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SPM 1993
1. Given the function f :x 3 4x and
functiong:x x2
1, find (a)f -1
(b)f -1g(3) [5 marks]
2. Given the functionsf,g and h as a
f: 2
g: 2
3
,x! 2
h: "x2 2
(i) dete#$ine functionf h(x)
(ii) find the va%ue ofg
-1
(-2)[& marks]
3. 'unction miven that m:x 5 3x2. fpis a anothe# function and mpiven
that mp:x -1 3x2, find functionp.
[3 $a#*s]
SPM 1994
1. Given the functionsf(x) + 2 x and
functiong(x) + kx
2
n. f the co$ositefunctiongf(x) + 3x2 12x , find
(a) the va%ues of kand n [3 $a#*s](b) the va%ue ofg2(/) [2 $a#*s]
2. 0he function f is defined as
f:x23
-
++
, fo# a%% va%ue ofxecet
x+ handpis a constant.
(i) dete#$ine the va%ue of h (ii) the va%ue of 2 $as b itse%f unde#
functionf. 'ind(a) the va%ue ofp(b) the va%ue of anothe#xhich is
$aed onto itse%f
(c) f-1(-1) [& $a#*s]
SPM 1995
1. Given the functionf(x) + 3x cand
inve#se functionf-1(x) + mx34 . 'ind
(a) the va%ue of mand c [3 $a#*s]
(b) (i)f(3)
(ii)f-1f(3)[3 $a#*s]
2. Given the functionf:x mx n, g:x (x 1)2 4 and
fg:x 2(x 1)2 5. 'ind(i) g2(1)(ii)
the va%ues of mand n(iii) gf-1
[5 $a#*s]
SPM 1996
1. Given the functionf:x2
*h
+
,x!2
and inve#se functionf -1:x3
52
,x!3
'ind
(a) the va%ues of hand k [3 $a#*s](b) the va%ues ofxhe#ef(x) + 2x [3 $a#*s]
2. Given the functionf:x 2x 5 andfg:x13 2x, 'ind
(i) functiongf
(ii) the va%ues of cifgf(c2 1) + 5c - "[5 $a#*s]
SPM 1997
1. Given the functionsg: xpx q andg2 :x 25x 4 (a) 'ind the va%ue of p and q (b) ssu$e thatp/, find the
va%ue ofxso that 2g(x) +g(3x 1)
b
1
CHAPTER 1: FUNCTIONS
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SPM 1998
1. Given the functions h(t) + 2t 5t2andv(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 functionsf(x) + "x 5 and
g(x) + 2x 3 , find
(a) f g-1(x)
(b) the va%ue ofxso thatgf(-x) + 25
SPM 1999
1. Given the functionf:x k mx. 'ind (a)f-1() in te#$s of kand m [2 $a#*s]
(b) the va%ues of kand m, iff-1(14) + - 4andf(5) + -13 [4 $a#*s]
2. (a) 0he functiongis defined as
g:xx 3. Given the function
fg:xx2"x &. 'ind
(i) functionf(x)
(ii) the va%ue of kiff(2k) + 5k [& $a#*s]
SPM 2000
1. Given the functiong-1(x) +3
*5and
f(x) + 3x2 5. 'ind(a) g(x) [2 $a#*s]
(b) the va%ue of k heng(x2) + 2f(-x)[3 $a#*s]
2. Given the functionf:x 4 3x. (a) 'ind
(i) f2(x)
(ii) (f2)-1(x)
(iii) (f-1)2 [" $a#*s]
SPM 2001
1. Given the functionf:x ax b, a / and f 2:x 6x
'ind
(a) the va%ues of aand b [3 $a#*s](b) (f-1)2(x) [3 $a#*s]
2. Given the functionf-1(x) +-
1
,x!p
andg(x) + 3 x. 'ind
(a)f(x) [2 $a#*s]
(b) the va%ue ofpifff-1(p21) +g[(2-p)2]
( c) #ane of va%ue ofpso thatfg-1(x) +x
no #ea% #oots
[5 $a#*s]
SPM 2002
1. Given the functionf(x) + 4x-2 andg(x) + 5x3. 'ind
(i) fg-1(x)
(ii) the va%ue ofxso thatfg-1(2
) +
5
2
[5 $a#*s]
2. (a) Given the functionf:x3x 1, find
f
-1
(5) [2 $a#*s]
(b) Given the functionf(x) + 5-3xandg(x) + 2ax b, he#e aand bis a
constants. f fg(x) + 3x, find the
va%ues of aand b[3 $a#*s]
2
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SPM 2003
1. 7ased on the above info#$ation, the#e%ation beteen 8 and 9 is defined b set
of o#de#ed ai#s (1,2), (1,4), (2,"), (2,);.
ia#a$ 1 shos the #e%ation beteen set8 and set 9
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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 functionx
xmxh
:
, /x , he#e mis a constant
>G@A 2
'ind the va%ue of m
[2 $a#*s]
Paper 2
1. Given that 23: xxf and
15
: +x
xg , find
(a) )(1
xf
[1 $](b) )(
1 xgf [2 $]
( c) )(xh such that "2)( += xxhg[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 aand b[3$]
SPM 2008
Paper 1
1. >ia#a$ 1 shos the #ah of thefunction 12)( = xxf , fo# the
do$ain 5/ x .
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SPM 1994
1. f and a#e the #oots of the Cuad#aticeCuation 2x2 3x " + /, fo#$ anothe#
Cuad#atic eCuation ith #oots3
and
3
[4 $a#*s]
SPM 1995
1. Dne of the #oots of the eCuation
x2px 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 kif the
eCuation /322 =++ kkxx has no #ea% #oots
[3 $a#*s]
4. 8#ove that the #oots of the eCuation
(1 p)x2x p+ / has a #ea% and
neative #oots if / FpF 1 [5 $a#*s]
SPM 1996
1. Given that aand ba#e the #oots of the
eCuationx2 (a b)x ab+ /.f mand n a#e the #oots of the eCuation
(2x 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 $a#*s]
SPM 1997
1. Given that m 2 and n- 1 a#e the #oots
of the eCuationx2 5x+ -4. 'ind theossib%e va%ue of mand n.
SPM 1998
1. 0he eCuation of px2px 3q+ 1 2x
have the #ootsp
1and C
(a) 'ind the va%ue of pand q
(b) Het, b usin the va%ue ofpand qin (a)
fo#$ the Cuad#atic eCuation ith #ootspand -2q
SPM 1999
1. Dne of the #oots of the eCuation 2x2 6x+ 2k- 1 is doub%e of the othe#
#oot, he#e kis a constant. 'ind the #ootsand the ossib%e va%ues of k.
[4 $a#*s]
2. Given the eCuationx2 "x & + h(2x 3)
have to eCua% #ea% #oots. 'ind the va%ues
of h.[4 $a#*s]
3. Given that anda#e the #oots of theeCuationx2 2x k+ /, hi%e 2I and 2J
a#e the #oots of the eCuationx2mx6+/.
'ind the ossib%e va%ues of kand m.
[" $a#*s]
SPM 2000
"
CHAPTER 2: QUADRATIC EQUATIONS
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1. 0he eCuation 2x2px q+ / has the
#oots -" and 3. 'ind
(a) the va%ues ofpand q [3 $a#*s] (b) the #ane of va%ues of kif the
KCuation 2x2px q+ khas no #ea%
#oots [2 $a#*s]
SPM 2001
1. Given that 2 and ma#e the #oots of theeCuation (2x-1)(x 3) + k(x 1), he#e k
is a constant.
'ind the va%ues of mand k [4 $a#*s]
2. f and a#e the #oots of the Cuad#atic
eCuation /132 2 =+ xx , fo#$ anothe#Cuad#atic eCuation ith #oots
3I 2 and 3J 2.[5 $a#*s]
SPM 2002
1. Given the eCuationx2 3 + k(x 1) has
the #ootspand q, he#e kis a constant,
find the #ane of va%ue of kif the eCuation has to diffe#ent #ea% #oots.
[5 $a#*s]
2. Given that2
and
2
a#e the #oots of the
eCuation kx(x 1) + 2mx.
f + " and ! 3, find the va%uesof kand m.
[5 $a#*s]
SPM 20031.
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SPM 1993
1. Given the Cuad#atic eCuation
f(x) + "x 1 3x2.(a) K#ess Cuad#atic eCuationf(x) in the
fo#$ k m(x n)2, he#e k, mand n
a#e constants. >ete#$ine hethe# the
functionf(x) has the $ini$u$ o#$ai$u$ va%ue and state the va%ue of
the $ini$u$ o# $ai$u$ va%ue.
(b)
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1. 9uad#atic functionf(x) + 2[(x m)2 n],
ith mand na#e constants, have a$ini$u$ oint ("t,3t2).
(a) state the va%ue of mand nin te#$s of t
(b) if t+ 1, find the #ane of va%ue of ksothat the eCuationf(x) + khas a distinct
#oots
2. 'ind the #ane of va%ues ofxif
(a) 2(3x2x) M 1 x
(b) 4y 1 + 5xand 2y 3 x
3. Given thaty+x2 2kx 3k has a
$ini$u$ va%ue 2.
(a) Eithout usin diffe#entiation $ethod,
find to ossib%e va%ue of k.(b) 7 usin the va%ue of k, s*etch the
#ahy+x2 2kx 3k in the sa$eais
(c)
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has to distinct #oots.
2. Given thaty+p qxx2+k (x h)2fo# a%% va%ues ofx(a) 'ind
(i) h(ii) k
in te#$s ofpandOo# q(b) the st#aiht %iney+ 3 touches the
cu#vey+p qxx2
(i) statepin te#$s of q
(ii) if q+ 2, state the eCuation
of the ais of s$$et# fo#the cu#ve.
Het, s*etch the #ah fo#
the cu#ve
SPM 2003 (paper 2)
1. 0he functionf(x) +x2 4kx 5k2 1has a $ini$u$ va%ue of r2 2k, he#e r
and ka#e constants.
(a) 7 usin the $ethod of co$%etinsCua#e, sho that r+ k-1
[4marks]
(b) Nence, o# othe#ise, find the va%ues
of kand rif the #ah of the functionis s$$et#ica% aboutx+ r2- 1
[4 marks]
SPM 2004 (paper 1)
1. 'ind the #ane of va%ues ofxfo# hich
x(x 4) M 12 [3 $a#*s]
2. >ia#a$ 2 shos the #ah of the
functiony+ -(x k)2 2, he#e kis aconstant.
'ind(a) the va%ue of k
(b) the eCuation of the ais of s$$et#
(c) the coo#dinates of the $ai$u$ oint
[3 $a#*s]
SPM 2005 (paper 1)
1. 0he st#aiht %iney+ 5x 1 does not
inte#sect the cu#vey+ 2x2xp.
'ind the #ane of va%ues ofp[3 $a#*s]
2. >ia#a$ 2 shos the #ah of a
Cuad#atic functionsf(x) + 3(xp)2 2, he#epis a
constant.
0he cu#ve
y+f(x) has the $ini$u$ oint
(1, q), he#e qis a constant. ia#a$ 2 shos the #ah of a Cuad#aticfunctionf(x)+3(x p)2 2, he#e is a
constant
>ia#a$ 2
0he cu#ve + f() has the $ini$u$ oint(1,C), he#e C is a constant.
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c) the eCuation of the ais of
s$$et# [3 $]
SPM 2006
1. >ia#a$ 3 shos the #ah of Cuad#atic
function )(xfy= . 0he st#aiht %ine4=y is a tanent to the cu#ve )(xfy=
a) #ite theeCuation
of the
ais of
s$$et# of the cu#ve
b) e#ess )(xf in the fo#$
cbx ++ 2)( , he#e band ca#econstants.
[3 $a#*s]
3. 'ind the #ane of the va%ues ofxfo#xxx +>+ 4)4)(12(
[2 $a#*s]
SPM 2007(paper 1)
1. 'ind the #ane of va%ues ofxfo#
hich xx +12 2
[3 $a#*s]
2. 0he Cuad#atic function
42)( 2 += xxxf can be e#essed
in the fo#$ nmxxf += 2)()( ,he#e mand n a#e constants.
'ind the va%ue of mand of n[3 $a#*s]
nse# m+PPPP.. n+PPPP..
SPM 2008 (paper 1)
1. 0he Cuad#atic function
rqxpxf ++= 2)()( , he#ep, qand ra#e constants, has a $ini$u$ va%ue of
-4. 0he eCuation of the ais of s$$et#isx+ 3
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SPM 1993
1. ia#a$ 2 shos the net of an oenedbo 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 vand w.
SPM 1999
12
1 $
1 $ 1$
1 $
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1. Given the cu#vey2+ (1 x) and the
st#aiht %inex
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 hand Oo# k
(i) %o m 6
(ii) %o " 24
(b)
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2 %o 3 (xy) + 2 %o 3x %o 3 y,
sho thatx 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 (4x 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 + $and %o y 5 +y
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1. (a) Given that %o 35 + k. f 5 12 + 15,
'ind in te#$s of k
(b)
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SPM 1993
1. 34$ti3ns t3 this q$esti3n by sca4e
drawing wi44 n3t be accepted8oint5and oint have a coo#dinate of
(4,1) and (2, 4). 0he st#aiht %ine *is
e#endicu%a# to5cuttin -ais at oint *. 'ind
(a) the #adient of5
(b) the eCuation of st#aiht %ine *( c) the coo#dinates of*
SPM 1993
1. '#o$ the above dia#a$, oint&(1, /)and oint(-2, /) a#e the to fied oints.
8oint5$oves such that5&: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 oint5
so thatdivides the %ine'5in the #atio 2 : 3. 'ind
the coo#dinates of5
SPM 1994
2. (a)0he above dia#a$, 8, 9 and @a#e th#ee oints a#e on a %ine
42 = xy 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 < $oves such that its distance
1
CHAPTER !: COORDINATE EOMETR"
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f#o$ to fied oints K(-1, /) and '(2, ")
in the #atio 2
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SPM 1997
1. n the dia#a$, 7 and 7L a#e tost#aiht %ines that e#endicu%a# to each
othe# at oint 7. 8oint and oint 7 %ie on
x-ais andy-ais #esective%. Given the
eCuation of the st#aiht %ine 7 is
/623 =+ xy (a) 'ind the eCuation of 7L [3$](b) f L7 is #oduced, it i%% inte#sect thex-
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 thex-ais
andy-ais #esective%. 9 is the $idoint of8 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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3. n the
dia#a$,5(2, 6), (5, &) and*
3,2
14 a#e
$idoints of st#aiht %ines%&,&and%
#esective%, he#e%5*fo#$s aa#a%%e%o#a$.
(a) 'ind
(i) the eCuation of the st#aiht %ine%& (ii) the eCuation of the e#endicu%a#
bisecto# of st#aiht %ine%
[5$]
(b)
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2. 0he dia#a$ shos a t#aeTiu$#"+,.
Given the eCuation of#"is /123 = xy'ind
(a) the va%ue of k [3$]
(b) the eCuation of#,and hence, find
the coo#dinates of oint# [5$](c) the %ocus of oint5such that t#ian%e
"5,is a%as e#endicu%a# at5
[2$]
SPM 2001
1. Given the oints5(, /) and (/, -"). 0he
e#endicu%a# bisecto# of5inte#sects the
aes at#and".'ind
(a) the eCuation of#" [3$]
(b) the a#ea of #7" , he#e 7is theo#iin. [2$]
2. 34$ti3ns t3 this q$esti3n by sca4edrawing wi44 n3t be accepted.
8/10/2019 SPM QUESTIONS3[2]
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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 =+ xy 8oint,%ies on thex-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 += xy . 0his st#aiht %ineis tanent to cu#ve /252 =++ pyx atoint". Given"divides the st#aiht %ines#+in the #atio 1 : 2.
'ind
(a) the va%ue ofp [3$](b) the coo#dinates of oints"and +
[4$]
(c) the eCuation of the st#aiht %ine thatasses th#ouh oint"and is
e#endicu%a# to the st#aiht %ine#+
[3$]
3. Given#(-1, -2) and"(2, 1) a#e to fied
oints. 8oint5$oves such that the #atioof #5and5"is 1 : 2.
(a)
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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#esspin 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$]
(c) oint5$oves such that its
distance f#o$ oint#is a%as 5units.
'ind the eCuation of the %ocus of5
[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 = xy
'ind the coo#dinates of"
[3 $a#*s]
25
%& : kpxy +=
* : pxky += )2(
he#epand ka#e constant
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P2#$%&'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 of5 [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 %ine5.
(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$](b) t is iven that oint5and oint
%ie on the %ocus of 9. La%cu%ate
(i) the va%ue of k,(ii) the coo#dinates of
[5$]
(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,yis 5
(a) sho that 12=+ yx
(b) hence, find the $ode fo# the nu$be#s
hen
(i) yx=
(ii)yx
(c) if standa#d deviation is 3&2
1, find
the va%ues ofx
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 ofxif $oda%
$a#* is 2(b) the $ini$u$ va%ue ofxif $ean
$a#* $o#e than 3
(c) the #ane of va%ue ofxif $edian$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#$s of m
(b) 'ind the ossib%e va%ues f m(c) 7 usin the va%ues of mf#o$ (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 $edian
is5m .
'ind
(a) the va%ues of mand k [4$]
(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#
yis added to these fou# intee#s,
the $ean beco$es 1/. 'ind theva%ue ofy
[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 ,,,,, xxxxxx 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 $a#*s, 2x
[3$]
(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$be#s is 1//
and the standa#d deviation is 3kK#ess min te#$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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SPM 2006
paper 1
1. set of ositive intee#s consists of 2, 5
and m. 0he va#iance fo# this set of
intee#s is 14. 'ind the va%ue of m[3 $a#*s]
paper 2,section A
1. 0ab%e 1 shos the f#eCuencdist#ibution of the sco#es of a #ou of
ui%s in a a$e.
(a) t is iven that the $edian sco#e of
the dist#ibution is 42.
La%cu%ate the va%ue of k[3 $a#*s]
(b) 8se the graph paper t3 answer this
q$esti3nBsin a sca%e of 2 c$ to 1/ sco#es
on the ho#iTonta% ais and 2 c$ to
2 ui%s on the ve#tica% ais, d#a ahisto#a$ to #e#esent the
f#eCuenc dist#ibution of thesco#es.'ind the $ode sco#e
[4 $a#*s]
(c) Ehat is the $ode sco#e if the sco#e
of each ui% is inc#eased b 5W [1 $a#*]
SPM 2007
Paper 2
1. 0ab%e 1 shos the cu$u%ative f#eCuencdist#ibution fo# the sco#es of 32 students in a
co$etition
0ab%e 1
(a) 7ased on tab%e 1, co and co$%ete
0ab%e 2
0ab%e 2[1 $]
(b) Eithout d#ain an oive, find the
inte#Cuati%e #ane of the dist#ibution [5 $]
SPM 2007
S&or% N-m/%r o0 -i$1/-16 1
2/-26 2
3/-36
4/-46 12
5/-56 k
"/-"6 1
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Paper 1
1. set of data consists of five nu$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# kis 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 ofxand ofy. Nence, state the $oda%c%ass
["$]
SPM 1993
1. 0he dia#a$ shos to a#cs,5and *,
of to ci#c%es ith cent#e 7and ith #adii
7and 7*#esective%. Given the #atio7:*+ 3:1, 'ind
(a) the an%e
in #adian(b) the a#ea of the shaded #eion5*["$]
SPM 1994
1.
0he dia#a$ shos a se$ici#c%e ith
cent#e 7and dia$ete##7+. 'ind theva%ue of the an%e (in de#ees and
$inutes) so that the %enth of a#c of the
ci#c%e#"sa$e ith the tota% of dia$ete#
#7+and %enth of a#c of the ci#c%e"+
34
CHAPTER : CIRCULAR MEASU
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2. n the dia#a$, A and H a#e the cente#s
of to con#uent ci#c%es ith #adius rc$#esective%.
a.
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[5$]
2.
0he dia#a$ shos, D7 is a se$ici#c%eith cent#e > and K7 is a %enth of a#c of
the secto# ith cent#e L. 0he eCuation of 7
is 1"12=+
yx
La%cu%ate
(a) the a#ea of #"+(b) #+" in #adians(c) the a#ea of the shaded #eion
SPM 1997
1. (a) Lonve#t
(i) "4/2/X into #adians(ii) 4.3" #adians into
de#ees
[2$](b)
0he dia#a$ shos to secto#s75and 7*of to concent#ic
ci#c%e ith cent#e D. Given=756 #ad, the %enth of a#c5
is tice the %enth of #adius 7, and
the %enth of #adius 7+"
'ind(i) the va%ue of
(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#s5%'and ', ofto ci#c%es ith cent#e5and #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$%eendu%u$ that sins f#o$5to . f the
an%e57is / and the %enth of a#c5is
14.4 c$, 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 7and #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 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 7and #adius 5 c$. Given
the %enth of a#c5*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 7and #adius " c$.&7is 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 7is 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 ofthe a#c*is &.24 c$ and the e#i$ete# of
the secto#*7is 25 c$. 'ind the va%ue ofin #ad
[3$]
a%r 2#$%&'ion A)
1. >ia#a$ 1 shos the secto#57, cent#e
7ith #adius 1/ c$ 0he oint*on 75is such that
7* : 75+ 3 : 5
>ia#a$ 1
La%cu%ate(a) the va%ue of , in #ad,
[3$]
(b) the a#ea of the shaded #eion , inc$2. [4$]
SPM 2004a%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 + 3.142)
[3$]
a%r 2#$%&'ion ()
1. >ia#a$ 4 shos a ci#c%e5*, cent#e 7 and #adius 5 c$.%&is a tanent to the
ci#c%e at . 0he st#aiht %ines,%7and&7,
inte#sect the ci#c%e at5and*
#esective%. 75*is a #ho$bus.%&is
36
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an
a#c of
a
ci#c%e, cent#e 7La%cu%ate
(a) the an%e , in te#$s of
[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 + 3.142, find
(a) the va%ue of , 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#57of aci#c%e, cent#e 7. 0he oint %ies on 75,
the oint 7 %ies on 7and#"is
e#endicu%a# to 7.
0he %enth of 7#+ c$ and
"
=576 #adian
t is iven that 7#: 75+ 4 : &(Bse 142.3= )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 in #adian
(d) the a#ea in c$2, of the shaded
#eion
4/
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a%r 2
1. >ia#a$ 4 shos the %an of aa#den.5+is a se$ici#c%e ith
cent#e 7and has a #adius of $.
*#is secto# of a ci#c%e ith cent#e
and has a #adius of 14 $.
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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 atoint 9 and oint @, #esective%.
[use ]142.3=
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 that5, and*a#e oints
such that 75+5and 75*+ 6//,
[Bse ]142.3= 'ind
(a) 7*, in #adians(b) the a#ea, in c$2of 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 %ine5is a co$$on
tanent to the ci#c%e at oint5and oint
.
42
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[use ]142.3=Given that =5=* #adian,(a) sho that 3&.1= (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
xxf , findf X(x)
SPM 1994
1. (a) Given that 53 2 += xy , finddx
dy
usin
the fi#st #inci%e
(c) 'ind
+121
xdx
d
2. Given4
1"
xy= , find
dx
dyif 2=x . Nence,
esti$ate the va%ue of( ) 46.1
1"
SPM 1995
1. Given1
21)(
3
=x
xxf findf ? (x)
2. Given )3( xxy = , e#ess
43
CHAPTER 7: DIFFERENTATIO
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122
2
++dx
dyx
dx
ydy in te#$s ofx.
Nence, find the va%ue ofxthat satisf the
eCuation 122
2
++dx
dyx
dx
ydy
3. 'ind the coo#dinates at the cu#ve2)52( = xy he#e the #adient of the
no#$a% fo# the cu#ve is4
1
SPM 1996
1. >iffe#entiate &4 )31( xx + ith #esect tox.
2. 0he #adient of the cu#ve2x
khxy += at
the oint
2
&,1 is 2. 'ind the va%ues of
hand k
3. Given 32 = xp and 23
py = . 'ind
(a) the a#oi$ate chane inxif thK
#ate of chane inpis 3 units e#
second
(b)dx
dyin te#$s ofx
(c) the s$a%% chane in , henx
dec#eases f#o$ 2 to 1.6
SPM 1997
1. (a) 'ind the va%ue of
24
2
%i$ 2
n
n
n
(b) Given
5)32()(
= xxf
findf
(x)
2. >iffe#entiate 34=
xy usin the fi#st
#inci%e
3. (a)
0he
dia#a$ shos a containe# in the shae of a
#a$id. 0he sCua#e base of the #a$id has
an a#ea of 3" c$
2
and the heiht of the#a$id is 4 c$. Eate# is ou#ed into the
containe# so that its su#face a#ea is 4p2c$2and its heiht f#o$ the ve#te of the #a$id
is hc$.
(i)
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(b) La%cu%ate the va%ue ofxso that
the a#ea of the shaded #eion is a
$ini$u$
[5$]
SPM 1998
1. Given that ,)12(4)( 5= xxxf find)(
Xxf
2.
0he dia#a$ shos a ooden b%oc*
consistin of a cone on to of a c%inde#
ith #adius ofxc$. Given the s%ant heihtof the cone is 2xc$. and the vo%u$e of the
c%inde# is 24 c$ 3
a) 8#ove that the tota% su#face a#ea of theb%oc*, c$ 2 , is iven b
+
+x
x 1"
3 2 [3$]
b)La%cu%ate the $ini$u$ su#face a#ea of the
b%oc* [3$]
c) Given the su#face a#ea of the b%oc*
chanes at a #ate of 42c$ 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$]
SPM 1999
1. Given( )x
xxf
31
2)(
52
= , find
)/(Xf [4$]
2. Given 22tty = and 14 += tx
(a)'inddx
dy, in te#$s ofx
(b) fxinc#eases f#o$ 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$, "++ 3xc$, +,+ 4xand#+ 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= )(i) La%cu%ate the #ate of chane
in the #adius of the ci#c%e
(ii) Nence, ca%cu%ate the #adius ofthe ci#c%e afte# 4 second
SPM 2000
0. >iffe#entiate the fo%%oin e#essionsith #esect tox
(a)4
31 x+ [2$]
(b)3
524 ++x
x[2$]
2. Given "43 2 += xxy . Ehen 5=x ,xinc#eases b 2. 'ind theco##esondin #ate of chane ofy.
1. 'ind the eCuation of the tanent to
the cu#ve rxy += 22 at the ointkx= . f the tanent asses th#ouh
the oint (1,/), find rin te#$s of k
4.(a) 0he st#aiht %ine kxy =+4 is theno#$a% to the cu#ve ( ) 312 2 = xy atoint#.
'ind
(i) the coo#dinates of oint#and the
va%ue of k
(ii) the eCuation of the tanent atoint#
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 ofxand hence, find the
$ai$u$ a#ea of t#ian%e#"+
SPM 2001
1. Givenr
rrf
25
34)(
+= find %i$ited va%ue
of )(rf hen r
2. Given that #ah of function
2
3)(
x
khxxf += has #adient function
3
2 6"3)(Xx
xxf = he#e hand ka#e
constants,
'ind
a. the va%ues of hand kb. x-coo#dinate of the
tu#nin oint of the #ah
of the function3. (a)
0he dia#a$ 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+ 2xx2
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of dia$ete# /." $ and heiht
/.5$.Eate# is ou#ed into the
containe# at a constant #ate of /.2 $3s-1
La%cu%ate the #ate of chane of the heiht of
the ate# %eve% at the instant hen the heiht
of the ate# %eve% is /.4 $
(use
+ 3.142^ \o%u$e of acone + hr
2
3
1 ) [4$]
SPM 2004
1. >iffe#entiate 42 )52(3 xx ith#esect tox [3$]
2. 0o va#iab%esxandya#e #e%ated b the
eCuationxxy 23 += .
Given that inc#eases at a constant #ate of
4 units e# second, find the #ate of chaneofxhenx+ 2 [3$]
a%r 2#$%&'ion ()
3. 0he #adient function of a cu#ve hich
asses th#ouh (1, -12) is xx "3 2 . 'ind
(a) the eCuation of the cu#ve [3$]
(b) the coo#dinates of the tu#nin oints of the cu#ve and dete#$ine hethe# each
of the tu#nin oints is a $ai$u$ o#
a $ini$u$ [5$]
SPM 2005
1. Given that( )253
1)(
=x
xh , eva%uate
h@ )1( [4$]
2. 0he vo%u$e of ate#, /c$3, in a
containe# is iven b hh/ 31 3 += ,
he#e hc$ is the heiht of the ate# in
the containe#. Eate# is ou#ed into thecontaine# at the #ate of 1/ c$3s-1. 'ind
the #ate of chane of the heiht of ate#,
in c$ s-1, at the instant hen its heihtis 2 c$
[3$]
a%r2#$%&'ionA)
3. cu#ve has a #adient function xpx 42
, he#epis a constant. 0he tanent to the
cu#ve at the oint (1,3) is a#a%%e% to the
st#aiht %ine /5=+ xy . 'ind(a) the va%ue ofp
(b) the eCuation of the cu#ve
SPM 2006
Paper 1
1. 0he oint5%ies on the cu#ve2)5( = xy . t is iven that the
#adient of the no#$a% at5is 4
1
'ind the coo#dinates of5[3$]
2. t is iven that&
3
2$y= , he#e
53 = x$ . 'inddx
dyin te#$s ofx
[4$]
3. Given that 43 2 += xxy
(a) find the va%ue ofdx
dyhenx+1
(b) e#ess the a#oi$ate chane iny,
in te#$s ofp, henxchanes f#o$ 1
to 1 p, he#e is s$a%% va%ue
SPM 2007
Paper 2
4
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1. cu#ve ith #adient function2
22xx
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 )(xfy= 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 += xxy has a
$ini$u$ oint at px = , he#epis aconstant.
'ind the va%ue ofp[3 $]
SPM 2008
Paper 11. 0o va#iab%esxandya#e #e%ated b the
eCuation2
1"
xy= .
K#ess, in te#$s of h, the a#oi$atechane inyhenxchanes f#o$ 4 to 4
h, he#e his a s$a%% va%ue
[3$]
2. 0he no#$a% to the cu#ve xxy 52 = atoint5is a#a%%e% to the st#aiht %ine
12+= xy . 'ind the eCuation of theno#$a% to the cu#ve at oint5.
[4$]
SPM 1993
1.
0he dia#a$ shos a 5*
(a) La%cu%ate obtuse an%e5* [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#$s of
(a)
(b)
Nence, sho that cos2611=
2.
n the dia#a$,5*is a st#aiht %ine.
La%cu%ate the %enth of5
SPM 2001
1.
0he dia#a$ shos a #a$id ith at#ianu%a# base5*hish is on a ho#iTonta%
%ane. \e#te /is ve#tica%% above5. Given
5+ 4 c$,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$2and"+, 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# 1662b 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$ent 11" >ec#eases 5 4
#on 14/ Ho chane 2
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eihtaes. Given the #ice of 8 in the
ea# 166" is @A 12.// and inc#eases to
@A 13./ in the ea# 1666. 7 usin166 as the base ea#, ca%cu%ate the va%ue
ofx. Nence, find the va%ue ofyif the
co$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%uex
(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 ofy
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
ofxandy
[5$]
SPM 2003
1. 0he dia#a$ be%o sho is a ba# cha#t
indicatin the ee*% cost of the ite$s 8,
T.% o0i'%m
Pri&%#RM) in
177!
Pri&%#RM) in
177
;%ig'ag%=
&/ 1/5 >
7 / 1// =
L "/ "&.5/ 2x
5
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9, @, < and 0 fo# the ea# 166/. 0ab%e 1
shos the #ices and the #ice indices fo#
the ite$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 $ain in#edients in
the #oduction of a te of biscuits
(a) La%cu%ate
(i) the #ice of in the ea# 1663 if its
#ice in the ea# 1665 is @A 3&.&/(ii) the #ice inde of5in the ea# 1665
based on the ea# 1661 if its #ice
inde in the ea# 1663 based on theea# 1661 is 12/
[5$]
(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$ount of the in#edients 8, 9, @, and
4.// 4.4/
"/
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>ia#a$ " 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/