Math January 2010

(1)

JANUARY 2O1O

FORM

TP

2010015

CARIBBEAN

EXAMINATIONS

COUNCIL

SECONDARY

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 set

Mathemati cal tables (provided) Graph paper (provided)

DO NOT TURN

TIIIS

PAGE

UNTIL

YOU

ARE TOLD

TO DO SO.

I

I I I

I

I I

r

Copyright O 2009 Caribbean Examinations Council@.

All

rishts reserved.

(2)

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

an

annual commission of 2o/oon the _{TOTAL value of}_cars

stld

_for_the_year.

If

_the

,u-l"r-u1

sold 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. Calculate}

the number of cups of pancake mix he must

use.

(

2 marks)

Neisha used 5 cups of milk to make pancakes using the same instructions. How

many pancakes did she

make?

(

₃_marks)

(i)

(ii)

Total

ll

marks

0 1 234020/JANUARY/F 20 1 0

(3)

2. (a) Given that

a:

6, b

:

-

4

and c

:

8, calculate the value

of

ctz+b

c-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'.

(

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 inequality

in (c)

(i)

above.

( l

mark)

Total 12 marks

3.

o 1 23 4020 I JANUARY/F 20 1 0

(4)

(b)

Using a pencil, a

ruler,

and a

pair

of compasses only:

(i)

Construct, accurately, the triangle

IBC

shown below, where,

AC:

6cm

Z

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

0 123 4020 I IAN UARY/F 20 1 0

(5)

4.

(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 to

LM,

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

I The 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)

o t234020 / JANUARY/F 20 I 0

(6)

(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

of

the equation of this line.

Page 7

Calculate his average speed

(

3 marks)

Total

ll

marks

Determine

(

3 marks)

(

3 marks)

J

T

s.

(a)

(b)

(iD

(iii)

(iv)

(v)

(i)

Using a scale of 1 cm to represent

I

unit

on both axes, draw thetriangle ABC with 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 the

imageA'B'C'.

(

2marks)

Draw

a

new

triangle

A"B"C'with

vertices

A,,

(-7,4),8,,(-4,4)

and

e'e6,7).

(

lmark)

Name and describe

the

single ffansformation that maps triangle

ABC

onto

triangleA"B"C".

(

2marks)

Total12

marks

0 1 234020/JANUARY tF 2010

(7)

6. A class of 26 students each recorded the distance travelled to schooi. The distance. to the nearest

km, is recorded below:

6

t2

30 21 26 39 l1

l6

11 J

l7

22 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

61

2t

-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 to represent 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)

o 1 23 4020 I JANUARY/F 20 1 0

(8)

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 i

l""i-!_ i

1

l+-.+.. )

i-i'i-i-iiii r-."_:_t_

4ii'i-i

!

I i- r-j.-it

0

-f -i

..i. ll.i.l.

-+-r..i 1

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

:i:i*i

o 123 4020 / JANUARY/F 20 1 0

(9)

8. 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 3

Pattern 1

O

l

_Hexagon

The table below shows the

sticks used to make EACH

2 Hexagons

number of hexagons in pattern.

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 Number

of

hexagons

in

the

pattern

I 2 -t 4 5 20 n

Number of sticks used

for

the

pattern

6

ll

T6

x

v

_^s

01234A20 I JANUARY/F 20 1 0

(10)

-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 particle

is

(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)

places.

marks)

a)

b) c)

they-coordinate of A

the x-coordinate of C

the

x

andy-coordinates

ofB.

(

3 marks)

Totel 15 marks

0 1 234020IJANUARYiT 20 1 0

(11)

10. (a) The manager of apizza shop wishes'to makex smallpizzas and_r,large pizzas. His oven holds 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. The

manager 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 on

they-axis to represent 5 large pizzas, draw the graphs

ofthe

lines associated

with 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 the

sale 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 the

pflzas.

(

lmark)

(iD

Use the coordinates ofthe vertices given at (b)

(iii)

to determine the

MAXIMUM

profit.

(

3 marks)

Total 15 marks

(b)

0123 4020 I JANUARY/F' 20 I 0

(12)

T

(a)

11.

Page 13

GBOMETRY AND TRIGONOMETRY

The diagram below, not drawn to scale, shows three stations _{P, Q}and R, such that the

bearing of _{Q from R}is 116" and the bearing of P from R is

242".

The vertical line at R

shows 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 elevation of ,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

0 I 23 4020 I JANUARY/F 20 1 0

(13)

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.

The diagram below not drawn to scale, shows two circles. C is the centre ofthe smaller circle, 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

0 1 23 4020 / TANUARY/F 20 1 0

(14)

(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 vectors

lbt

-)

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 point

FI is such that EF

:

3EH.

The point G is such that Mtr

:

5

MG.

M is the midpoint

of

oE.

-+

->

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)

ot23 4020 / JANUARY/F 20 1 0

(15)

14.

(a)

L

andNare two matrices where

L:lz 2l

and.^/:

lr

3l

l.r

4)

l.0

2

Evaluate

L

-

N2.

END OF TEST

(

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.s

s.

."1

(

3 marks)

Determine the coordinates

of

the image

of

(8,5) under the combined transformation, (d)

(e)

R followed by 7.