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  • 7/29/2019 Modul Halus F4(Mid Year)

    1/34

    Name: ______________________________

    Form: _____________ F4_R1_2011

    1 (a) The relation of setA = {4, 5, 6, 7} to setB =

    {0, 1, 2} is defined by the ordered pairs of

    {(4,1), (5,2), (6,0), (7,1)}.

    State

    (i) the image of 6(ii) the object of 1

    (iii) the domain of the relation

    (iv) the codomain of the relation

    (v) the range of the relation

    (vi) the type of the relation

    [(i) 0, (ii) 4, 7, (iii) {4, 5, 6, 7}, (iv) {0, 1, 2}, (v) {0, 1, 2}, (vi) many to-one]

    1 (b) The relation of set C= {a, b} to setD = {1, 2,

    3, 4} is defined by the ordered pairs of {(a,1),

    (a,2), (b,3)}.

    State

    (i) the image ofb

    (ii) the object of 1

    (iii) the domain of the relation

    (iv) the codomain of the relation

    (v) the range of the relation

    (vi) the type of the relation

    [(i) 3, (ii) a, (iii) {a, b}, (iv) {1, 2, 3, 4} (v) {1, 2, 3}, (vi) one to-many]

    1 (c) The graph below shows the relation between

    setA and setB.

    State

    (i) the image of 3

    (ii) the object of 4

    (iii) the domain of the relation

    (iv) the codomain of the relation

    (v) the range of the relation

    (vi) the type of the relation

    [(i) 9, (ii) 2, (iii) {1, 2, 3}, (iv) {1, 4, 9} (v) {1, 4, 9} , (vi) one to-one]

    1 (d) The graph below shows the relation between

    setA and setB.

    State

    (i) the image of 2

    (ii) the object ofd

    (iii) the domain of the relation

    (iv) the codomain of the relation

    (v) the range of the relation

    (vi) the type of the relation

    [(i) b, (ii) 3, 4, (iii) {1, 2, 3, 4}, (iv) {a, b, c, d} (v) {a, b, d}, (vi) many to many]

    1 (e) The graph below shows the relation between

    setA and setB.

    State

    (i) the image of 4(ii) the object ofy

    (iii) the domain of the relation

    (iv) the codomain of the relation

    (v) the range of the relation

    (vi) the type of the relation

    [(i)y, z, (ii) 3, 4, (iii) { 2, 3, 4, 5}, (iv) {x, y, z} (v) {x, y, z}, (vi) many to many]

    Halus_2 1

    1 2 3

    1

    9

    4

    SetA

    SetB

    1 2 3

    a

    c

    b

    SetA

    SetB

    4

    d

    2 3 4

    x

    z

    y

    SetA

    SetB

    5

  • 7/29/2019 Modul Halus F4(Mid Year)

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    1 (f)

    The diagram shows the relation between SetA and

    SetB.

    State

    (i) the value ofm,

    (ii) the object of 3,

    (iii) the image of 4,

    (iv) type of relation,

    (v) the relation by using the function notation.

    [(i) 7, (ii) 1, (iii) 9, (iv) one to one , (v)f(x) = 2x +1]

    1 (g)

    The diagram shows the relation between SetPand

    Set Q.State

    (i) the image of 2,

    (ii) the object of 4,

    (iii) the range of the relation,

    (iv) type of relation,

    (v) the relation by using the function notation.[(i) 4, (ii) 2, 2, (iii) {0, 1, 4}(iv) many to one , (v)f(x) =x2]

    1 (h)

    The diagram shows the relation between SetA and

    SetB.

    State

    (i) the value ofp,

    (ii) the object of 11,

    (iii) the image of 2,

    (iv) type of relation,

    (v) the relation by using the function notation.

    [(i) 5, (ii) 6, (iii) 3, (iv) one to one , (v)f(x) = 2x 1]

    1 (i)

    The diagram shows the relation between SetPandSet Q.

    State

    (i) the image of 1,

    (ii) the object of 2,

    (iii) the range of the relation,

    (iv) type of relation,

    (v) the relation by using the function notation.[(i) 2, (ii) 1, 1, (iii) {1, 2, 5}(iv) many to one , (v)f(x) =x2+1]

    Halus_2 2

    1

    2

    3

    4

    3

    5

    m

    9

    SetA SetB

    SetP Set Q

    0

    1

    2

    3

    4

    2

    1

    0

    1

    2

    SetP Set Q

    1

    2

    5

    2

    1

    0

    1

    2

    SetA SetB

    0

    2

    p

    6

    1

    3

    9

    11

  • 7/29/2019 Modul Halus F4(Mid Year)

    3/34

    2(a) Given that the functionf:x |5x 7|, find

    the values ofx such thatf(x) = 3.[x=2,x= 4/5]

    f(x) = 3

    |5x 7| = 3

    5x 7 = 3

    5x 7 = 3 5x 7 = 3

    5x = 10 5x = 4

    x = 2 x = 4/5

    2(b) Given that the function h :x |mx +3|, find

    the values ofm such that h(2) = 11.[m=4, m= 7 ]

    2(c) Given that the functiong:x |2x n |, find

    the values ofn such thatg(3) = 5.[n=1, n= 11 ]

    2(d) Given that the functionf:x |2 + 3x|, find the

    values ofm such thatf(m) = 8.[m=2, m= 10/3 ]

    2(e) Given that the functiong:x |4 5x|, find

    the value ofn such thatg(2) = 2n.[n= 3]

    2(f) Given that the function h :x |8 + mx|, find

    the values ofm such that h( 1) = 5 .[m=3, m= 13 ]

    2(g) Given that the functionf:x |4 +x|, find the

    values ofw such thatf( 1) =2w .

    [w = 6 ]

    2(h) Given that the function h :x |w 2x|, find

    the values ofw such that h(2) = 2w .[w = 4, w = 4/3 ]

    Halus_2 3

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    3(a) Given that the function .13)( += xxh find (i))(1 xh ,

    (ii) )2(1h

    [(i)3

    1x, (ii) 1/3]

    (i) )(1 xh =

    3

    1x(ii) )2(

    1h =3

    12

    =31

    Or )2(1h =x

    )(2 xh=

    132 += x

    x31 =

    x=3

    1

    3(b) Given that the function .25)( = xxf find (i))(1 xf ,

    (ii) )3(1f

    [(i)5

    2+x, (ii) 1]

    3(c) Given that the function .23)( xxg = find(i) )(

    1 xg ,

    (ii) the value ofm if mg = )1(1

    [(i)2

    3+ x, (ii) 1]

    3(d) Given that the function .4

    35)(

    =

    xxg find (i)

    )(1 xg , (ii) the value ofn if 3)(1 = ng

    [(i)5

    34 +x, (ii) n= 3]

    3(e) Given that the function 5,5

    23)(

    +

    = xx

    xxh

    find

    (i) )(1 xh ,

    (ii) the value ofp if ph = )2(1

    [(i)3

    25

    +

    x

    x,

    3

    25

    +

    x

    x(ii) 8/5]

    3(f) Given that the function 3

    4,

    43)(

    +=x

    x

    xxf

    find

    (i) )(1 xf ,

    (ii) the value ofp if 2)(1 = pf

    [(i)13

    4

    + x

    x,

    13

    4

    x

    x, (ii) 1]

    3(g) Given that the function .3

    52)(

    x

    xxh

    = find (i)

    )(1 xh ,

    (ii)2

    1)(1

    =

    mh

    [(i)53

    2

    x,

    53

    2

    +x(ii) 3]

    Halus_2 4

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    4(a) Given the function3

    72)(

    +=

    xxf , find the

    value ofx for 13)( =xf .[x= 16]

    13)( =xf

    133

    72=

    +x

    2x + 7 = 39 2x = 32

    x = 16

    4(b) Given the function 3,3

    4)(

    = x

    x

    xxg , find the

    value ofx for 16)( =xg .[x= 4]

    4(c) Given the function 0,3

    52)(

    = x

    x

    xxh , find

    the value ofp for 2)( =ph .[p= 2]

    4(d) Given the function ,12

    )(nx

    xxh

    +

    = find the

    value ofn for 1)3( =h .[n= 2]

    4(e) Given the function ,4

    2)(

    =

    mxxf find the

    value ofm for 3)2( =f .[m= 7]

    4(f) Given the function ,2)( += mxxg find thevalue ofm if mg = )2([m= 2/3 ]

    4(g) Given the function ,32)( pxxg = find the

    value ofp if

    3

    )1(p

    g =

    [p = 3/5]

    4(h) Given the function xpxf 2)( = find thevalue ofp if 12)2( = pf[p = 5]

    Halus_2 5

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    5(a) Given that function 12)( = xxg and54)( += xxgf , find the function off.

    [f(x)= 2x+3]

    g 54)( += xxf2 ( ) 1 = 4x +5

    2 (f(x) ) 1 = 4x +5

    2 f(x) = 4x +5 + 1

    2 f(x) = 4x +6f(x) = 2x + 3

    5(b) Given that function xxf 23)( = and56)( = xxfg , find the function ofg.

    [g(x)= 3x+4]

    5(c) Given that function 34)( = xxg andxxgf 65)( += , find the function off.

    [f(x)= 2+ 3/2 x]

    5(d) Given that function xxg 56)( += and13)( += xxgf , find the function off.

    [f(x)=(3x 5)/5 ]

    5(e) Given that function 32)( += xxf and13)( = xxfg , find the function ofg.

    [g(x)= (3x 4)/2]

    5(f) Given that function 35)( = xxf andxxfg 72)( += , find the function ofg.

    [g(x)= (7x + 5)/5]

    5(g) Given that function xxg 41)( += and7)( =xxgf , find the function off.

    [f(x)= (x 8)/4]

    5(h) Given that function xxf 26)( += andxxfg =)( , find the function ofg.

    [g(x)= (x 6)/2]

    5(i) Given that function 32)( = xxg andxxgf 47)( += , find the function off.

    [f(x)= 5+ 2x ]

    5(j) Given that function 34)( = xxf andxxfg 89)( = , find the function ofg.

    [g(x)= 3 2x]

    5(k) Given that function 7)( += xxf and32)( = xxgf , find the function ofg.

    Halus_2 6

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    [g(x)= 2x 17]

    7)(1 = xxf

    3)7(2)( = xxg 3142)( = xxg 172)( = xxg

    5(l) Given that function 43)( += xxf and25)( += xxgf , find the function ofg.

    [g(x)= (5x 14 )/3]

    5(m) Given that function xxg 35)( = andxxfg 37)( = , find the function off.

    [f(x)=x+2]

    5(n) Given that function 73)( = xxf andxxgf 65)( += , find the function ofg.

    [g(x)= 2x+19]

    5(o) Given that function 12)( = xxg and54)( += xxfg , find the function off.

    [f(x)= 2x+7]

    5(p) Given that function 3)( += xxg and23)( += xxfg , find the function off.

    [f(x)= 3x 7]

    5(q) Given that function 15)( += xxf and43)( = xxgf , find the function ofg.

    [g(x)= (3x 23 )/5]

    5(r) Given that function 57)( += xxf and114)( += xxgf , find the function ofg.

    [g(x)= 2x 9]

    6 (a) Diagram below shows the graph of function

    43)( = xxf with the domain0 x 6.

    Halus_2 7

  • 7/29/2019 Modul Halus F4(Mid Year)

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    Find

    (i) the value ofp,

    (ii) The range off(x) for the domain 0 x 6.

    [(i)p = 4/3, (ii) 0 f(x) 14]

    (i) 43)( = xxf (ii) 4)6(3)( =xf

    430 = p 14= 430 = p 14=

    430 = p 0 f(x) 14

    p=3

    4

    6 (b) Diagram below shows the graph of functionxpxf 23)( = with the domain0 x 5.

    Find

    (i) the value ofp,

    (ii) The range off(x) for the domain 0 x 5.

    [(i)p = 1/3, (ii) 0 f(x) 9]

    6 (c) Diagram below shows the graph of function

    xxf 28)( = with the domain0 x 5.

    Find

    (i) the value ofp,

    (ii) The range off(x) for the domain 0 x 5.

    [(i)p = 4, (ii) 0 f(x) 8]

    6 (d) Diagram below shows the graph of function

    xxf 27)( = with the domain0 x 5.

    Halus_2 8

    p 6

    y

    xO

    4

    1/2 5

    y

    xO

    1

    y

    xp 5O

    y

    x

    p 5O

  • 7/29/2019 Modul Halus F4(Mid Year)

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    Find

    (i) the value ofp,

    (ii) The range off(x) for the domain 0 x 5.

    [(i)p = 7/2, (ii) 0 f(x) 7]

    6 (e) Diagram below shows the graph of function

    52)( += xxf with the domain 3 x 2.

    Find

    (i) the value ofp and k,

    (ii) The range off(x) for the domain 3 x 2.[(i)p = 5/2 , k= 5(ii) 0 f(x) 9]

    6 (f) Diagram below shows the graph of function

    13)( += xxf with the domain 3 x 1.

    Find

    (i) the value ofp and k,

    (ii) The range off(x) for the domain 3 x 1.

    [(i)p = 1/3, k= 1 (ii) 0 f(x) 8]

    6 (g) Diagram below shows the graph of function

    34)( = xxf with the domain 2 x 3.

    Find

    (i) the value ofm and n,

    (ii) The range off(x) for the domain 2 x 3.

    [(i) m = 3/4, n = 3 (ii) 0 f(x) 11]

    Halus_2 9

    p 2

    y

    xO 3

    k

    p 1

    y

    xO

    3

    k

    m 3

    y

    xO 2

    n

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    6 (h) Diagram below shows the graph of function

    xpxf 5)( = with the domain 2 x 2.

    Find

    (i) the value ofp and n,

    (ii) The range off(x) for the domain 2 x 2.

    [(i)p = 2, n = 2 (ii) 0 f(x) 12]

    7(a) Given the function 45)( += xxg Find thevalues ofx in which it maps onto itself.[x= 1 ]

    g(x) =x

    5x + 4 =x

    4x = 4

    x = 1

    7(b) Given the function xxf 23)( = Find thevalues ofx in which it maps onto itself.[x= 1 ]

    7(c) Given the function .2,2

    4)(

    +

    = xx

    xxg Find

    the values ofx in which it maps onto itself.[x= 4 , 1 ]

    7(d) Given the function .5,5

    7)(

    +

    = xx

    xxf Find

    the values ofx in which it maps onto itself.[x= 7 , 1 ]

    7(e) Given the function .4,4

    3)(

    +

    = xx

    xg Find

    the values ofx in which it maps onto itself.[x= 3 , 1 ]

    7(f) Given the function .0,54

    )( +

    = xx

    xxg Find

    the values ofx in which it maps onto itself.[x= 5 , 1 ]

    Halus_2 10

    2/5 2

    y

    xO 2

    n

    1

    1

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    7(g) Given the function .2

    1,

    12)(

    = x

    x

    xxf Find

    the values ofx in which it maps onto itself.[x= 0 , 1 ]

    7(h) Given the function .3

    4)(

    =

    xxf Find the

    values ofx in which it maps onto itself.[x= 2 ]

    8(a) Given the function 53)( = xxf . Find thevalue ofm and n if nmxf +=2 .[m = 9, n= 20]

    ff (x) = 3 ( ) 5= 3 (3x 5 ) 5

    = ( 9x 15 ) 5

    = 9x 20

    = m x + n

    m =9, n = 20

    8(b) Given the function xmxg 5)( = . Find thevalue ofm and n if 24

    2 = nxg .[m = 6, n= 25]

    8(c) Given the function 3)( += pxxf andqxf = 162 . Find the value ofp and q ifp 0

    [m = 2, n= 7]

    8(f) Given the function qxpxg +=)( and492 = xg . Find the value ofp and q ifq >0.

    [p= 1 , q= 3]

    Halus_2 11

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    8(g) Given the function nmxxf +=)( and1542 += xf . Find the value ofm and n ifm >

    0.[m = 2 , n = 5]

    8(h) Given the function 3)( = mxxg andnxg += 642 .Find the value ofm and n if

    m

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    9 (g) Given that 12)( += xxg , xxf = 3)( . Findthe value ofp iffg(p) = p + 1.[p= 1/3]

    9 (h) Given that 52)( = pxxf , 23)( += xxg .Find the value ofp ifgf(1) = 5p.[p= 13]

    10 (a)Solve the equations ).3(223 2 += xxx Giveyour answer correct to four significant

    figures.[x = 2.230, 0.8968]

    )3(223 2 += xxx6223 2 += xxx

    0643 2 = xx

    a = 3, b = 4, c = 6

    a

    acbbx

    2

    42 =

    )3(2

    )6)(3(4)4()4( 2 =x

    = 2.230, 0.8968

    10 (b)Solve the equations .2

    3752 2

    +=

    xx Give

    your answer correct to three decimal places.[x = 2.879, 1.129]

    10 (c)Solve the equations .3)2( 22 xxx = Giveyour answer correct to three significant

    figures.[x = 0.851, 2.35]

    10 (d)Solve the equations .2)3()13(2 2 =+ xx Give your answer correct to four significant

    figures.[x = 11.56, 0.4322]

    10 (e)Solve the equations .2

    37

    1 x

    x

    x

    x +=

    Give your

    answer correct to four decimal places.[x = 1.2718, 0.4718]

    Halus_2 13

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    10 (f)Solve the equations 3213

    5=

    +x

    x. Give

    your answer correct to three significant

    figures.[x = 1.88, 0.710]

    11 (a)Form the quadratic equation which has theroots 2 and 7. Give your answer in the

    form of 02 =++ cb xa x , where a, b and c are

    constants.[ 01452 =+ xx ]

    SOR = 2 + ( 7 ) POR = 2 ( 7 )

    = 5 = 14

    x2 SORx + POR = 0

    x2 ( 5) x + ( 14) = 0

    x2 + 5 x 14 = 0

    OR

    (x 2) (x +7) = 0 014722 =+ xxx

    01452 =+ xx

    11 (b)Form the quadratic equation which has theroots 5 and 4 . Give your answer in the

    form of 02 =++ cb xa x , where a, b and c are

    constants.[ 0202 =+ xx ]

    11 (c) Form the quadratic equation which has theroots 2 and

    4

    3. Give your answer in the

    form of 02

    =++ cb xa x , where a, b and c are

    constants.

    [ 02

    3

    4

    112=+ xx , 06114 2 =+ xx ]

    11 (d) Form the quadratic equation which has theroots 3 and

    5

    1 . Give your answer in the

    form of 02 =++ cb xa x , where a, b and c areconstants.[ 03165 2 =++ xx ]

    11 (e)Form the quadratic equation which has theroots 4 and

    3

    1. Give your answer in the

    form of 02

    =++ cb xa x , where a, b and c areconstants.

    [ 03

    4

    3

    112=+ xx , 04113 2 =+ xx ]

    11 (f) Form the quadratic equation which has theroots

    3

    1and

    2

    5 . Give your answer in the

    form of 02 =++ cb xa x , where a, b and c are

    constants.

    [ 06

    5

    6

    132=+ xx , 05136 2 =+ xx ]

    Halus_2 14

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    11 (g) Form the quadratic equation which has theroots 3 and

    5

    2. Give your answer in the

    form of 02 =++ cb xa x , where a, b and c are

    constants.

    [ 05

    6

    5

    132=+ xx , 06135 2 =+ xx ]

    11 (h) Form the quadratic equation which has theroots

    3

    1and

    2

    5 . Give your answer in the

    form of0

    2=++

    cb xa x, where a, b and c are

    constants.

    [ 06

    5

    6

    132=+ xx , 05136 2 =+ xx ]

    Name: ______________________________

    Form: _____________ F4_R2_2011

    12 (a) One of the roots of the equation

    pxx 3272 =+ is three times the other root.Find the values ofp.[ = 3, p = 4]

    pxx 3272 =+02732 =+ pxx

    , 3

    SOR= , 3 POR= (3 )

    3p = 4 27 = 3 2

    = 3,

    p =3

    )3(4

    9 = 2

    = 4 3=

    = 3,

    p =3

    )3(4

    = 4

    12 (b) One of the roots of the equation

    0)5(52 2 =+ pxx is half the other root.Find the values ofp.[ = 5/2, p = 3]

    12 (c) One of the roots of the equation

    042 =+mxx is four times the other root.Find the values ofm.[ = 1, m = 5]

    12 (d) One of the roots of the equation

    mxmx += )2(2 2 is twice the other root.

    Find the values ofm.[ = 1, , m = 1, 4]

    12 (e) One of the roots of the equation

    0482 =+ xmx is one third the other root.Find the values ofm, m 0.[ = 2/3 , m = 3]

    Halus_2 15

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    12 (f)p and 2p are roots of the equation

    xkx 3122 2 =+ . Find the values ofkand p.

    [p = 1/2 ,k= 1]

    13(a) The quadratic equationx2 4x =1 2p has

    equal roots. Find the value ofp.[p = 4, 4]

    x2 4x =1 2p

    x2 4x 1 + 2p=0a = 1, b = 4, c = 1+2p

    = 2p 1

    b2 4ac = 0

    ( 4)2 4(1)(2p 1) = 0

    16 4(2p 1) = 0

    16 8p + 4 = 0

    20 8p = 0

    20 = 8p

    20/8 =p

    5/2 =p

    13(b) The quadratic equation pxx 22 = has two

    different roots. Find the range of the value of

    p.[p > 1/8]

    13(c) The quadratic equation xkx 672 =+ , k 0has equal roots. Find the possible value ofk.

    [k= 9/7]

    13(d) Find the range of values ofkif the quadratic

    equation 34)3(2 +=+ xxk does not have real

    roots.[k< 13/3]

    13(e) Find the range of values ofp if the quadratic

    equation 6252 = xpx has two real and

    distinct roots.[p > 1/30]

    Halus_2 16

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    13(f) Find the range of values ofm if the quadratic

    equation 0522 = mxx has no real roots.[m < 1/5]

    14 (a)Diagram 14(a) shows the graph off(x) =p(x 3)2 + q, wherep, q and rare

    constants.

    State

    (a) The value ofp,

    (b) the value ofq,

    (c) the value ofr.[p = 2, q = 5, r= 3]

    (a)f(x) =p(x 3)2

    + 5 13 =p(0 3)2 + 5

    13 = 9p + 5

    18 = 9p

    2 =p

    (b) q = 5 (c) r= 3

    14 (b)Diagram 14(b) shows the graph off(x) =a(x + b)2 2, where a, b and c are

    constants.

    State

    (a) The value ofa,

    (b) the value ofb,

    (c) the value ofc.[a= 3, b = 1, c = 2]

    14 (c)Diagram 14(c) shows the graph off(x) = r(x 3)2 + 7, wherep, q and rare

    constants.

    State

    (a)The value ofp,

    (b)the value ofq,

    (c) the value ofr.[p = 3, q = 7, r= 1/3 ]

    14 (d)Diagram 14(d) shows the graph off(x) =2(x+p)2 + q, wherep, q and rare

    constants.

    State

    (a) The value ofp,

    (b)the value ofq,

    (c) the value ofr.[p = 1, q = 3, r= 1]

    14 (e)Diagram 14(e) shows the graph off(x) =a(x+ b)2 + c, where a, b and c are

    constants.

    Halus_2 17

    O

    y

    x

    4

    (p, q)

    O

    y

    x

    r

    (1, 3)

    )

    O

    y

    x

    5

    (2,

    3) )

    O

    y

    x

    13

    (r,5)

    O

    y

    x

    5

    (1, c)

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    State

    (a)The value ofa,

    (b)the value ofb,

    (c) the value ofc.[a = 2, b = 2, c = 3]

    14 (f)Diagram 14(f) shows the graph off(x) =2(x 1)2 5, wherep, q and rare

    constants.

    State(a)The value ofa,

    (b)the value ofb,

    (c) the value ofc.[a = 3, b = 1, c = 5]

    15 (a) Diagram 15(a) shows the graph of a quadraticfunctiony =f(x). The straight line y = 1 is a

    tangent to the curve.

    (a)Write the equation of the axis of symmetry of the

    curve.

    (b)Expressf(x) in the form ( )2

    x b+ + c, where b and care constants.

    [x =2,f(x) = (x 2)2+1]

    (a)

    2

    2

    4

    2

    31

    =

    =+

    (b) 1)2()(2 += xxf

    2=x

    15 (b) Diagram 15(b) shows the graph of a quadraticfunctiony =f(x). The straight line y = 4 is a

    tangent to the curve.

    (a)Write the equation of the axis of symmetry of thecurve.

    (b)Expressf(x) in the form ( )2

    x b+ + c, where b and care constants.

    [x =4 ,f(x) = (x 4)2 4]

    15 (c) Diagram 15(c) shows the graph of a quadraticfunctiony =f(x). The straight line y = 9 is a

    tangent to the curve.

    (a)Write the equation of the axis of symmetry of the

    curve.

    (b)Expressf(x) in the form ( )

    2

    x b+ + c, where b andc are constants.[x =1 ,f(x) = (x 1)2+9]

    15 (d) Diagram 15(d) shows the graph of a

    quadratic functiony =f(x). The straight line

    y = 8 is a tangent to the curve.Halus_2 18

    O

    y

    x

    a

    (b, c)

    y= f(x)

    O 1 3 x

    y = 1

    y

    y= f(x)

    O 2 6 x

    y = 4

    y

    y = 9

    y= f(x)

    O 2 4 x

    y

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    (a)Write the equation of the axis of symmetry of the

    curve.

    (b)Expressf(x) in the form a ( )2

    x b+ + c, where a, band c are constants.[x =3 ,f(x) = 2(x 3)2 8]

    15 (e) Diagram 15(e) shows the graph of a

    quadratic functiony =f(x). The straight line

    y = 1 is a tangent to the curve.

    (a)Write the equation of the axis of symmetry of the

    curve.

    (b)Expressf(x) in the form a ( )2

    x b+ + c, where a, band c are constants.[x =3,f(x) = (x 3)2+1]

    15 (f) Diagram 15(f) shows the graph of a

    quadratic functiony =f(x). The straight line

    y = 2 is a tangent to the curve.

    (a)Write the equation of the axis of symmetry of the

    curve.(b)Expressf(x) in the form a ( )

    2x b+ + c, where a, b

    and c are constants.[x =2 ,f(x) = 2(x 2)2 2]

    15 (g) Diagram 15(g) shows the graph of a

    quadratic functiony =f(x). The straight line

    y = 3 is a tangent to the curve.

    (a)Write the equation of the axis of symmetry of the

    curve.

    (b)Expressf(x) in the form ( )2

    x b+ + c, where band c are constants.[x =3,f(x) = (x 3)2+3]

    15 (h) Diagram 15(h) shows the graph of a

    quadratic functiony =f(x). The straight line

    y = 8 is a tangent to the curve.

    Halus_2 19

    y= f(x)

    O 1 5x

    y = 8

    y

    10

    y= f(x)

    O 2 4 x

    y = 1

    y

    8

    y= f(x)

    O 1 3x

    y = 2

    y

    6

    = x

    O 6

    y = 3

    y

    y= f(x)

    y

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    (a)Write the equation of the axis of symmetry of the

    curve.(b)Expressf(x) in the form ( )

    2x b+ c, where b and

    c are constants.[x =4 ,f(x) = (x 4)2 8]

    16 (a) Find the range of the values ofx for

    (x + 2) (x 5) < 0[ 2 3 ]

    16 (c) Find the range of the values ofx for

    (4 +x) (5 + 3x) 0[ 4 x 5/3]

    16 (d) Find the range of the values ofx for

    (2 3x) (1 + 2x) 0[ 1/2 x 2/3]

    16 (e) Find the range of the values ofx for

    x (x + 6) > 0[x < 6,x > 0 ]

    16 (f) Find the range of the values ofx for

    x (3 2x) 0[x 0,x 3/2 ]

    16 (g) Find the range of the values ofx for

    (x + 3)2 >x + 5[x < 4 , x > 1 ]

    16 (h) Find the range of the values ofx for

    x (x 5) 3x2 7[x 7/2,x 1 ]

    Halus_2 20

    2 5

    + +

    O 8

    y = 8

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    16 (i) Find the range of the values ofx for

    0532 2 + xx[x 5/2,x 1 ]

    16 (j) Find the range of the values ofx for

    062 >+ xx[ 3

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    [p = 5/2, q = 37/4]

    17 (h) The quadratic functionsf(x) =x2 + 3x + 7can be expressed in the form

    f(x) = (x +p )2 + q, wherep and q are

    constants. Find the value ofp and q.[p = 3/2, q = 19/4]

    17 (i) The quadratic functionsf(x) =x2 + 7x 2can be expressed in the form

    f(x) = (x +p )2 + q, wherep and q are

    constants. Find the value ofp and q.[p = 7/2, q = 57/4 ]

    17 (j) The quadratic functionsf(x) = x2 + 7x +4can be expressed in the form

    f(x) =p(x + q )2 + r, wherep, q and r are

    constants. Find the value ofp, q and r.[p = 1 , q = 7/2 , r= 65/4]

    17 (k) The quadratic functionsf(x) = 2x2 + 3x 1can be expressed in the form

    f(x) =p(x + q )2 r, wherep, q and r are

    constants. Find the value ofp, q and r.[p = 2 , q = 3/4 , r= 1/8]

    17 (l) The quadratic functionsf(x) = 3x2 5x +2can be expressed in the form

    f(x) =p(x + q )2 + r, wherep, q and r are

    constants. Find the value ofp, q and r.[p = 3 , q = 5/6 , r= 49/12]

    17 (m) The quadratic functionsf(x) = 2x2 + 4x 3can be expressed in the formf(x) = a(x + b )2 + c, where a, b and c are

    constants. Find the value ofa, b and c.[ a = 2 , b = 1, c= 1]

    Halus_2 22

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    17 (n) The quadratic functionsf(x) = 3x2 4x +5can be expressed in the form

    f(x) = a(x + b )2 c, where a, b and c are

    constants. Find the value ofa, b and c.[ a = 3 , b = 2/3, c= 19/3]

    18 (a) The straight line )2( pxpy = intersectsthe curve 54

    2 += xxy at two differentpoints, find the range of values ofp.[p < 9/4]

    )2( pxpy =

    22 ppx =

    54

    2

    += xxy 542

    22 += xxppx

    22 5)24(0 pxpx ++=a = 1, b = 4 2p, c = 5 +p2

    042 > acb0)5)(1(4)24( 22 >+ pp

    0)5(441616 22 >++ ppp042041616 22 >++ ppp

    03616 >+ p

    p1636 > p>

    4

    9

    18 (b) The straight line pxy 2= intersects thecurve 13

    22 ++= xxpy at two differentpoints, find the range of values ofp.[p < 3/4]

    18 (c) Find the values ofm if the straight line

    6= mxy is a tangent to the curve472 2 += xxy .

    [m = 3, 11]

    18 (d) The straight line 1= mxy is a tangent tothe curve

    2xy = , find the values ofm.[m = 2, 2]

    18 (e) The straight line 14 += xy does notintersect the curve pxxy 23

    2 ++= , find therange of values ofp.[p > 7/8]

    Halus_2 23

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    18 (f) The straight line kxy = does notintersect the curve

    233 xxy =++ , find therange of values ofk.[k> 10/3]

    19 (a) Diagram 19(a) shows the graph of the

    quadratic function 2)(2 += mxy , where m

    is a constant.

    Find

    (a) the value ofm,

    (b) the equation of the axis of symmetry,

    (c) the coordinates of the minimum point.[m = 3,x = 3, (3, 2)]

    (a) m = 3, (b)x = 3,(c) (3, 2)

    19 (b) Diagram 19 (b) shows the graph of the

    quadratic function 1)(2 += kxy , where k

    is a constant.

    Find

    (a) the value ofk,

    (b) the equation of the axis of symmetry,

    (c) the coordinates of the maximum point.[k= 2,x = 2, (2, 1)]

    19 (c) Diagram 19 (c) shows the graph of the

    quadratic function 5)(2 ++= mxy , where

    m is a constant.

    Find

    (a) the value ofm,

    (b) the equation of the axis of symmetry,

    (c) the coordinates of the maximum point.[m = 4,x = 4, (4, 5)]

    19 (d) Diagram 19 (d) shows the graph of the

    quadratic function 3)(2 = kxy , where kis

    a constant.

    Halus_2 24

    (5,6)(1,6)

    xO

    y

    31x

    O

    y

    (6, 1)(2, 1)

    xO

    y

    (4, 2)(2, 2)

    xO

    y

    1 x = 3 5

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    Find

    (a) the value ofk,

    (b) the equation of the axis of symmetry,

    (c) the coordinates of the minimum point.[k= 3,x = 3, (3, 3)]

    19 (e) Diagram 19 (e) shows the graph of the

    quadratic function 1)( 2 ++= mxy , where mis a constant.

    Find

    (a) the value ofm,

    (b) the equation of the axis of symmetry,

    (c) the coordinates of the minimum point.[m = 4,x = 4, (4, 1)]

    19 (f)Diagram 19 (f) shows the graph of the

    quadratic function 4)(2 ++= mxy , where

    m is a constant.

    Find

    (d) the value ofm,

    (e) the equation of the axis of symmetry,

    (f) the coordinates of the maximum point.[m = 2,x = 2, (2, 4)]

    19 (g) Diagram 19 (g) shows the graph of the

    quadratic function 5)( 2 += bxy , where mis a constant.

    Find

    (d) the value ofb,

    (e) the equation of the axis of symmetry,

    (f) the coordinates of the minimum point.[b = 4,x = 4, (4, 5)]

    19 (h) Diagram 19 (h) shows the graph of the

    quadratic function 3)(2 += bxy , where

    m is a constant.

    Halus_2 25

    (5, 2)(3, 2)

    xO

    y

    4x

    O

    y

    (6, 1)(2, 1)

    xO

    y

    xO

    y

    (1, 2)(3, 2)

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    Find

    (g) the value ofm,

    (h) the equation of the axis of symmetry,

    (i) the coordinates of the maximum point.[b = 2,x = 2, (2, 3)]

    20 (a) Solve the equation 24(8)3x = .

    [x = 1/9]

    24(8)3x =13x32 2)(22 =

    19x2 22 =+

    2 + 9x =1

    9x = 1

    x = 1/9

    20 (b) Solve the equation32

    1162x = .

    [x = 5/8]

    20 (c) Solve the equation 13(27)x = .

    [x = 1/3]

    20 (d) Solve the equation( )

    3

    2781

    2

    x

    x

    = .

    [x = 1/2]

    20 (e) Solve the equation 125(25)52x = .

    [x = 1/2]

    20 (f) Solve the equation2x1-x (216)(36) += .

    [x = 8]

    20 (g) Solve the equation 03437 3-2x = .[x = 3]

    Halus_2 26

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    20 (h) Solve the equation 0125)5(52x = .

    [x = 1]

    Name: ___________________________

    Form: ________________(F4_R3_2011)

    21 (a) Solve the equationyy 27)9(3 1 =+

    [y = 3]

    yy 27)9(3 1 =+

    yy 3)1(21 3)3(3 =+

    yy 3221 33 =++

    1 + 2y +2 = 3y

    2y + 3 = 3y

    3= y

    21 (b) Solve the equation )2(1641+= nn

    [n = 5]

    21 (c) Solve the equation xxx

    8

    1)2(4 4 =+

    [x = 2/3 ]

    21 (d) Solve the equation1

    1281

    3

    1 + =

    x

    x

    [x = 1/2]

    21 (e) Solve the equation 24)3(2 2 =xx [x = 3]

    21 (f) Solve the equation xx

    3

    1442 2 =+

    [x = 2]

    21 (g) Solve the equation( )( )

    172

    32 12=

    ++ xx

    [x = 1]

    Halus_2 27

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    21 (h) Solve the equation236)3)(2( = xxx

    [x = 4]

    22 (a) Solve the equation 3222 32x = ++ x .[x = 3]

    3222 32x = ++ x

    32)2)(2()2)((2 32x = x

    32)8)(()4)((y = y 324 = y 8=y 82 =x

    322 =x

    x = 3

    22 (b) Solve the equation 12x 292 + = x .[x = 2]

    22 (c) Solve the equation 08033 13y = + y .[y = 1]

    22 (d) Solve the equation 2055 2232x += ++ x .[x = 1/2 ]

    22 (e) Solve the equation 04277 21-m = m .[m = 3]

    22 (f) Solve the equation2322 12y =+ ++ y .

    [y = 2]

    22 (g) Solve the equation 455 21-y += y .[y = 2]

    Halus_2 28

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    22 (h) Solve the equation 01233 122x =+ x .[x = 1]

    23 (a) Solve the equation )3)((279 12x12 + = xx .[x = 2/3]

    )3)((27912x12 +

    =xx

    )3)((33123x)12(2 +

    =xx

    )(3312x3x

    )24(2

    1

    ++=

    x

    15x12 33 + =x

    2x 1 = 5x + 1

    2 = 3x

    2/3 =x

    23 (b) Solve the equation 2y1y 16)8)((2

    1+

    =y .

    [y = 1/6]

    23 (c) Solve the equation2-n

    1-2n

    16

    14 = .

    [n = 1]

    23 (d) Solve the equation 927

    1 2+=

    m

    m .

    [m = 1/2]

    23 (e) Solve the equation 255 12)2( + = xx .[x = 1/3]

    23 (f) Solve the equation 122

    27

    181

    +=

    x

    x.

    [x = 1/8]

    23 (g) Solve the equation 1)9(2732=

    x.

    [x = 3/4]

    Halus_2 29

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    23 (h) Solve the equation ( ) 1416

    1 13=

    +x.

    [x = 1/3]

    24 (a) Solve the equation xx 33 log1)2(log =+ .[x = 1,x = 3]

    xx33

    log1)2(log

    =+ 1log)2(log 33 =++ xx

    1)2(log3 =+ xx

    12log2

    3 =+ xx 12 32 =+ xx 0322 =+ xx 0)1)(3( =+ xx 1,3 == xx

    1=x , rejectx = 3 since logarithm of a

    negative number is not defined.

    24 (b) Solve the equation

    )43(log2)3(log 22 +=+ xx .[x = 16]

    24 (c) Solve the equation

    )3(log)12(log2 22 += yy .[y = 1]

    24 (d) Solve the equation 1)83(loglog 33 =++ yy .[y = 1/3]

    24 (e) Solve the equation

    )1(log)72(log3 22 += pp .[p = 5/2]

    24 (f) Solve the equation 1)13(log2log 55 += pp .[p = 1/7]

    24 (g) Given that 2loglog 22 =+ nm , express m interms ofn.

    [m = 4/n]

    Halus_2 30

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    24 (h) Given that 3loglog2 55 += pqp , expresspin terms ofq.

    [p = 125q]

    25 (a) Solve the equation 27)1(log

    3 2 =+x

    [x = 7]

    27)1(log3 2 =+x

    33

    )1(2

    log3 =

    +x

    3)1(2log =+x

    32)1( =+x 81 =+x 7=x

    25 (b) Solve the equation)12(log

    525

    13 = y

    [y = 5/9]

    25 (c) Solve the equation 8)34(log

    2 5 =x

    [x = 32]

    25 (d) Solve the equation3

    1)1(log3 2 =y

    [y = 1/2]

    25 (e) Solve the equation 827log

    2 =x

    [x = 3]

    25 (f) Solve the equation 364log

    6 =x

    [x = 2]

    25 (g) Solve the equation 343)1(log

    7 2 =+y

    [y= 7]

    Halus_2 31

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    25 (h) Solve the equation2

    1)12(log2 3 =y

    [y = 2/3]

    26 (a) Given that palog3 = ,express

    (i)2

    3log aa , (ii) a3log9

    in terms ofp.[(i)

    p

    p

    +1

    2, (ii)

    2

    1 p+]

    (i)2

    3log aa (ii) a3log9

    aa3log2= 9log

    3log

    3

    3 a=

    )3log

    log(2

    3

    3

    a

    a= 2

    3

    33

    3log

    log3log a+=

    )log3log

    log(2

    33

    3

    a

    a

    +=

    2

    1 p+=

    p

    p

    +=

    1

    2

    26 (b) Given that mylog2 = ,express

    (i)34log yy , (ii) y32log8

    in terms ofm.

    [(i)m

    m32+, (ii)

    3

    5 m+]

    26 (c) Given that py 3log = and qy 5log = ,express

    (i) y45log , (ii) 215log y

    in terms ofp and q.

    [(i)qp +2

    1, (ii) qp +

    2]

    26 (d) Given that kmlog3 = and hnlog3 =,express

    (i) nmm2

    9log , (ii) mnm 3log

    in terms ofkand h.

    [(i) k

    hk

    +

    +

    2

    2, (ii) k

    hk++1]

    26 (e) Given that px 2log = and qx 3log = ,express

    (i)

    3

    log2

    6

    x, (ii)

    3

    3 18log xx

    in terms ofp and q.

    [(i)qp

    q

    +

    2, (ii) 1

    32

    +

    ++

    q

    pq]

    26 (f) Given that mylog5 = and kxlog5 =

    ,express

    (i)

    325

    125log

    y, (ii)

    x

    yx

    25log

    in terms ofm and k.

    Halus_2 32

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    [(i)2

    )1(3 m, (ii)

    k

    km +2]

    27 (a) Given that and are roots of

    0342 =++ xx , form a quadratic equationwith roots

    (i) 2 and 2

    (ii) ( 3) and ( 3)

    (iii) 3

    and 3

    (iv) (2+1) and (2+1)

    [(i) 01282 =++ xx , (ii) 024102 =++ xx ,

    (iii) 0143 2 =++ xx , (iv) 0562

    =++ xx ]

    (i) SOR = 2 + 2 POR= (2)(2)

    = 2( +) = 4

    = 2( 4) = 4(3)

    = 8 = 12

    02 =+ PORSORxx01282 =++ xx

    27 (b) Given that and are roots of

    0452 =+ xx , form a quadratic equationwith roots

    (i) 2 and 2

    (ii) ( 3) and ( 3)

    (iii) 3

    and 3

    (iv) (2+1) and (2+1)

    [(i) 016102 =+ xx , (ii) 022

    =+xx ,

    (iii) 041592

    =+ xx , (iv) 027122

    =+ xx ]

    Halus_2 33

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    27 (c) Given that and are roots of

    0752 2 =+ xx , form a quadratic equationwith roots

    (i) 3 and 3

    (ii) ( 2) and ( 2)

    (iii)2

    and

    2

    (iv) (2 1) and (2 1)

    [(i) 063152 2 =+ xx , (ii) 0111322

    =++ xx ,

    (iii) 071082

    =+ xx , (iv) 0872

    =+ xx ]

    27 (d) Given that and are roots of

    0273 2 =+ xx , form a quadratic equationwith roots

    (i) 4 and 4

    (ii) ( 1) and ( 1)

    (iii)

    2and

    2

    (iv) (2+3) and (2+3)

    [(i) 032283 2 =+ xx , (ii) 023 2 =xx ,

    (iii) 0672

    =+ xx , (iv) 0773232

    =+ xx ]