2011 Prelim HY Exam Solns (1)

7/28/2019 2011 Prelim HY Exam Solns (1)

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QUESTION MARKS OUTCOME

QUESTION 1:

a) Express 3 531 as a fraction in simplest form and show all working.

b) Evaluate2 23.24 2.1

36 2.1

+correct to 3 significant figures.

c) Increase $159 by 8% then decrease that by 6%.d) If ( )2 21

3k m v u= find the value ofm correct to 2 decimal places

when k= 724, v = 142 and u = 7.8

e) Evaluate 6432

f) Calculate3 8

22

)1084.3(

)1076.1(

and write your answer in standard notation

correct to 3 significant figures.

QUESTION 2:

a) If 27 + 12 = A , findA.

b) Expand and simplify ( )2 3 3 3 6

c) Rationalise the denominator 10

22

giving an exact answer in simplest

form.

d) Find the exact value of 3 22 1t t + when 2 5t= .

e) Show that3 2 2 3

3 2 2 3

+

can be expressed in the form

a + b 6 and finda andb.

f) Simplify2 1

5 2 5 2

+

2

2

1

2

2

2

2

2

2

2

3

2

P3


2/8


3/8


QUESTION 5:

a) Find the exact value of tan( 150 )

b) If7

tan

8

= and cos 0 < , find the exact value of cosec

c) A vertical pole on level ground is supported by two wires PT

and QT, of unequal length and are connected to its top. PT is

12 m long and is inclined at 48o

15 m long.

to the horizontal and QT is

i) Copy the diagram and mark the relevantinformation on it.

ii) Find the height of the pole ST. (correct to 1decimal place)

iii) Find the angle at which QT is inclined to thehorizontal.

(correct to the nearest degree)

d) A surveyor takes bearings and measurements from a point M to

determine the distance across a lake XY.

X is 1200 m from M on a bearing of 340o and Y is 1500 mfrom M on a bearing of 060o

i) Copy the diagram and mark the relevant.

information on it.

ii) Find the distance XY across the lake to thenearest metre.

iii) Hence or otherwise find the bearing of Y from X

2

2

1

2

2

1

2

3

P4

T

QSP

NOT TO

SCALE

Y

X

M

NOT TO

SCALE


4/8


QUESTION 5: continued

e) Determine the area of this triangle correct to 2 decimal places.

f) Two yachts are both due east of a lighthouse, which is on top of

a cliff. The angle of elevation from Yacht A is 37 and from

Yacht B 23 . It is known that Yacht A is exactly 300m from

the lighthouse.

Find the distance from the ship B to the lighthouse correct to the

nearest metre.

g) Determine the value of x if cos(2 20) sin( 10)x x = +

h) Mohammed starts at town A, walks 3 kilometres due South turns andthen walks 7 kilometres due East and reaches town B. What is the bearing

of town A from town B.

1

2

2

3

P4

126

12.6 m

9.7m

NOT TO

SCALE

A B

300m x


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6/8

2011 PRELIMINARY MATHEMATICS HALF

YEARLYSOLUTIONS

a.Question 1

= 3.531 3.5313131Let = 3.5313131x

=

=

=

=

=

=

100 353.13131

3.53131

99 349.6

349.6

99

3496

990

1748 2633

495 495

x

x

x

x

b. = =+

2 2

3.24 2.1 0.9862422887 0.98636 2.1

c. $161.42 d. ( )= 2 21

3K m v u

( )

( )

=

=

=

=

2 21724 14.2 7.83

2172 140.8

2172

140.815.43

m

m

m

e. = =2 3 4 6 2 7 6 8 f. = 142.61787.... 4.26 10

a.Question 2

+ =27 12 A

+ = +

=

=

=

27 12 3 3 2 3

5 3

75

75A

b. ( ) = 2 3 3 3 6 18 2 18 = 18 6 2

c. = = =2 2 10 2 20 4 5 2 510 10 510 10

d. ( ) ( ) + = +3 2

3 22 1 2 5 2 2 5 1t t

= +

=

8 5 5 2 4 5 1

40 5 39

e. + =+

3 2 2 3 3 2 2 3 18 12 6 12

18 123 2 2 3 3 2 2 3

=

=

30 12 6

6

5 2 6

= = 5, 2a b

f. ( ) ( ) + =+

2 5 2 1 5 22 1

5 45 2 5 2

= 10 2 2 5 2

a.Question 3

( ) ( )+ = + 2 25 2 3 4 15 20 6 8x y x y x xy xy y

= 2 215 14 8x xy y

b. ( )( )( )

+ ++ =

+22 22

2 4 2 2

x x xx x

x x x x

( )( )

( )( )

+ +=

+

+=

+

2

2

2 4

2 2

2 5

2 2

x x x

x x

x x

x x

c. ( )( )

( ) ( )+ ++ + =

+

23 2

2 3

1 1 11

1 1

m m x x x m m m

x x x x x x m

( ) ( )( )

( )

+=

= +

2

2

1 1

1 1

1

m x x

x

m x

d. i) ( ) = 33 38 27 2 3d d ( )( )= + +

2

2 3 4 6 9d d d ii)

( ) ( )+ = + +2 2 2 2 2 2 2 2mx my nx ny m x y n x y

( )( )= + 2 2x y m n e. i) ( )( ) = +2 5 14 7 2x x x x

ii) ( ) ( ) = 2 2 21 16 1 4x x

( ) ( )

( )( )

= +

= +

1 4 1 4

5 3

x x

x x


7/8

f. ( ) ( )( ) = +3 22 2 4 4x x x x = + +

= +

3 2 2

3 2

4 4 2 8 8

6 12 8

x x x x x

x x x

a.

i)

Question 4

+=

2

23 2

a a

( )+ =

+ =

=

=

2 2 3 12

2 4 3 12

4 12

16

a a

a a

a

a

ii) ( ) ( ) = +6 1 3 8y y

= +

==

=

6 6 3 24

3 6 243 30

10

y y

yy

y

b. + =2 6 7 0x x

( )

=

=

=

+ =

=

2

2

4

2

6 6 4 1 7

2

6 8

2

6 8 6 8,

2 2

4.41, 1.59

b b ac x

a

c. + 15 4 3x +

3 15 4 3

18 4 12

14 32

x

x

x

d. + =3 7 2x y ( ) =

=

+ + =

=

=

=

2 8 2

4 2 16

3 2 2

7 14

2

4

x y

x y

x y

x

x

y

e. = = +26 5 6y x y x

( ) ( )

= +

+ =

+ =

= =

= =

2

2

6 5 6

6 5 6 0

3 2 2 3 0

2 3,

3 2

8 27

3 2

x x

x x

x x

x x

y y

a.Question 5

( ) = tan 150 tan 30

=1

3

b. =


8/8

ii) =sin4812

h

=

=

12 sin48

8.9 m

h

iii) =8.9

sin15

= 36 d. i )

X

M

Y

N

1500m

1200m20

0

600

200

N

ii)

= + 2 2 21200 1500 2 1 200 1 500 cos 80XY= 1751XY

iii)

=

1500 1751

sin sin 80

=

=

=

sin sin 80

1500 1751

1500 sin80sin

1751

58

Bearing is ( ) + 180 58 20

= 102

e. = 1 sin2

A ab C

=

= 2

112.6 9.7 sin126

2

49.44 m

f. i)

370

1430

x

230

300 m

A B

ii) =

300

sin143 sin23

x

=

=

300 sin143

sin23

462 m

x

g. ( ) ( ) = + cos 2 20 sin 10x x + + =

=

=

=

2 20 10 903 10 90

3 100

133

3

x xx

x

x

h.

A

B

3km

7km

N

N

=

=

3tan

7

23

Bearing is + = 270 23 293

2011 Prelim HY Exam Solns (1)

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