Add Mth Paper 2 2008

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    SULIT 1 3472/2

    2

    12 jam Dua jam tiga puluh minit

    Kertas soalan ini mengandungi 17 halaman bercetak.

    1. Calon dikehendaki membaca maklumat di halaman belakang kertas soalan ini.

    2. Calon dikehendaki menceraikan halaman 17 dan ikat sebagai muka hadapan bersama-samadengan jawapan anda..

    SEKTOR SEKOLAH BERASRAMA PENUH

    KEMENTERIAN PELAJARAN MALAYSIA

    PEPERIKSAAN PERTENGAHAN TAHUN TINGKATAN LIMA 2008

    3472/2ADDITIONAL MATHEMATICS

    Kertas 2

    Mei

    JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

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    The following formulae may be helpful in answering the questions. The symbols given are

    the ones commonly used.

    ALGEBRA2 4

    2

    b b ac

    x a

    =

    2 am an = a m + n

    3 am an = a m - n

    4 (am)n = a mn

    5 logamn = log am + logan

    6 logan

    m= log am - logan

    7 log amn = n log am

    8 logab = a

    b

    c

    c

    log

    log

    9 Tn = a + (n 1)d

    10 Sn = ])1(2[2

    dnan

    +

    11 Tn = arn-1

    12 Sn =r

    ra

    r

    ra nn

    =

    1

    )1(

    1

    )1(, r 1

    13r

    aS

    =

    1

    , r

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    STATISTICS

    3

    1 Arc length,s = r

    2 Area of sector,A = 21

    2r

    3 sin 2A + cos 2A = 1

    4 sec2A = 1 + tan2A

    5 cosec2A = 1 + cot2A

    6 sin 2A = 2 sinA cosA

    7 cos 2A = cos2A sin2A= 2 cos2A 1= 1 2 sin2A

    8 tan 2A =A

    A2tan1

    tan2

    TRIGONOMETRY

    9 sin (AB) = sinA cosB cosA sinB

    10 cos (AB) = cosA cosBsinA sinB

    11 tan (AB) =BA

    BA

    tantan1

    tantan

    12C

    c

    B

    b

    A

    a

    sinsinsin==

    13 a2 = b2 + c2 2bc cosA

    14 Area of triangle = Cab sin2

    1

    1 x =N

    x

    2 x =

    f

    fx

    3 =N

    xx 2)( =

    2_2

    xN

    x

    4 =

    f

    xxf 2)(=

    22

    xf

    fx

    5 m = Cf

    FNL

    m

    + 2

    1

    6 10

    100Q

    IQ

    =

    7

    =

    w

    w

    i

    iiI

    I

    8)!(

    !

    rn

    nPrn

    =

    9!)!(

    !

    rrn

    nCr

    n

    =

    10 P(A B)=P(A)+P(B)P(A B)

    11 P(X=r) = rnrrn qpC , p + q = 112 Mean, = np

    13 npq=

    14 z =

    x

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    Section A

    [40 marks]

    Answerallquestions in this section .

    1 Solve the simultaneous equations 12

    1=+ yx andy2 10 = 2x . [5 marks]

    2 A curve has a gradient function kx2 4, where kis a constant. The tangent to the curve at

    the point ( 1 , 3 ) is parallel to the straight liney + x 5 = 0.

    Find

    (a) the value of k, [3 marks]

    (b) the equation of the curve. [3 marks]

    3 A piece of rope with a length ofPcm is cut into 40 parts such that the length of each part

    form an arithmetic progression. Given that the length of the longest part is 324 cm and the

    difference between two consecutive parts is 8 cm.

    Find(a) the length of the shortest part, [2 marks]

    (b) the value ofP, [2 marks]

    (c) the part that has a length of 172. [2 marks]

    4 (a) Prove that cos sin sin2 = coscos 2. [2 marks]

    (b) Sketch the graph ofy = 2 cos 2x for x0 . Hence, using the same axes,

    draw a suitable straight line and state the number of solutions to the equation

    xx =+ 2cos2 for x0 .

    4

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    [6 marks]

    5

    y

    Diagram 1 shows a straight lineLRM. Given thatRM= 2LR.

    (a) Find the value ofh. [2 marks]

    (b) Given thatRPis perpendicular toLM. Find the equation of the straight line RP.[3 marks]

    (c) Given thatPRMQ is a parallelogram, find the area ofPRMQ.

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    Diagram 1

    x

    0

    L(-6, 12)

    M(12, 0)

    R(0, h)

    P

    [3 marks]

    5

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    6 The weights of 60 students to the nearest kg are represented in the histogram with

    given midpoints as shown in Diagram 2.

    (a) Calculate the mean and the standard deviation of the weights. [4 marks]

    (b) Subsequently, it was discovered that the reading of the weighing machine was 0.5 kg

    less than the true weight. What is the correct mean and standard deviation?

    [3 marks]

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    Weights (kg)

    Number of students

    O 30 40 50 60 70

    8

    1011

    1516

    Diagram 2

    6

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    Section B

    [40 marks]Answerfour questions from this section.

    7 Use graph paper to answer this question.

    Table 1 shows the values of two variables, x andy, obtained from an experiment.

    Variablesx andy are related by the equation ( )BxA

    y +=2

    , whereA and B are

    constants.

    x 0 10 0 25 0 50 0 75 1 00 1 25

    y 9 04 15 35

    29 54

    48 34

    71.74 99 75

    Table 1

    (a) Plot y againstx, using a scale of 2 cm to 0 2 unit on thex-axis and

    2 cm to 1 units on the y -axis. Hence, draw the line of best fit.[4 marks]

    (b) Use your graph from (a) to find the value of

    (i) A

    (ii) B [6 marks]

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    8 Diagram 3 shows a quadrilateral PQRS. The point Tlies onPQ.

    Diagram 3

    It is given that vuPQ 27 += , vuSR 42 = , vuPS 87 += , PQhPT = andSRkST = .

    (a) Find RQ in terms ofu and v. [2marks]

    (b) Express PT in terms of

    (i) h , u and v ,

    (ii) k, u and v . [3 marks]

    (c) Hence, find the values ofh and k. [4 marks]

    (d) Express TS in terms ofu and v. [1 mark]

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    P

    Q

    R

    S

    T

    8

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    9 Diagram 4 shows a sectorOPTR, centre O with radius 11 cm.PQR is a semicircle,

    centre Swith diameter 20 cm. OSTis a straight line. (Use 3.142 = )

    Diagram 4

    Calculate

    (a) POR, in radian, [2 marks]

    (b) the perimeter, in cm, of the shaded region, [3 marks]

    (c) the area, in cm2

    , of the triangle OPR, [2 marks]

    (d) the area, in cm2, of the shaded region. [3 marks]

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    O

    P

    Q

    R

    T

    S

    9

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    10 Diagram 5 shows the arrangement of the first three of an infinite series of similar

    rectangles. The length and width of the first rectangle arex cm andy cm respectively.

    The measurements of the length and the width of each subsequent rectangle are half ofthe measurement of its previous one.

    (a) Show that the areas of the rectangles form a geometric progression and state

    the common ratio.

    [3 marks]

    (b) Given thatx = 160 cm and y = 80 cm.

    (i) Determine which rectangle has an area of 28

    13 cm .

    (ii) Find the sum of the areas of 4th to 10th rectangles.

    (iii) Find the sum to infinity of the areas, in cm2, of the rectangles.

    [7 marks]

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    x cm

    y cm

    Diagram 5

    10

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    11 Diagram 6 shows the straight liney = 7 x intersecting the curvey =x24x +7 at the

    pointsA and B.

    Diagram 6

    (a) Find the area of shaded region. [3 marks]

    (b) By using differentiation, find the minimum point of the curve. [3 marks]

    (c) The region bounded by the curve, thex andy axes and the axis of symmetryof the curve is revolved through 360about thexaxis. Find the volume

    generated, in terms of . [4 marks]

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    y

    0x

    y =x2 4x +7

    y = 7x

    A(0,7)

    B(3,4)

    11

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    Section C

    [20 marks]Answertwo questions from this section.

    12

    Diagram 7 shows ABCwith BAC= 34, AB = 7 cm, AC= 9 cm andBC= tcm.

    (a) Find the value of t. [2 marks]

    (b) Calculate the value of BCA. [3 marks]

    (c) Find the area of ABC. [2 marks]

    (d) IfACis extended to D such that the area of ABD is twice the area of ABC,

    find the length ofBD. [ 3

    marks]

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    Diagram 7

    7 cm

    9 cm

    t cm

    B

    C

    34A

    12

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    Diagram 8 shows a tetrahedron VPQR. The vertex Vis vertically aboveR.

    Given that PQR is obtuse, VR = 5 cm,PR = 12 cm , QR = 10 cm and QPR = 45.

    Find,

    (a) PQR, [3 marks]

    (b) the length ofPQ, [2 marks]

    (c) PVQ, [3 marks]

    (d) the area of trianglePVQ. [2 marks]

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    V

    10 cm

    5 cm

    R12 cm

    P45

    Q

    Diagram 8

    13

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    14 A particular type of cake is made by using four ingredientsK, L, Mand N.

    Table 2 shows the price indices for the four ingredients and the percentages of usage of

    the ingredients in making the cake.

    Ingredients Price index for the year 2003 basedon the year 2001

    Percentage of usage (%)

    K 115 30L x 20M 130 10N 110 40

    Table 2

    (a) Calculate

    (i) the price per kg ofMin the year 2001 if its price per kg in the year 2003

    was RM 3.25,

    (ii) the price index ofKin the year 2003 based on the year 1999 if its price

    index in the year 2001 based on the year 1999 was 109.

    [5 marks]

    (b) The composite index number for the cost of making the cake in the year 2003 based

    on the year 2001 was 112.5.

    Calculate

    (i) the value ofx,

    (ii) the cost of making a cake in the year 2001 if the corresponding cost

    in the year 2003 was RM 6.30.

    [5 marks]

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    15 Table 3 shows the price indices of five components A,B, C,D andEthat are needed to

    make an electronic equipment for the year 2003 based on the year 2001. The changes in theprice indices of the components from the year 2003 to the year 2004 and their weightages are

    shown.

    Components Price index in 2003based on 2001

    Change in price indexfrom 2003 to 2004

    Weightage

    A 140 Increase 10% 3B 130 No change 2C 125 No change 6

    D 110 Increase 20% 5E x Decrease 5% 4

    Table 3

    (a) If the price of componentB was RM8 in the year 2001, calculate its price in 2003.

    [2 marks]

    (b) Given that the prices of component E in the year 2001 and 2003 were RM5.50

    and RM6.60 respectively, find the value ofx. [2 marks]

    (c) Calculate the composite index for the cost of making the electronic equipment in

    the year 2004 based on the year 2001. [3 marks]

    (d) (i) If the cost of making the electronic equipment in the year 2001 is RM 2986,

    calculate the cost of the electronic equipment in the year 2004.

    (ii) The cost of making the electronic equipment is expected to increase by 30%

    from the year 2004 to the year 2009. Find the expected composite index for theyear 2009 based on the year 2001.

    [3 marks]

    END OF QUESTION PAPER

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    Nama:..

    Kelas:...........................................................

    Arahan Kepada Calon

    1 Tulis nama dan kelas anda pada ruang yang disediakan.2 Tandakan ( ) untuk soalan yang dijawab.3 Ceraikan helaian ini dan ikat sebagai muka hadapan bersama-sama dengan buku

    jawapan.

    Bahagian Soalan SoalanDijawab

    Markah Penuh Markah Diperolehi(Untuk Kegunaan Pemeriksa)

    A 1 5

    2 63 64 85 86 7

    B 7 108 109 1010 1011 10

    C 12 1013 1014 1015 10

    Jumlah

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    INFORMATION FOR CANDIDATES

    MAKLUMAT UNTUK CALON

    1. This question paper consists of three sections: Section A, Section B and Section C.Kertas soalan ini mengandungi tiga bahagian:Bahagian A, BahagianB dan

    Bahagian C.

    2. Answer all questions in Section A, four questions from Section B and two questionsfrom Section C.

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    Jawab semuasoalan dalamBahagian A, empatsoalan daripadaBahagian B dan duasoalan daripadaBahagian C.

    3. Show your working. It may help you to get marks.

    Tunjukkan langkahlangkah penting dalam kerja mengira anda. Ini boleh membantuanda untuk mendapatkan markah .

    4. The diagrams in the questions provided are not drawn to scale unless stated. Rajah yang mengiringi soalan tidak dilukis mengikut skala kecuali dinyatakan.

    5. The marks allocated for each question and sub-part of a question are shown in brackets.Markah yang diperuntukkan bagi setiap soalan dan ceraian soalan ditunjukkan dalamkurungan.

    6. A list of formulae is provided on pages 2 to 3.Satu senarai rumus disediakan di halaman 2 hingga .

    7. Graph papers are provided.Kertas graf disediakan.

    8. You may use a non programmable scientific calculator. Anda dibenarkan menggunakan kalkulator saintifik yang tidak boleh diprogram.

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