Other School 1 EM P1

7/29/2019 Other School 1 EM P1

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This paper consists of22 printed pages, including the cover page and formula sheet.

Class Index Number

Candidates Name: _______________________________

Preliminary Examination

Exam Date: Duration: 2 hours

Subject MATHEMATICS PAPER 1 (4016 / 01)

LevelSecondary 4 Express

Setter

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.You may use a soft pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid.

Answer allquestions.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 possible.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 decimalplace.

For , use either your calculator value or 3.142, unless the question requires the answerin terms of.

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

Use


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Mathematical Formulae FExam

U

Forxaminers

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Compound Interest

Total amount 1100

nr

P = +

Mensuration

Curved surface area of a cone rl=

Surface area of a sphere 24 r=

Volume of a cone21

3 r h=

Volume of a sphere 34

3r=

Area of triangle1

sin2

ABC ab C=

Arc length r= , where is in radians

Sector area 21

2r = , where is in radians

Trigonometry

sin sin sin

a b c

A B= =

C

A

2 2 2 2 cosa b c bc= +

Statistics

Meanfx

f=

2

Standard deviation

22fx fx

f f

=


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Answerall the questions.Forxaminers

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1 Evaluate

(a) , giving your answer in standard form,)1067.1()1068.6( 52

(b)24.50

78.0)02.347.2( +, giving your answer to 2 significant figures.

Answer: (a) [2]

(b) [1]

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2 Paul bought a limited edition book in 2010. In 2011, he sold it for $310 at a profit of150%. Find the price at which he bought the book in 2010.

Answer $ [2]

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3


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3 (a) Express as a single fraction in its lowest form)(

)(22 ba

bc

ba

acab

+

.For

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(b) Simplifyqp

r

p

pqr32

2 3

)3(

)( .

Answer: (a) ..

[2]

(b) .. [2]

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4 12 men, each working 6 hours, can produce 2400 gift packs in one day. How manyhours must 16 men work in order to produce the same number of gift packs?

Answer: h [2]

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4


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5 Find the greatest integerxthat satisfies the inequality 2054

7x .Forxaminers

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Answer: x= [2]

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6 Three MRT trains take 1 minute, 1 minute 12 seconds and 1 minute 24 seconds to movefrom one station to another. The three trains started together at 1100 hrs from the samestation. Find the time when the three MRT trains will next meet.

Answer: hrs [2]

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7 Two dice are thrown at the same time. Find the probability that(a) the two dice show the same number,(b) the sum of the 2 numbers is less than 10.

Answer: (a) [1]

(b) [2]

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5


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8 Given that x{= is an integer: 1


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10The table below shows the number of students who ran certain distances.Forxaminers

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UDistance (km) 5 10 15 20 25Number of students 4 3 1 4 x

(a) Given that the modal distance ran is 25 km, find the smallest possible value ofx.

(b) Given that the median is 20 km, find the largest possible value ofx.

Answer: (a) Smallestx= [1]

(b) Largestx = [2]

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11The surface area of one face of a solid cube is units2.36244 2 ++ yy

(a) Factorise completely.36244 2 ++ yy(b) Hence, find in terms ofy, the length of each side of the cube, leaving your answer in

its simplest form.

Answer: (a) . [2]

(b) .units [1]

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12 In the diagram, POQ is a sector of a circle with centreO and radius 10 cm. Given thatPRis perpendicular to OQ andPR =6 cm,

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Find(a) POQ in radians,

8

(b) the area of the shaded region.

10 cm6 cm

P

O R Q

Answer: (a) rad [2]

(b) ..cm2 [3]

_____________________________________________________________________________

13 In the diagram, ABCDE is a regular pentagon. The sidesABandCD are produced tomeet at F. Find

(a) m

,(b) n.

m n

D

EC

A FB

Answer: (a) m=. [2]

(b) n=.. [2]

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14 In the diagram, ABC is a right-angled triangle whereAB =28 cm, AC =35 cm andBC isproduced toD. Express as a fraction, the value of

Forxaminers

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U(a) ,ACBsin

3528

B

A(b) .ACDcos

C D

Answer: (a) sinACB=

[1]

(b) cosACD=. [2]

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15 The graphs of 2 linear equations are shown below. It is given that the coordinates ofQare (0, 8), the coordinates ofR are (6, 0) and the area of PQR is 32 cm2. The equationof linePQ isy=ax+b. Find

y

R (6, 0)

Q (0, 8)

P

(a) the coordinates ofP,(b) the values ofaandb.

x

Answer: (a) P (, ) [2]

(b) a=

b= [2]

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16 Car A travelled for 3 hours at an average speed ofukm/h, and then for 5 hours at anaverage speed ofvkm/h. It travelled a total distance of 462 km.Car B travelled for 2 hours at an average speed ofukm/h, and then 7 hours at anaverage speed ofvkm/h. It travelled a total distance of 528 km.

(a) Form two equations in terms ofuandv.

(b) Hence, find the value ofuand ofv.

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Answer: (a)

[1]

(b) u=.

v= [3]

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17 The diagram below shows the graph of a quadratic equation which cuts the x-axis atP(1, 0) andQ(7, 0).

10

xP Q

(a) State the line of symmetry of the graph.(b) Find the equation of the curve.(c) Find the greatest value ofy.

Answer: (a) Line of symmetry:

[1]

(b) Equation of curve:

. [1]

(c) Greatesty=. [2]

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18 The frequency, f, of a radio wave varies inversely as its wavelength, W.WhenW=2000 m, f=50 kHz.

(a) Find the equation connectingWand f.

(b) Find the frequency when the wavelength is 1250 m.

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Answers : (a) [2]

(b) .. kHz [1]

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19 The masses (in kg) of 20 students are shown in the stem and leaf diagram below.

11

Key: 3 | 0 means 303 0 2 2 4 7 94 5 7 8 9 9 95 2 4 6 6 96 1 4 6

Draw a box-and-whiskers plot below to represent the distribution. [2]

4030 50 60 70


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20 The first 5 terms of a sequence are 3, 7, 11, 15, 19.

(a) Write down the next 2 terms of the sequence.

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Answer: (a) and ... [1]

(b) Write down an expression for thenth term of the sequence.

Answer: (b) ... [1]

(c) The first 5 terms of another sequence are22222 5

23,

4

19,

3

15,

2

11,

1

7.

Write down an expression for thenth term of the sequence.

Answer: (c) ... [2]_____________________________________________________________________________

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Forxaminers

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U21 The diagram below shows Car A travelling at a uniform speed of 60 m/s. Car B

accelerated uniformly from rest to reach a speed of 80 m/s in 10 s and maintained thisspeed untilT s.

13

80

60

Speed(m/s)

Time (s)10 T

Car A

Car B

0

Find(a) the acceleration of Car B from 0 to 10s,

Answer: (a) m/s2 [1]

(b) the time when both cars were travelling at the same speed,

Answer: (b) s [2]

(c) the distance travelled by Car B ints, given that t>10,

Answer: (c) m [2]


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Forxaminers

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U(d) the time when Car B overtook Car A.

Answer: (d) s [2]

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22 The diagram below shows the positions of 3 ships,A, B andC. C is due south ofA andB andC are equidistant fromA. The bearing of B from A is 040.

14

Find the bearing of(a) B fromC,

A

C

BN

Answer: (a) . [2]

(b) C fromB.

Answer: (c) . [2]

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23 (a) Draw a fully labelled triangleABC such thatAB =12 cm, BC=13 cm and. The line AB has been drawn for you. [2]= 60CAB

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U(b) Measure and state the length ofAC, in cm.(c) Construct the angle bisector ofABC. [1](d) Construct the perpendicular bisector of the lineAC. [1](e) The angle bisector ofABC meets the perpendicular bisector of the lineAC at the

pointP. Measure and state the length ofAP, in cm.

Answer (a), (c), (d)

A B

Answer: (b) AC =..cm [1]

(b) AP =..cm [1]

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~ End of Paper ~

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Examinations Paper 1 AnswersForxaminers

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FExam

UQuestion No. Solutions Marks Remarks

1. (a) )1067.1()1068.6( 52

=4000

= 3104

M1

A1 or B2

(b) 0.085 B1

2. Price at which Paul bought book

= 310250

100

=$124

M1

A1

3. (a)

)(

)(

22 ba

bc

ba

acab

+

=)(

)(

))((

)(

cb

ba

baba

cba

+

+

=ab

a

M0.5

M0.5

A1

Factorisation

For (b-c)

Accept

ba

a

(b)

qp

r

p

pqr32

2 3

)3(

)(

=qp

r

p

rqp32

222 3

9

=3

3

3p

qr

M1

A1

Expansion

4. Time taken by 1 man =6 12 =72h

Time taken by 16 men =72 16 =4.5h

OR

Time taken by 16 men =

5.4

16

126

=

M1

A1

M1

A1

16


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Question No. Solutions Marks Remarks

5.20

75

4x

7

214

254

7

x

x

Greatest integerx=14

M1

A1

6 (a) LCM of 60, 72, 84 =2520s =42 mins

Time when train next meet =11 42hrs

M1

A1

0.5m for

not

converting to

mins

7 (a) P(both dice show same no.)

=636

=16

B1

No marks ifnot in lowest

form

(b) Possible outcomes of sum10:

( 5, 5), (6, 4), (4, 6), (5, 6), (6, 5), (6, 6)

P(sum of numbers less than 10)

=36

61

=6

5

M1

A1

Or B2

No marks if

not in lowest

form

8 (a) 2 B1

(b) 9, 15 B2

(1m each)

0.5m for

each wrong

answer

9 (a)Length of road = =

1001000200003.15

3.06 kmA1

(b) 1cm: 200m

1cm2 : 40000m2 A1

(c)Area of polyclinic =

40000

60000=1.5 cm2

A1


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10 (a) 5 B1

(b) Largest possible x

=4 +3 +1 +3

=11

M1

A111 (a) 36244 2 ++ yy

= or)94 + )3)(124(6( 2 + yy + +yy

2)3+y

=4(

M1

A1

(b) Length of each side =2(y+3) A1

12 (a)

10

6sin =POQ

643.0=POQ

rad

M1

A1

(b) Shaded area =

218.2 cm=

22 sin)10(2

1)64350.0()10(

2

1POQ=

Or

2cm

2

18.2

)10)(6(2

1)64350.0()10(

2

1

=

=

M2

A1

M1 for each

segment

13 (a)m=

5

)180(3

=108

M1

A1

(b) CBF =180 108 =72

n=2

)272(180 or 108-72 (ext of triangle)

=36

M0.5**

M0.5

A114 (a)

5

4or 0.8

A1 No marks if

fraction not

in lowest

terms


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(b) BC = 212835 22 =

5

3

35

21cos ==ACD or 0.6

M1

A1

No marks if

fraction not

in lowest

terms15 (a)

PR = 81

32=

82

P (2, 0)

M1

A1

(b)a=Gradient of PQ = 4

)2(0=

08

b =8

A1

B1

16 (a)

662 = uv

52872

46253

=+

=+

vu

vu

B1 0.5m each

for any 2

equations

(b) Elimination or Substitution Method

60=v

54=u

M1

A1

A1

17 (a)

3=2

)1(7 +

=x B1

(b) )1)(7( += xxy

)1)(7( += xxy

Or

Or

762 ++= xxy

B1

(c)16=

)13)(73( +=y M1**A1

Also accept

completing

the square

method


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18 (a)

W=f

k

k

W

kf

100000100000

200050

=

=

=

Orf

W or100000

= 100000=fW

M1 (k)A1

(b)80

1250

100000==f kHz

A1

19 M0.5 for

correct range

(30, 66)

M0.5 for

median (49)

M0.5 each

for upper and

lower

quartile(38, 55)

20 (a) 23 and 27 B1 0.5 m each

(b) Tn =4n 1 B1

(c)Tn = 2n

34n+

B2 1m for

numerator

and 1m for

denominator

21 (a) 8 m/s2 A1

(b)

sx 5.7=

x 608 =

M1

A1

4030 50 60 704938 56 66


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(c) Distance travelled

40080 = t

)80)(10(80102

1+= t

M1

A1

(d)

st

t

tt

20

40020

6040080

=

=

=

M1

A1

22 (a) (i)

Bearing of B from C =

= 0202

40(ext angle of tri)

M1

A1

-0.5m for not

giving

reason / not

leaving

bearings in 3

digits

(ii) Bearing of C from B

=360 20 (180 40)

=200

M1

A1

23 (a) Arc forBC =13cm

Marking for = 60CAB

Joining all pointsLabelling of triangle

M0.5

M0.5

M0.5M0.5

(b) AC =13.9 0.2 cm B1

(c) 3 arc markings M1

(d) 4 arc markings M1

(e) AP =8.7 0.2 cm B1


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Other School 1 EM P1