CHAPTER 1: FUNCTIONS SPM 1993
SPM 1995
1. Given the function f : x → 3 – 4 x and function g function g : x : x → x2 – 1, find (a) f (a) f -1 (b) f (b) f -1 g (3) [5 marks] marks]
1. Given the function f function f ( x) x) + 3 x x c and 4
inve#se function f function f -1( x) x) + mx mx 3 . 'ind (a) the va%ue of m and c [3 $a#*s] (b) (i) f (i) f (3) (3) (ii) f (ii) f -1 f (3) (3) [3 $a#*s]
2. Given the functions f functions f , g and h as a f : : → 2 3
g : : → − 2 , x ! x ! 2 2 h : → " x – 2 (i) (ii)
2. Given the function f function f : x : x → mx mx n, 2 g : x : x → → ( x x 1) – 4 and fg : x : x → → 2( x x 1)2 – 5. 'ind (i) g 2(1) (ii) the va%ues of m and n (iii) gf -1 [5 $a#*s]
dete#$ine function f h( h( x) x) -1 find th the va%ue of of g (-2) [& marks] marks]
3. 'unction m iven that m : x : x → → 5 – 3 x2 . f p f p is is a anothe# function and mp iven mp iven 2 that mp : mp : x x → → -1 – 3 x , find function p function p.. [3 $a#*s]
SPM 1996
+ * 1. Given the function f function f : x : x → → − 2 , x!2 x!2 2 − 5 and inve#se function f function f -1 : x : x → → − 3 , x!3 x!3 h
SPM 1994
1. Given the functions f functions f ( x) x) + 2 – x – x and 2 function g function g ( x) x) + kx n. f the co$osite function gf function gf ( x) x) + 3 x2 – 12 x x , find (a) the va%ues va%ues of k and and n [3 $a#*s] 2 (b) the va%ue va%ue of of g (/) [2 $a#*s]
'ind (a) the va%ues of h and k [3 $a#*s] (b) the va%ues of x of x he#e he#e f f ( x) x) + 2 x [3 $a#*s] 2. Given the function f function f : x : x → → 2 x x 5 and fg : x : x →13 →13 – 2 x, x, 'ind (i) function gf (ii) the va%ues of c if gf if gf (c2 1) + 5c 5c - " [5 $a#*s]
2. 0he function f is defined as - +
f : x : x → → 3 + 2 , fo# a%% va%ue of x of x ecet ecet x + x + h and p and p is is a constant. (i) dete#$ine the va%ue of h (ii) the va%ue of 2 $as b itse%f unde# function f function f . 'ind (a) the va%ue of p of p (b) the va%ue va%ue of anothe# anothe# x x hich hich is $aed onto itse%f (c) f -1(-1) [& $a#*s]
SPM 1997
1. Given the functions g: functions g: x → x → px px q and 2 g : x→ x→ 25 x x 4 (a) 'ind the va%ue of p and q (b) ssu$e that p that p/, /, find the va%ue of x of x so so that 2 g ( x) x) + g + g (3 (3 x x 1) b
1
SPM 1998
SPM 2001
1. Given the functions h(t ) + 2t 2t 5t 5t 2 and v(t ) + 2 6t 6t 'ind (a) (a) the the va%ue 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 function f function f : x : x → → ax ax b, a / 2 and f : x : x → 6 x – x – 'ind (a) the va%ues of a and b [3 $a#*s] (b) ( f f -1)2( x) x) [3 $a#*s]
−1 ( x) x) + - − , x ! x ! p p
1. Given the functions f functions f ( x) x) + " x x 5 and g ( x) x) + 2 x x 3 , find -1 (a) f g ( x) x) (b) the va%ue va%ue of of x so x so that gf that gf ((- x) x) + 25
2. Given the function f function f -1 and g and g ( x) x) + 3 x x.. 'ind (a) f (a) f ( x) x) [2 $a#*s] -1 2 (b) the va%ue of p of p if if ff ff ( p p –1) + g + g [(2[(2- p) p)2] ( c) #ane of va%ue of p of p so so that fg that fg -1( x) x) + x + x no #ea% #oots [5 $a#*s]
SPM 1999
1. Given the function f function f : x : x → → k – – mx. mx. 'ind -1 (a) f (a) f () in te#$s of k and and m [2 $a#*s] (b) the va%ues of k and and m, if f if f -1(14) + - 4 and f and f (5) (5) + -13 [4 $a#*s]
SPM 2002
1. Given the function f function f ( x) x) + 4 x -2 x -2 and g ( x) x) + 5 x 3. x 3. 'ind (i) fg -1( x) x)
2. (a) 0he function g function g is is defined as : x → → x g : x x 3. Given the function fg : x : x → → x x2 " x x &. 'ind (i) function f ( x) x) (ii) the va%ue of k if f if f (2k (2k ) + 5k 5k [& $a#*s]
(ii)
the va%ue of x so x so that fg that fg -1( 2 ) + 5 [5 $a#*s]
2. (a) Given the function f function f : x : x →3 →3 x x 1, find -1 f (5) [2 $a#*s] (b) Given the function f function f ( x) x) + 5-3 x and x and g ( x) x) + 2ax 2ax b, he#e a and b is a constants. f fg ( x) x) + – 3 x, x, find the va%ues of a and b [3 $a#*s]
SPM 2000 5 − * 3 and 1. Given the function g function g -1( x) x) + 2 f ( x) x) + 3 x – 5. 'ind (a) g ( x) x) [2 $a#*s]
(b) the va%ue va%ue of of k hen g hen g ( x x2) + 2 f ((- x) x) [3 $a#*s]
2
2. Given the function f function f : x : x → → 4 – 3 x. x. (a) 'ind (i) f 2( x) x) 2 -1 (ii) ( f f ) ( x) x) -1 2 (iii) ( f ) [" $a#*s]
2
"
SPM 2003
3. Given the function h( x) x) + , x ! x ! / and the co$osite function hg ( x) x) + 3 x, x, find (a) g ( x) x) (b) the va%ue va%ue of of x so x so that gh that gh(( x) x) + 5 [4 $a#*s]
8 + 1, 2, 3; 9 + 2, 4, ", , 1/; 1. 7ased on the above info#$ation, the #e%ation beteen 8 and 9 is defined b set of o#de#ed ai#s (1,2), (1,4), (2,"), (2,);.
SPM 2005
1. n >ia#a$ 1, the function h $as x $as x to to y y and the function g function g $as y $as y to to z z
2. Given that g that g : x : x → → 5 x x 1 and 2 h : x : x → → x x – 2 x 3, x 3, find -1 (a) g (a) g (3) (b) hg ( x) x) [4 $a#*s] SPM 2004
1. >ia#a$ 1 shos the #e%ation beteen set 8 and set 9
>ete#$ine (a) h-1(5) (b) gh (b) gh((2)
d ? ?
f ?
? ? ? ?
e ?
w x y z
[2 $a#*s]
2. 0he function is defined as 5
w( x) x) + 2 − , x ! x ! 2. 'ind -1 (a) w ( x) x) -1 (b) w (4) [3 $a#*s] 3. 0he fo%%oin info#$ation #efe#s to the functions h and g. and g.
Diagram 1
h : x : x → → 2 x – x – 3 g : x : x → → 4 x x - 1
2. Given the function h : x : x → → 4 x x m and
'ind gh 'ind gh-1
5
[3 $a#*s]
h-1 : x : x → → 2hk 2hk . , he#e m and k a#e constants, find the va%ue of m and of k . [3 $a#*s]
3
SPM 2006 Paper 1 1. n dia#a$ 1, set 7 shos the i$ae of ce#tain e%e$ents of set
Paper 2
1. Given that f : x → 3 x − 2 and g : x
x
→ +1 5
, find
−1
(a) (b)
f ( x)
[1 $]
f −1 g ( x)
[2 $]
( c) h( x) such that hg ( x) = 2 x + " [3 $]
SPM 2007 Paper 1 1. >ia#a$ 1 shos the %inea# function h.
>G@A 1 (a)
ia#a$ shos the function h : x
→
m − x x
, x
≠ / , he#e m is a
constant
(a)
−1
2.
2
Given the function
f : x → x − 3
, find the va%ue of
x such that f ( x) = 5
>G@A 2
[2$] 'ind the va%ue of m [2 $a#*s]
4
(b) the #ane of f ( x) co##esondin to the iven do$ain [3 $]
3. 0he fo%%oin info#$ation is about the function h and the co$osite function h2
2. Given the function g : x → 5 x + 2 2 and h : x → x
2
h : x → 3" x − 35
− 4 x + 3 , find
−1 (") g a)
b) hg ( x) [4$]
3. Given the functions f ( x) = x − 1 and g ( x) = kx + 2 , find
'ind the va%ue of a and b [3$]
a) f (5) b) the va%ue of k such that gf (5)+14 [3$]
SPM 2008 Paper 1
1. >ia#a$ 1 shos the #ah of the function
f ( x) = 2 x − 1
do$ain / ≤ x ≤ 5 .
, fo# the
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 to eCua% #ea% #oots. [2 $a#*s]
SPM 1994
SPM 1997 1. Given that m 2 and n - 1 a#e the #oots of the eCuation x2 5 x + -4. 'ind the ossib%e va%ue of m and n.
1. f α and β a#e the #oots of the Cuad#atic eCuation 2 x2 – 3 x – " + /, fo#$ anothe# β α
SPM 1998
Cuad#atic eCuation ith #oots 3 and 3 [4 $a#*s]
1. 0he eCuation of px2 px 3q + 1 2 x
1
SPM 1995
have the #oots p and C (a) 'ind the va%ue of p and q (b) Het, b usin the va%ue of p and q in (a) fo#$ the Cuad#atic eCuation ith #oots p and -2q
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 $a#*s] 1
SPM 1999
2. Given that 2 and -5 a#e the #oots of the Cuad#atic eCuation. E#ite a Cuad#atic eCuation in a fo#$ ax2 bx c + / [2 $a#*s]
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 #oots and the ossib%e va%ues of k. [4 $a#*s]
3. 'ind the #ane of va%ue of k if the eCuation x #ea% #oots
2
+ kx + 2k − 3 = / has no
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 $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]
3. Given that α and β a#e the #oots of the eCuation 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 1996
1. Given that a and b a#e the #oots of the eCuation x2 – (a b) x ab + /.
SPM 2000
"
2. 0he Cuad#atic eCuation x( x 1) + px - 4 has to distinct #oots. 'ind the #ane of va%ues of p [3 $a#*s] SPM 2004 1. 'o#$ the Cuad#atic eCuation hich has
2
1. 0he eCuation 2 x 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 $a#*s]
1
the #oots -3 and 2 . Give ou# anse# in the fo#$ ax2 bx c +/, he#e a, b and c a#e constants [2 $a#*s] SPM 2005
SPM 2001
1. Given that 2 and m a#e the #oots of the eCuation (2 x -1)( x 3) + k ( x – 1), he#e k is a constant. 'ind the va%ues of m and k [4 $a#*s]
1.
2. f α and β a#e the #oots of the Cuad#atic eCuation 2 x + 3 x − 1 = / , fo#$ anothe# Cuad#atic eCuation ith #oots 3I 2 and 3J 2. [5 $a#*s] 2
SPM 2006
1. Cuad#atic eCuation
x 2 + px + 6 = 2 x has to eCua% #oots. 'ind the ossib%es va%ues of p. [3 $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] α
SPM 2007
1. (a)
+ 5x − 2 = /
β
2. Given that 2 and 2 a#e the #oots of the eCuation kx( x – 1) + 2m – x. f α + β + " and αβ 3, find the va%ues of k and m. [5 $a#*s]
(c) 0he Cuad#atic eCuation
SPM 2003 1.
SPM 2008 1. t is iven that -1 is one of the #oots of the Cuad#atic eCuation
hx 2 + kx + 3 = /, he#e h and k a#e constants, has to eCua% #oots K#ess h in te#$s of k [4 $a#*s]
x 2
− 4 x − p = /
'ind the va%ue of p [2 $a#*s]
&
2. (a) 'ind the #ane of va%ue of x if 5 x ≥ x2 [2 $a#*s] (b) 'ind the #ane of va%ue of p if x2 – ( p 1) x 1 – p2 + / has no #ea% #oots. [3 $a#*s]
SPM 1993
SPM 1995
1. Given the Cuad#atic eCuation f ( x) + " x – 1 – 3 x2. (a) K#ess Cuad#atic eCuation f ( x) in the fo#$ k m( x n)2, he#e k , m and n a#e constants. >ete#$ine hethe# the function f ( x) has the $ini$u$ o# $ai$u$ va%ue and state the va%ue of the $ini$u$ o# $ai$u$ va%ue.
1. Eithout usin diffe#entiation $ethod o# d#ain #ah, find the $ini$u$ o# $ai$u$ va%ue of the function y + 2(3 x – 1)( x 1) – 12 x – 1. 0hen s*etch the #ah fo# the function y. [5 $a#*s] 2. Given that 3 x 2 y – 1 + /, find the #ane of va%ues of x if y F 5. [5 $a#*s]
(b) <*etch the #ah of function f ( x) (c ) 'ind the #ane of va%ue of p so that the eCuation " x – 4 - 3 x2 + p has to diffe#ent #ea% #oots. [1/ $a#*s] SPM 1994
3. 'ind the #ane of va%ues of n if 2n2 n ≥ 1 [2 $a#*s] SPM 1996
1. f ( x) + / is a Cuad#atic eCuation hich has the #oots -3 and p. (a) #ite f ( x) in the fo#$ ax2 bx c [2 $a#*s] (b) Lu#ve y + kf ( x) cut y-ais at the oint (/,"/). Given that p + 5, 'ind (i) the va%ue of k (ii) the $ini$u$ oint [4 $a#*s]
1. n the dia#a$ 1, the $ini$u$ oint is (2, 3) of the function y + p( x h)2 k . 'ind (a) the va%ues of p, h and k (b) the eCuation of the cu#ve hen the #ah is #ef%ected on the x-ais [2 $a#*s]
2. 'ind the #ane of va%ues of x if (a) x( x 1) F 2 [2 $a#*s] −3
(b) 1 − 2 x ≥ x
2. (a) Given that f ( x) + 4 x2 – 1 'ind the #ane of va%ue of x so that f ( x) is a ositive (b) 'ind the #ane of va%ue of x that satisf ineCua%it ( x – 2)2 F ( x – 2)
[3 $a#*s] SPM 1997
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 so that the eCuation f ( x) + k has a distinct #oots
SPM 1999
1. (a) 'ind the #ane of va%ue of x so that 6 2 x 3 and 16 3 x 4 (b) Given that 2 x 3 y + ", find the #ane of va%ue of x hen y F 5 2. 'ind the #ane of va%ue of x if ( x – 2)(2 x 3) ( x – 2)( x 2)
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
SPM 2000 1. Eithout usin diffe#entiation $ethod o# d#ain #ah, dete#$ine the $ini$u$ o# $ai$u$ oint of the function y + 1 2 x – 3 x2. Nence, state the eCuation of the ais of s$$et# fo# the #ah. [4 $a#*s]
2
3. Given that y + x 2kx 3k has a $ini$u$ va%ue 2. (a) Eithout usin diffe#entiation $ethod, find to ossib%e va%ue of k . (b) 7 usin the va%ue of k , s*etch the #ah y + x2 2kx 3k in the sa$e ais (c)
2. 0he st#aiht %ine y + 2 x k does not inte#sect the cu#ve x2 y2 – " +/ . 'ind the #ane of va%ues of k [5 $a#*s]
SPM 1998 1.
SPM 2001
1.(a)
1 2. Given that f -1 ( x) + p − x , x ! p and g ( x) + 3 x. 'ind the #ane of va%ue of p so that f -1 g ( x) + x has no #ea% #oots SPM 2002
1. Given the Cuad#atic eCuation 6
x2 3 + k ( x 1), he#e k is a constant, hich has the #oots p and q. find the #ane of va%ues of k if p and q has to distinct #oots.
(b) the eCuation of the ais of s$$et# (c) the coo#dinates of the $ai$u$ oint [3 $a#*s]
2. Given that y + p qx – x2 + k – ( x h)2 fo# a%% va%ues of x (a) 'ind (!) h (!!) k in te#$s of p andOo# q (b) the st#aiht %ine y + 3 touches the cu#ve y + p qx – x2 (i) state p in te#$s of q (ii) if q + 2, state the eCuation of the ais of s$$et# fo# the cu#ve. Het, s*etch the #ah fo# the cu#ve
SPM 2005 (paper 1)
1. 0he st#aiht %ine y + 5 x – 1 does not inte#sect the cu#ve y + 2 x2 x p. 'ind the #ane of va%ues of p [3 $a#*s] 2. >ia#a$ 2 shos the #ah of a Cuad#atic functions f ( x) + 3( x p)2 2, he#e p is a constant.
SPM 2003 (paper 2) 1. 0he function f ( x) + x2 – 4kx 5k 2 1 has a $ini$u$ va%ue of r 2 2k , he#e r and k a#e constants. (a) 7 usin the $ethod of co$%etin sCua#e, sho that r + k -1 [4 marks] (b) Nence, o# othe#ise, find the va%ues of k and r if the #ah of the function is s$$et#ica% about x + r 2 - 1 [4 marks]
0he cu#ve y + f ( x) has the $ini$u$ oint (1, q), he#e q is a constant. ia#a$ 2 shos the #ah of a Cuad#atic function f ( x)+3( x p)2 2, he#e is a constant
SPM 2004 (paper 1) 1. 'ind the #ane of va%ues of x fo# hich x( x – 4) M 12 [3 $a#*s]
2. >ia#a$ 2 shos the #ah of the function y + -( x – k )2 – 2, he#e k is a constant.
'ind (a) the va%ue ofDiagram k 2
>ia#a$ 2 0he cu#ve + f() has the $ini$u$ oint
1/
a) b)
(1,C), he#e C is a constant.
SPM 2008 (paper 1)
1. 0he Cuad#atic function
f ( x) = p ( x + q ) 2 + r , he#e p, q and r
1. >ia#a$ 3 shos the #ah of Cuad#atic
a#e constants, has a $ini$u$ va%ue of -4. 0he eCuation of the ais of s$$et# is x + 3
function y = f ( x) . 0he st#aiht %ine
y
= −4 is a tanent to the cu#ve y = f ( x)
a)
#ite the eCuation the ais of s$$et# the cu#ve e#ess
of of b)
2. 'ind the #ane of the va%ue of x fo#
( x − 3) 2
f ( x) in
the
( x + b)
< 5 − x . [3 $]
fo#$ 2
SPM 2008 (paper 2) 1. >ia#a$ 2 shos the cu#ve of a
+ c , he#e b and c a#e
2 ( ) = − f x x Cuad#atic function
constants. [3 $a#*s]
0he cu#ve has a $ai$u$ oint at "(2, p) and inte#sects the f ( x)-ais at oint #
3. 'ind the #ane of the va%ues of x fo# (2 x − 1)( x + 4)
+ kx − 5 .
> 4 + x
[2 $a#*s] SPM 2007(paper 1)
>ia#a$
2
a)
the
1. 'ind the #ane of va%ues of x fo# 2 hich 2 x
≤ 1 + x
[3 $a#*s] 2. 0he Cuad#atic function
f ( x) = x 2 + 2 x − 4 can be e#essed 2 ( ) = ( + ) −n, f x x m in the fo#$
he#e m and n a#e constants. 'ind the va%ue of m and of n [3 $a#*s] nse# m+PPPP.. n+PPPP..
coo#dinates of [1$] b) 7 usin the $ethod of co$%etin sCua#e, find the va%ue of k and of p. [4$] c) dete#$ine the #ane of va%ues of x, if f ( x) ≥ −5
[2$]
11
SPM 1993 1.
SPM 1997 1. Given that (3k , -2 p) is a so%ution fo# the si$u%taneous eCuation x – 2 y + 4 and 2
SPM 1994 1.
3
x 2 y +1. 'ind the va%ues of k and p
2. >ia#a$ 2 shos a #ectanu%a# ond %&' and a Cua#te# a#t of a ci#c%e &' ith cent#e ' . f the a#ea of the ond is 1/ π $2 and the %enth %& is %one# than
2. >ia#a$ 2 shos a #ectanu%a# #oo$. shaded #eion is cove#ed b e#i$ete# of a #ectanu%a# ca#et hich is %aced 1 $ aa f#o$ the a%%s of the #oo$. f the a#ea and the e#i$ete# of the ca#et a#e
the %enth of the cu#ve & b π $, 'ind the va%ue of x.
3
4 $2 and 12 $, find the $easu#e$ents of the #oo$. 1$ 1$
1$ 1$
SPM 1998 1.
Diagram 2
x 3
SPM 1995 1.
2 y + 4 , x " y + 3
2. >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#face a#ea is 135 c$3, La%cu%ate the ossib%e va%ues of v and w.
2. cuboids aCua#iu$ $easu#ed $ c$ Q w c$ Q $ c$ has a #ectanu%a# base. 0he to a#t of it is uncove#ed hi%st othe# a#ts a#e $ade of %ass. Given the tota% %enth of the aCua#iu$ is 44/ c$ and the tota% a#ea of the %ass used to $a*e the aCua#iu$ is "3// c$2. 'ind the va%ue of $ and w SPM 1996 1. Given that (-1, 2k ) is a so%ution fo# the eCuation x2 py – 26 + 4 + px – xy , he#e k and p a#e constants. >ete#$ine the va%ue of k and p
12
22
[use π +
&
]
SPM 2002
SPM 1999 1. Given the cu#ve y2 + (1 – x) and the y st#aiht %ine x + 4. Eithout d#ain the #ah, ca%cu%ate the coo#dinates of the inte#section fo# the cu#ve and the st#aiht %ine. 2.
1. Given that x y – 3 + / is a st#aiht %ine cut the cu#ve x2 y2 – xy + 21 at to diffe#ent oint. 'ind the coo#dinates of the oint
2.
x
" y 2 x 3 y + 6 and x R y + R1 SPM 2000 1.
a$
2.
8a* $in has a #ectanu%a# shaes of %and. Ne %anted adi and a$ on the a#eas as shon in the above dia#a$. 0he a$ is %anted on a #ectanu%a# shae a#ea. Given the a#ea of the %and %anted ith adi is 115 $2 and the e#i$ete# of %and %anted ith a$ is 24 $. 'ind the a#ea of %and %anted ith a$.
SPM 2001 1. Given the fo%%oin eCuation: ' + 2 x R y + 3 x 1 * + xy R 'ind the va%ues of x and y so that 2 ' + + *
4. >ia#a$ 2 shos, #", is a iece of ae# in a #ectanu%a# shae. ts a#ea is 2 c$2. #"- is a se$i-ci#c%e shae cut off f#o$ the ae#. the e#i$ete# %eft is 2" c$. 'ind the intee# va%ues of x and y
SPM 2003 1.
13
1.
x
2 y +
1 and y2 R 1/ + 2 x
SPM 2006 1.
Give ou# anse# co##ect to th#ee deci$a% %aces [5 $] SPM 2007 1.
2 x
2
− 1/ x + y + 6 = / [5 $]
SPM 2008 1.
x − 3 y + 4 = / x 2
+ xy − 4/ = / [5$]
14
a#e constants. y + 4 hen x + 2 and y + hen x + 5. 'ind the va%ues of k and m SPM 1995 1.
(a) 1(2&
2x
)+1
t
(b) 5 + 2".3 t
r
2. (a) Given that m + 2 and n + 2 , state in te#$s of r andOo# t SPM 1993
1. f 3 R %o x + 2%o 1/ y, state x in te#$s
(i)
mn3 32 , %o
(ii)
%o m R %o 4 n
2
1/
of y b) 0he te$e#atu#e of a $eta% inc#eased /
m
x
(i) %o 6 (ii) %o 24
m
"
/
/
1
4
2x
eCuation 0 + 3/(1.2) hen the $eta% is heated fo# x seconds. La%cu%ate (i) the te$e#atu#e of the $eta% hen heated fo# 1/.4 seconds (ii) ti$e, in second, to inc#ease the te$e#atu#e of the $eta% f#o$ 3/ L
(b)
/
f#o$ 3/ L to 0 L acco#din to the
2. (a) f h + %o 2 and k + %o 3, state in te#$s of h and Oo# k m
to 15// L
+ 32 SPM 1996
(ii)
%o
x
1" R %o
x
2 + 3
n+ 2
SPM 1994 1.
x + 2
(c)
(b)
1 x
(i) 4
x + 3
+4 2
s
3
) in a
R5+/
+5 x
(ii) 2 . 3 + 5
x +1
(b) Given that %o 5 3 + /."3 and
1
r
%o 2 x x
2. (a) Given that %o n + of n
n −1
2. (a)
(a) %o 3 x %o 6 3 x + R1 x + 4
n
1. (a) K#ess 2 R 2 1/(2 si$%if te#$s
, find the va%ue
%o 5 & + 1.2/6. ithout usin a ca%cu%ato# scientific o# fou#-fiu#e tab%e , ca%cu%ate
t
(b) Given that 2 + 3 + " . K#ess t in te#$s of r and s m
( c) Given that y + kx he#e * and $ 15
2. (a) Given that %o a 3 + x and %o a 5 + y. 45
(i) %o 5 1.4 (ii) %o & &5
K#ess %o SPM 1997
%o 6 y
+
3 a in te#$s of x and y
(b) 'ind the va%ue of %o 4 %o r r (c ) 0o ee#i$ents have been conducted to et #e%ationshi beteen to va#iab%es x and y. 0he eCuation
1.
a
x
y
3(6 ) + 2& and %o 2 y + 2 %o 2 ( x – 2) e#e obtain f#o$ the fi#st and second ee#i$ent #esective%
2
SPM 1999
%o 3 &
2.(a) 'ind the va%ue of 3 ithout usin a scientific ca%cu%ato# o# fou# fiu#e tab%e.
1. Given that %o 2 3 + 1.55 and %o 2 5 + 2.322. Eithout usin scientific ca%cu%ato# o# fou#-fiu#e $athe$atica% tab%es, 'ind
(b)
(a) %o 2 45
5 %o x 3 2 %o x 2 - %o x 324 + 4 and ive ou# anse# co##ect to fou# sinificant fiu#es.
6 5
(b) %o 4 2. (a) Given that x + %o 3, find the va%ue of 2
3. (a) Given that
x
2 %o 3 ( x y) + 2 %o 3 x %o 3 y, 2
y
4 . Nence find the va%ue of 4 if y + 1 x
2
sho that x y + & xy
(b) Given that %o a 3 + /.&624. Eithout usin scientific ca%cu%ato# o# fou#fiu#e $athe$atica% tab%es
(b) Eithout usin scientific ca%cu%ato# o# fou#-fiu#e $athe$atica% tab%es, so%ve the eCuation
#ove that %o a 2&a + 3.3&&2 so%ve the eCuation
(i) (ii)
%o 6 [%o 3 (4 x – 5)] + %o 4 2
3 Q a n−1 + 3
(c ) fte# n ea# a ca# as bouht the SPM 2000
n
& #ice of the ca# is @A "/ ///
1. (a)
. La%cu%ate afte# ho $an ea#s i%% the ca# cost %ess than @A 2/ /// fo# the fi#st ti$e
(b) f 3
x %o a
SPM 1998
1. Given that %o
3 y
x
2x
%o 2 x
+ (2
+ 1 3x
6 +
), #ove that
%o a
%o 12 46 × %o "4 12 %o & 1" 2. (a)
4 + $ and %o y 5 + y
in te#$s of $ andOo# w
Eithout usin scientific ca%cu%ato# o# fou#-fiu#e $athe$atica% tab%es
1"
1. Given that %o 2 . − %o 4 v = 3 , e#ess in te#$s of / [4 $a#*s]
2
(b) Given that 3 % xy + 4 2% y - % x ith the condition x and y is a ositive intee#.
2 x −1 2.
[4 $a#*s]
(c) 0he tota% savins of a cooe#ation afte# n ea#s is iven as
SPM 2004,P1
n
4 x x +" 1.
2///(1 /./&) . La%cu%ate the $ini$u$ nu$be# of ea#s #eCui#ed fo# the savins to eceed @A 4 ///.
%o 5 2 = m
2. Given that SPM 2001
1. Given that %o 2 k + p and
%o 3
k
e#ess
2. (a) Given that %o %o 1/
y
%o 5 4.6
and
in te#$s of m and p
x + 4 1.
− 2 x+3 = 1 [3 $a#*s]
+ 2 and
2.
%o 3 4 x − %o 3 ( 2 x − 1) = 1
+ -1, sho that
xy – 1// y + 6
[3 $a#*s]
(b)
+ 24 3
x
3.Given that
x
%o m 2 = p
and
%o m 3 = r
,
2&m 4 in te#$s of p and r
%o m
(ii) %o 3 +%o 6 ( 5 x + " ) e#ess SPM 2002
3
1. (a) Given that %o 5 + k . f 5
2 λ −1
+ 15,
'ind λ in te#$s of k
SPM 2006
2 x −3
(b)
1.
=
1 4 x + 2
[3 $a#*s]
2 %o x − 4 %o 1" y = 3
2. Given that
4 2. (a) Given that
%o 2 xy
=
2 + 3 %o 2 x − %o 2 y ,
e#ess y in te#$s of x [3 $a#*s]
(b)
and 5 × 125 he#e m and k a#e constants
2 m −1 × 32 k + 2
= 1"
,
SPM 2005,P1
2
(i) 3
%o 5 & = p
+ r
'ind %o k 1 in te#$s of p and r 1/ x
= & x
3− k
=1 3.
2 + %o 3 ( x + 1) = %o 3 x
SPM 2003,P1
1&
[3 $a#*s] SPM 1993 SPM 2007
1. Given that %o 2 = x and %o 2 c
=
y,
b 4 c in te#$s of x and e#ess %o y [4 $a#*s]
6(3 2. Given that
n −1)
= 2& n [3 $a#*s] 1. '#o$ the above dia#a$, oint & (1, /) and oint (-2, /) a#e the to fied oints. 8oint 3 $oves such that 3& : 3 + 1:2 (a)
SPM 2008(paper 1) 1.
= 4 x [3 $]
x 2 + y 2
2. Given that %o 4 x = %o 2 3 , find the va%ue of x. [3 $]
− 4x = /
(b) have a coo#dinates (2, 2), (5, 3), (4, -1) and (, C) #esective%. Given that 7L> is a a#a%%e%o#a$, find (a) the va%ue of and C (b) a#ea of 7L>
SPM 1993 1. 01$t!1ns t1 th!s q$est!1n by scae draw!ng w! n1t be accepted 8oint 3 and oint 4 have a coo#dinate of (4,1) and (2, 4). 0he st#aiht %ine 4* is e#endicu%a# to 34 cuttin -ais at oint *. 'ind (a) the #adient of 34 (b) the eCuation of st#aiht %ine 4* ( c) the coo#dinates of *
SPM 1993
1
in the #atio 2
SPM 1995 1. 01$t!1ns t1 th!s q$est!1n by scae draw!ng w! n1t be accepted.
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#$ of inte#cets (b) ' is etended to oint 3 so that divides the %ine '3 in the #atio 2 : 3. 'ind the coo#dinates of 3 SPM 1994
G#ah on the above sho that the st#aiht %ine SAH 'ind (a) the va%ue of r (b) the eCuation of the st#aiht %ine assin th#ouh oint S and e#endicu%a# ith st#aiht %ine SAH 2. 0he st#aiht %ine y = 4 x − " cuttin the 2 = y x cu#ve
(a) ca%cu%ate (i) the coo#dinates of oint 8 and oint 9 (ii) the coo#dinates of $idoint of 89 (iii) a#ea of t#ian%e D89 he#e 9 is a o#iin
2. (a)0he above dia#a$, 8, 9 and @ a#e th#ee oints a#e on a %ine 2 y − x
'ind (i) (ii)
(iii)
− x − 2 at oint 8 and oint 9
= 4 he#e 89 : 9@ + 1:4 the coo#dinates of oint 8 the eCuation of st#aiht %ine assin th#ouh the oint 9 and e#endicu%a# ith 8@ the coo#dinates of oint @
(b) oint < $oves such that its distance f#o$ to fied oints K(-1, /) and '(2, ")
(b) Given that the oint @(3, k ) %ies on st#aiht %ine 89 (i) the #atio 8@ : @9 (ii) the va%ue of k
SPM 1996
16
SPM 1997
1. n the dia#a$, the st#aiht %ine y
1. n the dia#a$, 7 and 7L a#e to st#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
= 2 x + 3 is the e#endicu%a# bisecto# of
st#aiht %ine hich #e%ates oint 3 (5, &) and oint 4(n, t ) (a) 'ind the $idoint of 34 in te#$s of n and t (b) E#ite to eCuations hich #e%ates t and n ( c) Nence, find the distance of 34
3 y + 2 x − 6 = /
(a) 'ind the eCuation of 7L [3$] (b) f L7 is #oduced, it i%% inte#sect the xais at oint @ he#e @7 + 7L. 'ind the coo#dinates of oint L [3$]
2. 0he dia#a$ shos the ve#tices of a #ectan%e /7 on the La#tesian %ane (a) 'ind the eCuation that #e%ates p and q b usin the #adient of /7 (b) sho that the a#ea of ∆./7 can p −
5
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 $idoint of 8< (a) 'ind (i) the coo#dinates of oint 9 (ii) the a#ea of Cuad#i%ate#a% D89@ [4$] (b)Given 9@:@0 + 1:3, ca%cu%ate the coo#dinates of oint 0
q + 1/
2 be e#essed as ( c) Nence, ca%cu%ate the coo#dinates of oint / , iven that the a#ea of #ectanu%a# /7 is 5 units 2 (d) 'ine the eCuation of the st#aiht %ine in the inte#cet fo#$
2/
(c) oint $ove such that its distance 1
f#o$ oint < is 2 of its distance f#o$ oint 0. (i) 'ind the eCuation of the %ocus of the oint (ii) Nence, dete#$ine hethe# the %ocus inte#sects the x-ais o# not
3. n the dia#a$, 3 (2, 6), 4(5, &) and * 4 1 ,3 2 a#e $idoints of st#aiht %ines %& , & and % #esective%, he#e %34* fo#$s a a#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)
SPM 1998
1. n the dia#a$, L> and 7LK a#e st#aiht %ines. Given L is the $idoint of >, and 7L : LK + 1:4 'ind (a) the coo#dinates of oint L (b) the coo#dinates of oint K (c ) the coo#dinates of the oint of inte#section beteen %ines 7 and K> #oduced [3$] 2. 8oint 8 $ove such that distance f#o$ oint 9(/, 1) is the sa$e as its distance f#o$ oint @(3, /). 8oint < $ove so that its distance f#o$ 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)
(c ) La%cu%ate the a#ea of ∆ 34* and hence, find the a#ea of ∆ %&) [3$] SPM 1999
1. Given oint #( −2,−4) and oint " ( 4,) . 8oint 3 divides the %ine se$ent #" in the #atio 2 : 3. 'ind (a) the coo#dinates of oint 3 (b) the eCuation of st#aiht %ine that is e#endicu%a# to #" and asses th#ouh 3 .
2 2 x + y − " x − 4 y + 4 = / oint < is
produced=diperpanjangkan
( c) La%cu%ate the coo#dinates of the oint of inte#section of the to %ocus (d) 8#ove that the $idoint of the st#aiht %ine 90 is not %ie at %ocus of oint <
21
2. 0he dia#a$ shos the cu#ve
y 2 = 1" − x that inte#sects the x the x--
ais at oint 7 and the y the y-ais -ais at oint # oint # and and , ,..
1. 0he dia#a$ shos a t#ian%e #" t#ian%e #" he#e # he#e # is is on the y the y-ais. -ais. 0he eCuations of the st#aiht %ine #, %ine #,
2. 0he dia#a$ shos a t#aeTiu$ #", t#aeTiu$ #",..
Given the eCuation of #" of #" is is 3 y − 2 x − 1 = / 'ind (a) (a) the the va%ue va%ue of k [3$] (b) the eCuat eCuation ion of of #, and #, and hence, find the coo#dinates of oint # oint # [5$] (c) the %ocu %ocuss of oin ointt 3 such such that t#ian%e "3, is "3, is a%as e#endicu%a# at 3 at 3 [2$] SPM 2001 1. Given the oints 3 oints 3 (, (, /) and 4(/, -"). 0he e#endicu%a# bisecto# of 34 of 34 inte#sects inte#sects the aes at # at # and and " ".. 'ind (a) the eCuati eCuation on of of #" [3$] #5" , he#e 5 is the (b) the a#ea a#ea of ∆ #5" o#iin.
and ", and ", a#e a#e y − 3 x + 1 = / and 3 y + x
−& =
2. 01$t!1ns t1 th!s q$est!1n by scae draw!ng w! n1t be accepted.
/ #esective%.
'ind (a) the coo#di coo#dinates nates of oint oint , , (b) (b) the the #atio #atio #, #, : : , ,
[2$]
'ind (a) the coo#din coo#dinates ates of oints oints # # and and " " [1$] (b) the eCuation eCuation of the st#aiht st#aiht %ines %ines # # and " and " [5$] (c) the coo#dinates coo#dinates of oint [2$] (d) the a#ea a#ea of t#ian%e t#ian%e #" [2$]
1. 0he dia#a$ shos a t#ian%e 7L ith an a#ea 1 units2 . the eCuation of the st#aiht %ine " is " is y − x + 1 = /. 8oint , 8oint , %ies on the x the x-ais -ais and divides the st#aiht %ine " in " in the #atio m : n. 'ind (a) the coo#dinates coo#dinates of oint " oint " (b) m : n
3. n the dia#a dia#a$, $, the eCuatio eCuation n of ", of ", is is y
= −" . oint 8 $oves such that its
2. # 2. #(1, (1, 3), " 3), " and and a#e a#e th#ee oints on the st#aiht %ine y
1
= 2 x + 1 . 0his st#aiht %ine
2 + 5 y + 2 p = / at x is tanent to cu#ve
distance f#o$ # f#o$ # is is a%as 2 the distance of # of # f#o$ f#o$ the st#aiht %ine " %ine " . 'ind (a) the eCuation eCuation of the the %ocus of 3 of 3 (b) the the x-coo#dinates x-coo#dinates of the oint of inte#section of the %ocus and the x-ais x-ais [5$] SPM 2002
oint " oint ".. Given " Given " divides divides the st#aiht %ines # in in the #atio 1 : 2. 'ind (a) (a) the the va%ue va%ue of of p [3$] (b) the coo#dinate coo#dinatess of oints oints " " and and [4$] (c) the eCuation eCuation of the the st#aiht st#aiht %ine that that asses th#ouh oint " oint " and and is e#endicu%a# to the st#aiht %ine # [3$] 3. Given # Given #(-1, (-1, -2) and " and "(2, (2, 1) a#e to fied oints. 8oint 3 8oint 3 $oves $oves such that the #atio of #3 and 3" and 3" is is 1 : 2. (a)
[2$] (b)
23
[2$] (c) 'ind the the eCuation eCuation of the st#aiht st#aiht %ine %ine # [3$] (d) Given the st#aiht st#aiht %ine # %ine # inte#sects inte#sects the %ocus of oint 3 oint 3 at at oint , oint ,.. 'ind the coo#dinates of oint , oint , [3$]
draw!ng w! n1t accepted. oint 3 oint 3 $oves $oves a%on the a#c of a ci#c%e ith cent#e # cent#e #(2, (2, 3). 0he a#c asses th#ouh 4(-2, /) and * and *(5, (5, k ). ). (a) (a) 'ind 'ind (i) the eCuation of the %ocus of the oint 3 oint 3 (ii) the va%ues of k ["$]
SPM 2003(P1) 2003(P1)
1. 0he oints #( 2h, h) , " ( p, t ) and
(b) 0he tanent tanent to the ci#c%e ci#c%e at oint oint 4 inte#sects the -ais at oint . 'ind the a#ea of t#ian%e 54 [4$]
+ ( 2 p,3t ) a#e on a st#aiht %ine. " %ine. " divides divides
# inte#na%% inte#na%% in the #atio 2 : 3 K#ess p K#ess p in in te#$s of t [3$] 2. 0he eCuations of to st#aiht %ines a#e y
+
x
=1
and 5 y = 3 x + 24 . >ete#$ine hethe# the %ines a#e e#endicu%a# to each othe# 5
3
[3$] 3. x and x and y y a#e a#e #e%ated b the eCuation
y = px
2
+ qx , he#e p he#e p and and q a#e
SPM 2004(P1) 2004(P1)
1. >ia#a$ 3 shos a st#aiht %ine #ah of of y x aainst x aainst x
constants. st#aiht st#aiht %ine is obtained y b %ottin x aainst x aainst x,, as shon in >ia#a$ 1. y x
y x
2 y x x " = − Given that , ca%cu%ate the va%ue
>ia#a$ 1
of k and and of h [3$]
La%cu%ate the va%ues of p and p and q [4$] P2"#$%&ion '( 1. s1$t!1ns 1. s1$t!1ns t1 th!s q$est!1n by scae
2. >ia#a$ 4 shos a st#aiht %ine 89 ith
24
x
+
y
(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 each othe#.
=1
the eCuation 2 3 . 0he oint 8 %ies on the x-ais and the oint 4 %ies on the yais
%&
:
y
=
px + k
*
:
y
= (k − 2) x +
p
he#e p and k a#e constant 'ind the eCuation of the st#aiht %ine e#endicu%a# to 34 and assin th#ouh the oint 4 [3$]
K#ess p in te#$s of k
[2$]
3. 0he oint # is (-1, 3) and the oint " is (4, "). 0he oint 3 $oves such that 3# : 3" + 2 : 3. 'ind the eCuation of the %ocus of 3 [3$]
P2"#$%&ion A( 4. >i#a$ 1 shos a st#aiht %ine , hich $eets a st#aiht %ine #" at the oint , . 0he oint %ies on the y-ais
P2"#$%&ion '( 2. 01$t!1ns t1 th!s q$est!1n by scae draw!ng w! n1t accepted.
(a) #ite don the eCuation of #" in the fo#$ of inte#cets [1$] (b) Given that 2 #, + ,", find the coo#dinates of , [2$]
(a) 'ind (i) (ii)
25
the eCuation of the st#aiht %ine #" 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 3 $oves such that its distance f#o$ oint # is a%as 5 (i) units. 'ind the eCuation of the %ocus of 3 (ii) [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 "
(b) Given that L:L7 + 3:2, find the coo#dinates of L (c) oint 8 $oves such that its distance f#o$ oint is a%as tice its distance f#o$ oint 7 'ind the eCuation of the %ocus of 8 Nence, dete#$ine hethe# o# not this %ocus inte#cets the -ais
0he eCuation of the st#aiht %ine " is y
= 2x − 1
'ind the coo#dinates of " SPM 2007
[3 $a#*s] P2"#$%&ion '( 1. 01$t!1ns t1 th!s q$est!1n by scae draw!ng w! n1t be accepted
n dia#a$ 1, the st#aiht %ine #" has an
>ia#a$ 3 shos the t#ian%e D7 he#e D is the o#iin. 8oint L %ies on the st#aiht %ine 7
eCuation y + 2 x + = / . #" inte#sects the -ais at oint and inte#sects the -ais at oint "
(a) La%cu%ate the a#ea, in unit2, of t#ian%e D7
2"
>ia#a$ 13 (a) E#ite don the eCuation of the st#aiht %ine 0 in the fo#$ x a
y b
=1
(b) oint 8( x, y) $oves such that 30 + 3 . 'ind the eCuation of the %ocus of 3 [4 $]
y + 2 x + = /
2. 0he oints (/,3), (2,t ) and (-2,-1) a#e the ve#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 $]
>ia#a$ 1 8oint 3 %ies on 7 such that #3 : 3" + 1:3 'ind (a) the coo#dinates of 3 [3 $] (b) the eCuations of the st#aiht %ine that asses th#ouh 3 and e#endicu%a# to #" [3 $] SPM 2007 (ae# 1) x
+
+ y = 1
1. 0he st#aiht %ine " h has a - inte#cet of 2 and is a#a%%e% to the
st#aiht %ine y + kx = / .>ete#$ine the va%ue of h and of k [3 $a#*s] 2. 0he ve#tices of a t#ian%e a#e #(5,2), "(4,") and (,-2). Given that the a#ea of 2
the t#ian%e is 3/ unit , find the va%ues of p. [3 $a#*s] SPM 2008(paper 1) 1. >ia#a$ 13 shos a st#aiht %ine assin th#ouh 0 (3,/) and (/,4)
SPM 2008 ia#a$ shos a t#ian%e 534. 8oint 0 %ies on the %ine 34.
2&
(a) oint 7 $oves such that its
2
1
distance f#o$ oint 0 is a%as 2 units. 'ind the eCuation of the %ocus of 7 [3$] (b) t is iven that oint 3 and oint 4 %ie on the %ocus of 7 . La%cu%ate (i) the va%ue of k , (ii) the coo#dinates of 4 [5$] 2 (c) Nence, find the a#ea, in unit , of t#ian%e 534 [2$]
SPM 1993 1. 0he $ean fo# the nu$be#s ", 2, ", 2, 2, 1/, x, y is 5
(a) sho that x + y = 12 (b) hence, find the $ode fo# the nu$be#s hen
2
(i)
x = y
(ii) x ≠ y
1
(b) f anothe# nu$be# is added to the 1/ nu$be#s in set , the $ean does not (c) if standa#d deviation is 2 , find chane. 'ind the standa#d deviation the va%ues of x of these nu$be#s. ["$] 2. 0he be%o tab%e shos the $a#*s obtained b a #ou of students in a $onth% SPM 1995 1. (a) Given a %ist of nu$be#s 3, ", 3, . test . 'ind the standa#d deviation of these nu$be# Aa#*s 1-2/ 21-4/ 41-"/ "1-/ 1-1// (b) 'ind a ossib%e set of five intee#s Hu$be he#e its $ode is 3, $edian is 4 and # of 5 12 11 4 student $ean is 5. s Hu$be#s of c%asses Hu$be#s of ui%s " 35 2. (a) (a) Dn a #ah 5 3" 0he tab%e shos the #esu%ts ae#, d#a a 4 3/ of a su#ve of the nu$be# histo#a$ of ui%s in seve#a% c%asses in a schoo%. and use it to esti$ate the $oda% $a#* 'ind (b) 7 ca%cu%atin the cu$u%ative (i) the $ean f#eCuenc, find the $edian $a#*, (ii) the standa#d deviation, ithout d#ain an oive of the nu$be# of ui%s in each c%ass (c) La%cu%ate the $ean $a#* (b) e Hu$be#s of vi%%ae#s SPM 1994 1-2/ 5/ 1. 0he be%o tab%e shos the $a#*s 21-4/ &6 obtained b a #ou of students in a $onth% 41-"/ 4& test . "1-/ 14 1-1// 1/ Aa#*s 1 2 3 4 5 Hu$be # of student s
4
"
2
x
3&
1
'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.
0he tab%e shos the ae dist#ibution of 2// vi%%ae#s. Eithout d#ain a #ah, ca%cu%ate (i) the $edian (ii) the thi#d Cua#ti%e of thei# aes SPM 1996
1. 0he %ist of nu$be#s x − 2, x + 4, 2 x + 5, 2 x − 1, x + & and x − 3 has a $ean of &.'ind (a) the va%ue of x
26
(b) the va#iance
to #e#esent the state based on thei# consistenc [3$] (b) $se a graph paper t1 answer th!s q$est!1n 0he data in the tab%e shos the $onth% sa%a# of 1// o#*e#s in a
["$] 2. Senth ($$) Hu$be#s of fish 2/-26 2 3/-36 3 4/-46 & 5/-56 12 "/-"6 14 &/-&6 6 /-6 3 0he tab%e shos the %enth of nu$be#s of 5/ fish (in $$) (a) ca%cu%ate the $ean %enth (in $$) of the fish (b) d#a an oive to sho the dist#ibution of the %enth of the fish (c) f#o$ ou# #ah, find the e#centae of the nu$be#s of fish hich has a %enth $o#e than 55 $$
Aonth%
SPM 1997 1. 0he tab%e shos a set of nu$be#s hich has been a##aned in an ascendin o#de# he#e m is a ositive intee#
1
m-1
5
m3
1/
1
3
1
2
2
1
(i)
(ii)
Hu$be#s of o#*e#s 1/ 12 1" 22 2/ 12 " 2
7ased on the data, d#a an oive to sho dist#ibution of the o#*e#sU $onth% sa%a# '#o$ ou# #ah, esti$ate the nu$be# of o#*e#s ho ea#n $o#e than @A 3 2//
(a) e#ess $edian fo# the set nu$be# in te#$s of m (b) 'ind the ossib%e va%ues f m (c) 7 usin the va%ues of m f#o$ (b), find the ossib%e va%ues of $ode
2.
SPM 1998 1. 0he $ean of the data 2, k , 3k , , 12 and 1 hich has been a##aned in an ascendin o#de#, is m. f each e%e$ent of the data is #educed b 2, the ne $edian
(a) 0he fo%%oin data shos the nu$be# of ins *noc*ed don b to %ae#s in a #e%i$ina# #ound of bo%in co$etition. 8%ae# : , 6, , 6, , " 8%ae# 7: &, , , 6, &, 6 Bsin the $ean and the standa#d deviation, dete#$ine the bette# %ae#
5m
is . 'ind
3/
(a) the va%ues Aa#*s F1/ F2/ F3/ F4/ F5/ F"/ F&/ F/ of m and k Hu$be [4$] # of 2 21 42 " & 6 1// student (b) the s va#iance of the ne data [2$] test 2.
Σ x and Σ x
Aa# /-6 *s '#eCuenc
Aa#*s
2/-26
Hu$be# of ui%s
3/-36
2
14
4/-46
5/-56
35
5/
2
1&
(a) 7ased on the tab%e above, co co$%ete the tab%e &/-&6 be%o 2 [2$]
(b) Eithout d#ain an oive, esti$ate the inte#Cua#ti%e #ane of this dist#ibution. [4$]
(b) nu$be# of sco#es tota%in 1/ ith a $ean of " and the su$ of the sCua#es of these sco#es of 1 2//, is ta*en out f#o$ set V. La%cu%ate the $ean and va#iance of the #e$ainin sco#es in set V. [&$] SPM 1999 1. 0he set of nu$be#s intee# ositive 2, 3, ", &, 6, x, y has a $ean of 5 and a standa#d deviation of ". 'ind the ossib%e va%ues of x and y
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 $ean [4$] the $edian [3$] the standa#d deviation [3$] of the dist#ibution
SPM 2001 1. (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 the va%ue of y [2$] (b) 'ind the standa#d deviation of the set of nu$be#s be%o: 5, ", ", 4, &
2. 0he f#eCuenc dist#ibution of $a#*s fo# 3/ ui%s ho too* a additiona% $athe$atics test is shon in the tab%e Mar*# 2/-36 46-56 "/-&6 /-66
"/-"6
SPM 2000 1. 0he tab%e shos the #esu%ts 1// students in a
Fr$+,$n%" 5 14 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 $a#* [3$]
[3$] 2. 0he tab%e shos the f#eCuenc dist#ibution of the $a#*s obtained b 1// ui%s Mar*#
31
N,m.$r o/ 0,0i#
"-1/ 11-15 1"-2/ 21-25 2"-3/ 31-35 3"-4/ (i) La%cu%ate the va#iance
1. set of ea$ination $a#*s
12 2/ 2& 1" 13 1/ 2
x1 , x 2 , x 3 , x 4 , x5 , x"
has a $ean of 5 and a standa#d deviation of 1.5 'ind (i) the su$ of the $a#*s, ∑ x (ii) the su$ of the sCua#es
(a)
[3$]
2 of the $a#*s, ∑ x [3$] Kach $a#* is $u%ti%ied b 2 and then is added to it. 'ind, fo# the ne set of $a#*s, (i) the $ean (ii) the va#iance [4$]
(ii) Lonst#uct a cu$u%ative f#eCuenc tab%e (b) and d#a an oive to sho the dist#ibution of thei# $a#*s. '#o$ the oive, find the e#centae of ui%s ho sco#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 SPM 2004,p2 #$%&ion A sco#es a#e a##aned in an ascendin o#de#. 1. set of data consist of 1/ nu$be#s. the Given the $ean sco#e is and the thi#d su$ of the nu$be# is 15/ and the su$ of the Cua#ti%e is 11. sCua#es of the data is 2 4&2. (a) 'ind the $ean and va#iance of the 1/
≤5
≤ 1/
≤ 15
≤ 2/
≤ 25
≤ 3/
≤ 35
Hu$be # of ui%s
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
SPM 2005,
32
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
S%or$ 1/-16 2/-26 3/-36 4/-46 5/-56 "/-"6
N,m.$r o/ 0,0i# 1 2 paper 2,section A 12 1. 0ab%e 1 shos the k f#eCuenc 1 dist#ibution of the sco#es of a #ou of ui%s in a a$e.
[3] paper 2,section A
(a) t is iven that the $edian sco#e of the dist#ibution is 42. La%cu%ate the va%ue of k [3 $a#*s] (b) se the graph paper t1 answer th!s q$est!1n Bsin 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 a histo#a$ to #e#esent the f#eCuenc dist#ibution of the sco#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 $a#*]
1. >ia#a$ 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$]
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 $a#*s]
SPM 2007
33
Paper 2
F1/
F2/
F3/
F4/
F5/
2
21
42
"
1. 0ab%e 1 shos the
SPM 2008(Paper 1) 1. set of seven nu$be#s has a $ean of 6 Mar*# 1/-16 2/-26 3/-36 4/-46 5/-56
cu$u%ative f#eCuenc dist#ibution fo# the sco#es of 32 students in a co$etition
N,m.$r o/ %ania&$# 4 x y 1/
(a) 'ind ∑ x (b) Ehen a nu$be# k is added to this set, the ne $ean is .5 [3$] 0ab%e 1 SPM 2008(Paper 2) 1. 0ab%e 5 shos the $a#*s obtained b 4/ candidates in a test.
(a) 7ased on tab%e 1, co and co$%ete 0ab%e 2 Aa#*s '#eCue nc
/-6
1/-16
2/-26
3/-36
4/-46
Given that the $edian $a#* is 35.5, find the va%ue of x and of y. Nence, state the $oda% c%ass ["$]
0ab%e 2 [1 $] (b) Eithout d#ain an oive, find the inte#Cuati%e #ane of the dist#ibution [5 $] SPM 2007 Paper 1
1. set of data consists of five nu$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 $]
34
SPM 1993
2. n the dia#a$, A and H a#e the cente#s of to con#uent ci#c%es ith #adius r c$ #esective%. 1. 0he dia#a$ shos to a#cs, 30 and 4*, of to ci#c%es ith cent#e 5 and ith #adii 50 and 5* #esective%. Given the #atio 50 :0* + 3:1, 'ind
/ ∠ 3'4 = 12/
b. 'ind, in te#$s of π and r , the a#ea of the shaded #eion ["$] SPM 1995
(a) the an%e θ in #adian (b) the a#ea of the shaded #eion 34*0 ["$] SPM 1994 1.
0he dia#a$ shos a se$ici#c%e #", ith cent#e ,. 8oint 3 $ove such that 3" + " + "#. Socus fo# the oint 8 is a ci#c%e ith cent#e ". (a) 'ind the distance of " (b)
0he dia#a$ shos a se$ici#c%e ith cent#e 5 and dia$ete# #5 . 'ind the va%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# #5 and %enth of a#c of the ci#c%e "
2 2 x y + is
= 4 x + " y + 3&
(c) (i) 'ind the a#ea of $a=o# secto# "#3 in te#$s of π (ii) Nence, sho that the a#ea of the shaded #eion is 25(π + 1) unit 2
35
the secto# ith cent#e L. 0he eCuation of 7 x
+
y
is 12 " La%cu%ate
=1
(a) the a#ea of ∆ #"+ (b) ∠#+" in #adians (c) the a#ea of the shaded #eion
SPM 1997 (a) Lonve#t
2. 0he dia#a$ shos a iece of i#e in the shae of a secto# 534 of a ci#c%e ith cent#e 5 . 0he %enth of the i#e is 1// c$. Given the %enth of a#c 34 is 2/ c$, find
(i) (ii)
"4/2/X into #adians 4.3" #adians into de#ees [2$]
(b)
0he an%e θ in #adians (b) the a#ea of secto# 534
(a)
[5$] SPM 1996
1.
0he dia#a$ shos to secto#s 534 and 5*0 of to concent#ic ci#c%e ith cent#e D. Given ∠534 = θ #ad, the %enth of a#c 34 is tice the %enth of #adius 54, and the %enth of #adius 50 +" 'ind
0he dia#a$ shos a iece of ca*e ith a unifo#$ c#oss-section in the shae of a secto# 534 of a ci#c%e ith cent#e 5 and #adius 2/ c$. 0he %enth of a#c 89 is 15 c$ and the thic*ness of the ca*e is c$. 'ind (a) the an%e of this secto# in #adians (b) the tota% su#face a#ea of the ca*e [5$]
(i) (ii)
2.
2.
0he dia#a$ shos, D7 is a se$ici#c%e ith cent#e > and K7 is a %enth of a#c of
3"
the va%ue of θ the e#i$ete# of the shaded #eion [4$]
0he dia#a$ sho se$ici#c%e 89@ ith cent#e D and secto# 9<0 of a ci#c%e ith cent#e <. Given <0 + 5, D@ + 4 c$, and the %enth of a#c 90 + 4.5 c$. 'ind (a) (b)
to ci#c%es ith cent#e 3 and 4 #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# 3%' [2$] (d) the a#ea of the shaded #eion [4$]
∠40. in #ad
the a#ea of the shaded #eion [4$]
SPM 1998
SPM 1999 1.
1.
0he dia#a$ shos to secto#s 534* and 50 of to concent#ic ci#c%e ith cent#e 5 havin the sa$e a#ea. Given 530 and 54 a#e st#aiht %ines, ∠ 354 = /." #ad, 5* = c$, and the %enth of a#c 34 sa$e as that a#c 4* 'ind (a) the %enth of 30 (b) the %enth of a#c 0
0he dia#a$ shos the osition of a si$%e endu%u$ that sins f#o$ 3 to 4. f the an%e 354 is / and the %enth of a#c 34 is 14.4 c$, find (a) the %enth of 54 [3$] (b) the a#ea of #eion set b the endu%u$ [2$] 2.
2.
0he dia#a$ shos a t#aditiona% Aa%a *ite, au bu%an, that has an ais of s$$et# 5*. Given that #3" is an a#c of a ci#c%e ith cent#e 5 and #adius 25 c$. #"4 is a
0he dia#a$ shos a secto# '%& ith cent#e ' and to secto#s 3%' and 4', of
3&
se$ici#c%e ith cent#e H and dia$ete# 3/ c$. 40 is an a#c of ci#c%e ith cent#e * and #adius 1/ c$. Given that the %enth of a#c ,8 is 1.&5 c$. La%cu%ate (a) ∠ #5"
[4$]
(b) the a#ea of se$ent #9" ( c) the a#ea of the shaded #eion (Bse π = 3.142 )
SPM 2000 1.
SPM 2001 1.
0he dia#a$ shos to secto#s 5#3 and 5"4, of to concent#ic ci#c%e ith cent#e 5. Given ∠ #53 = /.5 #ad, 5" = 3 #5 , and
0he dia#a$ shos a secto#, 534* of a ci#c%e ith cent#e 5 and #adius 5 c$. Given the %enth of a#c 34* is &." c$, find (a) ∠ 35* in #adians [2$]
the #atio of the %enth of a#c #3 to the %enth of a#c "4 is 2:3 La%cu%ate ∠ "54 in de#ees
(b) the a#ea of the shaded #eion [4$]
[5$] 2.
(a) (b) (c) (d)
2.
0he dia#a$ shos se$ici#c%e ,#-8 ith cent#e 4 and #ho$bus 4#3 . La%cu%ate the #adius of se$ici#c%e ,#-8 [1$] the an%e θ in #adians [3$] the a#ea of secto# 4#- the a#ea of the shaded #eion
0he dia#a$ shos a ci#c%e, &#", ith cent#e 5 and #adius " c$. &5 is an a#c of a ci#c%e ith cent#e . Given #" is a#a%%e% to &, #" + " c$ and ∠ &5) + 12//
[2$]
(a) 'ind ∠ #5" [1$] (b) La%cu%ate the a#ea of se$ent 70
3
[4$] (c)
(b) the e#i$ete# of the shaded #eion [4$] ( c) the a#ea of the shaded #eion [4$]
shaded #eion is " 3 + " + 2π [5$]
SPM 2002 1.
SPM 2003 0a0$r 1 1 >ia#a$ 1 shos a secto# *50 ith cent#e 5
>ia#a$ 1 0he dia#a$ shos to secto#s 5#" and 5, of to concent#ic ci#c%es ith cent#e 5 , he#e #5, and "5 a#e st#aiht %ines. Given 5" + (k 2) c$, 5, + k c$ and e#i$ete# of the fiu#e is 35 c$. 'ind (a) the va%ue of k [3$] (b) the diffe#ence beteen the a#eas of secto# 5#" and 5, [2$] 2.
0he %enth of the a#c *0 is &.24 c$ and the e#i$ete# of the secto# *50 is 25 c$. 'ind the va%ue of θ in #ad [3$] 0a0$r 2"#$%&ion A( 1. >ia#a$ 1 shos the secto# 354, cent#e 5 ith #adius 1/ c$ 0he oint * on 53 is such that 5* : 53 + 3 : 5
n the dia#a$, #", is a #ectan%e and 5#-, is a secto# of a ci#c%e ith cent#e 5 and #adius " c$. Given 5 is the $idoint of # .La%cu%ate (a) ∠ #5, in #adians [2$]
>ia#a$ 1 La%cu%ate (a) the va%ue of θ , in #ad,
36
[3$]
(b) the a#ea of the shaded #eion , in c$2. [4$]
La%cu%ate the an%e
α ,
in te#$s of π [2$]
the %enth, in c$, of the a#c %& [4$] 2
the a#ea, in c$ , of the shaded #eion [4$] SPM 2005 0a0$r 1 1. >ia#a$ 1 shos a ci#c%e ith cent#e 5
SPM 2004 0a0$r 1 1. >ia#a$ 1 shos a ci#c%e ith cent#e 5
0he %enth of the $ino# a#c #" is 1" c$ and the an%e of the $a=o# secto# #5" is 26// . Bsin π + 3.142, find
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)
(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$]
[3$] 0a0$r 2"#$%&ion '(
1. >ia#a$ 4 shos a ci#c%e 34* , cent#e 5 and #adius 5 c$. %4& is a tanent to the ci#c%e at 4. 0he st#aiht %ines, %5 and &5, inte#sect the ci#c%e at 3 and * #esective%. 534* is a #ho$bus. %& is an a#c of a ci#c%e, cent#e 5
0a0$r2 "#$%&ion '( 1. >ia#a$ 1 shos a secto# 354 of a ci#c%e, cent#e 5. 0he oint %ies on 53 , the oint 7 %ies on 54 and #" is e#endicu%a# to 54.
4/
>ia#a$ &
0he %enth of 5# + c$ and
Given that D7 + 1/ c$, Z + 4 c$, ∠ ;#< = 1.1 #adians and the %enths of 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
∠ 354 =
π
" #adian
t is iven that 5# : 53 + 4 : & (Bse π = 3.142 ) La%cu%ate (a) the %enth, in c$, of #3 (b) the e#i$ete#, in c$, of the shaded #eion, (c) the a#ea, in c$2, of the shaded #eion SPM 2006 0a0$r 1 1. >ia#a$ & shos secto# 5#" ith cent#e D and secto# #;< ith cent#e a
0a0$r 2 1. >ia#a$ 4 shos the %an of a a#den. 34 is a se$ici#c%e ith cent#e 5 and has a #adius of $. *#4 is secto# of a ci#c%e ith cent#e and has a #adius of 14 $.
41
SPM 2007
Paper 1
1. >ia#a$ 4 shos a secto# 7DL of a ci#c%e ith cent#e D t is iven that #, + c$ and "# + #5 + 5, + , + 5 c$ 'ind the %enth, in c$, of the a#c " 2
the a#ea, in c$ , of the shaded #eion [4 $] SPM 2) 0a0$r 2
∠+54 = 1.65"
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%.
#adians
[use π = 3.142] La%cu%ate (a) the a#ea, in $2 of the %an [2$] (b) the %enth, in $, of the fence #eCui#ed fo# fencin the f%oe# bed [4$] 2 (c) the a#ea, in $ , of the f%oe# bed [4$]
[use
π =
3.142]
La%cu%ate (a) the %enth, in c$, of the a#c 7 [5 $] 2
(b) the a#ea in c$ , of shaded #eion [5 $] SPM 2 0a0$r 1 1. >ia#a$ 1 shos a ci#c%e ith cent#e 5 and #adius 1/ c$.
42
1. >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 34 is a co$$on tanent to the ci#c%e at oint 3 and oint 4.
Given that 3 , 4 and * a#e oints such that 53 + 34 and
[Bse 'ind
π =
∠ 53* + 6/ , /
3.142]
(a) ∠ 45*, in #adians (b) the a#ea, in c$2 of the co%ou#ed @eion [4$]
[use
π =
3.142]
∠ 3;* = θ #adian, sho that θ = 1.3& (to to
Given that (a)
deci$a% %aces) [2$] (b) ca%cu%ate the %enth, in c$ of the $ino# a#c 4* [3$] 2 (c) ca%cu%ate the a#ea, in c$ , of the co%o#ed #eion. [5$]
SPM 2 0a0$r 2
43
y = 3x
2
1. (a) Given that usin the fi#st #inci%e
+ 5 , find
dy dx
d 1
dx 2 x + 1
(c) 'ind
2. Given
y =
dy
1"
x , find dx if x 4
= 2 . Nence,
1" 4
esti$ate the va%ue of (1.6) SPM 1995
f ( x ) = 1. Given
1 − 2 x 3
x − 1 find f = ( x)
2. Given y = x(3 − x) , e#ess
d 2 y
+ x
dy
+ 12 dx dx 2 in te#$s of x. Nence, find the va%ue of x that satisf the eCuation 2 d y dy y 2 + x + 12 dx dx y
3. 'ind the coo#dinates at the cu#ve
y = (2 x − 5) 2 he#e the #adient of the no#$a% fo# the cu#ve is 1 4
SPM 1996 SPM 1993
f ( x) = 1 Given that
1 − 2 x
4 & ( 1 3 ) + x x 1. >iffe#entiate ith #esect to
2
x.
4 x − 3 , find f X( x)
SPM 1994
44
y
= hx +
a#ea is 4 p2 c$2 and its heiht f#o$ the ve#te of the #a$id is h c$. (i)
k 2
x at 2. 0he #adient of the cu#ve − 1,− & 2 is 2. 'ind the va%ues of the oint h and k
3. Given p
= 2 x − 3 and
y
=−
3 p 2 . 'ind
(a) the a#oi$ate chane in x if thK #ate of chane in p is 3 units e# second
(b)
dy
(b) (c)
dx in te#$s of x
the s$a%% chane in , hen x dec#eases f#o$ 2 to 1.6 SPM 1997
n 2 − 4 n → 2 n − 2 (a) 'ind the va%ue of %i$
1.
0he dia#a$ shos a #ectan%e YSA insc#ibed in a ci#c%e. Given Y + x c$ and S + " c$ sho that the a#ea of the shaded #eion, # c$2, is iven b
5 = − f x x ( ) ( 2 3 ) (b) Given find f ′′ ( x)
y
2. >iffe#entiate #inci%e 3.
=
4 x
# =
−3
usin the fi#st
x 2
π
4
− " x + 6π
(b) La%cu%ate the va%ue of x so that the a#ea of the shaded #eion is a $ini$u$
(a)
[5$] SPM 1998 5 = − f x x x ( ) 4 ( 2 1 ) , find f ( x) 1. Given that X
0he dia#a$ 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$. Eate# is ou#ed into the containe# so that its su#face
45
y = t − 2t and x = 4t + 1 2. Given 2
dy
(a)'ind dx , in te#$s of x (b) f x inc#eases f#o$ 3 to 3./1, find the co##esondin s$a%% inc#ease in t . [2$]
2.
3 (a)
0he dia#a$ shos a ooden b%oc* consistin of a cone on to of a c%inde# ith #adius of x c$. Given the s%ant heiht of the cone is 2 x c$. and the vo%u$e of the 3 c%inde# is 24 π c$ a) 8#ove that the tota% su#face a#ea of the
0he dia#a$ shos a bo ith a unifo#$ c#oss section #",- . Given #" + -, + (3/-" x) c$, " + 3 x c$, , + 4 x and #8 + 2 c$ (i)
2
b%oc*, c$ , is iven b 1" 3π x 2 + x +
[3$]
b)La%cu%ate the $ini$u$ su#face a#ea of the b%oc* [3$]
3
c) Given the su#face a#ea of the b%oc* −1 2 chanes at a #ate of 42 π c$ s . 'ind
c$ , is iven b eCuation / = 3// x − 4 x 2
(ii)
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$. find the a#oi$ate inc#ease in the su#face a#ea of the b%oc* [2$]
3 (b) iece of i#e "/ c$ %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$ s −1 (use π = 3.142 )
SPM 1999
(i)
( x − 2) 2
f ( x) =
1. Given f X (/)
La%cu%ate (b) the va%ue of that $a*es / a $ai$u$ (c) the $ai$u$ va%ue of /
5
(ii)
1 − 3 x , find [4$]
SPM 2000
4"
La%cu%ate the #ate of chane in the #adius of the ci#c%e Nence, ca%cu%ate the #adius of the ci#c%e afte# 4 second
1. >iffe#entiate the fo%%oin e#essions ith #esect to x (a)
1 + 3x 4
SPM 2001 f ( r )
1. Given
[2$]
of f (r ) hen
= 4 + 3r 5 − 2r find %i$ited va%ue r → ∞
2 x + 5 4 (b) x
+3
[2$]
2.
y = 3 x Given
3.
inc#eases b 2. 'ind the co##esondin #ate of chane of y. 'ind the eCuation of the tanent to
2
− 4 x + " . Ehen x = 5 , x 2. Given that #ah of function
2 2 y = x + r at the oint x = k . f the cu#ve
3 f ( x) = hx
the tanent asses th#ouh the oint (1,/), find r in te#$s of k 4.(a) 0he st#aiht %ine 4 y + x
=
y = ( 2 x − 1)
f X ( x)
k is the 2
+
k 2
x has #adient function 6"
= 3 x 2 −
x 3 he#e h and k a#e
constants, 'ind
−3
no#$a% to the cu#ve at oint #. 'ind (i) the coo#dinates of oint # and the va%ue of k (ii) the eCuation of the tanent at oint #
a. the va%ues of h and k b. x-coo#dinate of the tu#nin oint of the #ah of the function 3. (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 of the #ectan%e. Given the e#i$ete# of 7L> is 4/ c$ a.
4.(b) 0he dia#a$ shos a to in the shae of a se$ici#c%e ith cent#e 5. >ia$ete# #" can be ad=usted so that oint hich %ies on the ci#cu$fe#ence can $ove such that # " + 4/ c$. Given that # + x c$ and the a#ea of t#ian%e #" is # d) c$, find an e#essions fo# dx in te#$s of x and hence, find the $ai$u$ a#ea of t#ian%e #"
4 + π y 2 4 Bsin π = 3.142 , find the 2/ y −
b.
%enth and idth of the #ectan%e that $a*e the a#ea of the shaded #eion a $ai$u$
4&
1 $$ fo# eve# " second, find the #ate of chane in the a#ea of in* b%ot at the instant hen its #adius is 5 $$ (ii) Bsin diffe#entiation, find the a#oi$ate va%ue of the a#ea in* b%ot at the instant hen its #adius is 5./2 $$
3 2 y = x − x + & , find the 2 5 Given
(b)
dy
va%ue of dx at the oint (2, 3). Nence, find (i) the s$a%% chane in x, hen y dec#eases f#o$ 3 to 2.6 (ii) the #ate of chane in y, at the instant hen x + 2 and the #ate of chane in x is /." unit e# second [5$] SPM 2002 p
1. Given
SPM 2003 0a0$r2"#$%&ion A(
= (1 + t ) 3 + 5 t 2
Given that y = 14 x (5 − x) , ca%cu%ate (a) the va%ue of x hen y is a $ai$u$ (c) the $ai$u$ va%ue of y [3$]
2
dp
'ind dt and hence find the va%ues of t dp
he#e dt
=&
y = x 2 + 5 x
2.
2. Given that , use diffe#entiation to find the s$a%% chane in hen inc#eases f#o$ 3 to 3./1 [3$] dy
= 2x + 2 dx 3. (a) Given that and y = " hen x = −1 , find y in te#$s of x
2
y + 2 x – x
[3$] (b) Nence, find the va%ue of x if (a) 0he dia#a$ shos the cu#ve
x
y = 3 x − x that asses th#ouh the 2
2
d 2 y dx
2
+ ( x − 1)
dy dx
+ y= [4$]
o#iin. Given st#aiht %ines 7 and 89 touch the cu#ve at oint D and oint @ #esective%, he#e 7 and 89 a#e e#endicu%a# to each othe#. 'ind the coo#dinates of oint @
0a0$r2"#$%&ion'( 4. (a) >ia#a$ 2 shos a conica% containe# of dia$ete# /." $ and heiht /.5$.Eate# is ou#ed into the containe# at a constant #ate of /.2 $3 s-1
[4$]
(b) d#o of in* fa%%s on a iece of ae# and fo#$s an eandin in* b%ot in the shae of a ci#c%e. (i) f the #adius of the in* b%ot inc#eases at a constant #ate of
4
/ =
cone + 3
π r 2 h
)
3
h
3
+ h
containe# is iven b , he#e h c$ is the heiht of the ate# in the containe#. Eate# is ou#ed into the containe# at the #ate of 1/ c$3 s-1. 'ind the #ate of chane of the heiht of ate#, in c$ s-1, at the instant hen its heiht is 2 c$ [3$]
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 π + 3.142^ \o%u$e of a 1
1
0a0$r2"#$%&ionA(
[4$]
px
SPM 2004
− 4 x
st#aiht %ine y + x − 5 = / . 'ind (a) the va%ue of p (b) the eCuation of the cu#ve SPM 2006 Paper 1 1. 0he oint 3 %ies on the cu#ve
2 4 3 ( 2 5 ) − x x 1. >iffe#entiate ith
#esect to x
[3$]
2. 0o va#iab%es x and y a#e #e%ated b the y
=
3 x +
2
x . eCuation Given that inc#eases at a constant #ate of 4 units e# second, find the #ate of chane of x hen x + 2 [3$]
y = ( x − 5) 2 . t is iven that the #adient of the no#$a% at 3 is 'ind the coo#dinates of 3
asses th#ouh (1, -12) is 3 x − " x . 'ind (a) the eCuation of the cu#ve [3$] (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$] 2
2. t is iven that
4
y =
2 3
$&
, he#e
dy
$
= 3x − 5 . 'ind
dx in te#$s of x
[4$] 2 = 3 y x 3. Given that
+x−4 dy
SPM 2005
h( x) = 1. Given that
−1
[3$]
0a0$r 2"#$%&ion '( 3. 0he #adient function of a cu#ve hich
h> (1)
2
3. cu#ve has a #adient function , he#e p is a constant. 0he tanent to the cu#ve at the oint (1,3) is a#a%%e% to the
(a) find the va%ue of dx hen x +1 (b) e#ess the a#oi$ate chane in y, in te#$s of p, hen x chanes f#o$ 1 to 1 p, he#e is s$a%% va%ue
1
( 3 x − 5) 2
, eva%uate
[4$]
2. 0he vo%u$e of ate#, / c$3, in a
SPM 2007 Paper 2
46
1. cu#ve ith #adient function has a tu#nin oint at (k , )
2 x −
2 x 2
(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 y = f ( x) is such that dy
3kx + 5 , he#e k is a constant. 0he #adient of the cu#ve at x = 2 is 6 dx +
'ind the va%ue of k [2 $] 2. 0he cu#ve
y = x
2
− 32x + "4 has a
SPM 1993 1.
= $ini$u$ oint at x p , he#e p is a constant. 'ind the va%ue of p [3 $]
SPM 2008 Paper 1 1. 0o va#iab%es x and y a#e #e%ated b the y =
1"
0he dia#a$ shos a ∆ 34* (a) La%cu%ate obtuse an%e 34*
2
x . eCuation K#ess, in te#$s of h, the a#oi$ate chane in y hen x chanes f#o$ 4 to 4 h, he#e h is a s$a%% va%ue [3$]
(b) <*etch and %abe% anothe# t#ian%e hich is diffe#ent f#o$ t#ian%e 34* in the dia#a$, he#e the %enths of 34 and 4* as e%% as an%e 34* a#e $aintained. [1$]
2 y = x − 5 x at 2. 0he no#$a% to the cu#ve
oint 3 is a#a%%e% to the st#aiht %ine y
[2$]
= − x + 12 . 'ind the eCuation of the
(c ) f the %enth of 8@ is #educed hi%e the %enth of 89 and an%e 89@ a#e $aintained, ca%cu%ate the
no#$a% to the cu#ve at oint 3 . [4$]
%enth of 8@ so that on% one ∆ 34* can be fo#$ [2$]
5/
2.
0he dia#a$ shos a #a$id ith ∆ #"+ as the ho#iTonta% base. Given that #" + 3 c$, " + 4 c$ and ∠ #"+ = 6/ and ve#te , is 4 c$ ve#tica%% above ", ca%cu%ate the a#ea of the s%antin face. [5$] /
0he dia#a$ shos a %and fo#$ t#ian%e, #" , divide b th#ee a#ts. #,", "8 , and #-9 is a st#aiht %ine ∠ "#+ =
12
13 Given that sin (a) if the fence ant to bui%d a%on the bounda# " , ca%cu%ate the tota% %enth is needed
(b) La%cu%ate
SPM 1995 1.
∠ "+#
(c ) Given that the a#ea of ∆ 8+9 sa$e ith the a#ea ∆ #,- . La%cu%ate the %enth of 9
SPM 1994 1.
∠ #,+ =
4
5 he#e n the dia#a$, sin ∠ #,+ is an obtuse an%e. La%cu%ate
(a) the %enth of L co##ect to to deci$a% %aces [3$] (b) ∠ #"+ [2$]
n the dia#a$, ", is a st#aiht %ine, ca%cu%ate the %enth of ,
SPM 1996 1.
2.
51
0he dia#a$ shos a cuboid. La%cu%ate (a)
∠ %4)
(b) the a#ea of ∆ %4)
[4$] [2$]
2.
1. n the dia#a$, ", + 5 c$, " + &c$, , + c$ and #- + 12 c$, ",- and #, a#e a st#aiht %ines. 'ind (a) ∠ ",+ (b) the %enth of #,
n the dia#a$, oints #, ", , , and - %ie on a f%at ho#iTonta% su#face. Given ", is a st#aiht %ine, ∠ #+" is an obtuse an%e and
2.
the a#ea of ∆ #,- + 2/ c$2, ca%cu%ate (a) the %enth of >
(b)
∠ ,#-
SPM 1997
0he dia#a$ shos a #a$id /#", ith a sCua#e base #",. /, is ve#tica% and base #", is ho#iTonta%. La%cu%ate (a) ∠/.6
1. 0he dia#a$ 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 $aintained
(b) the a#ea of %ane /
[3$]
SPM 1999
SPM 1998
52
1. n the dia#a$, 34* is a st#aiht %ine. La%cu%ate the %enth of 30
0he dia#a$ shos a t#aeTiu$ #", La%cu%ate (a) ∠+", (b) the %enth of st#aiht %ine #
SPM 2001 1.
2. %& is a t#ian%e ith side %& + 1/ c$. Given that sin ∠ &%) = /.45" and sin ∠ %&) = /.3" , La%cu%ate (a) ∠ %)& (b) the a#ea of ∆ %&) SPM 2000
0he dia#a$ shos a #a$id ith a t#ianu%a# base 34* hish is on a ho#iTonta% %ane. \e#te / is ve#tica%% above 3 . Given 34 + 4 c$, 3/ + 1/ c$, /* + 15 c$ and
∠/4* = // La%cu%ate (a) the %enth of 4* (b) the a#ea of the s%antin face
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#$s of (a)
SPM 2002 1.
α
(b) β α
Nence, sho that cos
=
11 26
2.
53
0he dia#a$ 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 (a) ∠ "#, (b) the %enth of ", ( c) the %enth of "
Given that /" + / + 2.2 $ and #" + # + 2." $, ca%cu%ate (a) the %enth of " if the a#ea of the base is 3 $2 (b) the %enth of #/ and the base is 25/ (c ) the a#ea of t#ian%e /#"
2.
SPM 2004
0he dia#a$ shos a #is$ ith a unifo#$ t#ianu%a# c#oss-section 30 . Given the vo%u$e of the #is$ is 315 c$3. 'ind the tota% su#face a#ea of the #ectanu%a# faces
1. 0he dia#a$ shos a Cuad#i%ate#a% 7L> such that ∠ #"+ is acute
[5$] SPM 2003
1. 0he dia#a$ shos a tent \7L in the shae of a #a$id ith t#ian%e 7L as the ho#iTonta% base. \ is the ve#te of the tent and the an%e beteen the inc%ined %ane \7L and the base is 5//
54
(a) La%cu%ate (i) ∠ #"+
(a) La%cu%ate the %enth, in c$, of # [2$]
(ii) ∠ #,+ (iii) the a#ea, in c$2, of Cuad#i%ate#a% #", [$]
(b) Cuad#i%ate#a% #", is no fo#$ed so that # is a diaona%,
∠ #+, = 4/ / and #, + 1" c$ La%cu%ate the to ossib%e va%ues of ∠ #,+ [2$]
(b) t#ian%e #="== has the sa$e $easu#e$ents as those iven fo# t#ian%e #" , that is, #== + 12.3 c$, ="= + 6.5
(c ) 7 usin the acute ∠ #,+ f#o$ (b), ca%cu%ate (i) the %enth, in c$, of , (ii) the a#ea, in c$2, of the Cuad#i%ate#a% #",
c$ and ∠ " X#== + 4/.5/, but hich is diffe#ent in shae to t#ian%e #" (i) <*etch the t#ian%e #="== (ii)
∠ #="== [2$]
cc%ic Cuad#i%ate#a%+sisi e$at *ita#an ve#tica% + $encancan ho#iTonta% + $enufu* obtuse an%e + sudut ca*ah s%antin face + e#$u*aan condon acute + ti#us fo#$ed + dibentu* diaona% + een=u#u SPM 2006
SPM 2005
1. >ia#a$ 5 shos a Cuad#i%ate#a% 7L>
1. 0he dia#a$ shos t#ian%e #"
55
the %enth, in c$, of #
>ia#a$ 5 0he a#ea of t#ian%e 7L> is 13 c$2 and ∠ "+, is acute La%cu%ate (a) ∠ "+, (b) the %enth, in c$, of ",
(b)
∠ #+" [4 A]
ii. 8oint U %ies on L such that #U " + #"
[2 $] [2 $]
(c) ∠ #", [3 $] 2 (d) the a#ea, in c$ , Cuad#i%ate#a% #", [3 $]
(i) (ii)
s*etch ∆ # U " ca%cu%ate the a#ea, in 2
c$ , of
∆ # U "
[" A]
SPM 2006
SPM 1993 1. 0he tab%e be%o shos the $onth% eenses of %iUs fa$i%
1. >ia#a$ & shos Cuad#i%ate#a% #",
5"
7$ar
166
1662
E90$n#$# 'ood 0#anso#tation @enta% K%ect#icit _ ate#
@A 32/ @A / @A 2/ @A 4/
@A 34 @A 3 @A 322 @A 4/
'ind the co$osite inde in the ea# 1662 b usin the ea# 166 as the base ea#. Nence, if %iUs I&$m Pri%$ In$9 $onth% inco$e
Mon&$90$n#$# 'ood Nouse #enta% Knte#tain$ent L%othin Dthe#s
Pri%$ In$9
13/ 115 11/ 115 13/
;$ig&ag$ " 2 4
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 the ea# 1663, co##ect to the nea#est #init, if the tota% $onth% eenses of the Zus#isU househo%d in the ea# 166/ is @A 5/
SPM 1995 1. 0he tab%e be%o shos the #ice indices and eihtaes of fou# ite$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/
dist#ibution of the $onth% eenses in the ZusnisU househo%d in the ea# 166/. 0he tab%e that fo%%os shos the #ice indices in the ea# 1663 based on the ea# 166/
SPM 1996 1. (a) n the ea# 1665, the #ice and #ice inde of a *i%o#a$ of a ce#tain #ade of #ice a#e @A 2.4/ and 1"/. Bsin the ea# 166/ as the base ea#, ca%cu%ate the #ice of a *i%o#a$ of #ice in the ea# 166/. [2$]
5&
In$9 n,m.$r< I i ;$ig&ag$#< W i
1/5 5 ? x
(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 #esective%. I&$m
Pri%$ In$ 9 1664
Cang$# &o Pri%$ in$9 /rom 1664 &o 166!
;$ig&ag$#
Eood
1/
nc#eases 1/
5
Le$en t #on
11"
>ec#eases 5
4
14/
Ho chane
2
124
Ho chane
1
64 x
12/ 4
(a) the va%ue of m [2$] (b) the va%ue n [2$]
SPM 1998 1. 0he #ice inde of a ce#tain ite$ in the ea# 166& is 12/ hen 1665 is used as the base ea# and 15/ hen 1663 is used as the base ea#. Given the #ice of the ite$ in the ea# 1665 is @A 3"/, ca%cu%ate its #ice in the ea# 1663 SPM 1999
1. 0he co$osite inde nu$be# of the data in the be%o tab%e is 1/
'ind the va%ue of x
[4$]
SPM 2000
La%cu%ate the co$osite #ice inde in the ea# 166" [3$] SPM 1997
Foo i&$m 'ish 8#an Lhic*en 7eef Lutt%efish
Pri%$ in$9
1. 0he be%o 14/ tab%e shos the 12/ #ice indices 125 and 115 eihtaes of 13/ th#ee ite$s in the ea# 1665 based on the ea# 166/.Given the #ice of ite$ @ in the ea# 166/ and 1665 a#e @A 3/ and @A 33 #esective%, and the co$osite #ice inde in the ea# 1665 is 13/. I&$m 3 4 *
Pri%$ in$9 12/ 15/ m
1. 0he tab%e be%o shos the #ice indices and eihtaes of 5 tes of food ite$s in the ea# 166 usin the ea# 166" as the ;$ig&ag$# base ea#. Given the co$osite #ice inde in the 166, usin the ea# 166" as 4 the base ea#, is 12&. 2 4 3 ;
La%cu%ate (a) the va%ue of x [4$] (b) the #ice of a *i%o#a$ of chic*en in the ea# 166 if the #ice of a *i%o#a$ of chic*en in the ea# 166" is @A 4.2/ ["$]
;$ig&ag$ 2 n 3
La%cu%ate
5
T-0$ o/ i&$m
SPM 2001
1. 0he tab%e be%o shos the #ices indices of ite$s #, ", and ith thei# #esective eihtaes. Given the #ice of 8 in the ea# 166" is @A 12.// and inc#eases to @A 13./ in the ea# 1666. 7 usin 166 as the base ea#, ca%cu%ate the va%ue of x. Nence, find the va%ue of y if the co$osite #ice inde is 113 te$ 7 L
8#ice inde x 6 123
Eeihtae 5 y 14 - y
(@A)
A B C
Zea# 1666 55 40 80
Zea# 2/// 66 x 100
8#ice nde (7ase ea# 1666) 120 150 125
;$ig&ag$ =
<
7
/
1//
;
L
"/
"&.5/
2 x
(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 of these ite$s in the ea# 166 based on the ea# 166" is 14/, find the va%ues of x and y [5$]
1. 0he tab%e be%o shos the #ices, #ice indices and the nu$be# of th#ee ite$s
8#ice
Pri%$ "RM( in 166 1/5
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
SPM 2002
te $
Pri%$ "RM( in 166! &/
Hu$be#
of ite$s 200 500 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
56
0
z
12/
(a) 'ind the va%ue of (i) x (ii) y (iii) z
SPM 2003
1. 0he dia#a$ be%o sho is a ba# cha#t indicatin the ee*% cost of the ite$s 8, 9, @, < and 0 fo# the ea# 166/. 0ab%e 1 shos the #ices and the #ice indices fo# the ite$s. te$
(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"
8#ice inde fo# the ea# 1665 based on the ea# 1663 135 x 1/5 13/
3 4 * 0 ) M R ( T S O C Y L K E E W
@A 2.5/
8e#centae of usae ()
(d) 0he cost of the ite$s 4/ inc#eases b 2/ 3/ f#o$ the ea# 1665 to 1/ the ea# 2///. 2/ 'ind the co$osite inde fo# the ea# 2/// based on the ea# 166/
35 30 25 20 15 10
SPM 2004
5 0 P
Q
R
S
T
1. 0he tab%e be%o shos the #ice indices and e#centae of usae of fou# ite$s, 8, 9, @ and < hich a#e the $ain in#edients in the #oduction of a te of biscuits
ITEMS
(a) La%cu%ate (i) the #ice of 0 in the ea# 1663 if its #ice in the ea# 1665 is @A 3&.&/ (ii) the #ice inde of 3 in the ea# 1665 based on the ea# 1661 if its #ice inde in the ea# 1663 based on the ea# 1661 is 12/ [5$] I&$m#
Pri%$ in 166
Pri%$ in 1665
8
x
@A /.&/
Pri%$ In$9 in 1665 .a#$ on 166 1&5
9 @ <
@A 2.// @A 4.// @A ".//
@A 2.5/ @A 5.5/ @A 6.//
125 y 15/
(b) 0he co$osite inde nu$be# of the %ost of biscuits #oduction fo# the ea# 1665 based on the ea# 1663 is 12. La%cu%ate the va%ue of
"/
(ii)
the #ice of a bo of biscuits in the ea# 1663 if the co##esondin #ice in the ea# 1665 is @A 32 [5$]
(a) 'ind the va%ue of , and T [3$] (b) (i) ca%cu%ate the co$osite inde fo# the cost of $a*in these biscuits in the ea# 2//4 based on the ea# 2//1 (ii) Nence, ca%cu%ate the co##esondin cost of $a*in these biscuits in the ea# 2//1 if the cost in the ea# 2//4 n#edient 3 4 * 0
8#ice e# *i%o#a$ (@A) Zea# 2//4 Zea# 2//5 5.// 7 2.5/ 4.// x < 4.// 4.4/
as @A 265
(c) 0he cost of $a*in these biscuits is eected to inc#ease b 5/ f#o$ 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 2005 1. 0he tab%e be%o shos the #ices and the #ice indices fo# the fou# in#edients 8, 9, @ and < used in $a*in biscuits of a a#ticu%a# *ind. >ia#a$ be%o shos a ie cha#t hich #e#esents the #e%ative a$ount of the in#edients 8, 9, @, and < used in $a*in these biscuits
n#edients
3 4 * 0
8#ice e# * Zea# Zea# 2// 2//4 1 /./ 1.// 2.// y /.4/ /."/ z /.4/
[5$]
SPM 2006 1. a#ticu%a# *ind of ca*e is $ade b usin fou# in#edients 3 , 4, * and 0 . 0ab%e shos the #ices of the in#edients
8#ice inde fo# the ea# 2//4 based on the ea# 2//1 x 14/ 15/ /
(a) 0he inde nu$be# of in#edient 3 in the 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
"1
