# Math–May 2005 Past Paper

CARIBBEAN EXAMINATIONS COUNCIL

SECONDARY EDUCATION CERTIFICATE

EXAMINATION

MATHEMATICS

Paper 02—General Proficiency

2 hours 40 minutes

26 May 2005

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=∏r2h 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=∏r2 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.

If ax2 + bx + c = 0; then x = -b + √b2 – 4ac/ 2a

Trigonometric Ratios

Sin Ѳ = Opposite 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) 5a2b + ab2

(2 marks)

(ii) 9k2-1

(2 marks)

(iii) 2y2 – 5y + 2

(2 marks)

b) Expand and simplify:

(2x + 5)(3x – 4)

(2 marks)

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

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)

(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) = a2 + b2

(3 marks)

Total 10 marks

Section II

Answer TWO questions in the section

ALGEBRA AND RELATIONS, FUNCTIONS AND GRAPHS

9 (a) Write 5x2 + 2x – 7 in the form a(x + b)2 + c, where a, b and c are real numbers.

(4 marks)

(b) Hence or, otherwise, determine

(i) the minimum value y = 5x2 + 2x – 7

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

(3 marks)

(c) Find the values of x for which 5x2 + 2x – 7=0

(3 marks)

(d) Sketch the graph of y = 5x2 + 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

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