qs026-1 sem2 2010-2011

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    OS026/1MathematicsPaper 1Semester IISession 20JO/20J J2 hours

    OS026/1Matematik

    Kertas 1Semester II

    Sesi 2010/20112 jam

    PERPUSTAKAANKGI f J "'AfRIKULA,: HKNIKAI l'AHo1l"IGKf "'E NTtRIAN PENDIDII

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    QS026/1

    ARAHAN KEPADA CALON:

    Kertas ini mengandungi 10 soalan.Jawab semua soalan di dalam buku jawapan yang disediakan.Gunakan muka surat baru bagi nombGr soalan yang berbcza.Markah pcnuh yang diperuntukkan bagi setiap soalan atau bahagian soalan ditunjukkandalam kurungan pada penghujung soalan atau bahagian soalan.Semua langkah kerja hendaklah ditunjukkan dengan jelas.Kalkulator saintifik yang tidak boleh diprogramkan sahaja yang boleh digunakan.Jawapan berangka boleh diberi dalam bentuk Jr . e. surd, pecahan atau sehingga tiga angkabercrti. di mana-mana yang sesuai. kecuali jika dinyatakan dalam soalan.

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    QS026/1

    INSTRUCTIONS TO CANDIDATE:This question booklet consists of 10 questions.Answer all questions in the answer booklet provided.

    Usc a new page for each question.The full marks for eaeh question or section shown in the bracket at the end of the questionor section.All steps must be shown clearly.Only non-programmable scientiJic calculators can be used.Numerieal answers may be given in the form of Jr , e. surd, tlactions or up to threc significantfigures. where appropriale. unless stated otherwise in the question.

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    QS026/1

    Trigonometri

    Had

    Hipcrbolik

    SENARAI RUMUS MATEMATIK

    sin (A B) = sin A kos B kos A sin Bkos (A B) = kos A kos B =+= sin A sin Btan (A B) = tan A tan B_. 1=+= tan A tcm B

    . . B 2 . A+B k A -Bsm A + sm sm.--- os--2 2

    . . "k A+B . A -BsmA - sm B = L os - - - - sm-2 2A+B A -Bkos A + kos B = 2 k o s - - ~ kos-

    2 2. A+B . A -BkosA - kos B = - 2 sm - -- sm-2 2

    had sin h = 1h ~ O hhad 1- kos h = 0

    h ~ O h

    sinh (x + y) = sinh x kosh y + kosh x sinh ykosh (x + y) = kosh x kosh y + sinh x sinh ykosh2x "- sinh2 x = 11 - tanh2 x = sekll xkoth2 x-I = kosekh2 xsinh 2x = 2sinh x kosh xkosh 2x = kosh2 X + sinh2 x

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    QS026/1

    SENARAI RUMUS MATEMATIK

    Pcmbczaan dan Pcngamiran

    /(x) j'(x)kotx - kosck2 Xsek x 3ek x tan x

    - kosck x kot x

    koth x - kosckh2 Xsekh x - sckh x tanh x

    kosekh x - kosckh x koth x

    f /' (x) I I-iT0- dx f (x) +c:c 1nf II dl' = UV - f \' Ju

    4._- S = 47(1 '2/' = 7(1'Sfera 31 S=1T r sV - - II ,.2hon mcmbu!at tcgak 3

    SiJindcr mcmblliat h ~ g a k s = '2 7( rh

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    Q5026/1

    LIST OF MATHEMATICAL FORMULAE

    Differentiation and Integration

    f'(x)1cot x - cosec- xsec x sec x tan x

    cosec x - cosec x cot x

    coth x -- coscch2 Xscch x - sech x tanh x

    cosech x - cosech x coth x

    J ,h In I/(x) I +cJ == uv J lb - v du

    4 'V == _.- n r"Sphere 31 2 S= n r sV==-n rhRight Circular Cone 3

    Right circular cylinder S == '2 n rh

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    OS026/1

    d1'Jika x = sek f) clan ) ' = 1. tan f} . ca n dalam sebutan e.dx[5 markah]

    2 Dapatkan persamaan bulatan yang mdaJui titik (1,4). (2,2) dan (-1,3). Seterusnya,dapatkan jejari bulatan tersebut.

    [6 markah]

    3 Dibcri tiga vcktor g = 2i + [)i ! 4k. 12 = i - 3k dan s:. = 5i + 6i + 2k Cari nilai scdcmikian hingga

    (a) a scrcnj ang dengan b.[3 markah]

    (b) axb=c .[4 markahJ

    4 Buktikan bahawa kosh 2x - sinh 2 x = 1. Seterusnya, cari nilai tanh x jika. I -'SI n 1 x = --.4

    [7 markah]

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    - -

    QS026/1

    Jl)1 If x = sec e and )' = 2tan e. tInd in terms of e.dx[5 marks]

    2 Find an equat ion of the circle that passes through the points 0,4), (2.2) and (- L3).Hence, find the radius of the circle.

    [6 marks]

    3 Given three vectors = 2i + + 4k, 12 = i - 3k and = 5i + 6i + 2k. Find the valueof [3 such that

    (a) is perpendicular to b.[3 marks]

    (b) a x b = c.[4 marks]

    4 Prove that cosh 2 x - sinh 2 x = 1. I knee, find the value of tanh x if sinh x = 4[7 marks]

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    Q5026/1

    5 (a) Gunakan prinsip pcrtama rerbitan untuk menunjukkan bahavvad. k-(Sll1X)= 05X .(b:

    r5 l1Iarkah J

    d 2dv v(b) Dibcri .1/ = sin(J."). can da n - - ' , dalam scbutan x. Sererusnya. alaud, dr'd 2y dy;dcngan cara lain. tunjllkkan bahawa x-- - - - + 4x \' = 0dx 2 dx ' . .

    [5 markah]

    . 9,2 - 366 Dibcri f (x) =--,-.. x ..

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    Q5026/1

    5 (a) Use the first principle cd'derivative to show thatd .--(Sl l1x) = cos.\".dx

    [5 marks]

    2. 1 1-1nc1 ely and d y . ,(b) (liven y = SIl1CC ) . .1 111 terms at x. Hence, ordx dx 2

    d 2Y dy :;otherwise. show that x- - - -- + 4x y = 0. dx 2 dx .[5 marks]

    2. f" 9x - 366 (Jwen (x) = - ~ ~ -" . XC - 9 .

    (a) Detemline the vertical and horizontal asymptotes of f.13 marks]

    (b) Determine the interval of x on which f is increasing and f is decreasing.[5 marks]

    (c) Sketch the graph of f.[3 marksl

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    7 Diberi titik A(L 3,1), B(4, -1. 2), C(12, 0,1) dan D(O, 2. 0).

    Cari

    (a) pcrsamaan vektor bagi garis AB.[3 markah]

    (b) persamaan bagi satah ABC dalam bentuk Cartesian.[5 markahJ

    (c) sudut tirus antaru satah ABC dan satah ABD.[5 markah]

    8 Bulatan C mclalui asalan dan berpusat pada titik (3,-3).

    (a) Dapatkan persamaan bulatan C.[3 markahJ

    (b) Jika garis y = x - 6 bertemu bulatan C pada titik P dan Q, tcntukankoordinat P dan Q.

    [5 markah1(c) Cari koordinat bagi titik-tit ik pada bulatan C di mana tangennya adalah selari

    dengan garis PQ.[5 markah1

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    7 Given the points A( L 3, 1), B(4, -1, 2), C(12, 0, 1) and D(O, 2, 0).

    Find

    (a) a vector equation of the line AB.l3 marks]

    (b) an equation of the plane ABC in the Cartesian form.[5 marksJ

    (c) the acute angle between the plane ABC and the plane ABD.[5 marks]

    8 A circle C passes through the origin and has its centre at the point (3,-3).

    (a) Obtain the equation of the circle C.[3 marksJ

    (b) If the line y = x - 6 meets the circle C at the points P and Q, determinethe coordinates of P and Q.

    l5 marks]

    (c) Find the coordinates of the points on the circle C where the tangents areparallel to the line PQ.

    l5 marks]

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    OS02611

    -1- _ 19 Lengkung Y = X + ax' + hr mempunyai titik lengkuk balas pada (-2,0).

    (a) Cari nilai a dan n,[5 markah]

    (b) Tllnjukkan bahawa satu lagi titik lengkuk ba1as bagi lengkung ia1ah

    [4 markah]

    (c) Gunakan lljian terbitan kedua untuk mencari koordinat ekstremum tempatanlcngkung tcrsebut.

    [4 markah]

    10 (a) Buktikan bahawa kos3x = 4kos 'x - 3kosx.14 markah]

    (b) Gunakan idcntiti di atas untuk

    (i) mencari semua penyelesaian da1am se1ang -180 < x::::: 180 bagipcrsamaan 2kos 3x + kos 2x + 1= O.

    [7 markah]

    (ii) tunjukkan bahmva kus' 2x = l (kos 6x + 3kos 2x). Seterusnya.,7

    nilaikan f (, kus' 2x dx."[4 markah]

    , KERTAS SOALA;\ TAMAT

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    QS026f1

    9 The curve t ==.'(4 + ax' + bx c has a point of inflection at (-2.0).

    (a) Find the values of a and h.r5 marks]

    ,h I h . f" 1 1 ' f' I . ( i 15 '\(b) S ow t 1at anot er pomt 0 m ectlOn 0 t1e curve IS - -. - I.\ 216 ) [4marksj

    (c) Usc the second derivative test to find the coordinates of the local extremum orthe curve.

    14 marks J

    10 (a) Prove that cos 3x == 4 co:' 1 - 3 cos x.[4 marks I

    (b) Use the above identity to

    (i) find ail the solutions in the interval -180 0 < x:::: 1800 of the equation2 cos 3x + cos 2x + 1 == O.

    [7 marks]

    (i i) show that cos 1 2x == L(cos 6x + 3cos 2x). Hence. evaluate4-

    (. cos; 2xdx.fu[4 marks]

    END OF QVESTION BOOKLET