TEST CODE OI234O2OJANUARY 2O1OFORM TP 2010015

CARIBBEAN EXAMINATIONS COUNCILSECONDARY ED UCATION CERTIFICATE

EXAMINATION

MATHEMATICS

Paper 02 - General Proficiency

2 hours 40 minutes

05 JANUARY 2010 (a.m.)

INSTRUCTIONS TO CANDIDATES

AnswerALL questions in Section I, andANY TWO in Section II.

Write your answers in the booklet provided.

All rvorking must be clearly shown.

A list of formulae is provided on page 2 of this booklet.

Bxamination Materials

Electronic calculator (non-programmable)Geometry setMathemati cal tables (provided)Graph paper (provided)

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

I

III

I

I

II

r

Copyright O 2009 Caribbean Examinations Council@.All rishts reserved.

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I

(a)

SECTION I

AnswerALL the questions in this section.

All working must be clearly shown.

Using a calculator, or otherwise, calculate the exact value of

(i) his fixedsalary for the year

(ii) the amount he received in commission for the year

(iii) his TOTAL income for the year.

(c) The ingredients for making pancakes are shown in the diagram below.

l.

Page 3

( 3 marks)

( l mark)

( l mark)

( l mark)

(b)

2.76

0s + 8'72

In a certain company, a salesman is paid a fixed salary of $3 140 per month plus anannual commission of 2o/oon the TOTAL value of cars stld for the year. If the ,u-l"r-u1sold cars valued at$720 000 in 2009, calculate

Ingredients for making 8 pancakes

2 cups pancake mix

1

15 cups milk

Ryan wishes to make 12 pancakes using the instructions given above. Calculatethe number of cups of pancake mix he must use. ( 2 marks)

Neisha used 5 cups of milk to make pancakes using the same instructions. Howmany pancakes did she make? ( 3 marks)

(i)

(ii)

Total ll marks

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2. (a) Given that a: 6, b : - 4 and c : 8, calculate the value of

ctz+bc-b

Simplify the expression:

(i) 3(x - y) + 4(x + 2v)

(ii) 4x2 x 3xa

6x3

(a) T and E are subsets of a universal set, U, such that:

U : {7,2,3,4,5,6;7,8,9,10, 11, 12 }

T: {multiplesof3 I

E : { even numbers }

(i) Draw a Venn diagram to represent this information.

(ii) List the members of the set

a) TaE

b) (TwD'.

Page 4

( 3 marks)

1 2 marks)

( 3 marks)

( 4 marks)

( l mark)

( l mark)

(b)

(c) (D Solve the inequality

x-3 ( 3 marks)

(ii) Ifx is an integer, determine the SMALLEST value of-r that satisfies the inequalityin (c) (i) above. ( l mark)

Total 12 marks

3.

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(b) Using a pencil, a ruler, and a pair of compasses only:

(i) Construct, accurately, the triangle IBC shown below, where,

AC: 6cmZ ACB : 60o

ICAB: 60"

Complete the diagram to shon' the kite, ABCD, in which lD

Measure and state the size of I DAC.

(ii)

(iiD

Page 5

( 3 marks)

:5cm.( 2 marks)

( l mark)

Total12 marks

6cm

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4.

Page 6

(a) The diagram below, not drawn to scale, shows a triangle LMN with LN : 12 cm,NM :x cm and I NLM : 0 o. The point K on LM is such that i/K is perpendicular toLM, NK:6 cm, and KM: 8 cm.

(b)

Calculate the value of

(Dx(iD e.

The diagram below shows a map of a playing field drawn on a grid of IThe scale of the map is I : 1 250.

( 2 merks)

( 3 merks)

cm squar€s.

(D

(ii)

Measure and state, in centimetres, the distance from S to F on the map.( lmark )

Calculate the distance, in metres, from ,S to F on the ACTUAL playing field.( 2 marks)

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(iii) Daniel ran the distance from ^Sto

F in9.72 seconds.in

a) m/s

b) km/h

giving your answer correct to 3 significant figures.

A straight line passes through the point T (4,1) and has a gradient ofthe equation of this line.

Page 7

Calculate his average speed

( 3 marks)

Total ll marks

Determine( 3 marks)

( 3 marks)

J

Ts. (a)

(b)

(iD

(iii)

(iv)

(v)

(i) Using a scale of 1 cm to represent I unit on both axes, draw thetriangle ABCwith vertices A (2;3), B (5,3) and C (3,6).

On the same axes used in (b) (D, draw and label the line y : 2.( l mark)

Draw the image of triangle ABC vnder a reflection in the line y : 2. Label theimageA'B'C'. ( 2marks)

Draw a new triangle A"B"C'with vertices A,, (-7,4),8,,(-4,4) ande'e6,7). ( lmark)

Name and describe the single ffansformation that maps triangle ABC ontotriangleA"B"C". ( 2marks)

Total12 marks

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6.

Page 8

A class of 26 students each recorded the distance travelled to schooi. The distance. to the nearestkm, is recorded below:

6

t230

21

26

39

l1

l611

J

l722

22'34

24

32

25

l6

22

8

13

18 28

19 14

23

(a) Copy and complete the frequency table to represent this data.

Distance in kilometres Frequency

I -5 1

6- 10 2

lt - l5 4

t6-20 612t -25

26-30

31 - 35

36-40( 2 marks)

(c)

(d)

Using a scale of 2 cm to represent 5 km on the horizontal axis and a scale of I cm torepresent 1 student on the vertical axis, draw a histogram to represent the data.

( 5 marks)

Calculate the probability that a student chosen at random from this class recorded the

distance travelled to school as 26 km or more. ( 2 marks)

The P.T.A. plans to set up a transportation service for the school. Which average, mean,

mode or median, is MOST appropriate for estimating the cost of the service? Give a

reason fbr your answer. ( 2 marks)

Total 11 marks

(b)

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7. The graph shown below represents a function of the form: f(x): ax2 * bx + c.

Using the graph above, deterrrine

(D the value of f(x) when x: 0

(ii) the values of x whenf(x) : 0

(iii) the coordinates of the maximum point

(iv) the equation of the axis of symmetry

(v) thevalues of xwhen.f(x):5

(vi) theintervalwithinwhichrlieswhenfx) > 5.

Page 9

' ( lmark)

( 2 marks)

( 2 marks)

( 2 marks)

( 2 marks)

Write your answer in the forrn a < x < b.( 2 marks)

Total 11 marks

.-r- i- f-1-

:i.'j'i-'i-r

-|I.i....i....i...i

i....i-..;...i.

-i-i-i.-i

i-t +l.+-i.-i.f -i.--i...i...i.

ti..i...i..i..r..1...i...r....1

..t...+...i....i.iiit

u....i...i il""i-!_ i

1

l+-.+..

) i-i'i-i-iiii

4ii'i-ir-."_:_t_ !

I i- r-j.-it

0-f-i..i. ll.i.l.

-+-r..i 1

':i..'i...i...1

:i:i*i

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8.

Page 10

Bianca makes hexagons using sticks of equal length. She then creates patterns by joining the

hexagons together. Pattems 1,2 and 3 are shown below:

Pattem2 Pattern 3Pattern 1

Ol Hexagon

The table below shows thesticks used to make EACH

2 Hexagons

number of hexagons inpattern.

3 Hexagons

EACH pattem created and the number of

(a) Determine the values of

(i) x

(ii) v

(iii) z.

Write down an expression for S in terms of n. where S represents

used to make a pattern of n hexagons.

Bianca used a total of 76 sticks to make a pattern of ft hexagons.

of h.

(b)

(c)

( 2 marks)

( 2 marks)

( 2 marks)

the number of sticks( 2 marks)

Determine the value( 2 marks)

Total 10 marks

Numberof

hexagonsin the

pattern

I 2 -t 4 5 20 n

Numberof sticksused for

thepattern

6 ll T6 x v ^s

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-T

(a)9.

Page 11

SECTION II

Answer TWO questions in this secfion.

RELATIONS, FUNCTIONS AI\D GRAPHS

The relationship between kinetic energy, E,mass, m, and velociry v, for a moving particleis

(b)

l"E::l|lV .2

(i) Express v in terms of E and m. ( 3

(ii) Determinethevalueofywhen E:45and,m:13. ( 2

Given S6) : 3*-Bx+2,

(i) witeg(x) intheform a(x+b)2 *c,where a,bandc e R ( 3

(iD solve the equation S(x):0, writing your answer(s) correct to 2 decimal(4

(iii) A sketch of the graph of g(x) is shown below.

Copy the sketch and state

marks)

marks)

marks)

places.marks)

a)

b)

c)

they-coordinate of A

the x-coordinate of C

the x andy-coordinates ofB. ( 3 marks)

Totel 15 marks

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10. (a)

Page 12

The manager of apizza shop wishes'to makex smallpizzas and_r,large pizzas. His ovenholds no more than 20 pizzas.

(D Write an inequality to represent the given condition. ( 2 marks)

The ingredients for each small pizza cost $ 15 and for each large pizza S30. Themanager plans to spend no more than $450 on ingredients.

(ii)

(i)

Write an inequality to represent this condition. ( 2 marks)

Using a scale of 2 cm on the x-axis to represent 5 small pizzas and 2 cm onthey-axis to represent 5 large pizzas, draw the graphs ofthe lines associatedwith the inequalities at (a) (i) and (a) (ii) above. ( 4 marks)

(iD Shade the region which is defined byALL of the following combined:the inequalities written at (a) (i) and (a) (ii)the inequalities x > 0 andy > 0 ( I mark)

(iii) Using your graph, state the coordinates of the vertices of the shaded region.( 2 marks)

(c) The pizza shop makes a profit of $8 on the sale of EACH small pizza and S20 on thesale of EACH large pizza. All the pizzas that were made were sold.

(i) Write an expression in r and y for the TOTAL profit made on the sale of thepflzas. ( lmark)

(iD Use the coordinates ofthe vertices given at (b) (iii) to determine the MAXIMUMprofit. ( 3 marks)

Total 15 marks

(b)

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T

(a)11.

Page 13

GBOMETRY AND TRIGONOMETRY

The diagram below, not drawn to scale, shows three stations P, Q and R, such that thebearing of Q from R is 116" and the bearing of P from R is 242". The vertical line at Rshows the North direction.

(i)

(i i)

Show that angle PR.Q: 126". ( 2 marks)

Given that PR: 38 metres and QR : 102 metres, calculate the distance PQ,giving your answer to the nearest metre. ( 3 marks)

(b) K, L and M are points along a straight line on a horizontal plane, as shown below.

KLM

A vertical pole, ,S1(, is positioned such that the angles of elevation of the top of the pole,S from L and M are 2I" and 14o respectively.

The height of the pole,,S1(, is 10 metres.

(i) Copy and complete the diagram to show the pole SK and the angles of elevationof ,S from L and M.

Calculate, correct to ONE decimal place,

( 4 marks)

(ii)

a)

b)

the length of KL

the length of LM. ( 6 marks)

Total 15 marks

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t2. (a)

Calculate, giving reasons for your answer, the measure of angle

(D GFH

(iD GDE

(iii) DEF.

(b) Use r :3.14 in this part of the question.

Given that GC:4 cm,calculate the area of

-,(1) ttiangle GCH

(ii) the minor sector bounded by arc GH andradli GC and HC

*(iD the shaded segment.

Page 14

The diagram below not drawn to scale, shows two circles. C is the centre ofthe smallercircle, GH is a common chord and DEF is a triangle.

Angle GCH:88" and angle GHE : 126".

( 2 marks)

( 3 marks)

( 2 marks)

( 3 marks)

( 3 marks)

( 2 marks)

Total 15 marks

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(a)

Page 15

VECTORS AND MATRICES

The figure beloq not drawn to scale, shows the points O (0,0), A (5,0) and B (-1,4)which are the vertices of atriangle OAB.

o (0,0)

(i) Express in the ar- lfl the vectorslbt

-)a) OB

-) -+

A (5,0)

b) OA+OB (3marks)

(ii) If M (xy) is the midpoint of AB, determine the values of x and y.( 2 marks)

In the figure below, not drawn to scale, OE, EF and MF are straight lines. The pointFI is such that EF : 3EH. The point G is such that Mtr : 5 MG. M is the midpoint ofoE.

-+ ->The vector OM: v and EH: tt.

(b)

(i) Write in terms of a and/or y, an expression for:_>

a) HF-+

b) MF-)

c) OH

Showthat i": I (z,*r\5\ /

Hence, prove that O, G and lllie on a straight line.

( l mark)

( 2 marks)

( 2 marks)

( 2 marks)

( 3 marks)

Total 15 marks

(iD

(iii)

B (-1,4)

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14. (a) L andNare two matrices where

L:lz 2l and.^/: lr 3ll.r 4) l.0 2

Evaluate L - N2.

END OF TEST

Page 16

( 3 marks)

( 4 marks)

Total 15 marks

(b)

(c)

The matrix, M, isgiven as M :'I 12 ] . Cut.olate the values ofx for which Mis

singular. t3 x) (2marks)

A geometric transformation, R, maps the point (2,1) onto (-1,2).

Giventhat n : [0 {l ,"ul"rrtutethevaluesofpandq. ( 3marks)lq ol'

[ .']A translation, T --l'_l maps the point (5,3) onto (1,1). Determine the values of r and.

l.ss. ."1( 3 marks)

Determine the coordinates of the image of (8,5) under the combined transformation,

(d)

(e)R followed by 7.

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