SPM QUESTIONS3[2]

8/10/2019 SPM QUESTIONS3[2]

1/61

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


2/61

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


3/61

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


4/61

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)


5/61

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 .


6/61

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


7/61

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.


8/61

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)


9/61

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)


10/61

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.


11/61

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


12/61

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 $


13/61

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.


14/61

1.


15/61

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)


16/61

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


17/61

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

'ind in te#$s of k

(b)


18/61

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"


19/61

f#o$ to fied oints K(-1, /) and '(2, ")

in the #atio 2


20/61

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)


21/61

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)


22/61

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.


23/61

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)


24/61


25/61

(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 ()


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


26/61

P2#$%&'ion ()


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


27/61

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&


28/61

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


29/61

'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.


30/61

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.


31/61

"/-&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.


32/61

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$]


33/61

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


34/61

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


35/61

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.


36/61

[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


37/61

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#;"


38/61

(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)


39/61

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


40/61

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/


41/61

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 $.


42/61

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


43/61

[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


44/61

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)


45/61

(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)


46/61

(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"


47/61

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


48/61

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


49/61

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)


50/61

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/


51/61

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


52/61

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


53/61

(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


54/61

(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


55/61

>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


56/61

"%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%


57/61

#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


58/61

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


59/61

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


60/61

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/

"/


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/

Documents

SPM QUESTIONS3[2]