l. i r+ · 2009. 10. 28. · peperiksaan percubaan stpm 2009 matematik t kertas 2 masa: 3jam...

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l. 2. 3. 4. 5. 6. 7 ! I . (a) .If X c Y and Z'c Y' . show that Xc Z (b) Find the solution set of the inequality - x 2 + 6 > I x I r+ a x::; 0 Given I(x) = bx 2 +c o <x<2 2x+2 Determine the value of a if lill! f(x) = - 2. Find the values of b and c x-+o if limf(x) and limf(x) exist. x-+2 x-+o Solve the equation z z* - 2 z + 2 z * = 5 - 4 i, where z and z * are complex numbers and its conjugate respectively. Given that 4x 2 -1 is a factor for the polynomial 4X4 + of + 7x 2 - 2x + b, filld the values of the constants a and b. Factorise the polynomial completely. ( 3 2 J (1 OJ 2 If M = , 1 = , prove that 2M -M = I. Hence fmd the -2 -1 ° 1 matrix ofM- 1 Show that M3 = 3M 21 and deduce the matrix M3 . 211 Show that can be expressed as - - -- . x(x+2) x x+2. By using the method of differences, show that n 3 9 3(2n+3) x(x+2) = 4" - 2( n+1)(n+2) Hence, find the sum of the series _3_ + _3_ + _3_ + ..... + 3 lx3 2x4 3x5 10x12 Prove that, if x is so small that the terms in x 3 and higher powers may be neglected, then -- = 1 - x + - x 2 H¥x -x 1 1 +x 2 By substituting x = 0.125, sho; that J7 2 83 128 2 [3] [3] [5] [5] [7] [6] [2] /. [4] / [2] [3] [2] '/ papercollection

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  • l.

    2.

    3.

    4.

    5.

    6.

    7

    !

    I .

    (a) .If X c Y and Z'c Y' . show that Xc Z

    (b) Find the solution set of the inequality - x 2 + 6 > I x I

    r+ a x::; 0 Given I(x) = bx2 +c o

  • 8. Find the equation of the tangents to the circle x 2 + y2 == 2

    (a) -at (-1, -1)

    (b) that passes through the point (3, 1)

    9. (alif x = t -! and y = 2t + !, where t is a non-zero parameter, ~' t t

    10.

    dy 3 prove that - = 2 - -2-- .

    dx t + 1

    Deduce that -1 < dy < 2 dx

    d 2 d M.Given y == e- 2x cos3x , prove that -f+ 4..z + 13y = 0

    /- dx dx

    (a) Show that f 3;'+4 dx = In 4J7 2X2 - 3x -2 3

    [3]

    [8]

    [4]

    [3]

    [5]

    [6]

    (b) By using the substitution t = tan 0 or otherwise, fmd the value of [8]

    tr

    _ 6f 1 dO 2cos2 0-1 -o

    11. Show that the equation 2 In x = ~ has only one real root that lies x

    between x = 2 and x = 3. [3]

    5 Use the Newton Raphson method to find the root of 2Inx = - [5]

    x correct to 3 decimal places by using 2.5 as the first approximation.

    3

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  • 12. Find the coordinates of the turning points of each of the curves [5]

    3 1 y=-x +3x+2 and y=- ----x 3 -3x-2

    Sketch the curves in separate diagrams and give the equations ofthe [6]

    asymptotes of the second curve.

    Find the set of values of k where the equation x 3 - 3x - 2 = - ~ has [2]

    (i) one real root

    (ii) three real and different roots

    Prepared by :

    Teh Siew Hong

    Checked by:

    h Guat Beng Head athematics Department Penang Chinese Girls' High Schn· 1\

    4

    Approved by : .

    '~ r.,VolreCilew

    Guru K.- SIins ella Matetnalik SMJK ~amtA P.PINANG

    shteh pcghs270809

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  • No. :

    Date:

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  • SMJK Perempuan China Pulau Pinang Peperiksaan Percubaan STPM 2009

    Matematik T Kertas 2 Masa: 3jam

    Instructions to candidates: Answer all questions All necessary working should be shown clearly. Non exact numerical answers may be given correct to three significant figures, or one decimal place in the case of angles, unless a different level of accuracy is specified in the question. A list of mathematical formulae is provided .

    / Prove that 8 58

    . cos 0 + cos 20 + cos 30 + cos 40 = 4 cos-cosO cos-· 22

    Hence find all the values of e between O{l and 180{l such that cos8 + cos2B+cos3B = -cos4B [6 marks]

    2. Find all the values of x, where 0 ° < x < 360°, that ;atisfy the equation cosec x + cot x = 3. Give your answers correct to the nearest 0.1 D.

    [6 marks]

    3. In the given diagram, AP and AQ are tangents. QBR is a straight line. Prove that RP is parallel to AQ.

    p [6 marks]

    1

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  • 4 ,/p, Q and R are three points such that PQ = 2 i + 3 j and QR = 4 i - j . Fimr~"~lePQR. [3 marks]

    (b) The force R = 6 i + 2 j is a resultant of force P which is parallel to i and force Q which is para]]el to i + j. Find the magnitudes ofP and Q.

    (3 marks]

    5. A §hlp. which was at a position vector of 12 i + 21 j nautical miles reJative to the position of a destroyer, was traveling with a velocity of 15 i + 6 j knots. The captain of the ship realized that the destroyer was pursuing it with a velocity of 24 i + u j knots. If the destroyer managed to intercept the ship, find the value of u and also find the actual velocity of the destroyer. Find the time taken by the destroyer in its successful pursuit of the ship.

    [7 marks]

    6. The rate of change in the temperature of water is given by the differential equation

    dx

    dt = - k( x - fJ) where x is the temperature of water at the time t

    ( measured in hours), f} the room temperature and k a positive constant. Boi!ing water at the temperature of 100°C is left to coo1 down in the room temperature of 30° C. It takes 1 hour for the boiling water to cool down to 80° C.

    7 Show that k= In - [ 4 marks]

    5 When the temperature of the water is 60°C, it is placed in-the freezer of

    temperature - 8 dc. Find the time taken for the water to form ice at 0 DC. [6 marks]

    7. (4 company has two factories X and Y. Factory X employs 30 workers ari{'fu~;ory y employs 40 workers. The mean-pay of the workers in X is RM 1250, with standard deviation ofRM 280. The mean pay of the workers in Y is RM 1190, with standard deviation ofRM 290. (i) Calculate the mean pay of all workers in the company, [2 marks] (ii) Calculate the standard deviation of the pay of all workers in the company

    [2 marks]

    2

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  • 7 (b) The table below shows the number of hours per week spent reading magazines by 18 students in a class.

    I 25 I 31 8 48

    50 24 47 32 34 I 42 48 40 34 40 40 44

    I 16 45 L--

    (i) ./ (ii)

    (iii) (iv)

    ~

    Construct a stemplot for the data above Find the median and interquartile range Calculate the mean and standard deviation Draw a boxplot to represent the data Comment on the skewness of the distribution

    [2 marks] [2 marks] [2 marks] [2 ;narks] [l mark]

    &(f( factory produces ball bearings using machines A, Band C. The percentages At-bearings produced using machines A, Band Care 20 %, 35 % and 45 %

    r

  • (a) Determine the value of a (b) Write down the cumulative distribution function of X (c) Find P ( jX - E(X)j < 1)

    (2 marks] [3 marks] [4 marks]

    11. (/Pive patients need to undergo an operation that has a success rate of 0.8. / . 'By assuming that each operation is independent of one another, find

    (i) the probability that none of the operations fail { 2 marks] (j) the probability that at least 4 patients will be saved [ 2 marks]

    (b) In a factory which produces an electronic component on a large scale, the production of each unit ofthe component is independent of one another and the probability of producing a unit that is defective is 0.0006. By using the Poisson approximation, find (i) the probability that from 500 units of the component produce.,d, none

    is defective, [3 marks] (ii) the probability that from 500 units of the components produced,

    ~ there are, at mest, only 3 defective units. [3 ma;~~

    12. The probability of getting a rotten apple in a supermarket isJl&5'. By usi g the Normal Distribution as an approximation, find the probability that out of 500 apples in the supermarket, (i) more than 80 apples are rotten, (ii) at most 50 apples are rotten

    [3 marks] [3 marKs]

    ',,-(b) The mass of an apple follows a normal distribution with me~n 50 g and .,,--standard deviation 5 g. Find the gn:lbahility that 4 apples havera-total mass of more than 180 g. . l •

    [6 marks]

    Prepared by: Lau YF Verified by:

    Approved by: f\t1 Tay Yoke Che\"

    Guru KIIIIIl Sains dan M ·temalik SMJK PEREMPUAN CHINA ,'.PINANG

    4

    C Beng He?d of hematics Department

    Penang Chinese Girls' High School

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    STPM Trials 2009 Math T Paper 1 (Penang Chinese Girls High School) wit ansIMG_0001IMG_0002IMG_0003IMG_0004IMG_0005IMG_0006IMG_0007

    STPM Trials 2009 Math T Paper 2 (Penang Chinese Girls High School) wit ansIMG_0004IMG_0005IMG_0006IMG_0007IMG_0001IMG_0002IMG_0003