TEST CODE 01234020

FORM TP 2016017 JANUARY 2016

CAR I B B EAN E XAM I NAT I O N S C O U N C I L

CARIBBEAN SECONDARY EDUCATION CERTIFICATE®

EXAMINATION

MATHEMATICS

Paper 02 – General Pro ciency

2 hours 40 minutes

READ THE FOLLOWING INSTRUCTIONS CAREFULLY.

1. This paper consists of TWO sections: I and II.

2. Section I has EIGHT questions and Section II has THREE questions.

3. Answer ALL questions in Section I and any TWO questions from Section II.

4. Write your answers in the booklet provided.

5. Do NOT write in the margins.

6. All working MUST be clearly shown.

7. A list of formulae is provided on page 4 of this booklet.

8. If you need to rewrite any answer and there is not enough space to do so on
the original page, you must use the extra page(s) provided at the back of this
booklet. Remember to draw a line through your original answer.

9. If you use the extra page(s) you MUST write the question number clearly in
the box provided at the top of the extra page(s) and, where relevant, include
the question part beside the answer.

Required Examination Materials

Electronic calculator
Geometry set

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

Copyright © 2014 Caribbean Examinations Council

All rights reserved.
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LIST OF FORMULAE

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Volume of a prism 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.
1
Volume of a right pyramid V = — Ah where A is the area of the base and h is the perpendicular height.
3

Circumference C = 2 r where r is the radius of the circle.

θ
Arc length S = —— × 2 r where θ is the angle subtended by the arc, measured in
360
degrees.

Area of a circle A = r2 where r is the radius of the circle.

θ
Area of a sector A = —— × r2 where θ is the angle of the sector, measured in degrees.
360

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

Roots of quadratic equations If ax2 + bx + c = 0,

–b + √ b2 – 4ac
then x = ——————
2a

opposite side
Trigonometric ratios sin θ = —————
hypotenuse

adjacent side
cos θ = —————
hypotenuse

opposite side
tan θ = —————
adjacent side

1
Area of Δ = — bh where b is the length of the base and h is the perpendicular

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Area of triangle
2
height.
1
Area of Δ ABC = — ab sin C
2
Area of Δ ABC = √ s (s – a) (s – b) (s – c)
a+b+c
where s = ————
2
a b c
Sine rule —— = —— = ——
sin A sin B sin C

Cosine rule a2 = b2 + c2 – 2bc cos A

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SECTION I
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Answer ALL questions in this section.

All working must be clearly shown.

1. (a) Using a calculator, or otherwise, calculate the EXACT value of

( 3.6 + )
51.84 ÷ 3.75

(2 marks)
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(b) The diagram below, not drawn to scale, shows two jars of peanut butter of the same
brand.

Which of the jars shown above is the BETTER buy?

Show ALL working to support your answer.
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(3 marks)

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(c) Thomas invested $1498 at 6% simple interest per annum.

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Calculate:

(i) The interest, in dollars, earned after six months

(2 marks)

(ii) The TOTAL amount of money in his account after 3 years

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(2 marks)

(iii) How long it will be before his investment earns $449.40

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E

(2 marks)

Total 11 marks

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2. (a) (i) Solve for x, where x is a real number.

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8 – x ≤ 5x + 2

(3 marks)

(ii) Show your solution to (a) (i) on the number line below.
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(1 mark)

(b) Expand and simplify

2x (x + 5) − 3(x – 4).
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(2 marks)

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(c) Simplify

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3x2 × 4x3 .
2x

(2 marks)

(d) Write as a single fraction, in its lowest terms,

x +1 5− x .
+
2 5

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(2 marks)

(e) Factorize completely

4x2 − 4

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(2 marks)

Total 12 marks

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3. (a) The Venn diagram below shows the number of students in Form 5A who have visited
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Canada (C) or Dominica (D).

(i) How many students have visited Dominica ONLY?

.............................................................................................................................
(1 mark)

(ii) Write an expression, in terms of x, to represent the TOTAL number of students

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who have visited Canada.

..............................................................................................................................
(1 mark)

(iii) Given that there are 25 students in Form 5A, calculate the value of x.

(2 marks)
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(iv) Hence, write down the number of students in each of the following subsets:

• C D ....................................................................................................

• C D ....................................................................................................

• (C D) ................................................................................................
(3 marks)

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(b) (i) Using a ruler, a pencil and a pair of compasses, construct accurately, the

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square EFGH where EF = 6 cm.

(Show ALL construction lines and curves.)

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(4 marks)

(ii) Measure, and state in centimetres, the length of the diagonal FH.

FH = ............................................................................................................. cm
(1 mark)

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Total 12 marks

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4. (a) The diagram below shows a map of an island drawn on a grid of 1 cm squares.

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(i) State, in cm, the length of LM as shown in the diagram.

LM = .......................................................................................................... cm
(1 mark)

(ii) Estimate, by counting squares, the area of the map shown in the diagram.

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(1 mark)

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(iii) On the island, the actual distance LM is 20 km. Complete the following
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statement:

On the map, 1 cm represents ................................................... km.

(1 mark)

(iv) Write the scale of the map in the form 1 : x.

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(1 mark)

(v) What distance on the island will be 3 cm on the map?

(1 mark)

(vi) What area on the island will be represented by 3 cm2 on the map?
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(2 marks)

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(b) The diagram below, not drawn to scale, shows a prism with cross section PQRST and

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length 20 cm. PQRST is made up of a rectangle PQRT and a semicircle RST such that
PQ = 6 cm and QR = 5 cm.

Use = 3.14

(i) Calculate the area of the cross section PQRST.

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(2 marks)
(ii) An engineer needs a similar prism whose volume is NOT more than 900 cm3.
Estimate, in cm, the length of the longest prism he can use.

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(2 marks)

Total 11 marks

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5. (a) In the diagram below, not drawn to scale, ST = 6 m, WR = 11.2 m, WT = 14.8 m and
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angle WRS = 90°.

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Calculate, giving your answer to 1 decimal place

(i) the length RS

(2 marks)
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(ii) the measure of angle RTW.

(2 marks)

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(b) The graph below shows a triangle ABC and its image A'B'C' after undergoing a single

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transformation.

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(i) Write down the coordinates of the vertices of ∆ABC.

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(1 mark)

(ii) Write down the coordinates of the vertices of ∆A'B'C'.

(1 mark)
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(iii) Describe FULLY the transformation that maps triangle ABC onto triangle A'B'C'.

..............................................................................................................................

..............................................................................................................................

..............................................................................................................................
(2 marks)

(iv) On the graph on page 16, draw the line x = 1 AND the triangle A˝B˝C˝, the
image of triangle ABC after a re ection in the line x = 1. (3 marks)

(v) State ONE geometrical relationship among ∆ABC, ∆A'B'C' and ∆A˝B˝C˝

............................................................................................................................
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..............................................................................................................................

..............................................................................................................................
(1 mark)

Total 12 marks

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6. (a) The table below gives the number of cars sold in a country, in hundreds, from 2010 to

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2014.

Year 2010 2011 2012 2013 2014

Cars sold
19 10 26 16 30
(in hundreds)

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(i) Complete the line graph on page 18 to represent the given information.
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(1 mark)

(ii) Between which two consecutive years was there the GREATEST increase in
cars sold?

.............................................................................................................................
(1 mark)

(iii) What was the TOTAL number of cars sold in the ve year period 2010 to 2014?
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(2 marks)

(iv) The mean number of cars sold from 2010 to 2015 was 22.5 hundred. How
many cars were sold in 2015?

(2 marks)
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(b) (i) A line JK has equation 2y = 5x + 6. Determine the gradient of JK.

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Gradient of the line JK is .............................................................................. .
(2 marks)

Another line GH, is perpendicular to JK and passes through the point (5, –1).

(ii) State the gradient of the line GH.

Gradient of the line GH is ............................................................................. .

(1 mark)

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(iii) Determine the equation of line GH.

Equation of the line GH is ............................................................................ .

(2 marks)

Total 11 marks

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7. The table below shows how the minutes taken by all students to complete a science experiment

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were recorded and grouped.

Number of Students who

Time (minutes) Completed Cumulative Frequency
(Frequency)

1–5 1 1

6–10 2 3

11–15 5

16–20 7

21–25 10

26–30 15

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31–35 8

36–40 2

(a) Complete the cumulative frequency column in the table. (2 marks)

(b) On the grid on page 23, using a scale of 2 cm to represent 5 minutes on the x-axis
and 2 cm to represent 5 students on the y-axis, draw a cumulative frequency curve to
represent the information in the table. (5 marks)
Using the graph, estimate

(c) (i) the median time taken to complete the experiment

(2 marks)

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(ii) the probability that a student, chosen at random, took 30 minutes or less to
complete the experiment.

(2 marks)

Show on your graph, using broken lines, how these estimates were obtained.

Total 11 marks

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8. The diagram below shows the rst three gures in a sequence of gures.

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Figure 1 Figure 2 Figure 3

(a) Draw the fourth gure in the sequence.

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(2 marks)

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(b) The table below shows the number of dots and lines in each gure. Study the pattern in
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the table and complete the table by inserting the missing values in the rows numbered
(i), (ii), (iii) and (iv).

Figure Number of Dots Number of Lines

1 4 6

2 7 11

3 10 16

(i) 4 ...................................... ....................... (2 marks)

Entries omitted for

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Figures 5–9

(ii) 10 ...................................... ....................... (2 marks)

Entries omitted for

some Figures

(iii) ............ 49 ....................... (2 marks)

Entries omitted for

some Figures
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(iv) N ...................................... ....................... (2 marks)

Total 10 marks

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SECTION II

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Answer TWO questions in this section.

ALGEBRA AND RELATIONS, FUNCTIONS AND GRAPHS

9. (a) The diagram below shows the graph of three lines and a shaded region, S, de ned by
three inequalities associated with these lines.

The inequality associated with the line y = 3 is y ≥ 3.

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(i) State the other TWO inequalities which de ne the shaded region.

...................................................................................................................................
(2 marks)

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The function P = 5x + 2y – 3 satis es the solution set represented by the closed triangular
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region.

(ii) Identify the three pairs of (x, y) values for which P has a maximum or a minimum
value.

....................................................................................................................................

....................................................................................................................................

....................................................................................................................................
(2 marks)

(iii) Which pair of (x, y) values makes P a maximum?

Justify your answer.

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P is a maximum at ............................................................................................... .
(3 marks)
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(b) The function f(x) and g(x) are de ned as follows

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3
f(x) = and g(x) = x2
2x + 1

(i) Evaluate EACH of the following:

–1
• g —
2

–1
• fg —
2

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(4 marks)

(ii) Write an expression in x for f –1(x).

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(4 marks)

Total 15 marks

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MEASUREMENT, GEOMETRY AND TRIGONOMETRY

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10. (a) The gure below, not drawn to scale, shows a circle with centre O. The radius of the
circle is 21 cm and angle HOK = 40°.

22
Use =
7

Determine

(i) the area of the minor sector HOK

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(2 marks)

(ii) the area of triangle HOK

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(3 marks)

(iii) the area of the shaded segment.

(2 marks)

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(b) The diagram below, not drawn to scale, shows a circle with centre O. TAE is a tangent to

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the circle at point A and angle AOD = 72°.

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Calculate, giving the reason for each step of your answer, the measure of:

(i) ∠ADC = ............................................

(2 marks)

(ii) ∠ACD = ............................................

(2 marks)

(iii) ∠CAD = ............................................

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(2 marks)

(iv) ∠OEA ............................................

(2 marks)

Total 15 marks
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VECTORS AND MATRICES

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11. (a) The points A, B and C have coordinates A (–2,8), B (4,2) and C (0,9). M is the midpoint
of the line segment AB.

x
(i) Express EACH of the following in the form   :
 y

• OB = ..............................................

• AB = ..............................................
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• OM = ..............................................
(5 marks)

(ii) Using a vector method, show that AC and OB are parallel.

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(2 marks)

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 2 p −3 

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(b) The matrix M is de ned as M = 
1 
.
 4
Determine the value of p for which the matrix M is singular.

(2 marks)

 1 2  5 −1 
(c) A and B are two 2 × 2 matrices such that A =   and B =  .
 −4 3   0 3

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(i) Calculate 2A + B.

(2 marks)

(ii) Determine B –1, the inverse of B.

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(2 marks)

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 5 −1  x   9 
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(iii) Given that     =   , calculate the values of x and y.

 0 3   y   3
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(2 marks)

Total 15 marks
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END OF TEST

IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.

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