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TEST CODE
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FORM TP 2016282
02234020
-l
MAY/JLINE 2016
CARIBBEAN EXAMINATIONS COUNCIL CARIBBEAN ADVANCED PROFICIENCY EXAMINATION@ PURE, MATHEMATICS
UNIT 2 -Paper 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS 2 hours 30 minutes
READ THE F'OLLOWING INSTRUCTIONS CAREFULLY.
1.
This examination paper consists of THREE sections.
2.
Each section consists of TWO questions.
3.
AnswerALL questions from the THREE sections.
4.
Write your answers in the spaces provided in this booklet.
5.
Do NOT write in the margins.
6.
Unless otherwise stated in the question, any numerical answer that is not exact MUST be written corect to three significant flgures.
7
If you need to rewrite
any answer and there is not enough space to do so page, you must use the extra lined page(s) provided at the on the original
back of this booklet. Remember to draw a line through your original answer. 8
If
you use the extra page(s) you MUST write the question number clearly in the box provided at the top of the extra page(s) and, where relevant, include the question part beside the answer.
Examination Materials Permitted Mathematical formulae and tables (provided) - Revised 2012 Mathematical instruments Silent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO
SO
Copyright @ 2014 Caribbean Examinations Council All rights reserved.
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022340201CAPE 2016
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0223402003
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-4SECTION A
Module
1
Answer BOTH questions.
1.
(a)
+ bx + c : 0, where a, b, c e R. The complex roots A quadratic equation is given by "* of the equation are cr: 1 - 3i and B.
(i)
Calculate (cr + F) and (crB).
[3 rnarks] GO ON TO THE NEXT PAGE
02234020tC4P8 20t6
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till ilil
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5
(ii)
Hence, show that an equation with roots
Lo_,
u"O
fr
ir given by
lOx2+2x+l:0
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0223402005
tll] til
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-6(b)
Two complex numbers are given as u : 4 + 2i and v
(D
:
| + 2 \azi
Complete the Argand diagram below to illustrate z.
1
-2
12345678
-1 -1
-2
[1 mark]
(ii)
On the same Argand plane, sketch the circle with equationlz
- ul:3.
[2 marks]
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Calculate the modulus and principal argume nt of
z: [+]
...t+
,ris:
.:s-_---
'--ia':
::{it.
,-F,
r"!i+:
.-=. iE+l
'-'l*.
:-Gtr\l
-.1E.
:-_*-_. :--:l
..t=l 'r-'Q}l ,:.:E:,
i$,, ,.s.
[6 marks]
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0223402007
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A function f is defined by the parametric equations .1E[.
x:4 cos tandy:3 sin 2tfor0
:*.,
..'El
6-. :r{:!r+i
-* .:tii:. .ft' .-r!ri:
Determine the x-coordinates of the two stationary values of f.
,IiJ,
.-hi.
:+i+l
.,8{ -}\j
..=.: ...k$::
,.s
-------
.14, -E:' .Fir'Er' i-J-riii
-.-ir},:,
--Sr.
\
:-!(.' -_--s-:
--*.. .,t=.
.rG.,
..t. r:S:.
..i.-q.
[7 marks]
Total25 marks GO ON TO THE NEXT PAGE 022340201CAPE 2016
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-9(a)
(x. v): ,,,I l} \^rJ,,, In I vg go as w A tunction w is defined
Z:ll x- to I
Determine
0w 0x
.t.. .qi.
[4 marksl
.R. .-€i:
,.ftr
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0223402009
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= ;:= =
(b)
Determine
I ea sin e'dx.
i.i;4. = .
'l J*,t
:.f,d
..t5 .'Jr1
-liii :.H
k
..tt+
.I$:
.h, :-t-. jFS
:-E_.
:;in. :-Ft' :_A_-'
--\f,'
t
iA-.
:-\9-'
:-{' :::=
rla
'E
-!Ia-
-F+ -.'-2ii.
..+.+.
',*1, -.lii.
-.\-. :{Hi E'
i' ,t-+'
.o
s,
:::t
= ::.*:
r+
.€,
.t*
-:gd-l
:]rri-E+. .1*(-
_,___-t_1
ts.
.r!i.:Ell
.8, .-Jr+: F+,
-t-: 'rr\.-:
[6 marks] GO ON TO THE NEXT PAGE 022340201CAPE 2016
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lilt !ilI
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(c)
Let f(x)
(i)
: -212i-3
1o, 2
x
5
Use the trapezium rule with three equal intervals to estimate the area bounded by f and the linesy : 0, x:2 and x: 5.
[5 marksl GO ON TO THE NEXT PAGE 02234020tCAPB 2016
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Using partial fractions, show that f(r)
:
*
- ;i
.fti.l
.ei. *j -A:l
.\r.-.-
..Q..
[6 marks] GO ON TO THE NEXT PAGE 022340201CAPE 2016
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0223402012
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(iii)
5
Hence, determine the value
of
I
f(r)
dx
2
[4 marks]
Total25 marks GO ON TO THE NEXT PAGE 02234020/CAPE 2016
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0223402013
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-t4SECTION B Module 2 Answer BOTH questions.
::}i' ----!rl
3.
(a)
A sequence is defined by the recurrence relation and (u,)' is the derivative of u,. For example,u3: Given that
ur:
u2*r: ur* x(ur)': |
u
n+
1:
u, _.r
I x(u,)' , where u r: l, ur.: x
+ x.
l3x + I and that u,o: 34x + l, flnd (ar)'.
-\ .{*t .*
..t{ ,.% -_.lri.
.,E .h;
.Ql ,.Q.
iisf
l
rJ.
..f(. ----Eaj!al tri j*.,--. .-.trr', .1\::
=., EJI .t*. :-\:
::-^i(,
iix.j 1l:
.:e. ..k: :_rar
:'\Jr
[4 marksl
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(b)
n
The n'h partial sum of a series,
Sn,
is given
'l*t..: ..ift-
(i)
Show that
Sn :
n(nz
UV
S,: I r(r - l).
- l)
J
ts: +
[7 marks]
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20
Hence, or otherwise, evaluate
I
r(r
l0
- l).
[5 marks] GO ON TO THE NEXT PAGE 022340201CAPE 2016
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0223402016
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(c)
(i)
Given that"Pr
,C
-
: , nl ,. show that ':P 'P (n - r)! tr
-l is equal to the binomial coefficient
[4 marks] GO ON TO THE NEXT PAGE
02234020/0APE 2016
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(ii)
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18 -
Determine the coefficient of the term in x3 in the binomial expansion (3x + 2)s.
of
[5 marks]
Totat 25 marks
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o2234020lcAPE 2016
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-19-
(a)
The function f is defined as
(i)
Show that
f(-r):
*4x+ I for-l
: (t + Z4!
(x)
ih.. ..G.
:*-
[3 marks]
GO ON TO THE NEXT PAGE
o2234020tCAPE 2016
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0223402019
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The series expansion of (1 +,r)k is given as
t+tGa k(k-l)x2 1 k(k-l)(k-2)x3 a k(k-l)(k-2)(k-3)xa ".. 4l 3! 2t whereteRand-1
(ii)
Determine the series expansion of f up to and including the term in xa
(iii)
Hence, approximate f(O.a) correct to 2 decimal places
[5 marksl
[3 marks] GO ON TO THE NEXT PAGE 022340201CAPE 2016
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(}:. :itt:
r
-2t (b)
..r:E:.
The function h(x)
:
x3 + x
i:-I$-:.
:fr..
(i)
:-_--_-'
=6. .
:0
Show that h(x)
-
-l
-
1 is deflned on the
interval [0,
l].
has a root on the interval [0, 1].
--trr+--'
..-E
h.
,.l'k" !+ i.r-r}: .'-'Ff:
rlsa: .:-^ :
i+ia{!i-j i.=.: l.^:' --\J-
:a: ,.n=
i'E ..:lr},
.-ts, ..1$..
,:lic' :l!\a-
i-t:' ''-i+r rts. i-!i+1
.{*. n:ti+i
[3 marks]
'-'-\--
.$: .:--.lir--l
l:'I.;' l.e} L:Er
Io, I.s. +-------
I..
l-_r
-1--------
t--
___;
j......-l
J..-:.. t---r'r
fN 1.:€'l I.;i, 1-----\--'
l--Ezr--:
t$ ;:$
].i+j.:
=f,;-: i:l=ii
a.S
--ts,-.
':it.r $ Q
rlsr :4l:.
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0223402021
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(ii)
Use the iteration
xn+1
with initial estimate *n'*l
x: I
0.7 to estimate the root
of h correct to 2 decimal places.
2.
.l-i:.
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0223402022
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s..
nIIT..
,ft'.
'---i*--'
vl-..
:_Ei:_-'
j\i:.*i'
.h{
{: rl+
t lj,
+.
s
.--\r--.
i+:-: t;::-:
F1
..eil
:-:t. ..€,.
:e[.
..E. ,rr+
..S.
..:{.
.#. .!ri-: .E -t+. ._r> 1l=
i-!El:,
..trr-
..1Er'
!q: r+':r+, ::c}.
:
.
1:l
t.-i
iE, t\{:. Fr .tii-
'!t"' 1*, t
it\:f :rA-.E::::
[6 marksl
.-s. .ri,,
GO ON TO THE NEXT PAGE o2234020lCAPE 2016
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0223402023
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-24(c)
IJse two iterations of the Newton-Raphson method with initial estimate x, : 1 to approximate the root of the equation g(x) : - 4 in the interval ll,2). Give your "4x-3 answer correct to 3 decimal places.
.rt?. -IIf--
El S+-
ts. l+. ,tia:. .{ii-: .\--1
s{. :E--j .t=,. --s*.'.
E., .9-.1
:S.,
[5 marks]
Total2S marks GO ON TO THE NEXT PAGE 02234020|CAPE 2016
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0223402024
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SECTION C
.:.-:. .Itt-,
fr. --'
Module 3
.U+. -.lFi-j
Answer BOTH questions.
I{ij
--Et-,
>..
.:riii-.
(a)
5.
A bus has 13 seats for passengers. Eight passengers boarded the bus before it left the terminal.
(i)
Determine the number of possible seating arrangements of the passengers who boarded the bus at the terminal.
--'
"rd
.s .!l
[2 marksl
(iD
At the first stop, no passengers will get off the bus but there are eight other persons waiting to board the same bus. Among those waiting are three friends who must sit together. Determine the number of possible groups of five of the waiting passengers that can
join the bus.
[4 marks] GO ON TO THE NEXT PAGE 022340201CAPE 2016
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0223402025
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-26(b)
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Gavin and his best friend Alexander are two of the five specialist batsmen on his school's cricket team.
,N.
.ft.
i::*-'
Given that the specialist batsmen must bat before the non-specialist batsmen and that all five specialist batsmen may bat in any order, what is the probabiliry that Gavin and Alexander are the opening pair for a given match?
.'I4:-:H.
.:-H-.
'::rE:
..-.\a_
..h{ "1I+.i,lsij
-.-ng:
.'i*r
..s
r'Ei .-A r*..
.ts, !!+i
-,I+
-tr'
.\:*: '*.-.
-E
.-tr+.
.G..
-4.
.,Q.
.S:
'.i]ri-.r
.r*--
-rlrii ._ttil tri-.'
.=; '\i-'
[5 marks] GO ON TO THE NEXT PAGE 02234020/CAPE 2016
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0223402026
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(c)
-l
-
A matrix A is given as
(i)
2 0
I
-1
4
J
-1
6
0
Find the lAl, determinant of A
.%..
E: tr=i.
[4 marks] GO ON TO THE NEXT PAGE 022340201CAPE 2016
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tilt
ilil tilil iltil ilfl tilt ffi 0223402027
tilt illt
il]
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-28
(ii)
-l
-
Hence, or otherwise, find A-r, the inverse of A.
.I\[. rE' ..tzf-.-JEil
H:,
r-F*-:'
_-a-l rELl
rt. ''A-il .-_\d
r
,.,f,:
'-
-'':
l=!l :rI}-
.g. ..X.,
::#'. ._-AXir
'I+' .b*. E. i.l-ii .-I*1.
Ei.
.ft. ..1Et-
:ii-
'---E-_
.t =G' it
r.o .Q
[10 marks]
Total25 marks GO ON TO THE NEXT PAGE 022340201CAP8
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0223402028
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-29-
(a)
Two fair coins and one fair die are tossed at the same time
(i)
Calculate the number of outcomes in the sample space.
[3 marks]
(ii)
Find the probability of obtaining exactly one head.
[2 marksl GO ON TO THE NEXT PAGE
02234020|CAPE 2016
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0223402029
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(iii)
Calculate the probability of obtaining at least one head on the coins and an even number on the die on a particular attempt.
--F['
,l!\-i.
.tl
..S,
:ci:. A\--.t+-
[4 marks] GO ON TO THE NEXT PAGE 022340201CAPE 2016
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lilll
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lil
0223402030
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:-:!1:
-l
- 31 Determine whethery
:
Cr*
* C{'is
a solution to the differential equation
i*ii
Ei
ji!$j
*'2"r" - xy' + y:
Ii i5+. :..t
:H. --.t-.-
o, where
C,
and Crarcconstants
::Iq' .'!$. --E:r_j _ _
'-F1'.
:.€:.
.:k.i
i.s. ----r\-
--r!!rl--
.-ICj
.S. .:ir1: ".:$:.{i+-l
l'.E-S+--
.*j j!i+l
ifrl-, !-.t+. i-\t-l
,.9--
-N,.
:= :::ii
..rs,
r.t(-r -!:+ ._-
-
:_-_---Y -
rr-th-:
.rit.-lt .t\a.-.
;= i.E+-l
:.I+J:,
-t-i:. :-\-:
''I(n
j+:j
..t:. R
[6 marks]
t:-s:
..E:
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ilfl til
til
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(c)
(i)
Show that the general solution to the differential equation
:X[.
+j.
n4:
3(f+i*:2y(r+2x)
t:
.r^-
is
:}rA--
5i-:' .'J{+: .E'.-
.k"
Y:C '.'l 1* * ry, where c e R
.-.r\i...
.Fr.. --FLl
s; .t-..Q.ieil 1A.i .--\J-: .
-!*l-:
+Et:
:s.-
:.{ta. --tri '-lra .
E:l
{..i..
kj
-:t!ia-:_
.-ft).,
.ty.-1\--. -l(-. rl:
-i\:
iHi. ..G,:
.E-. .(!rr r+t--.
.r\1, .--8. ..:=S., -t t--
,:Ei-
-f-i:
.t=.
'=i
rEi-j
-.*): ..t-i{-,
:-It
'
rd,
:-t-.
[7 marks] GO ON TO THE NEXT PAGE 022340201CAPE 2016
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tffi illil
llill ril
lfl
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(ii)
Hence, given thaty(1)
:
1, solve 3
(f + x)Y : 2y (l + 2x).
..itl .r*t
,f;(-r
.8.
[3 marks]
Total25 marks END OF TEST IF YOU FINTSH BEFORE TIME IS CALLED, CHECK YOUR WORI( ON THIS TEST.
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= 02234020/CAPE 2016
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0223402033
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BXTRA SPACE
If you
use this extra page, you MUST
write the question number clearly in the box provided
,k.
Question No.
:r=i
02234020/CAPE 2016
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