin.

(iii) Write down the 2×1 matrix, T which represents a translation of -3 units parallel to the x-axis and 5 units parallel to the y-axis.

(iv) The point P(6,11) undergoes the following combined transformations such that:

RN(P) maps P onto P’

NT(P) maps P onto P’

Determine the coordinates of P’ and P”.

**(8 marks)**

**Total 15 marks**

**END OF TEST**