hihs prelim 2011 sec 4 exp 5 n(a) paper 1 [turn over · 2016. 1. 5. · 9 hihs prelim 2011 sec 4...

1

HIHS Prelim 2011

Sec 4 Exp 5 N(A) Paper 1 [Turn over

2

HIHS Prelim 2011


3

HIHS Prelim 2011


Answer all questions.

1 (a) Find the fraction that is exactly between 11

5 and

11

6.

(b) Express %2

1105 as a fraction in its simplest form.

Ans: (a) ……………………….….……. [1]

(b)

……………………….….…….

[1]

2 Calculate

71.4

7

6)2.3(129

2

5.13

.

Give your answer correct to 4 decimal places.

Ans: ……………………….….……. [2]

3 Simplify 2

27

3

)( 36

3

5 y

x

xy

.

Ans: ……………………….….……. [2]

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HIHS Prelim 2011


4 In the construction of a ship, an alloy of copper, iron and titanium is used.

Every 130 kg of iron is mixed with 20 kg of copper, and every 100 kg of iron is mixed with 15 kg of

titanium. Assuming that the ratio of copper to iron to titanium in the alloy is represented by the

ratio x : y : z, find the ratio x : y : z.

Ans: ……… : ……… : ……… [2]

5 If the height of a triangle is decreased by 20% while its area remained unchanged, find the percentage

change in the length of the base.

Ans: ……………………………% [2]

5

HIHS Prelim 2011


6 List the integer values of such that 53

154

x.

Ans: x = ………………………… [3]

7 It is given that 22 532300 , when expressed as a product of its prime factors.

(a) Express 180 as a product of its prime factors.

(b) Find the largest integer that is a factor of both 180 and 300.

(c) Find the smallest integer value of x such that the lowest common multiple of 180, 300 and x

is 1800.

Ans: (a) ……………………………. [1]

(b)

……….……………….........

[1]

(c)

x = ……………………….…

[1]

6

HIHS Prelim 2011


8 Tap A takes 5 minutes to fill up a tank. It takes 4 minutes for tap A and tap B to fill up the same tank

together. Calculate the amount of time it would take for tap B to fill up the tank by itself.

Ans: ...…………..……………min [3]

9 (a) Shade in the diagram below the region representing )'(' ABB .

(b) Given that

x : x is an integer and 4 < x < 15}

P = { x : x is an even number}

Q = { x : x is a prime number }

Find

(i) )'( QPn ,

(ii) )'( QP .

[1]

(b)(i) …………..…………………. [1]

(b)(ii)

…………..………………….

[1]

A

ε

B

Ans: (a)

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HIHS Prelim 2011


10 The diagram shows a circular disc, with centre O. An arrow is attached to the disc and pointed initially

to N.

A fair die is thrown and if 1, 2, 3, 4 or 5 is shown, the arrow is rotated 90° clockwise, otherwise, the

arrow is rotated 90° anti-clockwise.

Find the probability that the arrow is pointing to

(a) position E after one throw,

(b) position N after two throws.

(c) position S after three throws.

Ans: (a) ……………………….….……. [1]

(b)

……………………….….…….

[1]

(c)

…..……………………….…

[1]

11 A microchip is made up of transistors printed on one side.

Each transistor is a flat square of side 51 nanometres.

(a) Express 51 nanometres in metres.

(b) Find the surface area of each transistor in square meters.

(c) The microchip is in the shape of a square of side 0.102 cm.

Assuming that the surface of the microchip is completely filled with transistors, calculate the total

number of transistors on the microchip.

Express all your answers in standard form.

Ans: (a) ……………………….….…m [1]

(b)

..……………………….....m2

[1]

(c)

..………………………….…

[2]

N

E W O

S

8

HIHS Prelim 2011


12 The diagram below shows a right-angle triangle ABC.

(a) Using compasses and ruler only, construct the

(i) perpendicular bisector of AB,

(ii) bisector of angle ABC.

(b) (i) State the radius of the circle passing through A, B and C.

(ii) Explain your reason for the answer to part (b)(i).

Ans: (a)(i), (a)(ii)

Ans: (b)(i) …………..……………… cm [1]

(b)(ii)………………………………………………………………………………………...

………………………………………………………………………………………………

………………………………………………………………………………………………

[1]

A B

C

[2]

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HIHS Prelim 2011


13 The prices of tickets for a KPOP concert is given below.

Child - $40; Adult - $80; Senior citizen - $65.

(a) Represent the information as a column matrix P.

(b) The number of tickets sold on one weekend can be represented by the matrix

Sunday

Saturday

30200315

25150200

Citizen

SeniorAdultChildren

N

Evalute the product NP.

(c) Explain what the matrix NP represents.

Ans: (a) ………..……………………. [1]

(b)

NP =…..……………….........

[2]

(c) …………………………………………………………………………………………..

..……….……………………………….……………………………………………………

[1]

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HIHS Prelim 2011


14 (a) Simplify )1(

3

)1(3

2

xx

.

(b) Make v the subject of the formula

y

xvyx

2

.

Ans: (a) ……………………………. [2]

(b)

v = …………………….........

[2]

15 (a) Factorise 672 2 xx completely.

(b) Using your result from above, solve the equation 062118 2 yy .

Ans: (a) ……………………………. [2]

(b)

y = .………………...…….

[2]

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HIHS Prelim 2011


16 The diagram shows the speed-time graph of a car.

(a) Calculate the distance travelled by the car when it was travelling at constant speed.

(b) Calculate the deceleration of the car at t = 37 min.

(c) Calculate the value of x if the average speed of the car in the last 25 minutes of the journey

was 53 km/h.

Ans: (a) .…………………………km [1]

(b)

……….………………km/h2

[1]

(c)

x = …………………...km/h

[2]

100 75 50 time (min)

40

x

100

Speed km/h

12

HIHS Prelim 2011


17 The marks obtained by the pupils in a mathematics test in two Secondary Three classes are shown in

the tables below.

Class A

Marks 45 – 49 50 – 54 55 – 59 60 – 64 65 – 69

No. of

pupils 2 3 8 23 6

Class B

Mean = 58 marks

Standard Deviation = 4 marks

(a) For Class A, calculate

(i) the mean,

(ii) the standard deviation.

(b) Compare and comment on the results of the two classes.

Ans: (a) ……………………………. [1]

(b)

……………………….........

[2]

(b) ..……….……………………………….………………………………………………...

..……….…………………………………………………………………………………..

[2]

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HIHS Prelim 2011


18 (a) (i) Express 322 xx in the form bax 2)( , where a and b are integers.

(ii) Hence state the coordinates of the turning point of 322 xxy .

Indicate if the turning point is a maximum or minimum point.

(b) The point (1,1) has been marked on the each diagram on the answer space.

On these diagrams, sketch the graphs of

(i) 3

1

yx ,

(ii) 13 xy .

Ans: (a)(i) ………………….……………. [1]

(a)(ii)

( .…. , ..… ) ;…..….......point

[2]

(b) (i)

(b) (ii)

x

y

x

y

[2]

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HIHS Prelim 2011


19 A series of diagrams of shaded and unshaded squares is shown below.

Diagram Number of shaded squares

(S)

Number of unshaded squares

(U)

Total number of squares (T)

1 1 0 1

2 6 3 9

3 15 10 25

4 28 21 49

(a) Find the number of shaded squares in Diagram 5.

(b) Write down, in terms of n, the total number of squares in the nth diagram.

(c) Calculate the total number of squares in Diagram 13.

(d) Write down the equation connecting S, U and T.

(e) Is it possible for a diagram to have a total of 2024 squares?

Explain your answer.

Ans: (a) ……………………………... [1]

(b)

…………...……………........

[1]

(c)

…………...……………........

[1]

(d)

..…………………………….

[1]

(e) ………..……….…………………………………………………………………………

……………..……….………………………………………………………………………

[1]

v v v v

v v

v v v

v v v

v v v v

v v

v v v

v v v

v v v

v v v

v v v

v v v

v v v

v v v

v v v

v v v

v v v

v v v

v v v

v v v

v v v

Diagram 1 Diagram 2 Diagram 3 Diagram 4

15

HIHS Prelim 2011


20 In the diagram, the equation of the line AC is 03023 xy .

The length of the line BC is 10 units.

(a) Find the coordinates of A and C.

(b) State the coordinates of B.

(c) Find the area of triangle ABC.

(d) Find the length of the perpendicular from B to AC.

Ans: (a) A = ( ..… , ….. ), C = ( ….. , ..… ) [2]

(b)

B = ( ..… , ..… )

[1]

(c)

…………………………….units2

[1]

(d)

…………………………….units

[2]

y

x O B

A

C

16

HIHS Prelim 2011


21 In the diagram, ACE is a triangle and BCDF is a parallelogram.

AB = x cm, BC = 40 cm, CD = 10 cm and DE = 20 cm.

(a) Prove that triangle ABF is similar to triangle FDE.

(b) Find x.

(c) Find BCDF

ABF

ramparallelog of area

triangleof area.

Ans: (a) ...…….……………………………………………………………………………..

………………………………………………………………………………………………

………………………………………………………………………………………………

[2]

(b)

x = …………………….......

[1]

(c)

……….……………….........

[2]

40

10 20

x

F

E D

A

B

C

17

HIHS Prelim 2011


22 In the diagram, the points A, B, C, D, and E all lie on the circumference of a circle.

DE is parallel to CB.

Given that angle ACE = 40°, angle CBE = 65° and angle CED = 60°,find

(a) angle ABE,

(b) angle ACB,

(c) angle BEC,

(d) angle AED.

Ans: (a) ∠ABE =……………………° [1]

(b)

∠ACB =…………...………°

[1]

(c)

∠BEC =………………...…°

[1]

(d)

∠AED =……………...……°

[2]

-------------------------- End of Paper 1--------------------------

A

B

C

D

E

65°

60°

40°

18

HIHS Prelim 2011


Answers

Question Answer

1(a)

1(b)

200

111

2 –0.1145

3

2

78 yx

4 40 : 260 : 39

5 Percentage increase = 25%

6 x = -1, 0, 1

7(a) 7(b) 60

7(c) 8

8 x = 20 min

9(b)(i) 5

9(b)(ii) {9}

10(a)

10(b)

18

5

10(c) 0

11(a) 11(b) 1510601.2

11(c) 8104

13(a)

13(b)

30550

21625

13(c) It represent the amount collected from the sale of tickets on Saturday and Sunday respectively.

14(a)

)1(3

11

x

14(b)

x

yxyv

)(

15(a)

19

HIHS Prelim 2011


15(b)

16(a)

16(b)

16(c) 66 km/h

17(a)(i) 60.3 marks or marks

17(a)(ii) 4.84 marks

17(b) Class A performed better as the mean marks was higher while the marks of the students had a lower spread in class B.

18(a)(i) 18(a)(ii) (1 , 2)

minimum point

19(a) 45

19(b) 19(c) 625

19(d) S + U = T

19(e) It is not possible as the number of squares should be a square term.

20(a) C = (-15 , 0)

20(b) B = (-5 , 0)

20(c) Area =

20(d) 5.55

21(a) ∠FAB = ∠EFD (corresponding angles) ∠AFB = ∠FED (corresponding angles, FB // DC) ΔABF = ΔFDE (AA property)

21(b) x = 20

21(c) Ratio =

22(a) ∠ABE = ∠ACE = 40o (∠s in the same segment)

22(b) ∠ACB = 60 o – 40o = 20o (alternate angles)

22(c) 180 o – 65 o – 60o = 55o (∠s in a triangle)

22(d) 135o

PRELIMINARY EXAMINATION 2011 SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC)

MATHEMATICS 4016/02 Paper 2

Name : _________________________ Date : 4 Aug 2011 Register No : _________________________ Duration : 2 h 30 min Class : _________________________

Additional Materials needed: 6 sheets of writing papers 1 sheet of graph paper

Instructions to Candidates

Write your index number and name on all the work you hand in. Write in dark blue or black pen. 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 , use either your calculator value or 3.142, unless the question requires the answer in terms of . The number of marks is given in brackets [ ] at the end of each question or part question. The total marks for this paper is 100.

Setter: Mrs Chang Poh Joo

This paper consists of 12 printed pages, inclusive of this cover page.

HOLY INNOCENTS’ HIGH SCHOOL

2

Holy Innocents’ High School Preliminary Examination 2011

Secondary Four Express / Five Normal (Academic) Mathematics Paper 2

Mathematical Formulae

Compound interest

Total amount

nr

P

1001

Geometry and Measurement

Curved surface area of a cone rl

Surface area of a sphere 24 r

Volume of a cone hr 2

3

1

Volume of a sphere 3

3

4r

Area of triangle ABC Cabsin2

1

Arc length r , where is in radians

Sector area 2

2

1r , where is in radians

Trigonometry

C

c

B

b

A

a

sinsinsin

Abccba cos2222

Statistics

Mean

f

fx

Standard deviation

22

f

fx

f

fx

3



Answer all the questions

1 The diagram shows a park ABCDE.

Given that angle BAE = angle BEC = 90o, angle BCE = 47o, AE = 30 m, BE = 52 m,

CD = 98 m.

(a) Calculate

(i) BC, [2]

(ii) the perimeter of the park. [3]

(b) C is due south of E.

Calculate the bearing of A from B. [3]

2 (a) Amos changed $3 600 Singapore dollars (S$) into Australian dollars (A$) for

his trip to Perth. Upon returning, he has A$68 left.

Given that the exchange rate is A$1 to S$1.25, calculate the amount he spent

for this trip, in Singapore dollars. [2]

(b) The cash price of a sofa set is $4 380. Basil paid for it by hire purchase, with downpayment of 15% and the remaining

amount over 18 equal monthly instalments, with interest charged at a flat rate of

5% per annum.

Find

(i) the interest charged by the hire purchase scheme, [2]

(ii) the amount of each instalment. [2]

(c) Cathy invested a sum of money in a bank at 6 % per annum compounded every half-yearly.

She received an interest of $11 798.38 at the end of 3 years.

Calculate the sum of money invested. [3]

A

C

B

D

E

30 52

98

47o 25o

4



3 (a) If x – y = 2 and x + y = – 2, find the value of 2222 2 yxyxyx . [2]

(b) Given that 4

32

yx

xy, find the value of

y

x. [2]

(c) A solid cone is cut into 2 parts, A and B, by a plane parallel to the base.

The heights of A and B are in the ratio 2 : 3. The volume of cone A is V cm3.

(i) Express the total volume of A and B in terms of V. [1]

(ii) If the volume of B is 245 cm3, calculate the volume of A. [2] (iii) If the base area of A is 30 cm2, calculate the base area of the

original cone. [2]

A

B

5



4 In the diagram, OABC is a parallelogram.

D is the mid-point of BC.

It is given that

OA = a and

OC = c.

E is the point on OC such that 5OE = 2OC.

F is the point on OD such that

FDOF2

1.

(a) Express, as simply as possible, in terms of a and/or c,

(i)

EC , [1]

(ii)

OD , [1]

(iii)

OF [1]

(b) Show that

FEAE 6 . [3]

(c) Given that the coordinates of F is (4, 2) and

12

3AE .

(i) Calculate

AE . [1]

(ii) Find the coordinates of E. [2]

(iii) Find the value of t, given that the vector

t

5.4 is parallel to

AE . [2]

A

O

B

C

F

E

a

c

D

6



5 (a) The diagram shows part of a regular polygon ABCDEF… , which has nine sides.

BC and ED are produced to meet at P.

(i) Show that angle PDC = 40o. [1]

Calculate

(ii) angle DCE, [1]

(iii) angle BCE, [1]

(iv) angle BEF. [1]

(b) GHJK is a quadrilateral. (i) Angle HGK= u o, angle GHJ= v o.

If GHJK is a parallelogram, write down the relationship between u and v.

[1]

(ii) Angle GJK= x o, angle JGK= y o.

If GHJK is a rhombus, write down the relationship between x and y.

[1]

(c) In triangle PQR, PQ = PR.

QX bisects angle PQR and RY bisects angle QRP.

Prove that triangles PQX and PRY are congruent. [3]

A

B

C

D

E

F

P

O

P

Y

Q

X R

7



6 The Adventure Society chartered an air-conditioned bus for $1500 to take a group of x

members to Malaysia for a trekking trip.

It was agreed that each member of the group would pay an equal share of this transport

fee.

(a) Write down an expression, in terms of x, for the amount of transport fee each member of the group had to pay. [1]

On the day of departure, three members of the group could not make it for the trip.

The Adventure Society decided that it would contribute $140 from its funds and that the

balance of the transport fee was to be shared equally by the remaining members.

(b) Write down an expression, in terms of x, for the transport fee that each remaining

member had to pay after the three members had withdrawn from the trip. [1]

(c) If each of the remaining members had to pay an additional $6 in order to cover the

transport fee, form an equation in x and show that it reduces to

3x2 + 61x – 2250 = 0. [3]

(d) Solve the equation 3x2 + 61x – 2250 = 0, giving the answers correct to three decimal places. [3]

(e) Hence find the transport fee that each member had to pay if all the members were present for the trip. [1]

8



7. The diagram below shows four points A, B, C and D on a piece of horizontal land.

It is given that AB = 22 m, AD = 33 m, BC = 24 m, angle BDC = 38o and

angle CBD = 54o.

Calculate

(a) BD, [3]

(b) angle .ABD [4]

(c) A tower TB stands vertically at B.

Given that the angle of elevation of T from A is 30o, find

(i) the height of the tower. [2]

(ii) the angle of depression of D from T. [2]

A

B

C

22

54o

D

33

24

38o

9



8 A cylindrical container with radius 30 cm and length 80 cm is partially filled with water.

Diagram I shows the cross-section of the container where the shaded area represents the

area in contact with water.

Angle AOB = 2.8 radians.

Diagram II shows the side view of the container.

(a) Find the area of the shaded region in Diagram I. [3]

(b) Find the volume of water in the cylindrical container. [1]

(c) Find the total surface area of container that is in contact with the water. [4]

The major sector AOBP in Diagram I is used to make a cone by joining

the edges OA and OB.

(d) Calculate the radius of the base of the cone. [2]

(e) Find the volume of the cone. [2]

A

O

B

a

B

2.8 rad 30 30

Diagram I

A B

O

Diagram II

80 cm

P

Q

10



9 Answer the whole of this question on a sheet of graph paper.

In a particular week, a tailor makes a profit of $y when he produces x shirts, where the

variables x and y are connected by the equation

xxy 10020310

1.

The table below shows the profit the tailor makes when he produces different number of

shirts.

(a) Suggest a reason for y = –200 when x = 0. [1]

(b) Calculate the value of p. [1]

(c) Using a scale of 2 cm to represent 10 shirts, draw a horizontal axis for 0 ≤ x ≤ 80.

Using a scale of 2 cm to represent $100, draw a vertical axis for –200 ≤ y ≤ 700.

On your axes, plot the points given in the table and join them with a

smooth curve. [3]

(d) Use your graph to estimate

(i) the profit made when 25 shirts are produced. [1]

(ii) the maximum possible profit of the tailor in a week, and the

corresponding profit per shirt when the profit is maximum. [2]

(iii) the gradient of the curve at the point where the number of shirts

produced is 40. [2]

(e) By drawing a suitable straight line, find the two values of x for which the average

profit per shirt is $9. [2]

Number of

shirts (x) 0 10 20 30 40 50 60 70 80

Profit ($y) –200 90 320 490 600 650 640 570 p

11



Cum

ula

tive

Fre

quen

cy

20 40 60 80 100

200

400

600

800

1000

0

Wages

10 The cumulative frequency graph shows the distribution of weekly wages of 800 workers

in Company A.

(a) Find

(i) the median wage, [1]

(ii) the interquartile range, [2]

(iii) the thirtieth percentile wage. [1]

(b) 15% of the workers earn more than $w. Find w. [1]

(c) If two workers are selected at random, find the probability that one worker

selected earns not more than $60 and the other earns more than $80. [2]

(d) If the company decides to increase the wages of each of the workers by 10%,

describe how the cumulative frequency curve will differ from the given curve.

[1]

12



(e) The weekly wages of 800 people in Company C is represented by the box and

whisker diagram below.

Wages

Compare the wages of the workers from Company A and C in two different ways.

[2]

~~ End of Paper 2 ~~

20 40 60 80 100 120

13



HIHS Prelim Exam 2011

Sec 4 Express Mathematics Paper 2

1a (i)

1a (ii)

1b

2a

2b(i)

2b(ii)

2c

3a

3b

3c(i)

3c(ii)

3c(iii)

4a(i)

4a(ii)

4a (iii)

4b

71.1 m

330 m

305.2o

S$3 515

$279.23

$222.35

$60 800.00

or $60 816.39

– 16

5

V8

125

16.8 cm3

187.5 cm2

c5

3

ca 2

1

ca3

1

6

1

cac

OEFOFE

5

2

6

1

3

1

ca15

1

6

1

ca

OEAOAE

5

2

)15

1

6

1(6 ca

FE6

FEAE 6

4c(i)

4c(ii)

4c(iii)

5a(ii)

5a(iii)

5a(iv)

5b(i)

5b(ii)

5c

6a

6b

6d

6e

7a

7b

7c(i)

7c(ii)

8a

8b

8c

8d

8e

9a

9b

9d(i)

9d(ii)

9d(iii)

9e

12.4 units

E = (4.5, 0)

t = –18

20o

120o

100o

u + v = 180

x = y

RPYQPX (common angle) [1 m]

PRYPQX ( PRQPQR are

base angles of isosceles triangle, base

angle bisected) [1 m]

PQ = PR (given)

∆PQX is congruent to ∆PRY (ASA)

[B1 for length & case]

x

50

3

1360

x

x = 19.046 or x = – 39.379 (rejected)

$78.76 or $78.95

39.0 m

57.8o

12.7 m

18.1o

1720 cm2

137 000 cm3

11 800 cm2

16.6 cm

7230 cm3

starting cost (rental of machine,

materials cost etc)

440

$410

profit = $660

profit per shirt = 45.12$53

660

gradient = 8 ± 1

x = 10 ± 0.5 or x = 67 ± 0.5

14



10a(i)

10a(ii)

10a(iii)

10b

10c

10d

10e

Median = $40

Q1 = 24, Q3 = 55, IQR = $31

P30 = $27

w = $62

799

66

The cumulative frequency curve will be less steep.

[Cumulative frequency curve will be stretched from x = 0 to x = 110.

i.e. cumulative frequency will reach 800 when wage = $110.]

Greater median for Company C. ($52)

Greater spread for Company C (IQR = 38)

hihs prelim 2011 sec 4 exp 5 n(a) paper 1 [turn over · 2016. 1. 5. · 9 hihs prelim 2011 sec 4...

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