MINISTRY OF EDUCATION, SINGAPORE
in collaboration with
CAMBRIDGE ASSESSMENT INTERNATIONAL EDUCATION
General Certificate of Education Ordinary Level
CANDIDATE
NAME
"\e'r
Pileirjs
•
INDEX
NUMBER
CENTRE
NUMBER
o
ADDITIONAL MATHEMATICS
4049/01
Paper 1
2 hours 15 minutes
Candidates answer on the Question Paper.
"c) ===== No Additional Materials are required.
*
READ THESE INSTRUCTIONS FIRST
Write your centre number, index number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE ON ANY BARCODES.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an approved scientific calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
The number of marks is given in brackets [ J at the end of each question or part question.
The total number of marks for this paper is 90.
This document consists of 17 printed pages and 1 blank page.
0
.1
PEAB
ciqSingapore Examinations and Assessment Board
14, Li.
UCLES & MOE 2019
C.ambridge Assessment
International Education
[Turn over
1. ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0,
x=
—b±44,/b2-4ac
2a
Binomial expansion
(a + M r' = an ± In)an---1b +(n]an-2b2 +... + n an-rbr ± . . . + bn ,
1
2
r
n (n-1 ...(
n!
) n— r + 1)
where n is a positive integer and [n =
r!
r r! (n rp=
2. TRIGONOMETRY
Identities
sin2A + cos2A = 1
sec2A = 1 + tan2A
cosec2A = 1 + cot2A
sin(A + B) = sinA cosB + cosA sinB
cos(A + B) = cosA cosB T- sinA sinB
tanA ± tanB
tan(A ± B) = 1 -I-- tanA tanB
sin2A = 2 sinA cosA
cos2A = cos2A — sin2A = 2 cos2A — I = I — 2 sin2A
tan2A =
2 tanA
1 tan2A
Formulae for MBC
a _ b _ c
sinA — sinB — sin C
a2 = b2 + c2 — 2bccosA
A=
C UCLES & MOE 2019
1
b c sinA
4049/01/SP/21
3
1
x
The line —a
+b
— = 1 , where a and b are positive constants, intersects the x-axis at S and the y-axis at T.
Given that the gradient of ST is
f-ty-
s
1
10 , find the value of a and of b.
and that the distance ST= V-21-116c. ?tAyvi-
Let
Le"\-
[5]
0
‘c)
S.7
bL
3
rzt
—ok
*'‘o
STC
-0 =
c12.0e
40
Tec.k.t.cvoydk coosairktk"%gl OurrcLecktA
-
UCLES & MOE 2019
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?0A--Viat
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4
2
The equation of a curve is y = 3 — 4 sin2x
(a) State the minimum and maximum values of y.
-
Ki•n\o•Nur.1
'1"\coCivvturct
3
[2]
-
(-‘
(b) Sketch the graph of y = 3 — 4 sin2x for 00 < x < 360°.
[3]
— co oft-k-- AccQ
- freA.4 tvt• • 41,7 sime, worts
twN
UCLES & MOE 2019
4049/01/SP/21
(,)tNm-'t Tch.i*s \ARA&
3
(a) Find the first 3 terms in the expansion, in ascending powers of x, of 2
simplest form.
. Give the terms in their
[3]
ks-k )-"4 -kiecvo ca re&
4-4•
t
..1
-k-Ns
—
)6
(b) Hence fmd the term independent of x in the expansion of 2 x (3
•-
r3
—
it+ - •
C5C--
[3]
ak'
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IS-X% "15k
=
=
-Tecvn \r\al2isa•aubit'
UCLES & MOE 2019
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4)-14
4049/01/SP/21
[Turn over
6
4
x2
The function f is defined by f(x) = 2 — 4 , x > 0 .
x +6
(a) Explain, with working, whether f is an increasing or a decreasing function.
[4]
10)
0.0
=
gin (Q.. Ci)
— ‘0
‘4. Qussfookk
oLo
tuziiked,
-04 o cw\dt
&k...7kovk
-1 - 4-' LI >
sk‘ cA) -7 0
C_
. _I
s
>0
aetvalL AkNak.
"cy1Z,n5 .1,k1^C_ ILO1\ 4b.r
eGotn c.V.t.s:(on)
Cka(2_
— C:-(12
(7c,
,-L'tck‘)
W%-k(b) A point P moves along the curve y = f(x) in such a way that the y-coordinate of P is increasing at a
rate of 0.05 units per second. Find the rate of increase of the x-coordinate of P when x =2.
[2]
.= 0
ckn
afi
IZASL
(hi
ety,
‘Ali*Itt s
art,
a-c,poit;,
,c
ck+,
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4049/01/SP/21
gCkt---
Wit k
7
5
On a certain date, 160 cases of influenza were recorded in a city. This number increased with time and after
t days the number of recorded cases was N. It is believed that N can be modelled by the formula N = 160e.
The number of cases recorded after 5 days was 245.
(a) Estimate the number of cases recorded after 7 days.
[4]
S- NI k-‘c
Ls‘ocin
000 e tet-s-V
/
r!F_
kv\
42— ,/
v)
(yr O. cAS
,&*\) accgkaA, .
QgkvNIVO-,
It‘O
ti•no" --t-z_ '•-- 1
sA0
-
= ‘6 0 t-
40%
1-
Z. (\tpeN‘Qt_Nr 4 cc4se_s, cv,.....aa
Ak9c 4-- dta* ‘ts
\kt.o
\
Influenza is declared an epidemic when the number of cases reaches 400.
(b) Estimate after how many days influenza is declared an epidemic.
[2]
I&V
z\-1/4
sr.:
44=1 7<t-N- 14
+30
v`^
5\v‘
s \Y\ k
V()
reZi,
k 600,3S
kAl\\\ -1CaNY-& CICON.:,kv•A
C UCLES & MOE 2019
4049/01/SP/21
[Turn over
cey
7E
dx =3 cosx —4 sin 2x. The curve passes through the point P(., 9 and the gradient
of the curve at P is 5. Find the equation of the curve.
[6]
For a particular curve
cY'rj
(v‘
d
ck3L
ddx.
kiA`tizAc... C.. its 0 co,6144.3t-
a (3s si--",
ktYSZA
rkkAttv)
cot ruk
40,6, G
) 45 •-= k
c=
-k-
^
:71
(3-)i
ke4
sivrA.
— 3c*. -A- go,
A efte-t-k-e.At
1Z)
w\k‘A
coertek
—3 It).) *. 0
kir‘ck
1.7
•
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7
4049/01/SP/21
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9
7
(a) Express each of 2x2 — 4x + 5 and —x2 — 4x —2 in the form a(x + b)2 + c, where a, b and c are
constants.
[4]
ourlemt
rAkkAk oa'r ,
—\
(b) Use your answers from part (a) to explain why the curves with equations y = 2x2 — 4x + 5 and
y = —x2 — 4x —2 will not intersect.
[3]
Q'kv\02.--
te_tviOr ed‘t_
0).--̀CV
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•
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aktve, u>ACctva
f
a 0,4)04v-,4,mfn \iCVVIZ_ feit
a-zt7-
NWAOL, I NNCVL Cu•CVe, A\ \ \
UCLES & MOE 2019
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4049/01/SP/21
[Turn over
10
8
Without using a calculator,
(a) show that cos 750 —
Csxs -A-S
—I
[2]
21,rf •
cos 0-s*-v.3O'
= css,A-e (9,S 3c)- -
tte cm 30'
s.\Thk.a
r
ny
(b) express sec2 75° in the form a + b \ 3 , where a and b are integers.
clubs- k-ktc\-.Nc"
UM.
sriwveCkk/ ko
isA•k& zIpitaZ.
UCLES & MOE 2019
4049/01/SP/21
[5]
11
XM
klt,
The diagram shows a triangular plot of ground, ABC, in which AB = 12m, AC = 16m and angle BAC = 900.
A gardener considers using a rectangular part, APQR, of the triangle, where P, Q and R lie on AB, BC and
AC respectively, for growing vegetables.
(a) Given that the length of AR is xm and the length of AP is ym, show that y = 12 —
A .ktc- ar.ck
1.m are 1...,Ackv- •
Uttfk) t4r\lk
-kt,e--
[3]
Vo
reovv,Pe. •QYD,1ki&\
(Aid:4N
k \c‘‘r
tt-
fit
Cc)ribt,5V1
(b) Given that x can vary, find the largest possible area of the vegetable plot.
[4]
vtae--kask- c=.-k
4CV$1, D-s'a
=
—
a2A -
2-
440
etrk
—
§ArPA)
viNAto
dflt
1U'1
N"
't.N
(vitt
r
rovoir,;(vv,kAirA
- ‘40.
r."`s‘
Vg)-3
a•52.)f,
vyN2-
C UCLES & MOE 2019
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[Turn over
12
10 The expression 2x3 — x2 + ax + b , where a and b are constants, has a factor of
x — 2 and leaves a remainder
of 12 when divided by x +2.
(a) Find the value of a and of b.
[4]
xiL -r-cm
"Tv,esym.v.‘
\bn
g\vo_
&LA, (k)
-
7- a
W,4 2
k- 3 2_
17-
4-4\-
a=
Rqrscv‘6Asc ThCoCt-Vi
3
Q4D A rz .-- k\ ivito 0-) 1
2.
k"2..
C"--)
(b) Using these values of a and b, solve the equation 2x3 — x2 + ax + b = 0 .
,
—
eir re.k
0
((-1) cx--\)
.1.
"
...s.
r
11(-
r•
=0
=7-0
015*.rne_
ettAGertz ,
kl°Lktiv-
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3'
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aski6ein
or 0.1.1041,1
cte-k*oevrvi
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— _L
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3 ov‘s &:4-fries1
c1/4 \ 1 "Ks cofrcece
UCLES & MOE 2019
[4]
4049/01/SP/21
13
11
In the diagram, A, B, C, D and E lie on a circle such that AB = BC and BA is parallel to CE. The tangent to
the circle at A meets CE produced at T. Angle TAE = 0.
(a) Show that CA bisects angle BCE.
[3]
4 -.&.tc.e\t,t. at&)
z_bvsc.
L_ ,Bc_p\
LckQS,
PitcZ._ ( k
)
S't n (ft_ z6(1\
Csserect
Z\v" eNsit- reN .
• CA NI4-cts I Bic-Z.%
ct..4 VAG1,-"
(b) Show that angle CDE =30.
[5]
Avo,1
4'. Lbw,. = LE.Lfs
C-014V--ek ccr-S14"c,
Ltpk
z tkA =
4CA-a
ktNi
—
a)
Cat tkce(WrIke-
S-IV**Vt* "kk\eccevv•)
-e -e (a.330,01vt\- atc&
&tvczttt niL)
= Mc;
4 CA)rrs
cv‘f\404_,s *i 1•N 0 mb-s-‘r*.._
tito° — 4 C-Pk..
( krEe - 3S)
tsoc' — tWo
33
cArtC e‘a:Vcct\cs cAvN 640
otaAkAcck •
7:- 3+3 oNrIcAATI)
C UCLES & MOE 2019
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[Turn over
14
12
(a) Prove the identity (cosecx cotx)(secx + 1) = tanx.
Ez,\vvx,
Cs-S
(PN
(555."
k
t_
x.
os.
t•f‘
cc,s'-)L
\•••\
UNN7u
C4 S. ')(.
UCLES & MOE 2019
4049/01/SP/21
15
(b) Hence solve the equation (cosecx — cotx)(secx + 1) =4 cotx for 00 <x < 1800 .
4.Cat:X-
C_cst.)
arczorzck-.\-- -4141tx.s.rn
-kom 4:C = t
o-t-
Abr•e\
[3]
\b°1Q corIc6k_
6S•43s° (36'0
'•X (to° 4-1 °
a
pt
= "k3 ' 1: °61:)
x. -zz C1-4-6 - 1 m:,•tc, 08) 17-)
(c) Show that there are no solutions to the equation (cosecx cotx)(secx + 1) = tan 2x
for 0° < x < 180° .
[2]
Ccostr—v,—
=
ow)
—
2.-k-cA"
csr-
tckY\
=
1—
k.k) - •
‘1.,\(.e._ 6 . C..<ktiZo'
AN'tr
10\0 SokiYikv‘
P43
over— vv, .sekakeri\
C UCLES & MOE 2019
4049/01/SP/21
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MtnCt "tavi
ks\lt..41N)
ccrak-LY--
[Turn over
16
13
In a race, a cyclist passes a point A at the top of a hill with a speed of 5 m/s. He then increases his speed and
passes the finishing post B, 10 seconds later, with a speed of 20 m/s. Between A to B, his velocity, vm/s, is
given by v = 0.1t2 +pt+q, where t is the time in seconds from passing A, andp and q are constants.
(a) Show that q= 5 and fmd the value ofp.
[3]
tr,( c
o
LP(VC-k 9.4°
(b) Find the acceleration of the cyclist when his speed is 11.6 m/s.
[4]
c
&
'1=
-t o)
12' `5
k
qc
CV= S
tOe
(51,10%An)
etogx-kluce<t -1
v
o• at,
G -S
S ar,j,
Icrc'M
•
kt,1 + 'S
v3AtAA V
k\-
a o ((;) ko-
S-
•. a. c.c..e-cke-cstorl = \ •
k\-(, o-\ -€)
covreirk
7=-0
co---A
-0-
a c.t.-
(yrje -k
UCLES & MOE 2019
• s-
4049/01/SP/21
errectkv •
17
(c)
Find the distance AB.
[3]
fsi
\-its. ‘,9\AcQx•Auilk
v c\tki
o • k-t,
k
374t-
0 't,
L. \)
akAto-'44K.
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3/47-t
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4049/01/SP/21
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'Ls co c-N•40-kizw*
Sk. "k"
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tv,
18
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4049/01/SP/21