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Nan Chiau High School Preliminary Examination (3) 2011

MATHEMATICS Paper 1 4016/01

Secondary 4 Express/5 Normal Academic 2 hours

12 September 2011, Monday

NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL NAN CHIAU HIGH SCHOOL

READ THESE INSTRUCTIONS FIRST

Write your name, class and index number on all the work you hand in. Write in dark blue or black pen in the spaces provided on the Question Paper. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid.

Answer all questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. Calculators should be used where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For pi, use either your calculator value or 3.142, unless the question requires the answer in terms of pi.

The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80.

Setter: Miss Sng Mui Yong

2

Mathematical Formulae

Compound interest

Total amount = n

rP

+

1001

Mensuration

Curved surface area of a cone = rlpi

Surface area of a sphere = 24 rpi

Volume of a cone = hr 231

pi

Volume of a sphere = 334

rpi

Area of triangle ABC = Cabsin21

Arc length = r , where is in radians

Sector area = 221

r , where is in radians

Trigonometry

Cc

Bb

Aa

sinsinsin==

Abccba cos2222

+=

Statistics

Mean = ffx

Standard deviation = 22

ffx

ffx

3

50

60 x

B

C D

Legends A: Students who sent less than 100 messages B: Students who sent at least 100 but less than

200 messages C: Students who sent at least 200 but less than

300 messages D: Students who sent more than 300 messages

A

Answer all the questions.

1 Evaluate

(a) 32.214.232.41

34.945.33 2

+,

(b) 35 1045.01093.8 , giving your answer in standard form.

Answer (a) [1]

(b) ... [1] _____________________________________________________________________________

2 A survey was conducted to find out the average number of SMS messages sent by a group of 180 students in August 2011. The result was then represented by a pie chart as shown below.

(a) Find the number of students who sent less than 200 messages.

(b) The same information is to be shown in a histogram. The height that represents the number of students who sent less than 100 messages is 3 cm. Find the height that represents the number of students who sent at least 200 but less than 300 messages.

Answer (a) [1]

(b) cm [1] _____________________________________________________________________________

4

B A

C

D

12 cm

13 cm

E

3 Anne and her friends went to Buddy Buddy Restaurant for lunch. They ordered 8 set meals which cost $11.90 each. A 10% service charge on the basic cost and 7% GST (Goods and Services Charge) are included in the bill. How much did Anne and her friends spend altogether?

Answer $ [2] _____________________________________________________________________________

4 In triangle ABC, = 90CBA and BC = 13 cm. D is a point on AC such that = 90BDA and BD = 12 cm. BA is produced to a point E.

Without the use of a calculator, find

(a) EAC cos ,

(b) AC.

Answer (a) [1]

(b) cm [2] _____________________________________________________________________________

5

5 The diagram shows a regular hexagon of side 8 cm. Using each vertex of the hexagon as the centre and the length of each side of the hexagon as the radius, 12 arcs are drawn. Find the area of the shaded region. Give your answer in the form piba + .

Answer ... cm [3] ____________________________________________________________________________

6 (a) The area of a plot of land was 2 hectares 450 square metres. Due to the construction of an expressway, it was then reduced by 1250 square metres. Find the final area in hectares. (1 hectare = 10 000 m2)

(b) Mr Tan makes a profit of 15% on every product that he sells. He sold a vacuum cleaner for $x. Calculate the profit that he made in terms of x.

Answer (a) hectares [1]

(b) $ ... [1] _____________________________________________________________________________

8 cm

6

7 Ruth and Rhoda went shopping. The matrices below show their purchases and the cost of each item purchased.

123

234killer- weedfertilizer pot

of Packets of Packets Flower

50.8

8

50.2unitper

Cost

(a) Find

50.8850.2

123234

.

(b) Explain what your answer to (a) represents.

(c) Multiplying your answer to (a) by a matrix, find the total amount of money spent by both Ruth and Rhoda.

Answer (a) [1]

(b) ....

.... [1]

(c) $ ... [1] _____________________________________________________________________________

Ruth

Rhoda

Flower pot

Fertilizer

Weed-killer

7

8 When written as a product of its prime factors, 44 321296 = and 532120 3 = . Find

(a) the highest common factor of 1296 and 120.

(b) the smallest possible integer value of k for which 1296k is a perfect cube.

(c) the smallest positive integer value of n for which 120n is a multiple of 1296.

Answer (a) [1]

(b) ... [1]

(c) ... [1] _____________________________________________________________________________

9 Mr Koh and Mr Lee started cycling at the same time to a park. Assume that both Mr Koh and Mr Lee cycled at constant speeds. In 30 minutes time, Mr Koh travelled 1 800 metres

less than Mr Lee. If Mr Koh increased his speed by 80 m/min, his new speed would be 67

of the original speed of Mr Lee. Find the original speed of Mr Koh.

Answer . m/min [3] _____________________________________________________________________________

8

10 (a) The surface area of the earth is approximately 0.51 billion square kilometers. Write 0.51 billion in standard form.

(b) A water molecule consists of 2 hydrogen atoms and 1 oxygen atom. If the mass of a hydrogen atom is approximately 1.7 10 27 kg and the mass of a oxygen atom is 2.7 10 26 kg, find the mass of a water molecule, expressing your answer in standard form.

Answer (a) [1]

(b) .. kg [1] ____________________________________________________________________________

11 (a) Simplify 3 232

16250 yx

yx

.

(b) Solve the equation ( ) 0827 31

32

= xx .

Answer (a) [2]

(b) ... [2] _____________________________________________________________________________

9

12 A line l cuts the curve ( )( )43 = xxy at the point A. Given that the point A is equidistant from the x- and y- axes and its x-coordinate is greater than 5.

Find

(a) the coordinates of the point A,

(b) the equation of the line l if it also passes through the point (8, 0).

Answer (a) [2]

(b) ... [2] _____________________________________________________________________________

10

13 (a) (i) Factorise completely 22 81012 yxyx .

(ii) Hence find the possible positive value of yx

32

if 081012 22 = yxyx .

(b) Simplify ( )22 23369 + xx .

Answer (a)(i) [1]

(ii) [1]

(b) ... [1] _____________________________________________________________________________

11

A

B C

14 (a) On the Venn Diagram shown in the answer space, shade the set ( )BAC IU ' .

(b) { }120 andnumber oddan is : = xxx .

(i) The sets P, Q and R each contains 2 elements and =RQP UU . Given that P = {1, 3} and ( ) { }9 5,'=QP U , list the elements in R.

(ii) Given that A ={x: x is a prime number}, B = {x: x is a perfect square}, C = {x: x is divisible by 3}. Find

(a) BAI .

(b) ( )CBn I .

Answer (a) [1]

(b)(i)..... [1]

(ii)(a).... [1]

(b).... [1]

_____________________________________________________________________________

12

15 As a result of global warming, the ice of some glaciers is melting. Tiny plants, called lichen, will grow on the rocks after the ice disappears.

Each lichen grows approximately in the shape of a circle. The radius of this circle, r mm, is directly proportional to the square root of )12( t where t represents the number of years after the ice has disappeared.

(a) If the the radius of the lichen is 3.5 mm 13 years after the ice has disappeared, find the relationship between r and t

(b) If the radius of the lichen is 7 mm T years after the ice disappeared, find the radius of the lichen 3T years after the ice disappeared.

(c) Give a reason to explain the significance of the number of years )12( t that appears in the relationship.

Answer (a) [1]

(b) mm [2]

(c) .....

.. [1] _____________________________________________________________________________

13

16 (a) The masses of 80 babies were taken and the results are shown in the box-and-whisker diagram below.

(i) Find the value of y given that 60 babies have mass less than or equal to y kg,

(ii) The left whisker is shorter than the right one. What does this mean?

(b) The masses of a group of 20 students were also taken. Their masses were shown below.

38 42 52 45 35 57 62 40 50 43

55 53 52 57 68 55 38 55 48 42

(i) In the answer space below, show the masses in an ordered stem-and leaf-diagram.

(ii) Find the median mass.

Answer (a)(i) [1]

(a) (ii) . [1]

(b)(i)

[1]

(b)(ii) kg [1] _____________________________________________________________________________

Masses in kg

14

A B

C

D E

60

A

B C

E

D

F

G

H

I J

K

L P

Q

17 (a) In the pentagon ABCDE, AEDEDCDCBCBA === and = 60EAB . Explain why the AB and ED are parallel.

(b) The diagram shows a regular hexagon ABCDEF and a regular octagon EFGHIJKL with a common side EF. AF and DE produced meet at P GF and LE produced meet at Q. Find the angle PEQ.

Answer (a) .....

.. [2]

(b) [2] _____________________________________________________________________________

15

18 (a) Find the smallest rational number of x which satisfies xx 352213 + .

(b) Two similar conical vessels have surface areas of 162 cm2 and 5 000 cm2. The larger vessel can hold 2 litres of water. Find the capacity of the smaller vessel in cm3.

Answer (a) [2]

(b) cm3 [2] _____________________________________________________________________________

16

19 The diagram shows a quadratic curve 2y ax bx c= + + .

(a) Find the values of a, b and c.

(b) The gradient of the graph at the point P is zero, find the coordinates of P.

Answer (a) a =

b =

c = .. [3]

(b) [1] _____________________________________________________________________________

y

-1 5

10

x

17

20 Brandon left home at 16 00 and cycled 16 km to his grandmothers home. The diagram below is Brandons distance-time graph.

(a) Calculate Brandons average speed on the journey from his home to his grandmothers home. Give your answer in km/h.

(b) How far was Brandon away from his grandmothers home at 16 20?

(c) Brandons sister, Anne, left their home at 17 00 and cycled to their grandmothers home at an increasing speed and reached there at 18 30. On the same axes, sketch the distance-time graph of Annes journey. [1]

(d) Brandon followed the same path to go home. He left his grandmothers home at 20 00 and cycled at a constant speed of 14 km/h. On the same axes, draw the graph to represent Brandons journey from his grandmothers home to his home.

[1]

Answer (a) km/h [1]

(b) km [1] _____________________________________________________________________________

16 00 17 00 18 00 19 00 20 00

4

8

12

16

Distance from home (km)

Grandmothers home

Time

Home 0 21 00

18

21 In the diagram, the lines AB and DC are parallel. DC = 3AB and E is a point on BD such

that 3DE = 2DB.

(a) Given that AB = 4a and AD = 2b, express the following vectors in terms of a and/or b.

(i) DB , (ii) AE , (iii) BC . (b) Explain why AE and BC are parallel. (c) Find the numerical value of area of ADE area of DBC

Answer (a)(i)..... [1]

(ii)..... [1]

(ii )...,,... [1]

(b) ..... [1]

(c) ... [1] _____________________________________________________________________________

E

C

B A

D

2b

4a

19

y

x O

56102 ++= xxy

1

2

3

1 0 2 3 x

y

22 (a) The diagram shows the graph of xkay = . State the value of

(i) k,

(ii) a.

(b) A ball is thrown from the top of a building. The vertical height above the ground level is represented by y metres x seconds after the ball is thrown. The diagram below shows the path travelled by the ball.

(i) Calculate the greatest horizontal distance travelled by the ball.

(ii) By expressing the equation in the form ( )2bxay += , find the greatest height above ground level when the ball is thrown.

20

(c) Sketch the graph of 1252 2 += xxy in the answer space below.

Answer (a)(i) k = .... [1]

(ii) a =.... [1]

(b)(i).... m [1]

(ii)..m [2]

(c ) [2]

_____________________________________________________________________________

21

N

A

B

C

D

E

F

23 The scale drawing in the answer space below shows a regular pentagon ABCDEF where B is due south of A.

(a) Find the bearing of D from F.

Answer (a) [1]

(b) The point P is on a bearing of 210 from A and is on the angle bisector of angle DEF. Find and label the position of the point P.

Answer (b)

[2]

(c) Given that AB = 60 km, find the shortest distance of P from AB.

Answer (c) .. km [1] _________________________________________________________________________

End of Paper