Ceramah AddMath

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Teknik Menjawab Addmath SPM Kertas 1 &2

Transcript of Ceramah AddMath

  • MENJAWAB SOALAN SPM

    ADDITIONAL MATHEMATICS

    Kertas 1 dan 2

    Skema Jawapan & PemarkahanSiri Ceramah Di:

    (i) SMK Dato Klana Putra, Lenggeng, Negeri Sembilan,

    (ii) SMK Engku Husain, Semenyih, Selangor,

    (iii) SMK Dengkil, Kuala Langat, Selangor,

    (iv) SMK Taman Jasmin 2, Kajang, Selangor,

    (v) SMK Jalan 4, Bandar Baru Bangi, Selangor,

    (vi) SMK Dato Ahmad Razali, Amapang, Selangor.

  • Q.1:

    (a) (a) --11, , 1 1 / / --1 1 and and 1 1 / {/ {--11, , 11} }

    .(1)reject: 1 or -1 / (-1, 1) / [-1, 1]

    (b) many(b) many--toto--one / one / .(1)many with one /many with one /

    many many one one

  • Q.2(a):

    (a)(a) f(x) = 4x + 2f(x) = 4x + 2

    f f --11(x) = y (x) = y

    f(y) = xf(y) = x

    = 4y + 2 = 4y + 2 -----------(1)

    x x = = 4y + 24y + 2

    x x -- 22y = y = ------------(2)

    44

  • Q.2(b)Q.2(b)::

    (b) (b) gfgf(2) = g[f(2)](2) = g[f(2)]

    = g[4(2) + 2] = g[4(2) + 2] ....(1)....(1)

    = g(10)= g(10)

    = 10= 102 2 3(10) 3(10) 11

    = 69 = 69 ..(2)..(2)

    f(2) = 4(2) + 2 f(2) = 4(2) + 2

    = 10 = 10 .(1).(1)

  • Q.3Q.3::

    gh(x) = g[h(x)]

    = mx2 + n 3 ..(P1)

    = 3x2 + 7

    Bandingkan:

    m = 3 m = 3 ...(1)...(1)

    n n 3 = 7 3 = 7 (P1)(P1)

    n = 10 n = 10 (2) (2) (Kedua-duanya)

  • Q.Q.44::

    2x2x2 +2 + p + 2 = 2px + xp + 2 = 2px + x22

    xx2 2 2px + p + 2 = 0 2px + p + 2 = 0 ..(P1)..(P1) aa = 1, b = = 1, b = -- 2p, c = p +22p, c = p +2

    bb22 -- 4ac > 04ac > 0

    ((--2p)2p)2 2 4(1)(p + 2) > 04(1)(p + 2) > 04p4p2 2 4p 4p 8 > 0 8 > 0 ..(P2)..(P2)pp2 2 p p 2 > 0 2 > 0

    Let: Let: pp2 2 p p 2 = 02 = 0(p (p 2)(p+ 1) = 02)(p+ 1) = 0p = 2 atau p = p = 2 atau p = -- 11

    Therefore: p > 2 atau p < Therefore: p > 2 atau p < --1 1 ...(3)

    -1 2x

  • Q.5Q.5::

    (x (x )(x + 4) = 0 )(x + 4) = 0 ..(P1) ..(P1)

    xx22 x x 4x 4x 8/3 = 08/3 = 0

    3x3x22 + 10x + 10x 8 = 0 8 = 0 (2)(2)

    OR:

    x2 ( - 4)x + ()(-4) = 0 (P1)

    3x2 + 10x 8 = 0 (2)

  • Q.6Q.6::

    (a) p = 3 (a) p = 3 (axis of symmetry) ...(1)

    (b) x = 3 ...(1)

    (c) (3, -1) ...(1)

    Note: Minimum value = -1

  • Q.7:

    32x + 1 = 4x

    (2x + 1)log10 3 = xlog10 4 ...(P1) 2xlog10 3 + log10 3 = xlog10 4

    2x(0.4771) x(0.6021) = - 0.4771 ...(P2) 0.9542x 0.6021x = - 0.4771 0.3521x = - 0.4771

    - 0.4771 x =

    0.3521

    = - 1.355 ...(3)

  • Q.8:

    2log2log22M = 2 + 4logM = 2 + 4log44NN

    4log4log22NN

    2log2log22M = 2 + M = 2 + .(P1)loglog2244

    loglog22MM22 = = loglog224 + 2log4 + 2log22N N ....(P2)

    loglog22MM22 = log= log224N4N

    2 2 ...(P3)

    MM22 = 4N= 4N2 2

    M = M = 4N4N22

    M = 2N M = 2N ......(4)

  • Q.9:

    (y 1) x = (5 + 2x) (y 1) ....(P1)

    y 1 x = 5 + 2x y + 1

    2y = 3x + 7 .(P2)

    3x + 7 y = .(3)

    2

  • Q.10(a):

    a = 108

    ar2 = 12

    ar2 = 12 .(P1)a 108

    r2 = /9

    r = /3 ...(2)

  • Q.10(b):

    108

    S = ..(P1)1 /3

    108=

    = 162 ...(2)

  • Q.11:

    x

    y =

    a + bx

    a + bx

    /y = x

    /y = a/x + b .(P1)

    a = gradient

    = - 6/4 = - 3/2 .(2)

    b = y-intercept

    = 6 .(1)

  • Q.12:

    xx/y /y -- yy/4 = 1/4 = 1

    4x 4x -- 2y = 8 2y = 8 ..(P1)

    2y = 4x 2y = 4x -- 88

    y = y = 44//22x x 44

    gradient, m = 2 m = 2 ..(P2)

    y y 7 = 2(x 7 = 2(x 2) 2)

    y = 2x + 3 y = 2x + 3 .(3)

  • Q.13:

    4m + 9s4m + 9s 2m + 6t2m + 6t . (P1)

    P(2m, m) P(2m, m) Q(s, t)Q(s, t) R(3s, 2t)R(3s, 2t)

    3 23 2

    5s 5s 9s = 4m, 9s = 4m, or 5t 5t 6t = 2m6t = 2m-- 4s = 4m, 4s = 4m, -- t = 2m t = 2m (P2)

    -- s = m, s = m,

    - 2s = - t

    s = t/2 ......(3)

    S = or t =5 5

  • Q.Q.1414::

    PA : PB = 2 : 1 PA : PB = 2 : 1 ..(P1)

    reject: PA/PB = 2/1

    PA = 2 (x 5)2 + (y 0) 2 ..(P1) or

    PB = (x + 2)2 + (y + 3)2 .(P1)

    PA2 = 4[(x 5)2 + y 2 ] =

    PB2 = (x + 2)2 + (y + 3)2 ....(P2) or

    4[(x 5)2 + y 2 ] = (x + 2)2 + (y + 3)2 ..(P2)

    3x2 + 3y2 - 44x + 6y + 87 = 0 ....(3)

  • Q.15

    A B

    C

    AC = AB + BC

    = AB - CB .........(P1)

    = (i + 2j) (-5i 6j)

    = 6i + 8j .(2)

    6i + 8jUnit vector AC = ...(P1)

    62 + 82

    6i + 8j 3i + 4j

    = or ..(2)10 5

    D

  • Q.16

    P(1, 2)

    O

    Q(8, -4)

    PQ = PO + OQ

    = - OP + OQ

    = - (i + j) + (8i + 4j)

    = 7i + 3j .(1)

    7i + 3j = ma + nb

    = m(i + j) + n(2i 3j)= (m + 2n)i + (m 3n)j ...(P1)

    Compare: m + 2n = 7

    m 3n = 3

    5n = 4 ..(P1)n = 4/5 )..(2 either)

    m = 27/5 )..(3)(both)

  • Q.Q.1717::

    sec sec = = 11//cos cos

    = = 11//t t (1) (1)

    cos (90 cos (90 ) = sin ) = sin .(1).(1)

    = 1 = 1 tt22 .(2).(2)

    1 t21

    t

  • Q.18Q.18:: 2 sec2 sec22 3tan3tan = 4= 4 2(1 + tan2(1 + tan22) ) 3tan3tan = 4 = 4 ....(1)....(1)

    2 + 2 tan2 + 2 tan22 3tan3tan = 4= 42tan2tan22 -- 3tan3tan 2 = 02 = 0

    (2tan(2tan + 1)(tan+ 1)(tan 2) = 0 2) = 0 ..(2)..(2) tantan = = -- atau atau tantan = 2= 2

    = (360= (360oo 2626o o 34), (18034), (180o o -- 2626o o 34)34)

    = 333= 333oo 26, 15326, 153o o 26 26 ...(3)...(3) = 63= 63o o 26, (18026, (180o o + 63+ 63o o 26)26)

    = 63= 630 0 26, 24326, 2430 0 26 26 ......(3)......(3) = 63= 630 0 26, 15326, 153o o 26, 24326, 2430 0 26, 33326, 333oo 26.26. ....(4)(4)

  • Q.19:

    BOC = BOC = 1 1

    = = 33..142 142 1 1

    = = 22..142 142 ..(P1)

    s = j

    BC = 15 x 2.142 ..(P2)

    = 32.13 ..(3)

  • Q.Q.2020::

    dy (x dy (x 4)(4x) 4)(4x) (2x(2x22

    + 3)(1)+ 3)(1) = = (P2)dx (x dx (x -- 4)4)22

    4x .(P1)

    2x2x22

    16x 16x -- 33= = ..(3)

    (x (x -- 4)4)22

  • Q.21Q.21::

    dy/dx = 4x dy/dx = 4x 3 3 ..(1) x = 0.01 x = 0.01 ..(1) x = 2x = 2

    yyxx

    dydydxdx

    yy dydydxdx (x)(x)

    = [4(2) = [4(2) 3](0.01) 3](0.01) (2) = 5(0.01)= 5(0.01)

    = 0.05 = 0.05 (3)

  • Q.22:

    3 3 33

    f(x) + (kx)dx = 20 f(x) + (kx)dx = 20 1 11 1

    33

    8 + [ kx8 + [ kx2 2 2 2 ] = 20 ] = 20 ..(P1)11

    33

    [ kx[ kx2 2 2 2 ] = 12 ] = 12 ..(P2) 11

    9k/2 k/2 = 12, 8k/2 = 12 ..(P3)

    k = 3 ..(4)

  • Q.Q.2323::

    6 46 4

    P or P P or P .(1)4 34 3

    6 46 4

    P x P = P x P = ((6 6 x x 5 5 x x 4 4 x x 33) x () x (4 4 x x 3 3 x x 22)) ..(P2)4 34 3

    = = 360 360 x x 2424

    = = 86400 86400 ...(...(33))

  • Q.Q.2424::99 55

    (a) C x C (a) C x C .(P1)66 44

    atauatau 9 x 8 x 79 x 8 x 7

    3 x 23 x 2

    = 84 = 84 (2)

    55 9 5 9 5 99 5 9 5 9 (b) C x C + C x C + C x C (b) C x C + C x C + C x C ....(P1)

    5 5 4 6 3 75 5 4 6 3 7

    = 948 = 948 (2)

  • Q.25:

    0.5 0.5 0.225 0.225