SPM Mid Year 2008 SBP Maths Paper 2 Question

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    SEKTOR SEKOLAH BERASRAMA PENUH

    BAHAGIAN SEKOLAH

    KEMENTERIAN PELAJARAN MALAYSIA

    PEPERIKSAAN PERTENGAHAN TAHUN

    TINGKATAN LIMA 2008

    MATEMATIK

    Kertas 2

    Dua jam tiga puluh minit

    Pemeriksa

    Bahagian SoalanMarkah

    Penuh

    Markah

    Diperoleh

    A

    1 3

    2 4

    JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU

    1. Kertas soalan ini mengandungi dua bahagian :

    Bahagian A dan Bahagian B. Jawab semua

    soalan daripada Bahagian A dan empat soalan

    dalam Bahagian B.

    2. Jawapan hendaklah ditulis dengan jelas dalam

    ruang yang disediakan dalam kertas soalan.

    Tunjukkan langkah-langkah penting. Ini boleh

    membantu anda untuk mendapatkan markah.

    3. Rajah yang mengiringi soalan tidak dilukis

    mengikut skala kecuali dinyatakan.

    4. Satu senarai rumus disediakan di halaman

    2 & 3.

    5. Anda dibenarkan menggunakan kalkulator

    saintifik yang tidak boleh diprogram.

    3 4

    4 4

    5 5

    6 6

    7 6

    8 5

    9 6

    10 5

    11 4

    B

    12 12

    13 12

    14 12

    15 12

    16 12

    Jumlah

    Kertas soalan ini mengandungi 23 halaman bercetak dan 1 halaman tidak bercetak

    SULIT

    1449/2

    Matematik

    Kertas 2

    Mei 2008

    212 jam

    1449/2

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    NAMA :

    TINGKATAN :

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    SHAPES AND SPACE

    1 Area of trapezium =2

    1 sum of parallel sides height

    2 Circumference of circle = d= 2r

    3 Area of circle = r2

    4 Curved surface area of cylinder = 2rh

    5 Surface area of sphere = 4r2

    6 Volume of right prism = cross sectional area length

    7 Volume of cylinder = r2

    h

    8 Volume of cone =3

    1r

    2h

    9 Volume of sphere =3

    4r3

    10 Volume of right pyramid =3

    1 base area height

    11 Sum of interior angles of a polygon = ( n 2) 180

    12arc length angle subtended at centre

    circumference of circle 360

    13area of sector angle subtended at centre

    area of circle 360

    14 Scale factor , k=PA

    PA'

    15 Area of image = k2

    area of object

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    Use

    Section A

    [52 marks]

    Answerall questions in this section.

    1 The Venn diagram in the answer space shows sets P, Q and R such that the universal set,RQP .

    On the diagram in the answer space, shade

    (a) the set QP .(b) the set ( P Q ) R .

    [3 marks]

    Answer:

    (a) (b)

    2 Calculate the value of v and ofw that satisfy the following simultaneous linear equations:

    112 wv

    733

    2 wv

    [4 marks]

    Answer :

    PQ

    R

    PQ

    R

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    Only3 Using factorisation, solve the following quadratic equation:

    2

    213

    x

    xx

    [4 marks]

    Answer :

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    Use 4 Diagram 1 shows a right prism with a horizontal rectangular base PQRS. VUQR is a

    trapezium.

    Identify and calculate the angle between the plane QWR and the base PQRS.

    [4 marks]

    Answer :

    T

    U

    W

    V

    P

    Q

    R

    S

    8 cm

    6 cm

    12 cm

    5 cm

    DIAGRAM 1

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    Only5 In Diagram 2, straight line TU is parallel to straight line VWand O is the origin .

    The equation of straight line VW is 02552 yx .

    Find:

    (a) the equation of the straight line TU.

    (b) the x-intercept of the straight line TU.

    [5 marks]

    Answer:

    (a)

    (b)

    DIAGRAM 2

    y

    x

    W

    U(1,5)

    V

    T

    O

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    Use

    6 Diagram 3 shows two sectors of circle ORQ and OPS with centre O.

    By using =7

    22, calculate

    (a) the perimeter, in cm, of the whole diagram.

    (b) the area, in cm2, of the shaded region.

    [ 6 marks ]

    Answer :

    (a)

    (b)

    OP

    Q

    R

    S

    1507 cm

    12 cm

    DIAGRAM 3

    7 cm

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    Only7 (a) State whether the following statements are true or false.

    (i) 2.02

    1and

    3 aaaa

    (ii) Number 5 is an odd number or number 2 is a prime number.

    (b) State the antecedent and consequent of the implication:

    (c) Complete the following argument.

    Premise 1 : If y > 3, then 5y > 15.

    Premise 2 : _________________________________________

    Conclusion : 5y > 15.

    [6 marks]

    Answer :

    (a) (i)

    (ii)

    (b) Antecedent : ______________________________

    Consequent : ______________________________

    (c) Premise 2 : ________________________________________________

    If y is less than zero, then y is a negative number.

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    Use 8 (a) Complete the following statement using the quantifier all or some, to make it a falsestatement.

    (b) Write down two implications based on the following statement:

    (c) Make a general conclusion by induction for the sequence of numbers 2, 14, 34,

    which follows the following pattern.

    2 = 4(1)2 2

    14 = 4(2)2 2

    34 = 4(3)2 2

    62 = 4(4)2

    2

    = ..

    [ 5 marks]

    Answer:

    (a) ____________ polygons have six sides.

    (b) Implication 1:

    ____________________________________________________________

    Implication 2 :

    _____________________________________________________________

    (c) _____________________________________________________________

    ______________________________________________________________

    ____________ polygons have six sides.

    x > y if and only if x < y

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    Only9 (a) The inverse matrix of

    45

    23is p

    35

    4 q

    Find the value of p and of q.

    (b) Hence, using matrices, calculate the value ofx and ofy that satisfy the following

    simultaneous linear equations :

    3x 2y = 8

    5x 4y = 13

    [6 marks]

    Answer :

    (a)

    (b)

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    Use

    10 Diagram 4 shows the speed-time graph of a particle for a period of 50 seconds.

    (a) Calculate the rate of change of speed, in ms-2

    , in the first 10 seconds.

    (b) Calculate the value of u, if the total distance traveled in the last 40 seconds is 750 m.

    [5 marks]

    Answer :

    (a)

    (b)

    Speed (ms-1

    )

    Time (s)0

    5

    10 30

    20

    50

    DIAGRAM 4

    u

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    Only11 Diagram 5 shows a solid formed by joining a prism to a right pyramid. Trapezium

    ABGFis the uniform cross-section of the prism.

    FG = GH= 7 cm. The height of the pyramid is 6 cm.

    Calculate the volume, in cm3

    , of the solid.

    [ Use = 722 ]

    [4 marks]

    Answer:

    DIAGRAM 5

    8 cm

    11 cmA B

    CD

    E

    F G

    H

    V

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    Use Section B

    [48 marks]

    Answer four questions from this section.

    12 (a) Complete Table 1 in the answer space for the equation y = 2x2

    + 4x + 16.

    x 4 3 2 1 0 1 2 3 4

    y 32 h 0 10 16 18 k 10 0

    Calculate the values of h and of k.

    [2 marks]

    (b) For this part of the question, use the graph paper on pag 15 .

    You may us flexible curve ruler.

    By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis ,

    draw the graph of y = 2x2

    + 4x + 16 for 4 x 4.[4 marks]

    (c) From your graph, find

    (i) the value ofy when x = 2.2

    (ii) the value of x when y = 5 [3 marks]

    (d) Draw a suitable straight line on your graph to find all the values of x which satisfy the

    equation 2x2

    + x 11 = 0 for 4 x 4.

    State these values ofx.

    [3 marks]

    Answer:

    (a) h =

    k=

    (b) Refer graph on page 15

    (c) (i) y =

    (ii) x = .

    (d) x =

    TABLE 1

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    Graph for Question 12

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    Use13 (a) Transformation T is a translation

    3

    2and transformation P is an anticlockwise

    rotation of 90o

    about the centre (5, 0).

    State the coordinates of the image of point (9, 4) under each of the following

    transformations:

    (i) P,

    (ii) TP,

    (iii) T2

    [4 marks]

    (b) Diagram 6 shows three quadrilaterals A, B and C, drawn on a Cartesian plane.

    (i) Cis the image ofA under the combined transformations WV.

    Descibe in full,:

    (a) the transformation V

    (b) the transformation W

    [5 marks]

    (ii) It is given that B represents a region of area 180 cm2.

    Calculate the area, in cm2, of the region represented by A.

    [3 marks]

    0 2 4 6 8 10 12x

    2

    4

    8

    y

    C A

    DIAGRAM 6

    B

    6

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    OnlyAnswer:

    (a) (i)

    (ii)

    (iii)

    (b) (i) (a)

    (b)

    (ii)

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    Use

    14 The data in Table 2 shows the mass, in kg, of a group of 32 members of Science and

    Mathematics societies.

    44 64 59 49 35 40 50 51

    60 50 40 43 47 45 57 5354 49 58 54 48 47 53 45

    51 52 62 59 56 46 51 56

    (a) Based on the data in Table 2 and by using a class interval of 5 kg,

    complete the following table.

    [4 marks]

    Answer:

    (a)

    Mass (kg) Frequency Midpoint

    35 39

    (b) Calculate the estimated mean of the mass of the members.

    [3 marks]

    (c) Using a scale of 2 cm to 5 kg on the x-axis and 2 cm to 1 member on the y-axis, draw

    a frequency polygon for the above data.

    [5 marks]

    Answer:

    (b)

    (c) Refer graph on page 19

    TABLE 2

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    Graph for Question 14

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    Use15 (a) Complete Table 3 in the answer space for the equation y =

    x

    2by writing down the

    values ofy when x = 0.5 and x = 2. [2 marks]

    (b) For this part of the question, use the graph paper provided on page 21.

    You may use a flexible curve rule.

    By using a scale of 2 cm to 1 unit on both the axes, draw the graph ofy = x

    2for

    4 x 4. [5 marks]

    (c) From your graph, find

    (i) the value ofx when y = 1.2 ,

    (ii) the value ofy when x = 2.3. [2 marks]

    (d) Draw a suitable straight line on your graph to find values of x which satisfies the

    equation x + 1 =x

    2for 4 x 4.

    State these values ofx.

    [3 marks]

    Answer:

    (a)

    x 4 3 2 1 0.5 0.5 1 2 3 4

    y 0.5 0.67 1 2 4 2 0.67 0.5

    TABLE 3

    (b) Refer graph on page 21

    (c) (i) x = .

    (ii) y = .

    (d) x = ............................................................................

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    Graph for Question 15

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    Use16 (a) Transformation U is a translation

    2

    1.

    Transformation V is a reflection in the straight line y = 1.

    State the coordinates of the image of point (3, -2) under each of the following

    transformations :

    (i) U,

    (ii) UV.

    [3 marks]

    (b) Diagram 7 shows three quadrilaterals ABCD, HGFE and KJIE on a

    Cartesian plane.

    J

    K

    I

    H

    G

    B

    DC

    E

    F

    A

    - 2

    - 4

    6

    - 8

    4

    - 10

    2

    - 6 - 4 - 2 6420

    DIAGRAM 7

    HGFEis an image of ABCD under transformation W and KJIE is an image of

    HGFE under transformation Z.

    (i) Describe in full,

    (a) the transformation W,

    (b) the transformation Z.

    [6 marks]

    (ii) Given that quadrilateral HGFE represents a region of area 26 cm2,

    calculate the area, in cm2, of the region represented by the shaded region.

    [3 marks]

    x

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