SPM Add Maths Pass Year Questions

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     SPM 1998

      1. Given the functions h(t ) + 2t   5t 2 andv(t ) + 2 6t  

    'ind(a)

    the va%ue of h(t ) hen v(t ) + 11/(b) the va%ues of t so that h(t ) + v-1(2)(c) function hv 

    1. Given the functions f ( x) + " x  5 and

     g ( x) + 2 x  3 , find

    (a)  f g -1( x)

    (b) the va%ue of x so that gf (- x) + 25

     SPM 1999

    1. Given the function f  : x → k  – mx. 'ind  (a) f  -1() in te#$s of k  and m  [2 $a#*s]

      (b) the va%ues of k  and m, if f  -1(14) + - 4and f (5) + -13 [4 $a#*s]

    2. (a) 0he function g  is defined as

     g  : x → x 3. Given the function

       fg  : x → x2 " x  &. 'ind

    (i) function f ( x)

    (ii) the va%ue of k  if f (2k ) + 5k   [& $a#*s]

     SPM 2000

    1. Given the function g  -1( x) +3

    *5− and

      f ( x) + 3 x2 – 5. 'ind(a)  g ( x) [2 $a#*s]

    (b) the va%ue of k hen g ( x2) + 2 f (- x)[3 $a#*s]

      2. Given the function f  : x → 4 – 3 x.  (a) 'ind

      (i)  f 2( x)

      (ii) ( f 2)-1( x)

      (iii) ( f  -1)2 [" $a#*s]

     SPM 2001

    1. Given the function f  : x → ax  b, a  /  and  f 2 : x → 6 x –

      'ind

    (a) the va%ues of a and b [3 $a#*s](b) ( f  -1)2( x) [3 $a#*s]

    2. Given the function f  -1( x) + -

    1

    −, x ! p

      and g ( x) + 3  x. 'ind

      (a) f ( x) [2 $a#*s]

      (b) the va%ue of p if ff  -1( p2 –1) + g [(2- p)2] 

    ( c) #ane of va%ue of p so that fg -1( x) + x

      no #ea% #oots

    [5 $a#*s]

      SPM 2002

    1. Given the function f ( x) + 4 x -2 and g ( x) + 5 x 3. 'ind

    (i)  fg  -1( x)

    (ii) the va%ue of x so that fg -1(2

    ) +

    5

    2

    [5 $a#*s]

    2. (a) Given the function f  : x →3 x  1, find

     f  

    -1

    (5) [2 $a#*s]

      (b) Given the function f ( x) + 5-3 x and g ( x) + 2ax  b, he#e a and b is a

    constants. f  fg ( x) + – 3 x, find the

    va%ues of a and b[3 $a#*s]

    2

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     SPM 2006 

     Paper 1

    1. n dia#a$ 1, set 7 shos the i$ae ofce#tain e%e$ents of set

    >G@A 1

    (a) ia#a$ shos the function x

     xm xh

      −→:

    , /≠ x , he#e m is a constant

      >G@A 2

    'ind the va%ue of m

    [2 $a#*s]

       Paper 2

      1. Given that 23:   −→   x x f    and

    15

    :   +→ x

     x g  , find

      (a) )(1

     x  f    −

    [1 $](b) )(

    1  x g   f    − [2 $]

      ( c) )( xh such that "2)(   +=   x xhg [3 $]

     SPM 2007 

      Paper 1

    1. >ia#a$ 1 shos the %inea#

    function h.

    (a)

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    3. 0he fo%%oin info#$ation is about the

    function h and the co$osite function2h  

    'ind the va%ue of a and b[3$]

     SPM 2008

     Paper 1

    1. >ia#a$ 1 shos the #ah of thefunction 12)(   −=   x x  f   , fo# the

    do$ain 5/   ≤≤  x .

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     SPM 1994

    1. f α and β a#e the #oots of the Cuad#aticeCuation 2 x2 – 3 x – " + /, fo#$ anothe#

    Cuad#atic eCuation ith #oots3

    β  and

    3

    α 

    [4 $a#*s]

     SPM 1995

    1. Dne of the #oots of the eCuation

       x2  px  12 + / is one thi#d of the othe#

    #oot. 'ind the ossib%e va%ues of  p.[5 $a#*s]

    2. Given that2

    1and -5 a#e the #oots of the

    Cuad#atic eCuation. E#ite a Cuad#aticeCuation in a fo#$ ax2 bx  c + /

    [2 $a#*s]

    3. 'ind the #ane of va%ue of k  if the

    eCuation /322 =−++   k kx x  has no  #ea% #oots

    [3 $a#*s]

    4. 8#ove that the #oots of the eCuation

    (1 – p) x2  x  p + / has a #ea% and

    neative #oots if / F p F 1 [5 $a#*s]

     SPM 1996 

    1. Given that a and b a#e the #oots of the

    eCuation x2 – (a  b) x  ab + /.f m and n a#e the #oots of the eCuation

    (2 x – 3)( x  4) k  + / and m + 4n, find

    the va%ue of k 

    [5 $a#*s]

    2. 'ind the va%ues of so that

    (3 – ) x2

     – 2( 1) x  1 + / has toeCua% #ea% #oots.

    [2 $a#*s]

     SPM 1997 

    1. Given that m  2 and n - 1 a#e the #oots

    of the eCuation x2  5 x + -4. 'ind the ossib%e va%ue of m and n. 

     SPM 1998

    1. 0he eCuation of  px2  px  3q + 1 2 x 

    have the #oots p

    1and C

    (a) 'ind the va%ue of  p and q

    (b) Het, b usin the va%ue of p and q in (a)

      fo#$ the Cuad#atic eCuation ith #oots p and -2q

     SPM 1999

    1. Dne of the #oots of the eCuation  2 x2  6x + 2k  - 1 is doub%e of the othe#

    #oot, he#e k  is a constant. 'ind the #ootsand the ossib%e va%ues of k.

    [4 $a#*s]

    2. Given the eCuation x2 – " x  & + h(2 x – 3)

      have to eCua% #ea% #oots. 'ind the va%ues

    of h.[4 $a#*s]

    3. Given that α and β  a#e the #oots of theeCuation x2 – 2 x  k  + /, hi%e 2I and 2J

    a#e the #oots of the eCuation x2 mx 6+/.

      'ind the ossib%e va%ues of k  and m.

      [" $a#*s]

     SPM 2000

    "

    CHAPTER 2: QUADRATIC EQUATIONS

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    1. 0he eCuation 2 x2  px  q + / has the

    #oots -" and 3. 'ind

    (a) the va%ues of p and q [3 $a#*s]  (b) the #ane of va%ues of k  if the

    KCuation 2 x2  px  q + k  has no #ea%

    #oots [2 $a#*s]

     SPM 2001

    1. Given that 2 and m a#e the #oots of theeCuation (2 x -1)( x  3) + k ( x – 1), he#e k 

      is a constant.

    'ind the va%ues of m and k    [4 $a#*s]

    2. f α and β a#e the #oots of the Cuad#atic

    eCuation /132   2 =−+   x x  , fo#$ anothe#Cuad#atic eCuation ith #oots

    3I 2 and 3J 2.[5 $a#*s]

     SPM 2002

    1. Given the eCuation x2  3 + k ( x  1) has

    the #oots p and q, he#e k  is a constant,

    find the #ane of va%ue of k  if the eCuation  has to diffe#ent #ea% #oots.

    [5 $a#*s]

    2. Given that2

    α and

    2

    β a#e the #oots of the

      eCuation kx( x – 1) + 2m – x.

      f α β  + " and αβ ! 3, find the va%uesof k  and m.

    [5 $a#*s]

     SPM 20031.

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    1. 9uad#atic function f ( x) + 2[( x – m)2  n],

    ith m and n a#e constants, have a$ini$u$ oint ("t ,3t 2).

    (a) state the va%ue of m and n in te#$s of t 

    (b) if t  + 1, find the #ane of va%ue of k  sothat the eCuation f ( x) + k  has a distinct

    #oots

    2. 'ind the #ane of va%ues of x if 

      (a) 2(3 x2 – x) M 1 – x

      (b) 4 y – 1 + 5 x and 2 y  3  x

    3. Given that y + x2  2kx  3k   has a

    $ini$u$ va%ue 2.

    (a) Eithout usin diffe#entiation $ethod,

    find to ossib%e va%ue of k .(b) 7 usin the va%ue of k , s*etch the

    #ah y + x2  2kx  3k   in the sa$eais

    (c)

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    c) the eCuation of the ais of

    s$$et# [3 $]

     SPM 2006 

    1. >ia#a$ 3 shos the #ah of Cuad#atic

    function )( x  f   y = . 0he st#aiht %ine4−= y is a tanent to the cu#ve )( x  f   y =

    a) #ite theeCuation

    of the

    ais of

    s$$et# of the cu#ve

     b) e#ess )( x  f    in the fo#$

    cb x   ++   2)( , he#e b and c a#econstants.

    [3 $a#*s]

    3. 'ind the #ane of the va%ues of x fo# x x x   +>+−   4)4)(12(

    [2 $a#*s]

     SPM 2007(paper 1)

    1. 'ind the #ane of va%ues of x fo#

    hich  x x   +≤12   2

    [3 $a#*s]

    2. 0he Cuad#atic function

    42)(  2 −+=   x x x f    can be e#essed

    in the fo#$ nm x x f     −+=   2)()( ,he#e m and n a#e constants.

      'ind the va%ue of m and of n[3 $a#*s]

    nse# m+PPPP..  n+PPPP..

     

     SPM 2008 (paper 1)

    1. 0he Cuad#atic function

    r q x p x f     ++=   2)()( , he#e p, q and r  a#e constants, has a $ini$u$ va%ue of

    -4. 0he eCuation of the ais of s$$et#is x + 3

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     SPM 1993

    1. ia#a$ 2 shos the net of an oened bo ith cuboids shae. f e#i$ete# of

    the net bo is 4 c$ and the tota% su#facea#ea is 135 c$3, La%cu%ate the ossib%e

    va%ues of v and w.

     SPM 1999

    12

      1 $

    1 $ 1$

      1 $

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    1. Given the cu#ve y2 + (1 – x) and the

    st#aiht %ine x

     y+ 4. Eithout d#ain the

    #ah, ca%cu%ate the coo#dinates of the

    inte#section fo# the cu#ve and the st#aiht

    %ine.2.

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    1.

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    2. (a) f h + %o  m  2 and k  + %o  m  3, state in

      te#$s of h and Oo# k 

      (i) %o  m  6

      (ii) %o "  24

    (b)

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      2 %o  3 ( x  y) + 2 %o  3 x  %o  3   y,

    sho that x  2  y  2 + & xy 

    (b) Eithout usin scientific ca%cu%ato# o#

    fou#-fiu#e $athe$atica% tab%es, so%ve

    the eCuation%o  6 [%o  3 (4 x – 5)] + %o  4 2

     

    (c ) fte# n ea# a ca# as bouht the

     #ice of the ca# is @A "/ ///n

       

      

    &.

    La%cu%ate afte# ho $an ea#s i%%the ca# cost %ess than @A 2/ /// fo#

    the fi#st ti$e

     SPM 1998

    1. Given that %o   x 4 + $ and %o   y 5 + y

     

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    1. (a) Given that %o   35  + k . f 5   12   −λ   + 15,

      'ind λ   in te#$s of k

    (b)

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     SPM 1993

    1. 34$ti3ns t3 this q$esti3n by sca4e

    drawing wi44 n3t be accepted8oint 5  and oint  have a coo#dinate of

    (4,1) and (2, 4). 0he st#aiht %ine * is

     e#endicu%a# to 5 cuttin -ais at oint   *. 'ind

      (a) the #adient of 5

      (b) the eCuation of st#aiht %ine *( c) the coo#dinates of *

     SPM 1993

     

    1. '#o$ the above dia#a$, oint & (1, /)and oint (-2, /) a#e the to fied oints.

    8oint 5  $oves such that 5& : 5 + 1:2

    (a) have acoo#dinates (2, 2), (5, 3), (4, -1) and (, C)

    #esective%. Given that 7L> is a

     a#a%%e%o#a$, find(a) the va%ue of and C

    (b) a#ea of 7L>

       SPM 1993

      2. 0he above dia#a$ sho, a

     a#a%%e%o#a$ &' .(a) 'ind the va%ue of  . Nence

    #ite don the eCuation of 

     & in the fo#$ ofinte#cets

    (b) ' is etended to oint 5  

    so that  divides the %ine '5  in the #atio 2 : 3. 'ind

    the coo#dinates of 5 

       SPM 1994

    2. (a)0he above dia#a$, 8, 9 and @a#e th#ee oints a#e on a %ine

    42   =−  x y  he#e 89 : 9@ + 1:4  'ind

    (i) the coo#dinates of oint 8(ii) the eCuation of st#aiht

    %ine assin th#ouh the

     oint 9 and e#endicu%a# ith 8@ 

    (iii) the coo#dinates of oint @ 

    (b) oint < $oves such that its distance

    1

    CHAPTER !: COORDINATE EOMETR"

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     SPM 1997 

    1. n the dia#a$, 7 and 7L a#e tost#aiht %ines that e#endicu%a# to each

    othe# at oint 7. 8oint and oint 7 %ie on

     x-ais and y-ais #esective%. Given the

    eCuation of the st#aiht %ine 7 is

    /623   =−+   x y  (a) 'ind the eCuation of 7L [3$](b) f L7 is #oduced, it i%% inte#sect the x-

    ais at oint @ he#e @7 + 7L. 'ind the

    coo#dinates of oint L [3$]

     2. 0he dia#a$ shos the st#aiht %ine

    #ahs of 89< and 9@0 on the La#tesian

     %ane. 8oint 8 and oint < %ie on the x-ais

    and y-ais #esective%. 9 is the $idoint of8<

    (a) 'ind

    (i) the coo#dinates of oint 9  (ii) the a#ea of Cuad#i%ate#a% D89@ 

    [4$]

    (b)Given 9@:@0 + 1:3, ca%cu%ate thecoo#dinates of oint 0

    (c) oint $ove such that its distance

    f#o$ oint < is2

    1 of its distance f#o$

     oint 0.

    (i) 'ind the eCuation of the

    %ocus of the oint(ii) Nence, dete#$ine hethe#

    the %ocus inte#sects the

     x-ais o# not

     SPM 1998

     1. n the dia#a$, L> and 7LK a#e st#aiht

    %ines. Given L is the $idoint of >, and

    7L : LK + 1:4'ind

    (a) the coo#dinates of oint L

    (b) the coo#dinates of oint K

    (c ) the coo#dinates of the oint ofinte#section beteen %ines 7 and K>

     #oduced

    [3$]2. 8oint 8 $ove such that distance f#o$

     oint 9(/, 1) is the sa$e as its distance

    f#o$ oint @(3, /). 8oint < $ove so thatits distance f#o$ oint 0(3, 2) is 3 units.

    Socus of the oint 8 and < inte#sects at

    to oints.

      (a) 'ind the eCuation of the %ocus of 8

      (b)

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    2. 0he dia#a$ shos a t#aeTiu$ #"+,.

    Given the eCuation of #" is /123   =−−   x y'ind

    (a) the va%ue of k  [3$]

    (b) the eCuation of #, and hence, find

    the coo#dinates of oint # [5$](c) the %ocus of oint 5  such that t#ian%e

     "5, is a%as e#endicu%a# at 5  

    [2$]

     SPM 2001

    1. Given the oints 5 (, /) and (/, -"). 0he

     e#endicu%a# bisecto# of 5 inte#sects the

    aes at # and ".'ind

    (a) the eCuation of #" [3$]

    (b) the a#ea of  #7"∆ , he#e 7 is theo#iin. [2$]

    2. 34$ti3ns t3 this q$esti3n by sca4edrawing wi44 n3t be accepted.

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     SPM 2002

    1. 0he dia#a$ shos a t#ian%e 7L ith

    an a#ea 1 units2 . the eCuation of thest#aiht %ine +" is   ./1 =+−  x y  8oint , %ies on the x-ais and divides the st#aiht

    %ine +" in the #atio m : n. 'ind(a) the coo#dinates of oint "

    (b) m : n 

    2. #(1, 3), " and +  a#e th#ee oints on the

    st#aiht %ine 12   +=   x y . 0his st#aiht %ineis tanent to cu#ve /252 =++   p y x  at oint ". Given " divides the st#aiht %ines #+  in the #atio 1 : 2.

      'ind

    (a) the va%ue of p [3$](b) the coo#dinates of oints " and + 

    [4$]

    (c) the eCuation of the st#aiht %ine that asses th#ouh oint " and is

     e#endicu%a# to the st#aiht %ine #+ 

    [3$]

    3. Given #(-1, -2) and "(2, 1) a#e to fied

     oints. 8oint 5  $oves such that the #atioof  #5  and 5" is 1 : 2.

    (a)

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    >ia#a$ 1

     La%cu%ate the va%ues of  p and q

    [4$]

    P2#$%&'ion ()

      1. s34$ti3ns t3 this q$esti3n by sca4e

    drawing wi44 n3t accepted.  oint 5  $oves a%on the a#c of a

    ci#c%e ith cent#e #(2, 3). 0he a#c asses th#ouh (-2, /) and *(5, k ).

    (a) 'ind

    (i) the eCuation of the %ocus of the  oint 5 

      (ii) the va%ues of k 

    ["$]

    (b) 0he tanent to the ci#c%e at oint

     inte#sects the -ais at oint  .'ind the a#ea of t#ian%e 7 [4$]

     SPM 2004(P1)

    1. >ia#a$ 3 shos a st#aiht %ine #ah of

     x

     y aainst x

    Given that2"   x x y   −= , ca%cu%ate the va%ue

    of k  and of h  [3$]

    2. >ia#a$ 4 shos a st#aiht %ine 89 ith

    the eCuation 132=+  y x

    . 0he oint 8 %ies

    on the x-ais and the oint  %ies on the y-ais

    'ind the eCuationof the st#aiht %ine e#endicu%a# to 5 and

     assin th#ouh the oint

    [3$]

    3. 0he oint # is (-1, 3) and the oint " is

    (4, "). 0he oint 5  $oves such that

     5# : 5" + 2 : 3.'ind the eCuation of the %ocus of 5 

    [3$]

    P2#$%&'ion A)

      4. >i#a$ 1 shos a st#aiht %ine +,hich $eets a st#aiht %ine #" at the

     oint  , . 0he oint +  %ies on the y-ais

    24

     x

     y

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    (a) #ite don the eCuation of #" in the

    fo#$ of inte#cets [1$](b) Given that 2 #, + ,", find the

    coo#dinates of , [2$]

    (c) Given that +, is e#endicu%a# to

     #", find the -inte#cets of +,[3$]

     SPM 2005(P1)

    1. 0he fo%%oin info#$ation #efe#s to the

    eCuations of to st#aiht %ines, %&  and

     * , hich a#e e#endicu%a# to eachothe#.

     

    K#ess p in te#$s of k  [2$]

    P2#$%&'ion ()

    2. 34$ti3ns t3 this q$esti3n by sca4e

    drawing wi44 n3t accepted.

    (a) 'ind

    (i) the eCuation of thest#aiht %ine #"

    (ii) the coo#dinates of "[5$]

    (b) 0he st#aiht %ine #" is etended to a

     oint , such that #" : ", + 2 : 3'ind the coo#dinates of ,

    [2$]

    (c) oint 5  $oves such that its

    distance f#o$ oint # is a%as 5units.

    'ind the eCuation of the %ocus of 5  

    [3$]

     SPM 2006(P1)

    1. >ia#a$ 5 shos the st#aiht %ine #" 

    hich is e#endicu%a# to the st#aiht %ine+" at the oint "

     

    0he eCuation of the st#aiht %ine +" is12   −=   x y

      'ind the coo#dinates of "

    [3 $a#*s]

    25

     %&  :   k  px y   +=

     *  :   p xk  y   +−=   )2(

    he#e p and k  a#e constant

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    P2#$%&'ion ()

    1. 34$ti3ns t3 this q$esti3n by sca4e

    drawing wi44 n3t be accepted  

    >ia#a$ 3 shos the t#ian%e D7 he#e D

    is the o#iin. 8oint L %ies on the st#aiht %ine7

    (a) La%cu%ate the a#ea, in unit2, of

    t#ian%e D7

    (b) Given that L:L7 + 3:2, find thecoo#dinates of L

    (c) oint 8 $oves such that its

    distance f#o$ oint is a%astice its distance f#o$ oint 7

    (i) 'ind the eCuation of the %ocus

    of 8(ii) Nence, dete#$ine hethe# o#

    not this %ocus inte#cets the

    -ais

     SPM 2007 

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    1. >ia#a$ 13 shos a st#aiht %ine assin

    th#ouh  (3,/) and  (/,4)

     

    >ia#a$ 13

    (a) E#ite don the eCuation of the

    st#aiht %ine   in the fo#$

    1=+

    b

     y

    a

     x

    (b) oint 8( x, y) $oves such that

     5  + 5 . 'ind the eCuation of the%ocus of 5  [4 $]

    2. 0he oints (/,3), (2,t ) and (-2,-1) a#e theve#tices of a t#ian%e. Given that the a#ea

    of the t#ian%e is 4 unit2, find the va%ues

    of t .[3 $]

     SPM 2008 ia#a$ shos a t#ian%e 75. 8oint   

    %ies on the %ine 5.

    (a) oint 9  $oves such that its

    distance f#o$ oint   is a%as

    2

    12  

    units. 'ind the eCuation of the %ocus

    of 9   [3$](b) t is iven that oint 5  and oint  

    %ie on the %ocus of 9 . La%cu%ate

    (i) the va%ue of k ,(ii) the coo#dinates of

    [5$]

    (c) Nence, find the a#ea, in unit2, oft#ian%e 75 [2$]

    2&

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     SPM 1993

    1. 0he $ean fo# the nu$be#s ", 2, ", 2, 2,

    1/, x, y is 5

      (a) sho that 12=+  y x

      (b) hence, find the $ode fo# the nu$be#s

    hen

      (i)  y x =

      (ii) y x ≠

      (c) if standa#d deviation is 3&2

    1, find

    the va%ues of x

    2. 0he be%o tab%e shos the $a#*s

    obtained b a #ou of students in a $onth%test .

    Aa#*s 1-2/ 21-4/ 41-"/ "1-/ 1-1//

     Hu$be# 

    students

    5 12 11 4

    (a) Dn a #ah ae#, d#a a histo#a$

    and use it to esti$ate the $oda% $a#*(b) 7 ca%cu%atin the cu$u%ative

    f#eCuenc, find the $edian $a#*,

    ithout d#ain an oive(c) La%cu%ate the $ean $a#*

     SPM 1994

    1. 0he be%o tab%e shos the $a#*s

    obtained b a #ou of students in a $onth%test .

    Aa#*s 1 2 3 4 5

     Hu$be# 

    of

    students

    4 " 2  x 1

    2

    CHAPTER *: STATISTICS

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    'ind

    (a) the $ai$u$ va%ue of x if $oda%

    $a#* is 2(b) the $ini$u$ va%ue of x if $ean

    $a#* $o#e than 3

    (c) the #ane of va%ue of x if $edian$a#* is 2

    2.

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    nu$be#s

    '#eCuenc 1 3 1 2 2 1

    (a) e#ess $edian fo# the set nu$be# inte#$s of m

    (b) 'ind the ossib%e va%ues f m(c) 7 usin the va%ues of m f#o$ (b),

    find the ossib%e va%ues of $ode

    2. (a) 0he fo%%oin data shos the nu$be# 

    of ins *noc*ed don b to %ae#s

    in a #e%i$ina# #ound of bo%inco$etition.

    8%ae# : , 6, , 6, , "

    8%ae# 7: &, , , 6, &, 6Bsin the $ean and the standa#d

    deviation, dete#$ine the bette# %ae# 

    to #e#esent the state based on thei#

    consistenc[3$]

    (b) $se a graph paper t3 answer this

    q$esti3n0he data in the tab%e shos the

    $onth% sa%a# of 1// o#*e#s in a

    co$an.

    (i) 7ased on the data, d#aan oive to sho

    dist#ibution of the

    o#*e#sU $onth% sa%a#(ii) '#o$ ou# #ah,

    esti$ate the nu$be# of

    o#*e#s ho ea#n $o#ethan @A 3 2//

     SPM 1998

    1. 0he $ean of the data 2, k , 3k , , 12 and

    1 hich has been a##aned in anascendin o#de#, is m. f each e%e$ent of

    the data is #educed b 2, the ne $edian

    is5m .

    'ind

    (a) the va%ues of m and k  [4$]

    (b) the va#iance of the ne data [2$]

    2.

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    "/-&6 14

    /-66 5

     

    (a) 7 usin a #ah ae#, d#a a

    histo#a$ and esti$ate the $oda%

    $a#* [4$](b) Eithout d#ain an oive, ca%cu%ate

    the $edian $a#* [3$](c) 'ind the $ean $a#* [3$]

     SPM 2000

    1. 0he tab%e shos the #esu%ts 1// students

    in a test

     

    (a) 7ased on the tab%e above, coco$%ete the tab%e be%o

    [2$]

    (b) Eithout d#ain an oive, esti$atethe inte#Cua#ti%e #ane of this

    dist#ibution.

    [4$]

     2. 0he tab%e shos the dist#ibution of $a#*s

    in a hsics test ta*en b 12/ ui%s.

    La%cu%ate

    (a) the $ean [4$]

    (b) the $edian [3$](c) the standa#d deviation [3$]

     of the dist#ibution

     SPM 2001

    0. (a) Given that fou# ositive intee#s

    have a $ean of 6.Ehen a nu$be#

     y is added to these fou# intee#s,

    the $ean beco$es 1/. 'ind theva%ue of y

    [2$]  (b) 'ind the standa#d deviation of the

    set of nu$be#s be%o:

      5, ", ", 4, &

    [3$]

    2. 0he tab%e shos the f#eCuenc

    dist#ibution of the $a#*s obtained b 1// ui%s

    Mar+$ N-m/%r o0 -i$

    "-1/ 12

    11-15 2/

    1"-2/ 2&

    21-25 1"

    2"-3/ 13

    31-35 1/

    3"-4/ 2(i) La%cu%ate the va#iance [3$]

    (ii) Lonst#uct a cu$u%ative f#eCuenc tab%e

    and d#a an oive to sho the

    dist#ibution of thei# $a#*s. '#o$ theoive, find the e#centae of ui%s hosco#ed beteen " to 24.

    [&$]

     SPM 2002

    1. 0he tab%e shos the dist#ibution of sco#es

    obtained b 6 ui%s in a co$etition. 0he

      sco#es a#e a##aned in an ascendin o#de#.Given the $ean sco#e is and the thi#d

    Cua#ti%e is 11.

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      nu$be# of ui%s in a CuiT. 0he nu$be# of 

      ui%s is 4/. 7 d#ain an oive, find

     7 d#ain an oive, find

    (a) 0he $edian(b) 0he e#centae of ece%%ent ui%s if

    the sco#e fo# the ece%%ent cateo# is

    31.5

     SPM 2003,p2 $%&'ion A

    1. set of ea$ination $a#*s

    "54321   ,,,,,   x x x x x x  has a $ean of 5 and a

    standa#d deviation of 1.5(a) 'ind

    (i) the su$ of the $a#*s,

     x∑

    (ii) the su$ of the sCua#es

    of the $a#*s, 2 x∑

    [3$]

    (b) Kach $a#* is $u%ti%ied b 2 and

    then is added to it.'ind, fo# the ne set of $a#*s,

    (i) the $ean

    (ii) the va#iance

    [4$]

     SPM 2004,p2 $%&'ion A

    1. set of data consist of 1/ nu$be#s. the

    su$ of the nu$be# is 15/ and the su$ of the

    sCua#es of the data is 2 4&2.(a) 'ind the $ean and va#iance of the 1/

    nu$be#s [3]

    (b) nothe# nu$be# is added to the setof data and the $ean is inc#eased b

    1

    'ind

    (i) the va%ue of this nu$be# (ii) the standa#d deviation of the set

    11 nu$be#s

    [4 $a#*s]

     SPM 2005,

     paper 1

    1. 0he $ean of fou# nu$be#s is m . 0he

    su$ of the sCua#es of the nu$be#s is 1//

    and the standa#d deviation is 3k  K#ess m in te#$s of k [3]

     paper 2,section A

    1. >ia#a$ 2 is a histo#a$ hich

    #e#esents the dist#ibution of the $a#*s

    obtained b 4/ ui%s in a test.

    (a) Eithout usin an oive, ca%cu%ate the

    $edian $a#* [3$]

     (b) La%cu%ate the standa#d deviation of the

    dist#ibution [4$]

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     Paper 1

    1. set of data consists of five nu$be#s.0he su$ of the nu$be#s is "/ and the su$

      of the sCua#es of the nu$be#s is //

      'ind fo# the five nu$be#s

    (a) the $ean

    (b) the standa#d deviation[3 $]

     SPM 2008(Paper 1)

     1. set of seven nu$be#s has a $ean of 6

      (a) 'ind  x∑

      (b) Ehen a nu$be# k  is added to this

    set, the ne $ean is .5[3$]

     SPM 2008(Paper 2)

    1. 0ab%e 5 shos the $a#*s obtained b 4/

    candidates in a test.

    Given that the $edian $a#* is 35.5, find the

    va%ue of x and of y. Nence, state the $oda%c%ass

    ["$]

     SPM 1993

    1. 0he dia#a$ shos to a#cs, 5  and *,

    of to ci#c%es ith cent#e 7 and ith #adii

      7  and 7* #esective%. Given the #atio7 :* + 3:1, 'ind

    (a) the an%eθ  

     in #adian(b) the a#ea of the shaded #eion 5* ["$]

     SPM 1994

    1.

      0he dia#a$ shos a se$ici#c%e ith

    cent#e 7 and dia$ete# #7+ . 'ind theva%ue of the an%e θ   (in de#ees and

    $inutes) so that the %enth of a#c of the

    ci#c%e #" sa$e ith the tota% of dia$ete#

     #7+  and %enth of a#c of the ci#c%e "+ 

    34

    CHAPTER : CIRCULAR MEASU

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    [5$]

    2.

    0he dia#a$ shos, D7 is a se$ici#c%eith cent#e > and K7 is a %enth of a#c of

    the secto# ith cent#e L. 0he eCuation of 7

    is 1"12=+

      y x

    La%cu%ate

    (a) the a#ea of  #"+ ∆(b)   #+"∠  in #adians(c) the a#ea of the shaded #eion

     SPM 1997 

    1. (a) Lonve#t

    (i) "4/2/X into #adians(ii) 4.3" #adians into

    de#ees

    [2$](b)

    0he dia#a$ shos to secto#s75 and 7*  of to concent#ic

    ci#c%e ith cent#e D. Givenθ =∠756 #ad, the %enth of a#c 5

    is tice the %enth of #adius 7, and

    the %enth of #adius 7  +"

    'ind(i) the va%ue of θ  

    (ii) the e#i$ete# of the shaded

    #eion

    [4$]2.

    0he dia#a$ sho se$ici#c%e 89@

    ith cent#e D and secto# 9

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    0he dia#a$ shos a secto# '%& ith

    cent#e '  and to secto#s 5%'  and ', ofto ci#c%es ith cent#e 5  and #esective%.

    Given the an%e of $a=o# Y ' is 3." #adians.

    'ind (a) the #adius of secto# '%&

    [2$](b) the e#i$ete# of the shaded

    #eion [2$]

    (c) the a#ea of secto# 5%'  [2$]

    (d) the a#ea of the shaded #eion[4$]

     SPM 1999

    1.

    0he dia#a$ shos the osition of a si$%e endu%u$ that sins f#o$ 5  to . f the

    an%e 57 is / and the %enth of a#c 5 is

    14.4 c$, find(a) the %enth of 7 [3$]

    (b) the a#ea of #eion set b the

     endu%u$[2$]

    2.

     

    0he dia#a$ shos a t#aditiona% Aa%a *ite,

    au bu%an, that has an ais of s$$et# 7*.

    Given that #5" is an a#c of a ci#c%e ithcent#e 7 and #adius 25 c$. #" is a

    se$ici#c%e ith cent#e H and dia$ete# 3/

    c$.   is an a#c of ci#c%e ith cent#e * and#adius 1/ c$. Given that the %enth of a#c

     ,+  is 1.&5 c$.

    La%cu%ate(a)  #7"∠(b) the a#ea of se$ent #;"

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    (b) the an%e θ   in #adians [3$]

    (c) the a#ea of secto# #-+  [2$]

    (d) the a#ea of the shaded #eion[4$]

     SPM 20011.

    0he dia#a$ shos a secto#, 75* of a

    ci#c%e ith cent#e 7 and #adius 5 c$. Given

    the %enth of a#c 5* is &." c$, find(a)  57*∠  in #adians

    [2$]

    (b) the a#ea of the shaded #eion[4$]

    2.

    0he dia#a$ shos a ci#c%e, ", ith

    cent#e 7 and #adius " c$. &7 is an a#c of a

    ci#c%e ith cent#e  . Given #" is a#a%%e% to

     &, #" + " c$ and  &7)∠ + 12//

    (a) 'ind  #7"∠ [1$](b) La%cu%ate the a#ea of se$ent 70

    [4$]

    (c)

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    n the dia#a$, #"+, is a #ectan%e and

    7#-, is a secto# of a ci#c%e ith cent#e 7 

    and #adius " c$. Given 7 is the $idoint of #+ .La%cu%ate

    (a)  #7,∠  in #adians

    [2$](b) the e#i$ete# of the shaded

    #eion [4$]

    ( c) the a#ea of the shaded #eion[4$]

     SPM 2003

    a%r 1

    1 >ia#a$ 1 shos a secto# *7  

    ith cent#e 7

    >ia#a$ 1

    0he %enth of the a#c *  is &.24 c$ and the e#i$ete# of

    the secto# *7  is 25 c$. 'ind the va%ue of θ   in #ad

    [3$]

    a%r 2#$%&'ion A)

    1. >ia#a$ 1 shos the secto# 57, cent#e

    7 ith #adius 1/ c$  0he oint * on 75  is such that

    7* : 75  + 3 : 5

    >ia#a$ 1

    La%cu%ate(a) the va%ue of θ  , in #ad,

    [3$]

    (b) the a#ea of the shaded #eion , inc$2. [4$]

     SPM 2004 a%r 11. >ia#a$ 1 shos a ci#c%e ith cent#e 7

    Given that the %enth of the $a=o# a#c #" is

    45.51 c$, find the %enth, in c$, of the

    #adius.(use π    + 3.142)

    [3$]

    a%r 2#$%&'ion ()

    1. >ia#a$ 4 shos a ci#c%e 5* , cent#e 7  and #adius 5 c$. %&  is a tanent to the

    ci#c%e at . 0he st#aiht %ines, %7 and &7,

      inte#sect the ci#c%e at 5  and * 

    #esective%. 75* is a #ho$bus. %&  is

    36

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      an

    a#c of  

    a

    ci#c%e, cent#e 7 La%cu%ate

    (a) the an%e α  , in te#$s of π  

    [2$]

    (b) the %enth, in c$, of the a#c %&[4$]

    (c) the a#ea, in c$2, of the shaded #eion

    [4$]

     SPM 2005

    a%r 1

    1. >ia#a$ 1 shos a ci#c%e ith cent#e 7

    0he %enth of the $ino# a#c #" is 1" c$ andthe an%e of the $a=o# secto# #7" is 26// .

    Bsin π    + 3.142, find

    (a) the va%ue of θ  , in #adians,

    (Give ou# anse# co##ect to fou#sinificant fiu#es)

    (b) the %enth, in c$, of the #adius of the

    ci#c%e [3$]

    a%r2 #$%&'ion ()

    1. >ia#a$ 1 shos a secto# 57 of aci#c%e, cent#e 7. 0he oint %ies on 75 ,

    the oint 7 %ies on 7 and #" is

     e#endicu%a# to 7.

      0he %enth of 7# + c$ and

    "

    π =∠ 576  #adian

    t is iven that 7# : 75  + 4 : &(Bse 142.3=π  )La%cu%ate

    (a) the %enth, in c$, of #5 

    (b) the e#i$ete#, in c$, of the shaded#eion,

    (c) the a#ea, in c$2, of the shaded #eion

     SPM 2006 

      a%r 1

      1. >ia#a$ & shos secto# 7#" ith

    cent#e D and secto# #=>  ith cent#e a

     >ia#a$ &

      Given that D7 + 1/ c$, Z + 4 c$,

    1.1=∠ =#>   #adians and the %enthsof

    a#c 7 + & c$, ca%cu%ate(c) the va%ue of θ   in #adian

    (d) the a#ea in c$2, of the shaded

    #eion 

    4/

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    SPM 266* a%r 2

    1. >ia#a$ 4 shos a ci#c%e, cent#e D and#adius 1/ c$ insc#ibed in a secto# 87

    of a ci#c%e, cent#e 8. 0he st#aiht %ines,

    8 and 87, a#e tanents to the ci#c%e at oint 9 and oint @, #esective%.

    [use ]142.3=π 

    La%cu%ate

    (a) the %enth, in c$, of the a#c 7

    [5 $]

    (b) the a#ea in c$  2 , of shaded #eion

    [5 $]

    SPM 266 a%r 1

    1. >ia#a$ 1 shos a ci#c%e ith cent#e 7

    and #adius 1/ c$.

      Given that 5 ,  and * a#e oints

    such that 75  + 5 and ∠ 75* + 6//,

    [Bse ]142.3=π   'ind

      (a) ∠ 7*, in #adians (b) the a#ea, in c$2 of the co%ou#ed

    @eion

    [4$]

    SPM 266  a%r 21. >ia#a$ shos to ci#c%es. 0he

    %a#e# ci#c%e has cent#e =  and #adius 12

    c$. 0he s$a%%e# ci#c%e has cent#e >  and

    #adius c$. 0he ci#c%e touch at oint *.0he st#aiht %ine 5 is a co$$on

    tanent to the ci#c%e at oint 5  and oint

    .

    42

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    [use ]142.3=π Given that θ =∠ 5=*  #adian,(a) sho that 3&.1=θ   (to to

    deci$a% %aces) [2$]

    (b) ca%cu%ate the %enth, in c$ of the

    $ino# a#c * [3$](c) ca%cu%ate the a#ea, in c$2, of the

    co%o#ed #eion. [5$]

     SPM 1993

    1 Given that34

    21)(

    2

    −−= x

     x x f   , find f X( x)

     SPM 1994

    1. (a) Given that 53   2 +=   x y , finddx

    dy 

    usin

    the fi#st #inci%e

    (c) 'ind    

      

    +121

     xdx

    2. Given4

    1"

     x y = , find

    dx

    dy if 2= x . Nence,

      esti$ate the va%ue of( )  46.1

    1"

     SPM 1995

    1. Given1

    21)(

    3

    −−

    = x

     x x f    find f ? ( x)

    2. Given )3(   x x y   −= , e#ess

    43

    CHAPTER 7: DIFFERENTATIO

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    (b) La%cu%ate the va%ue of x so that

    the a#ea of the shaded #eion is a

    $ini$u$

    [5$]

     SPM 1998

    1. Given that ,)12(4)(   5−=   x x x  f   find)(

    X x  f  

     

    2.

    0he dia#a$ shos a ooden b%oc*

    consistin of a cone on to of a c%inde#

    ith #adius of x c$. Given the s%ant heihtof the cone is 2 x c$. and the vo%u$e of the

    c%inde# is 24π    c$  3

    a) 8#ove that the tota% su#face a#ea of the b%oc*, c$  2 , is iven b

    +    

       + x

     x  1"

    3   2π   [3$]

     b)La%cu%ate the $ini$u$ su#face a#ea of the

      b%oc* [3$]

    c) Given the su#face a#ea of the b%oc*

    chanes at a #ate of 42π  c$  2 s   1− . 'ind

    the

    of chane of its #adius hen its #adius is

    4 c$. [2$]

    d) Given the #adius of the c%inde# inc#eases

    f#o$ 4 c$ to 4.//3 c$. find thea#oi$ate inc#ease in the su#face a#ea

    of the b%oc* [2$]

      SPM 1999

    1. Given( ) x

     x x f  

    31

    2)(

    52

    −−

    = , find

    )/(X  f   [4$]

    2. Given 22t t  y   −=  and 14   +=   t  x

      (a)'inddx

    dy, in te#$s of x

      (b) f x inc#eases f#o$ 3 to 3./1,

    find the co##esondin s$a%%inc#ease in t . [2$]

    3 (a)

    0he dia#a$ shos a bo ith a unifo#$

    c#oss section #"+,-  .Given #" + -, + (3/-" x)

    c$,  "+  + 3 x c$, +,+ 4 x and #  + 2 c$

    (i)

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    (b) the va%ue of that $a*es

    /  a $ai$u$

    (c) the $ai$u$ va%ue of / 

    3 (b) iece of i#e "/ c$ %on is bent to

    fo#$ a ci#c%e. hen the i#e is heated,its %enth inc#eases at a #ate of /.1 c$ s1−  (use 142.3=π  )(i) La%cu%ate the #ate of chane

    in the #adius of the ci#c%e

    (ii) Nence, ca%cu%ate the #adius of the ci#c%e afte# 4 second

     SPM 2000

    0. >iffe#entiate the fo%%oin e#essionsith #esect to x 

    (a)4

    31   x+ [2$]

    (b)3

    524 ++ x

     x[2$]

    2. Given "43   2 +−=   x x y . Ehen 5= x , x inc#eases b 2. 'ind theco##esondin #ate of chane of y.

    1. 'ind the eCuation of the tanent to

    the cu#ve r  x y   +=   22  at the ointk  x = . f the tanent asses th#ouh

    the oint (1,/), find r  in te#$s of k 

    4.(a) 0he st#aiht %ine k  x y   =+4  is theno#$a% to the cu#ve ( )   312   2 −−=   x y  at oint #.

    'ind

    (i) the coo#dinates of oint # and the

    va%ue of k 

    (ii) the eCuation of the tanent at oint #

    4.(b) 0he dia#a$ shos a to in

    the shae of a se$ici#c%e ith cent#e

    7. >ia$ete# #" can be ad=usted so that

     oint +  hich %ies on theci#cu$fe#ence can $ove such that

     #+   +" + 4/ c$. Given that #+  + x 

    c$ and the a#ea of t#ian%e #"+  is # c$, find an e#essions fo#

    dx

    d) in

    te#$s of x and hence, find the

    $ai$u$ a#ea of t#ian%e #"+ 

     SPM 2001

    1. Givenr 

    r r  f  

    25

    34)(

    −+=  find %i$ited va%ue

    of )(r   f    hen ∞→r 

    2. Given that #ah of function

    2

    3)(

     x

    k hx x f     +=  has #adient function

    3

    2   6"3)(X x

     x x f     −=  he#e h and k  a#e

    constants,

    'ind

    a. the va%ues of h and k b. x-coo#dinate of the

    tu#nin oint of the #ah

    of the function3. (a)

    0he dia#a$ shos a ci#c%e inside#ectan%e 7L> such that the ci#c%e

    is constant% touchin the to sides

    4"

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    of the #ectan%e. Given the e#i$ete#

    of 7L> is 4/ c$

    a. ia#a$ 2 shos a conica% containe#

    4&

     y + 2 x – x2

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    1. cu#ve ith #adient function2

    22 x x −  

    has a tu#nin oint at (k , ) 

    (a) 'ind the va%ue of * [3 $](b) dete#$ine hethe# the tu#nin oint is a$ai$u$ o# $ini$u$ oint

    [2 $]

    ( c) find the eCuation of the cu#ve

    [3 $] SPM 2007 

     Paper 1

    1. 0he cu#ve )( x  f   y =  is such that

    dx

    dy+ 53   +kx , he#e k   is a constant.

      0he #adient of the cu#ve at 2= x  is 6  'ind the va%ue of k 

    [2 $]

    2. 0he cu#ve "4322 +−=   x x y  has a

    $ini$u$ oint at  p x  = , he#e p is aconstant.

      'ind the va%ue of p[3 $]

     SPM 2008

     Paper 11. 0o va#iab%es x and y a#e #e%ated b the

    eCuation2

    1"

     x y = .

      K#ess, in te#$s of h, the a#oi$atechane in y hen x chanes f#o$ 4 to 4

    h, he#e h is a s$a%% va%ue

    [3$]

    2. 0he no#$a% to the cu#ve  x x y   52 −=  at oint 5  is a#a%%e% to the st#aiht %ine

    12+−=   x y . 'ind the eCuation of theno#$a% to the cu#ve at oint 5 .

    [4$]

     SPM 1993

    1.

    0he dia#a$ shos a ∆ 5*

    (a) La%cu%ate obtuse an%e 5* [2$]

    (b)

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    2.

    0he dia#a$ shos a %and fo#$ t#ian%e, #"+ , divide b th#ee a#ts. #,", "+ , and

     #-;+  is a st#aiht %ine

    Given that sin 13

    12

    =∠ "#+ (a) if the fence ant to bui%d a%on the

     bounda# "+ , ca%cu%ate the tota% %enth is

      needed

    (b) La%cu%ate  "+#∠  (c ) Given that the a#ea of  :+;∆  sa$e

    ith the a#ea  #,- ∆ . La%cu%atethe %enth of ;+  

     SPM 19941.

    n the dia#a$, "+, is a st#aiht %ine,ca%cu%ate the %enth of +,

    2.

    0he dia#a$ shos a #a$id ith  #"+ ∆as the ho#iTonta% base. Given that #" + 3

    c$, "+  + 4 c$ and /6/=∠ #"+   andve#te , is 4 c$ ve#tica%% above ",

    ca%cu%ate the a#ea of the s%antin face.

    [5$]

     

     SPM 1995

    1.

    n the dia#a$, sin5

    4=∠ #,+   he#e

     #,+ ∠  is an obtuse an%e. La%cu%ate(a) the %enth of L co##ect to to deci$a%

     %aces [3$]

    (b)  #"+ ∠ [2$]

     SPM 1996 

      1.

    5/

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    0he dia#a$ shos a cuboid. La%cu%ate

    (a)  %6)∠ [4$]

    (b) the a#ea of  %6)∆ [2$]

    2.

    n the dia#a$, oints #, ", + , , and -  %ieon a f%at ho#iTonta% su#face. Given "+, is a

    st#aiht %ine,  #+"∠  is an obtuse an%e andthe a#ea of  #,- ∆ + 2/

    c$2, ca%cu%ate

      (a) the %enth of >

      (b)  ,#- ∠  

     SPM 1997 

    1. 0he dia#a$ shos a t#ian%e #"+  

    La%cu%ate

      (a) the %enth of #"  (b) the ne a#ea of t#ian%e #"+  

    if #+  is %enthened hi%e the

    %enths of #", "+  and  "#+ ∠  a#e $aintained [3$]

     SPM 1998

    1. n the dia#a$, ", + 5 c$, "+  + &c$,

    +, + c$ and #-  + 12 c$, ",-  and

     #,+  a#e a st#aiht %ines. 'ind  (a)  ",+ ∠  (b) the %enth of #,

    2.

    0he dia#a$ shos a #a$id /#"+, itha sCua#e base #"+,. /, is ve#tica% and base

     #"+, is ho#iTonta%. La%cu%ate

    (a) /.8 ∠(b) the a#ea of  %ane /8  

     SPM 1999

    51

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    1.

    0he dia#a$ shos a t#aeTiu$ #"+,

    La%cu%ate(a) +",∠(b) the %enth of st#aiht %ine #+ 

    2. %& is a t#ian%e ith side %&  + 1/ c$.

    Given that sin   45"./=∠ &%)  andsin   3"./=∠ %&) ,

      La%cu%ate  (a)  %)& ∠  (b) the a#ea of  %&)∆

     SPM 2000

    1. 0he dia#a$ shos a cc%ic Cuad#i%ate#a%

    7L>. 0he %enths of st#aiht %ines >L

    and L7 a#e 3 c$ and " c$ #esective%.K#ess the %enth of 7> in te#$s of

    (a) α 

      (b) β 

    Nence, sho that cos2611=α 

    2.

    n the dia#a$, 5* is a st#aiht %ine.

    La%cu%ate the %enth of 5 

     SPM 2001

    1.

    0he dia#a$ shos a #a$id ith at#ianu%a# base 5* hish is on a ho#iTonta%

     %ane. \e#te /  is ve#tica%% above 5 . Given

     5 + 4 c$, 5/  + 1/ c$, /* + 15 c$ and//=∠/6*  

    La%cu%ate(a) the %enth of *(b) the a#ea of the s%antin face

     SPM 2002

    1.

    0he dia#a$ shos a Cuad#i%ate#a% #"+,.Given #, is the %onest side of t#ian%e

     #", and the a#ea of t#ian%e #", is 1/ c$2 

    La%cu%ate

    52

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    (a)  "#,∠(b) the %enth of ",

    ( c) the %enth of "+  

    2.

    0he dia#a$ shos a #is$ ith a unifo#$t#ianu%a# c#oss-section  5 . Given the

    vo%u$e of the #is$ is 315 c$3. 'ind the

    tota% su#face a#ea of the #ectanu%a# faces

    [5$]

     SPM 2003

    1. 0he dia#a$ shos a tent \7L in the

    shae of a #a$id ith t#ian%e 7L as theho#iTonta% base. \ is the ve#te of the tent

    and the an%e beteen the inc%ined %ane

    \7L and the base is 5//

    Given that /" + /+  + 2.2 $ and #" + #+  +

    2." $, ca%cu%ate

    (a) the %enth of "+  if the a#ea of the

     base is 3 $2

    (b) the %enth of #/  and the base is

    25/

    (c ) the a#ea of t#ian%e /#"

     SPM 2004

    1. 0he dia#a$ shos a Cuad#i%ate#a% 7L>  such that  #"+ ∠  is acute

    53

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    (a) La%cu%ate

      (i)  #"+ ∠  (ii)  #,+ ∠

      (iii) the a#ea, in c$2, of Cuad#i%ate#a% #"+,

    [$]

    (b) t#ian%e #?"?+?  has the sa$e

    $easu#e$ents as those iven fo# t#ian%e

     #"+ , that is, #?+?  + 12.3 c$, +?"?  + 6.5c$ and  "∠ X #?+?  + 4/.5/, but hich isdiffe#ent in shae to t#ian%e #"+

    (i) ia#a$ 5 shos a Cuad#i%ate#a% 7L>

    54

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      >ia#a$ 5

    0he a#ea of t#ian%e 7L> is 13 c$2 and "+,∠  is acute

    La%cu%ate

    (a)  "+,∠ [2 $](b) the %enth, in c$, of ", [2 $](c)  #",∠ [3 $](d) the a#ea, in c$2, Cuad#i%ate#a% #"+,

    [3 $]

     SPM 2006 

    1. >ia#a$ & shos Cuad#i%ate#a% #"+,

    i. La%cu%ate

    (a) the %enth, in c$, of #+ 

    (b)  #+"∠ [4 A]ii. 8oint U %ies on L such that

     #U " + #"

    (i) s*etch  #∆ U "+ (ii) ca%cu%ate the a#ea, in

    c$  2 , of  #∆ U "+ [" A]

     SPM 1993

    1. 0he tab%e be%o shos the $onth%

    eenses of %iUs fa$i%

    55

    CHAPTER 11: INDE8 NUM(ER

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      "%ar

    E9%n$%$

    177 1772

    'ood @A 32/ @A 34

    0#anso#tation @A / @A 3

    @enta% @A 2/ @A 322K%ect#icit _ ate#   @A 4/ @A 4/

    'ind the co$osite inde in the ea# 1662 b usin the ea# 166 as the base ea#.

    Nence, if %iUs $onth% inco$e in the ea#

    166 is @A //, find the $onth% inco$e#eCui#ed in the ea# 1662 so that the

    inc#eases in his inco$e is in %ine ith the

    inc#eases in his eenses

    [5$]

     SPM 1994

    1. 0he ie cha#t be%o shos thedist#ibution of the $onth% eenses in the

    ZusnisU househo%d in the ea# 166/. 0he

    tab%e that fo%%os shos the #ice indices inthe ea# 1663 based on the ea# 166/

     

    Mon'.

    %9%n$%$

    Pri&% In%9

    'ood 13/

    Nouse #enta% 115Knte#tain$ent 11/

    L%othin 115

    Dthe#s 13/

    La%cu%ate(a) the co$osite #ice inde, co##ect

      to the nea#est intee#, of the

    $onth% eenses in the ZusnisU

    househo%d

    (b) the tota% $onth% eenses in theea# 1663, co##ect to the nea#est

    #init, if the tota% $onth%

    eenses of the Zus#isU househo%din the ea# 166/ is @A 5/

     SPM 1995

    1. 0he tab%e be%o shos the #ice indices

    and eihtaes of fou# ite$s in the ea#

    1664 based on the ea# 166/. Given the

    co$osite #ice inde in the ea# 1664 is@A 114

     

    La%cu%ate

    (a) the va%ue of n(b) the #ice of a shi#t in 1664 if its

     #ice in 166/ is @A 4/

     SPM 1996 

    1. (a) n the ea# 1665, the #ice and #ice

    inde of a *i%o#a$ of a ce#tain #ade

    of #ice a#e @A 2.4/ and 1"/. Bsin theea# 166/ as the base ea#, ca%cu%ate

    the #ice of a *i%o#a$ of #ice in the

    ea# 166/.[2$]

    (b) 0he above tab%e shos the #ice indices

    in the ea# 1664 usin 1662 as the base

    ea#, chanes to #ice indices f#o$ the

    ea# 1664 to 166" and thei# eihtaes

    I'%m Pri&% In%9 ;%ig'ag%

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      #esective%.

    I'%m Pri&%

    In%9

    1774

    Cang%$ 'o

    Pri&% in%9

    0rom 1774 'o

    177!

    ;%ig'ag%$

    Eood 1/ nc#eases 1/ 5

    Le$ent 11" >ec#eases 5 4

    #on 14/ Ho chane 2

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      eihtaes. Given the #ice of 8 in the

    ea# 166" is @A 12.// and inc#eases to

    @A 13./ in the ea# 1666. 7 usin166 as the base ea#, ca%cu%ate the va%ue

    of x. Nence, find the va%ue of y if the

    co$osite #ice inde is 113

    te$ 8#ice inde Eeihtae

     x 5

    7 6  y

    L 123 14 - y

      SPM 2002

    1. 0he tab%e be%o shos the #ices, #ice

    indices and the nu$be# of th#ee ite$s

    te$

     

    8#ice (@A)

    8#ice

    nde Hu$be#

    Zea# 

    1666

    Zea#

    2///

    (7ase

    ea#

    1666) of ite$s

     A 55 66 120 200

    B 40 x 150 500

    C 80 100 125 y

    (a) 'ind the va%ue x

    (b) f the co$osite #ice inde of the th#ee

    ite$s in the ea# 2/// usin ea# 2/// as  the base ea# is 13".5, find the va%ue of y

    2. 0he tab%e be%o shos the #ices of th#ee

    ite$s , 7 and L in the ea# 166" and

    166, as e%% as thei# eihtaes

    (a) Bsin the ea# 166" as the base ea#,ca%cu%ate the #ice indices of ite$s ,

    7 and L

      (b) Given the co$osite #ice inde ofthese ite$s in the ea# 166 based on

    the ea# 166" is 14/, find the va%ues

    of x and y 

    [5$]

     SPM 2003

    1. 0he dia#a$ be%o sho is a ba# cha#t

    indicatin the ee*% cost of the ite$s 8,

    T.% o0i'%m

    Pri&%#RM) in

    177!

    Pri&%#RM) in

    177

    ;%ig'ag%=

    &/ 1/5 > 

    7 / 1//  = 

    L "/ "&.5/ 2 x

    5

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      9, @, < and 0 fo# the ea# 166/. 0ab%e 1

    shos the #ices and the #ice indices fo#

    the ite$s.

    I'%m$ Pri&% in

    1776

    Pri&% in

    1775

    Pri&% In%9

    in 1775

    /a$% on

    17768  x @A /.&/ 1&5

    9 @A 2.// @A 2.5/ 125

    @ @A 4.// @A 5.5/  y

    < @A ".// @A 6.// 15/

    0 @A 2.5/  z  12/

    (a) 'ind the va%ue of

    (i) x  (ii) y

      (iii) z 

    (b) La%cu%ate the co$osite inde fo# the

    ite$s in the ea# 1665 based on the ea#

    166/

    ( c) 0he tota% $onth% cost of the ite$s in

    the ea# 166/ is @A 45"

    (d) 0he cost of the ite$s inc#eases b 2/

    f#o$ the ea# 1665 to the ea# 2///.

    'ind the co$osite inde fo# the ea#

    2/// based on the ea# 166/

     SPM 2004

    1. 0he tab%e be%o shos the #ice indicesand e#centae of usae of fou# ite$s, 8, 9,

    @ and < hich a#e the $ain in#edients in

    the #oduction of a te of biscuits

    (a) La%cu%ate

    (i) the #ice of   in the ea# 1663 if its

     #ice in the ea# 1665 is @A 3&.&/(ii) the #ice inde of 5  in the ea# 1665

     based on the ea# 1661 if its #ice

    inde in the ea# 1663 based on theea# 1661 is 12/

    [5$]

    (b) 0he co$osite inde nu$be# of the %ostof biscuits #oduction fo# the ea# 1665

     based on the ea# 1663 is 12. La%cu%ate

    (i) the va%ue of

    (ii) the #ice of a bo of biscuits in theea# 1663 if the co##esondin

     #ice in the ea# 1665 is @A 32[5$]

     SPM 2005

    1. 0he tab%e be%o shos the #ices and the

     #ice indices fo# the fou# in#edients 8, 9,@ and < used in $a*in biscuits of a

    te$ 8#ice inde fo# the

    ea# 1665 based on

    the ea# 1663

    8e#centae of

    usae ()

     5  135 4/

    x 3/

     * 1/5 1/

      13/ 2/

    56

    0

    5

    10

    15

    20

    25

    30

    P Q R S

    ITEMS

     

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      a#ticu%a# *ind. >ia#a$ be%o shos a

     ie cha#t hich #e#esents the #e%ative

    a$ount of the in#edients 8, 9, @, and <used in $a*in these biscuits

    n#edients 8#ice e# * 8#ice inde fo# theea# 2//4 based on

    the ea# 2//1Zea# 

    2//1

    Zea#

    2//4

     5  /./ 1.//  x

    2.//  y 14/

     * /.4/ /."/ 15/

    z  /.4/ /

     

    (a) 'ind the va%ue of , and T [3$]

    (b) (i) ca%cu%ate the co$osite inde fo# the

    cost of $a*in these biscuits in theea# 2//4 based on the ea# 2//1

      (ii) Nence, ca%cu%ate the co##esondin

    cost of $a*in these biscuits in theea# 2//1 if the cost in the ea# 2//4

    as @A 265 [5$]

    (c) 0he cost of $a*in these biscuits iseected to inc#ease b 5/ f#o$ the

    ea# 2//4 to the ea# 2//&

      'ind the eected co$osite inde fo#the ea# 2//& based on the ea# 2//1

    [2$]

     SPM 2006 

    1. a#ticu%a# *ind of ca*e is $ade b

    usin fou# in#edients 5 , , * and  .0ab%e shos the #ices of the in#edients

     

    (a) 0he inde nu$be# of in#edient 5  inthe ea# 2//5 based on the ea# 2//4

    is 12/. La%cu%ate the va%ue of w

    [2$] 

    (b) 0he inde nu$be# of in#edient * in

    the ea# 2//5 based on the ea#2//4 is 125. 0he #ice e# *i%o#a$

    of in#edient * in the ea# 2//5 is

    @A 2.// $o#e than its co##esondin

     #ice in the ea# 2//4. 

    La%cu%ate the va%ue of x and of y

    [3$] 

    (c) 0he co$osite inde fo# the costof $a*in the ca*e in the ea#

    2//5 based on the ea# 2//4 is

    12&.5  La%cu%ate

    (i) 0he #ice of a ca*e in the ea# 

      2//4 if its co##esondin #ice  in the ea# 2//5 is @A3/."/

      (ii) the va%ue of $ if the

    Cuantities of in#edients 5 , ,

     * and   used a#e in the #atioof & : 3 : m : 2

    [3$]

     SPM 2007 

    1. 0ab%e 4 shos the #ices and the #ice

    indices of five co$onents, 8, 9, @, < and  0, used to #oduce a *ind of to

    n#edient

    8#ice e# *i%o#a$ (@A)

    Zea# 2//4 Zea# 2//5

     5  5.// 9 

    2.5/ 4.//

     * x > 

      4.// 4.4/

    "/

  • 8/9/2019 SPM Add Maths Pass Year Questions

    61/61

      >ia#a$ " shos a ie cha#t hich

    #e#esents the #e%ative Cuantit of

    co$onents used

    >ia#a$ "

    (a) 'ind the va%ue of and of

    [3$](b) La%cu%ate the co$osite inde fo# the

     #oduction cost of the tos in the

    ea# 2//" based on the ea# 2//4[3$]

    (c) 0he #ice of each co$onent

    inc#eases b 2/ f#o$ the ea#

    2//" to the ea# 2//Given that the #oduction cost of one to

    in the ea# 2//4 is @A 55, ca%cu%ate the

    co##esondin cost in the ea# 2//

    [4$]

    Comon%n' Pri&% #RM) 0or '%

    .%ar

    Pri&% in%9 0or '%

    .%ar 266! /a$% on

    '% .%ar 2664

    8 1.2/ 1.5/ 125

    9  x 2.2/ 11/

    @ 4.// ".// 15/

    < 3.// 2.&/  y

    0 2.// 2./ 14/