2009 ESpring Prelim Sec 4Ex EM P1

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  • 8/8/2019 2009 ESpring Prelim Sec 4Ex EM P1

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    J

    Name : _____________________________________________________ ( )

    Class : Sec 4/5____

    Mathematics

    Paper 1

    Friday 21th

    August 2009

    Additional materials:nil

    READ THESE INSTRUCTIONS FIRST

    Write your Name and Index Number 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.

    Answerall 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.You are expected to use an electronic calculator to evaluate explicit numerical

    expressions. If the degree of accuracy is not specified in the question, and if the answer is

    not, give the exact answer to three significant figures. Give answers in degrees to one

    decimal place.

    ForT, use either your calculator value or 3 142, unless the question requires the answer

    in term ofT.

    At the end of the examination, fasten all your work securely together.

    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.

    2 hour10:05 am 12:05 pm

    This question paper consists of17 printed pages including the cover page.

    4016/1

    East Spring Secondary SchoolTowards Excellence and Success

    Prelim Examination 2009Secondary 4 Express

    MARKS:80

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    Mathematical Formulae

    Compound interest

    Total amount = 1100

    nr

    P

    Mensuration

    Curved surface area of a cone = rlT

    Surface area of a sphere = 24 rT

    Volume of a cone =

    21

    3 r hT

    Volume of a sphere = 34

    3rT

    Area of triangle ABC=1

    sin2

    ab C

    Arc length = rU , where U is in radians.

    Sector area = 21

    2

    rU , where U is in radians.

    Trigonometry

    sin sin sin

    a b c

    A B C! !

    2 2 2 2 cosa b c bc A!

    Statistics

    Mean =fx

    f

    Standard deviation =

    22fx fx

    f f

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    AnswerALL questions

    1. Evaluate the following, expressing each answer in standard form.

    (a) 0.5 2m in 2km (b) 325 m/s in km/h

    Ans: (a) ___________ 2km [1]

    (b) __________km/h [1]

    2. A train leaves a station at 21 35and reaches its destination at 03 27.(a) Find the duration of the train journey in hour s and minutes.(b) Find the average speed of the train if it travels 322 km

    Ans: (a)____h ______min [1]

    (b) ___________km/h [1]

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    3. Solve

    (a) 65

    5.2 27!x

    (b) 152 1 !x

    Ans: (a) ______________ [1]

    (b) ______________ [1]

    4. Given thatA:B =3

    1:

    2

    1and B:C= 3:1 .

    Find the ratio ofA to Cin the formx:ywherexand yare integers.

    Ans:A:C= ___________[2]

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    5. Solve the following equations.

    (a) 16)32( 2 !x

    (b) 5)31(

    1

    )13(2

    1!

    yy

    Ans: (a) ____________[2]

    (b) ____________[2]

    6. June wants to buy a type of chocolates from a DUTY FREE shop before shedeparts from Changi Airport for a school trip. The chocolates are sold at 55

    cents a piece and packed into gift boxes that are sold at 80 cents each. Eachgift box can pack a maximum of 10 chocolates. If June has $35, find thenumber of pieces of the chocolates that she can buy at the most. .

    Ans: _____________ [3]

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    7. Express each of the following as a single matrix.

    (a)

    1

    22

    2

    3

    (b)

    21

    2323

    Ans: (a) _______________ [1]

    (b)_______________ [1]

    8. A straight line, P,passes through the point (2, 10). The gradient ofP is 2.

    (a)Write down the equation of the line P.

    (b) Given that Palso passes through the point (k, k+3), find the value of k.

    Ans: (a) ______________ [2]

    (b) ______________ [1]

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    9. A = _ a17,13,11,10,8,6,3,1

    B = _ a11,9,5,3,2

    Given that set C is a subset ofA, n(C) = 7 and CB = _ a3 .

    List the elements in set C

    Ans: C = _____________ a [2]

    10 Abby scores 65 marks and 79 marks for her first two tests. Find the lowestscore she needs to score in the third test if she wants to score an average of70 marks for the three tests.

    Ans: ___________marks [2]

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    11 Solve the following equation s simultaneously.0.3x+ 0.4y= 7

    . 1.5x 0.3y= 8

    Ans:x= ______, y= _______ [2]

    12 The following data shows the number of 10 -cent coins in the wallets of eightstudents.

    (a) The modal number of 10-cent coins is 11. Find the value ofx .

    Hence, using this value ofx ,(b) Find the median of the data.(c) Find the mean of the data.

    Ans: (a) ______________ [1]

    (b) ______________ [1]

    (c) ______________ [1]

    11, 12, 10, 9, 8, ,1x 11, 12

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    13 The numbers below shows a sequence.

    .

    (a)Write down the 50 th term in the sequence.(b)Write down an expression, in terms of n, for the nth term of the

    sequence

    Ans: (a) ______________ [1]

    (b) ______________ [1]

    14 (a) Factorise completely

    yxyx 15325 22

    (b) Express as a single fraction.

    2)62(2

    33

    y

    yy

    Ans: (a) ______________ [2 ]

    (b) ______________ [2]

    0, 3, 6, 9,

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    15 Given that yvaries inversely as 52 x and that2

    120!x when

    2

    1!y , find

    (a) yin terms ofx,

    (b) the value ofxwhen 5!y .

    Ans: (a) ______________ [2]

    (b) ______________ [1]

    16 The diagram shows the graph of ))(( xbxay ! .

    From the graph,

    find the value of a and b .

    Ans: a = ______, b = _______ [2]

    1 2

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    17 The diagram shows the speed - time graph of a cars journey. Calculate(a) the acceleration when t= 10,

    (b) the speed when t= 10,

    (c) the time taken by the car to travel 1.5 km.

    Ans: (a) __________ 2/sm [1]

    (b) ___________ sm / [1]

    (c) ____________sec [2]

    18 In the diagram, the straight line XYZ is parallel to PQ, and RY is parallel toQZ. r! 56XYR , r! 120QPR and r! 63YQZ . Calculate the value of

    (a) YZQ (b) YQP

    (c) YRP

    Ans: (a) _____________0 [1]

    (b) _____________0 [1]

    (c) _____________0 [1]

    240 40 60

    Time

    (tseconds)

    60

    Speed (metres per second)

    Y

    Z

    P

    Q

    R

    r56

    r63 r120

    X

    S

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    19 The diagram below shows 3 cones that are geometrically similar. The heightof coneA and cone B are 18 cm and 6 cm respectively.

    (a) If the base radius ofA is 6 cm, find the base radius of B.(b) If the base area ofA is 90 cm2, find the base area of B.

    (c) The volume ofCis

    64

    1the volume ofA. Find the height of C.

    Ans: (a) ____________cm [1]

    (b) ____________cm2[2]

    (c) ____________cm [2]

    20 A polygon has n sides. Three of its exterior angles are rrr 15and28,67 while.

    The remaining )3( n exterior angles are each equal to r50 . Find n.

    Ans: n= ____________[2]

    A B C

    18

    6

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    21 Using only ruler and compasses,(a) Draw and label clearly the triangle ABCwithAB = 7 cm,AC= 9 cm and

    BC= 8 cm. [2](b) Measure and write down ABC .(c) Construct and label the angle bisector of ABC . [1](d) Construct and label the perpendicular bisector ofAC. [1]

    .

    Ans: (b) ____________o [1]

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    22 The school has tasked the DNT department to design souvenirs for thevisitors from oversea during the YOUTH OLYMIPIC GAMES (YOG) event.The souvenir consists of a trapezoidal prism with a cylindrical hole of radius10 cm in the centre.AB = 35 cm, EF= 85 cm, FG = 28 cm, CF= DE= 84 cmand the height of the solid is 80 cm.

    Calculate the

    (ai) total volume of the souvenir,

    (ii) total surface area of the souvenir.

    (b) If the souvenir is melted and recast into a cylindrical solid of height 80 cm,find the radius of the solid.

    Ans: (a)(i)__________cm3 [2]

    (ii) __________ cm2 [2]

    (b) ___________cm [1]

    G

    F85 cm

    28 cm

    35 cm

    80 cm10 cm

    A

    D

    H

    E

    84 cm

    B

    C

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    23

    In the diagram above A, B, C and D are points on level ground. D is duewest ofC, CD = 11 m, BC= 14 m,AC= 15 m and r! 112ACD . It is giventhat the bearing of CfromA is 202o and the bearing ofA from B is 318o.

    Calculate(i) AD,(ii) the bearing ofA from C,(iii) AB

    ,(iv) shortest distance fromA to BC.

    Ans: (i) ______________m[2]

    (ii) ______________o[1]

    (iii) _____________o [2]

    (iv) ____________m [1]

    11 m C

    112o

    15 m

    14 m

    N

    A

    B

    D

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    24 The diagram shows a box-and-whisker plot for the following data set.

    (a) Find the value of 3x .

    (b) Find the interquartile range.(c) What does

    15 xx mean for the data set?

    Ans: (a) ____________ [1]

    (b) ____________[1]

    (c) ____________[1]

    25 The position vectors relative to an origin O, ofA and B are

    9

    14

    and3

    2

    respectively andAPB is a straight line.

    (a) Find AB .

    (b) Given further that ABAP4

    3! find the position vector ofP.

    Ans: (a) ______________ [1]

    (b) ______________[2]

    {12, 15, 18, 21, 22, 25, 29, 30, 31, 33, 38}

    1x 2x 3x 4x 5x

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    26 Chen Han and Dan Dan play a game of chinese chess everyday. For each

    game, the probability that Chen Han wins is8

    3and the probability that Dan Dan

    wins is5

    2.

    (a) Name the better player and explain why.(b) Calculate the probability that

    (i) Chen Han wins on both Monday and Tuesday(ii) Dan Dan wins on eitherWednesday or Thursday but not on both

    Wednesday and Thursday

    Ans:

    (a)____________________________________________________________ __ [1]

    (b)(i) ____________[1]

    (ii) ____________[2]

    ~End of Paper 1~

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    1a)

    1b)

    7105 v

    31017.1 v

    [B1]

    [B1]

    2a)

    2b)

    5 h 52 min

    54.9 km/h [ecf]

    [B1]

    [B1]

    3a)

    3b)

    3

    1

    [B1]

    [B1]

    4) A:B:C

    1

    1:

    3

    1:

    2

    1

    A:C

    2:11

    1:

    2

    1

    [M1]

    [A1]

    5a)

    5b)

    5.05.3

    07124

    169124

    2

    2

    !

    !

    !

    orx

    xx

    xx

    3.0

    930

    )13(101

    5)13(2

    21

    513

    1

    )13(2

    1

    !

    !

    !

    !

    !

    y

    y

    y

    y

    yy

    [M1]

    [A1]

    [M1]

    [A1]

    6)Cost of 10 chocolates with a box = 10(0.55) + 0.80 = $ 6.30

    Maxiumum number of boxes: $35 30.6$z = 5.555 [ecf]

    5} boxes

    = 50 or 54 chocolates

    [M1]

    [M1]

    [A1]

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    7a)

    7b)

    4

    7

    107

    [B1]

    [B1]

    8a)

    8b)

    6

    )2(210

    !

    !

    !

    c

    c

    cmxy

    62 !@ xy

    3

    6)(23

    !

    !

    k

    kk [ecf]

    [M1]

    [A1]

    [B1]

    9) 1,3,6,8,10,13,17 [B2 for all

    correct

    answer]

    [B1 for 1

    error]

    10)

    66

    3707965

    !

    v!

    x

    x

    [M1]

    [A1]

    11) X= 7.68

    Y= 11.7 [no ecf]

    [B2 for all

    correct

    answer]

    [B1 for 1

    error]

    12a)

    12b)

    12c)

    12

    11

    10.5

    [B1]

    [B1]

    [B1]

    13a)

    13b)

    147

    3n-3 or 3(n-1)

    [B1]

    [B1]

    14a) yxyxyx 5355

    355 yxyx

    [M1]

    [A1]

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    14b)2)3(4

    2

    3

    3

    y

    y

    y

    2)3(2

    718

    )3)(3(4

    1436

    )3)(3(42

    )3)(3(4)3)(4(3

    )3)(3(4

    2

    3

    3

    !

    !

    !

    !

    y

    y

    yy

    y

    yyy

    yyy

    yy

    y

    y

    [M1]

    [A1]

    15a)

    15b)

    35412

    1

    52

    !

    !

    !

    k

    k

    x

    ky

    52

    3

    !@

    xy

    3.4 [ecf]

    [M1]

    [A1]

    [B1]

    16) A= 1

    B= 2

    [B1]

    [B1]

    17a)

    b)

    c)

    2.5

    25 [ecf]

    37

    50242

    1500)242(30

    150060)24(2

    1

    !

    !

    !

    !

    t

    t

    t

    tt

    [B1]

    [B1]

    [M1]

    [A1]

    18a)

    b)

    c)

    56

    61 [ecf]

    116

    [B1]

    [B1]

    [B1]

    19a)

    19b)

    2

    10

    18

    6

    90

    2

    !

    !

    x

    x

    [B1]

    [M1]

    [A1]

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    19c)

    5.4

    1864

    13

    !

    !

    x

    x

    [M1]

    [A1]

    20) 00000 360503152867 !v n

    n=8

    [M1]

    [A1]

    21a)

    21b)

    21c)

    00 174 s

    [A1] ---

    Show construction line at ABC with accurate drawing

    [A1] ---

    Show construction line atAC with accurate drawing

    [A2] ---

    Show 2

    construction

    lines with

    accuratedrawing

    [A1] ---

    Show 2

    construction

    lines with

    accurate

    drawing

    [B1]

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    22a)

    (i)

    (ii)

    b)

    Vol of trap = 3400,134)28)(80(85352

    1cm!

    Total volume of souvenir =

    )28()10)(142.3(400,134 2 =125,602.4cm3 [ecf]

    Total surface area =

    2

    2

    12.795,18

    2820142.3

    )10142.34800(2)2885()2835()8428(2

    cm!

    vv

    vvvv

    22.4 [ecf]

    [M1]

    [A1]

    [M1]

    [A1]

    [B1]

    23)

    (i)

    (ii)

    (iii)

    (iv)

    Using cos rule,

    AD =)112cos(151121511 022 vvv

    = 21.7

    000 02290112 !

    000 644222 !!BA

    Using Sine Rule,

    0

    0

    4.74

    15

    sin

    14

    64

    !

    !

    AB

    AB

    Sin

    000 6.414.74.64180 !!A B [ecf]

    96.9

    156.41sin 0

    !

    !

    x

    x

    [B2]

    [B1]

    [M1]

    [A1]

    [B1]

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    24a)

    24b)

    24c)

    25

    13

    range

    [B1]

    [B1]

    [B1]

    25a)

    b)

    12

    12

    !

    !

    9

    9

    12

    12

    4

    3AP

    !

    !!

    6

    11

    9

    9

    3

    2APOAOP

    [B1]

    [M1]

    [A1]

    26a)

    (bi)

    (ii)

    (iii)

    Dan Dan. Her probability of winning is higher

    64

    9

    8

    3

    8

    3!v

    25

    12

    5

    2

    5

    3

    5

    3

    5

    2

    !

    vv

    40

    9

    5

    2

    8

    31 !

    [B1]

    [B1]

    [M1]

    [A1]

    [B1]