2005 Solutions

Suggested Solutions to Past CXC

Examination Papers

2005-2010

Paper

2 -

2010

80

Solution

Paper

2 -

2005

Page

1

Paper

2 -

2006

15

Paper

2 -

2007

31

Paper

2 -

2008

47

Paper

2 -

2009

63

Mathematics General Proficiency May 2005 1. a) 21 10 = 5 3 63-50 = 15 - 13/ - 15 b) i) A: 12.50

x

3

=

$37.50

_'v

B:

--

$33.90

=$

l

i

.

2

5

2

c:

--

$31.00

=5

$6.20 D: -15 X 108.28

=

$16.24 100 ii) 6

x

0.75=$4.50 6

x

0.40=$2.40 Totalmade

=

$6.90 Totalspent =$5.88 *NB:The exactprofit wasnot necessary in this case. (Difference: $6.90 - $5.88

=

$1.02)

Amanda made a

Rmfit

2 a) i) ab(5a

+

b) ii) (3k-l) (3k+ 1) iii) 2yL4y-y+ 2 2y(y - 2) -l(y - 2) (2y -1) (y - 2) b) 6xz - 8x

+

lSx -20 6xz + 7x - 20 c) i) 2(x - 3) = 2x-6 l ii) x

+

x - 3 + 2x -6=39 4x - 9

=

39 4x =48 x

=

12

·Solving the equation was not necessary

in this case.

3. a) i) 3x

+

x = 24

4x=24 x=6

~ students take both Music and Drama.

ii) Student who take drama only

=

7+ x

:. 7 + 6 =1.3.students take only Drama.

b) i) y- 5

=

-(x2

+

3) y=mx+c 3 2 2 y='3x+2+S or 5=-(-3)+c 3 2 5

=

-2

+

c y='3x

+

7 c=7 ii) 2x -3y = 0 Y = -x2

+

7 3 21Page

-3y =-2x -2x y =--3 2 Y

=

-X 3

m

=!. .:

for the line 2x - 3y

=

0 therefore it is parallel to y =

!

.

x

+

7

3 3

(Parallel lines have the same gradient)

4. a) Area of small pizza

=

rr(7.5)2 = 176.79 em-Area of medium pizza =rr(152)

=707.14em2

707.14

=

4 176.79

The medium pizza is more than twice as large as the small pizza. It isfour

times as large.

b) S·_Izeo f1_-0f_{me IUmplzza}di . _=--=707.14 23571_{. em}2

3 3

ratio of area to cost ofmedium pizza = 235.71 = 14.78 em2

/$

15.95

ratio of area to cost of small pizza

=

176.79

=

13.65 em2

/$

12.95

Therefore,it is better to buy a slice of medium pizza, rather than asmall

pizza.

5. The graph below shows the answer for 5(a), 5(b) (i) 5(b) (ii) - lern: 1uniton both axis. _J __ , c.

1;

;

;\ .

,

; . F '. , ;', . " , '. . , .;...., .' . .' , ~~" ..~, " , " ' ".. ., .. " .

.

..

' ..,....

.

'

F

'

;'

: .1.· .. · ~ I' .:'I· .' ! ;..~.;, : .' " " '

..

.

.

_. ._"/ ',: I' ," ."r:, I,·:' .i .-: , ,

'

,

.

.

': s:

.

.

'

\

'

. ",

,

:

.

.

'

,

'

,

y-

,

. ,;.:' ':~. :ii '" ; " .,: ,

.

.,

"

Et'

,'

,n ,·c. , :" ';', I, 1 ,l .'

.

~. I : J' I: .; .... " ." - ,,> "'" .• ,. '.

:

,

'

;

i

,

':

'

'I', 'i, ",: . 4\Page .... _{"., '} cT,

'

.

,

',:.. "

,

;t.

! ,

:

'

"

Hi) Composite transformation c) tan a =-1.8

2

a= tan"! 0.9

a= 42°

6. a) i) EAD

=

57° Reason: Alternate angles of parallel lines

Then x = 180 - (57 + 80) = 43°

ii) DAB=108-57

= 51°

Since angle of a quadrilateral add up to 360°. Then y = 360 - (51 + 57

+

90)

=360 -198

=162°

b) i) 32 + (-3)2 = 9 + 9

=18

ii) Ifx

=

Yzy + 5 x-5=Yzy y=2(x-5) r1(x) = 2x -10 rl(6) =2(6) -10

=

12 -10 =~ iii) fg(x) = 1fzX2 + 5 f g(2) = 1fz(22) + 5 =Yz(4)+5 =2+5

=Z

SIPage

7.

61Page

a) Cumulative Frequency Table showing the height of applicants

Height/ern Cumulative Frequency

$155 10 $160 65 $165 170 $170 280 $175 360 $180 390 $185 400 400 3!lO 300

J

250

'"

,

~

.

~ ₂₀₀ Q 150 100 50

Cumulative Frequency Curve of Heights of Applicants

y

!SS 16S 110 ,17,S; 1&0 lSS x H.ipt (cmj

b) i) 275 applicants ii) 167 em iii) 162.5 em iv)

-

95 400 8. a) i) (4 X 72)

+

(3 x 6)

+

2 216 ii)

I

~---~--- -103 (8 X 112) + (3 x 10) + 2 1000 iii) 1,--_n_3 --l.1_C(_n_-2_)x_C_n_+1_)2_+_(3_X_n~_2_1,-- __ n3__ b) (a - b) (a - b) (a + b) + ab(a + b)

=(a - b) (a- - b2)

+

a-b

+

ab-=

a3 - ab- - a-b + b3 + a2b + ab

-=a3- ab- + ab- - a-b + a2b + b3

=

a3

+

b3

Difference of squares

• (a - b)(a +b)=aL b'

9. a) 5xz + 2x-7 5(xZ+ ~x) - 7 5 5[Xl

+ ~

X

+

(!)Z ~ -7 - 5(!)2 5 5 5 1 1 Sex+_)2- 7-5 5 b) i) - 7-1 5

ii) WhenX +

5

1= 0,then x = -

'5

1

c) S(x

+

!)2 =36 5 5 (x

+

!)2 = 36 5 25 X + ~= ../36/25 1 6 x=--+_{5 -} -₅ E" h 1 6 1 6 It er x =--

+ -

or x=----5 5 5 5 =1 -7 5 d) v x 81Page

10. a) i) 40-0_'J£.77/2 - £".,J,LL m s 15-0 -

-Acceleration ==gradient of first portion of graph

ii) Distance travelled = 220(40

+

50) = 10(90)

=

9.illlm 20 b) i) 12km = 6km/h 2h

ii) He took a rest orstopped.

iii) ~=8km/h 1.5 c) i)

x=6

ii)

x~6

5 +5 y>--

x

- 8 1

y<-x+5

-6 li. _a) i) x 7.5

"evaluate ::-point inthe region for each line.

"Sine rule: "h absin C=Area of

triangle.

pL- ~ Q

1h(7.5) (4.5)Sin 0'

=

13.5 S. 13.5 InO' = --16.875 0"' = sino! 0.8 0"' = 53°

ii) 1h(15) (9) sin 53 = 67.5 sin 53

=

54 cm2

b) i) a) 180 - 136

=

44°

SJM =44 -2

=22° (base angles of an isosceles triangle)

b) SJM =

JMK

=

22° (alternate angles) :. JKM

=

180 - (124

+

22)

=

180-146

=

34° ii) a) MJ

--

50 sin 136 sin22 MJ =9.2..Zm b) --=--JK 92.7 sin 22 sin 34 JK

=Qllm

12. a) 17°N A-Antigua B - Belize G - Greenwich Meridian E-Equator G 10

I

P age

b) C= 2x 3.14x 6400

=

40192

=

40192 cos17

=

38423.6 1

=

38423.6 (~) 360 ::::2776km c) 1= 40192 (~) 360 ::::6140km 13. a) i)

AB=DC

Therefore.

AB

=

3K ii)

BD=BA+/iD

= -3K - 3y Hi) ---.-. 1----> DP=--BD 3 1

::::

-

"

3

(-3x - 3y)

=x+y

b)

AP=AD+DP

=-3y

+

x

+

y

=x

2y c)

PH=PD+DE

3 = -x - y

+ -

X 2 ::::Yzx-y

and

fiE

=

liP

+

PH

1 =X - 2y

+

-x - Y 2 3 ::::-x - 3y 2

=>

AP=2PH

Therefore.A,P and E are collinear.

d)

DA

== 3

G)

=

G)

IDAI

=

...

/3

2

+ 3

2

=

..

JI8

AB=AP+PE

=X - 2y

+

lhX - Y 3 =

2

"

X -

3y

= ~(~)-

3G)

=

G) - (D

==

(~3)

IABI

=

J

02

+

(-3)2 = ~ =3

DE

=~ (~) = (~)

IDEI

==

...

/3

2

+ 0

2 =~ =3 A o'---t+----" E 12

I

Page A

14. a) i) ii) iii) (2) (15) -(7) (5) =30 - 35

=

-5

=>

Misanon-singular matrix M-I =2.

(

15 -

5)

-5 -7 2 iv) (10 0)

(

X)

1 (15 1 Y

=

-5 -7

(

X)

= ~

(-45 - 85)

Y -5 21

+

34

(

X)

= ~ (-130) Y -5 55 X = 26 y=:.11

b) i)

R

=

(

-

1

1°)

•The general matrix of

0 rotationforiT'in the

anuclockwlse direction

about the

ii)

N

=

(

-

1

_-

_~)

origine:,:! -::::)

0

iii)

T=

(

~3

)

141 P age iv) RN=(~1 ~)(-~ _°1) = (~ -~) pI = (~ _~)

(161)

=

(-1~)

pl = (6.-11)

(161)

+

(-;3)

=

(136)

P"

=

(-g _~)

(136)

=

(

-

=-136)

P" = (-3.

-16)