-Lt
TEST CODE
FORM TP 2016281
O2234OIO
MAY/JUNE 20I6
CARIBBEAN EXAMINATIONS COUNCIL CARIBBEAN ADVANCED PROFICIENCY EXAMINATION@ PURE MATHEMATICS ANALYSIS, MATRICES AND COMPLEX NUMBERS
UNIT2-Paper0l
t
hour 30 minutes
0l JUNE 2016 (a.m.) READ THE FOLLOWING INSTRUCTIONS CAREF'ULLY. This test consists of 45 items. You will have t hour and 30 minutes to answer them. 2
In addition to this test booklet, you should have an answer sheet.
3
Do not be concerned that the answer sheet provides spaces for more answers than there items in this test.
4
Each item in this test has four suggested answers leftered (A), (B), (C), (D). Read each item you are about to answer and decide which choice is best.
5
On your answer sheet, find the number which conesponds to your item and shade the space having the same letter as the answer you have chosen. Look at the sample item below.
are
Sample Itern
The expression ( I
(A) (B)
4
(c)
l+3"6 4+2Jl
(D)
6' ' 7
ffi
8. 9'
t
+.6 )' is equivalent
to
Sample Answer
@
l0
The best answer to this itenr is *4 +
2.6,,, ,o (D)
If you want to change your answer,
erase it conrpletely before you
has been shaded.
When you are told to begin, turn the page and work as quickly and as carefully as you can retul.t'l to that itenr Iater.
You nray do any rough work in this booklet. The use of silent, non-prograrrnrable scientific calculators is allowed. Exanr ination Materials:
E
A list of ,rather,atical fornrulae and tables. (Revised 2olz)
-
-t
in your new cSoice.
If you canllot answer an itern, go on to the next one. You ntay
E
E E E
fill
DO NOT TURN THIS PAGE UNTrL YOU ARE TOLD TO DO 02234010/CAPE 2016
Copyri glrt A 20 I 4 Caribbean Exanr inations Counci All rights reserved.
I
SO.
-2I
The complex number z
=r|
-i
can
represented on an Argand diagram as
(A)
be
2
Which of the following is a sketch of the locus of the point represented by the complex number z, given that lz + 5ii: 3?
tm (A)
-t I zt7, i)
I
i
I
3
.r
Re
z
-3
(B)
Im (B)
Re
-I (c)
z (1, -1)
-3
3
fm
(c) z
Gl' l) 3
Re
(D)
fm
-3
I , I
T
&
(D)
I
-1
7'7
Re -3
02234010/CAPE 2016
3
GO ON TO THE NEXT PACE
,|, i i I
.l
-J-
I
i I
-1
Theexpresssion to
3
I I
i[(l + i)r-(l - i),] isequal
7
.
The derivative of ln
xl
is
l
I
(A) (B)
i
(c)
I
il
I t I
4
I
(A)
-.3X
(B)
-3x
-2 2 4
(D)
I
(c)
l,l
3x
t
4.
!l
ir
lf
xzy
- xt' =
10, then
lZ
(D)
is equal to
3x
li :i
rt
l0 2x-2y
(A)
rl I
li tl
+
(B)
I li il
il Ji
li 1,
ii
i j
I
8.
ff
*:Zxy,thenthe
value
of
*
atthe
point (1, 2) is
(c) x'-Zry 'i^-"
(A) (B)
(D)
(D)
6 8
(c)
Y'.-zxY
t2 16
2xy
I t,
I J
i,
9
I 1
5.
The value
of
(A)
t
cos-+rslt'l-
22
Ifi*=
IS
(A)
"'" (;)-'
-l
(B)
Ztan-t (2x) + c
2
(c)
tan-t (2x) + c
(D)
-tan 2
)_
,
lfT
-2
(B)
(c)
I
(D) {
I I
1
t I ''4
i
:
6
I
sec'x Ztan x
_ 10
-611s (A)
I
1lnlr".'*l+"
i
i n
lnltan xf+c
(c)
2lnf sec .xf +c
(D)
2lnf tan.rf+c
+c
A curve is given parametrically by the x = t2 - 2t, y = 12 + Zt. The
expression 2
(,
equations
i
(B)
-l
(A)
f",
*
is given by
t-1 t+l
il
ii
(B)
t+l r
I
(c)
-l
2t-l 2t+l
I
t
I tl t.
i I
(D)
2t
+l
2t
-l
i
I,
ti
GO ON TO THE NEXT PAGE
-4-
il
(A)
(B)
J{cosAx
+ coslx)dx
J(cos8,
- cos2x)dx
1
(c)
2
2'J
(cos8x
(A) (B)
-
4i
f(x,
y) is such that
af -e'-,.sin (x+y), then which of the
b=
following is TRUE?
- cos2x)dx
One square root of 3
lf
*ox = ,'[-sin (r+y) + cos (x + y)] and
I (cos8x + cos2x)dx
1
(D)
t2
14.
J(cosSx cos3x)dx=
(A)
f(x,y) =e'sin (x+y)
(B)
f
(c)
f (x, y)
(D)
f
is
Ji
+zi
15.
e'sin (x + y) +cos (x +y)
y):
e" cos (x +
y)
* tan2x, L *rr,,
fi 4
|(x-l)(x+3),d*=
(D)
cos (x + y)
4
4
22 II x-l 3(x+3)
(D)
4
16.
rI 3(x-l) (x+3) -+_
l3 ( _+-2(x-t)
l-1t
dx
22
,+-) ( 2(x-t)
n
(c)
J
(c)
d
tr
(A)
2!
(B)
:
Given secr.r= I
(B)
(A)
(x,
y):
Ji -zi
(c) 2- i (D) 2+i l3
(x,
2(x+3)
2(x+3)
dx
If the ternts of the sequence ur, u2, j ..,, un... satisfy the recurrence u
relation u,n, = dx
dx
u
,*
3, n
Zl, then the r,l, term
may be expressed as
* $77 * 3tt
(A) (B)
tt, u,
(c)
ut+3(n-l) u,+6(n-l)
(D)
GO ON TO THE NEXT PAOE
I 1
J
l 1
i
lj
-5-
i I
i
4
For-l
17.
21.
rn0
l-n I
(B)
1-2n 2n
(c)
2n
22.
l-2n
converges?
(A) (B)
u,=
(D)
n-3 n(n-l)
*r(t
*)=
3n
n+2
n+l n
n+l
-4 (c) n+1
n+l
,,=r( 4
fl-l
23.
i
fi::: :::'i':"*:il (A) (B)
1l
(A)
I
3
t2
9
(A) (B)
64 37
(c)
16
(D)
6
+.
. . is
k (k + l)--
24
^t
-s
m
+l
(B)
S.*,-{,
(c)
- S, S,, - S,t,
(D)
ryl ro infinity of the geornetric series Ilr. 16 + +
+ l)=g then, for m ( r?, i,t1t t_t
*-n+l
-t
(D)
Given tnat
I
il.'i j,:? i f;,:
-3
(c)
n
(D)
3u
20
3!
(B)
{
I9
(c)
u,,= Z(-l),,-,
(D)
I
2t
of the following
(C) u,=*
j
(A) (B)
(A)
Given thatuorepresents the nh term of a
sequence, which
j
be simplified
1+2n
(D)
t8
hcan
and written as
2
(A)
The express io"
tS,,
The equation e* -.xa = 0 has a root between
(A) 0 and (B) I and2 (C) 2 and 3 (D) 3 and 4 I
: I.
I
GO ON TO THE NEXT PAGE
-625.
By using the Newton-Raphson method with a first approximation x,, the second approximation 'x,*, for a root of the equation x5 = xj + 25 may be expressed as
28.
Arelay team of 5 teachers is to be chosen from a group of l5 teachers. In how many ways could this relay team be chosen?
(A)
(B)
(c)
x,,-xl,-il,+25 sxi
4{
- x: -
x,,
sxi
xl,
l0!
-zs
r0!
l5!
(c)
5
(D)
5!
-txj 29
26
tC, equals
The values ofx for which the expansion
L
tCr*'C,
(B)
'Cr*tC,
(A)
-l
8
(B)
-Z
(c)
ll --))
(D)
x < -2
Jltoo-sox;
,C,
X
3
(D)
of
isvalidare
(A)
(c)
g x 7c J
27
l5!
(D)
4rl, -Zxj +25
sxl
5!
l5!
(B)
-txj
- lx) -zs
+xl,
l5!
(A)
and
x>2
If the coefficient of .f in the expression of (6
-
ax)e is
-84, then the value of a is
30
Let f be a continuous function with and f (0.8) = 0.76.
-
f (0) = I (A) 36
The first approximation to the root in [0, 0.8], using linear interpolation, to 3
36
decimal places is
(c)
36
(D)
-36
(A) (B)
0.000 0.400
(c)
0.444
(D)
0.45s
(B)
GO ON TO THE NEXT PAGE
-7 31.
In how many ways can the letters P, Q,, R, and I be arranged so that P and Q, are always together, and .R and .S are always together? ,S
(A)
(B) (c) (D) 32
-
(t-z 33. IfP=
Q:t
s!
,O=
3l x2 x2
4 -14 -2
(B)
I
l4
-14
34
3
-7
jl
-14 -2 4
(D)
ti
I
t4
l,:
021
(PQT'
(B)
(QPT'
(D)
-2 -5
(c)
00 2t0
(A)
(c)
3 -7 -5 173
-7 -5 73 3
I
-7 -5
7
then
The matrix P-r equals
14
:
4
0
,
07
[0
173
I4
il
and
-7 6 -10 -14 3 -5 2t
4 -t4 -2 3 -'7 -5
(A)
2
7
st.z
A-r
r
[-,
3!
A is a 3 x 3 matrix with determinant If the matrix of cofactors of A is
then
lr
2t I
2t
Given that
P
a y
: 0 at .r = 0, the general
solution of the differential y"+6y'+9y=Qit
equation
(A) ,- s3r * Bx (B) != Bxe-3' (C) Y=e3*(A+Bx) (D) y: e-3'+.Br
3
li
{
l
GO ON TO THE NEXT PAGE
-835
A
sample space X consists solely of 3 mutually exclusive events, p, R and S. If P(8) = 0.3 and P(R) = 0.6, then P(S) =
(A)
(B) (c) (D) 36
Item 37 refers to the following table which
of males and females and their preferences for Drink A and Drink.B. shows the number
0.r
Male Female
0.2 0.6 0.e
school debating team comprising 3 teachers, 3 boys and 3 girls is to be chosen from 5 teachers, 4 boys and 6 girls. The number of ways in which this team can be
Drink.B
t2 20
18 r0
30
Total
32
28
60
Drinkl
A
37
Total 30
One person is randomly selected. What is the probability that this person is female
chosen is
and prefers Drink B?
(A)
27
(B)
54
(c)
182 800
(D)
I
(A)
6 I
(B)
3 5
(c)
l4 I
(D)
38
2
The FIRST ROW ofthe product PQ ofthe two3x3 matrices
P-
2
3
I
5
6
5
-l
2
J
0=[
1,,,
2
I
3
5
0
-l
-3
-2
4
rs
(A) (B)
())
(c)
(t6
(D)
(22
(16
-4 -5
-5)
0
7)
-35
-r)
t)
GO ON TO THE NEXT PAGE
-9 39 i"
r"l
.t '
,
.
42
Given that AB = J, BC = K, CD = L, ABC = p and BCD = Q, where J, K, L, p and e are matrices, the product of ABCD is
I
,.,
A, B, C and D are four3 x 3 matricies.
-
:,1,
,1
A suitable integrating factor for the solution of the differential equation
dy - +2v - =-l. ls dxxx
rf I,"1,'
,
(A) (B) (c) (D)
:,,:
40
JQ
JKL LJ PD
Two coins and a die with faces numbered I to 6 are thrown together once. Assuming that thedieand coins are fair, the probabi lity of obtaining 2 heads and a number tess than
ln2x
(c)
f
(D)
ek
sgL
24 'l
I
(B)
- 'i.,
(B)
I
(A)
:
b
Item 43 refers to the Venn diagram below which shows the probabilities associated with events .t( and L in a sample space S.
4is
,1"
(A)
8
0.45
2
(c)
J
',:
0.0s
i
(D)
J 4
41.
The general solution of the differential equation
(A)
y=d y= b
(D)
,5r:r:
*=:
(B)
(c)
I
43
*
The probability that occurs, is
(A) (B)
0.25 0.33
(c)
0.35 0.95
(D)
I
occurs, given
thatl(
y=x'lk )t=lnx+k
I
i::i::.
GO ON TN TI]tr NItr].T
D^ T:tr
_
44
l0
The number of possible values of x which satisfy the system of simuttaneous
_
45.
equations,
The matrix
represents a system of linear equations after some elementary row operations have been performed.
2x+3y*22=-5 4x + 6y
* 4z= -10
A
Ir
0
I
3
^=l;
I
I
2
0
0
2
6x+9y+62=-25 Which ofthe fo I lowi ng statements
rs
(A) (B)
0
(c)
2
(D)
3
(A) (B) (C) (D)
I
is
TRUE?
The solution is unique. There is'no solution. There are infinitely many solutions The solutions are dependent.
END OF TEST IF YOU FINISH BEF'ORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
m?14010/CAPE 2016
