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FORM TP 2017297
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TEST CODE
CARIBBEAN EXAMINATIONS
O2I34O2O
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MAY/JUNE 2OI7
COUNCIL
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION@
PURE MATHEMATICS
UNIT I -Paper02
ALGEBRA, GEOMETRY AND CALCULUS
2 hours 30 minutes
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
l.
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 correct to three significant figures.
7.
If you need to rewrite any answer and there is not enough
space to do so
on the original page, you must use the extra lined page(s) provided at the
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
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All rights reserved.
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0213402003
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SECTION A
Module
I
Answer BOTH questions.
1.
(a)
Let p and q be two propositions
p : It is raining.
q : John is sick.
Write EACH of the statements below in terms of p and q
(i)
It
is not raining or John is sick.
I
(ii)
If it is raining then John
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is not sick
[1 mark]
(b)
An operation * is defined on the set { I , 2,3, 4} as shown in the following table.
(i)
*
I
)
3
4
I
2
4
I
3
2
4
J
2
I
3
I
2
3
4
4
3
I
4
2
Prove that
*
is commutative
[1 markl
(ii)
Show that the identity element of
*
is 3
[2 marks]
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0213402004
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(c)
The polynomial f(x) : ox3 + 9x2 when divided by (x + 2).
(i)
Show that
a:2
and
b:
llx
+ 6 has a factor of (x
-
2) anda remainder of
12
-30
[4 marksl
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-6(ii)
Hence, solveax3 +9x2
- llx+b:0.
[9 marks]
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(d)
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Use mathematical induction to prove that
3+
16
+24+32 + ...+8n:4n(n + l) forall n e N.
[7 marksl
Total25 marks
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(a)
(i)
Given lhat a2 *
b2
:
l4ab, prove that ln
( a+b\
t,
t;l=,-{'n a+tnb)
[5 marksl
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0213402008
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9(i
i)
Solve the equation
2* + 312'1:
4.
[Your response may be expressed in terms of logarithms.]
[6 marksl
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0213402009
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-10(b)
The following diagram shows the graph of the function
f(x): I x+4
On the diagram,
(i)
(ii)
insert the asymptotes for the function
sketch the graph of
f
12
marksl
f-r, the inverse of f showing the asymptotes for f-r.
[4 marksl
I
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5
0
-20
5
2b
x
-1
I
-20
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0213402010
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Civen that o,, p and y are the roots of the equation
roots are 0T, oy and oB.
I
+ 3x + 2:0, form an equation whose
[8 marksl
Total25 marks
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---:
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Module 2
Answer BOTH questions.
3.
(a)
(i)
Prove the identity
tan(A) + tan(.B)
td.ttt,.7T
t)t:
| -tan(A)
tan(.B)
-
[4 marks]
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02134020/CAPE 2017
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(ii)
Given that sin
,A:|
express tan (A +
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13 -
andcos
B: I *nere
angle A is acute and angle .B is obtuse,
B) in the form a + b J1, where a and b are real numbers.
[6 marksl
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0213402013
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(b)
Solvetheequation
sin2
0-2cos2 0*3cos
0+5:0for0 <0<4n.
[6 marks]
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02134020/CAPE 2017
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0213402014
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r
-
(c)
(i)
Express
f
(A:6
cos
d+
-l
15 -
8 sin d in the form
r sin (d+ o) where 0 < a < 90'
[3 marks]
(ii)
Hence, or otherwise, find the general solution of
f 104:2
[6 marksl
Total25 marks
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4.
-l
-16(a)
(i)
The circle, C,, with equation
*' + f -
4x + 2y
-2:
0 and the circle C, have a
common centre. Given that C, passes through the point (-1, -2), express the
equation of C, in the form (* - h)' + O/ - k)' : k.
rtiti
[3 marks]
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021340201CAPE.2017
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(ii)
The equation of the Iine L, is x +
circle, C,, in a (i) on page 16.
3y: 3. Determine
whether L, is a tangent to the
[7 marksl
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(b)
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Let P(3, I , 2) and Q( l, -2, 4).
(i)
--)
Express the vector PQ in the form
xi
+
yi + zk.
[2 marks]
(ii)
Determine the Cartesian equation of the plane which passes through the point Q
and is perpendiculu. ro
fr.
[6 marksl
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-19(c)
The vector equations of two lines, Z, and Lr, are:.
L,: -i+ j - 2k + o (-2i +.1 - 3k)
Lr:-2i+j-4k+B(i-i+k)
(i)
Show that
L,
and
L, intersect.
[5 marks]
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0213402019
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(ii)
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Hence, determine the coordinates of the point of intersection of the two lines.
[2 marksl
Total25 marks
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0213402020
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SECTION C
Module 3
Answer BOTH questions.
5.
(a)
Determine the value of ft for which
r(x):
xt-I
x-l'
-.x*l
k,
x=1.
is continuous for all values of x.
[4 marks]
t$:fi
tr.gJlt
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0213402021
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-22 (b)
A curve, C, is described parametrically by the equations
x:
5t + 3 and
y: f - f
+2
dy
(i) Find;
in terms of t.
[3 marksl
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0213402022
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(ii)
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Hence, determine all points of C such
dy
that-:0.
[6 marks]
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0213402023
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(c)
(i)
Given that y:
2+2x2,
show that
dy
a)
b)
v: -2x:
'dx
d'y _,4 :
dtf
0
o
[9 marks]
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02134020/C,APE 2017
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0213402024
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(ii)
Hence, find the value
of
ff
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*n"n
*:
o
[3 marks]
Total25 marks
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6.
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-26(a)
Triangle PQRhas vertices P(0, I ) QQ,3) and R(4,2).
(i)
On the axes below, sketch triangle PQR.
v
x
I
(ii)
markl
Determine the equations of EACH of the following:
.PQ
.QR
.PR
.-S.'
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02134020/CAPE 2017
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0213402026
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[7 marks]
Flf.,'S
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021340201CAPB 2017
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0213402027
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(iii)
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-
Hence, use integration to determine the area of triangle PpR.
[7 marks]
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02t34020tcAPE 2017
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0213402028
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-29 -
(b)
The voftage in a circuit,
when /
:
4
satisfes the
equatio,
dVV
*
d, -:0.
0 seconds, write an expression for Z in terms of
Given
thatV:25 volts
r.
[5 marksl
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0213402029
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(c)
Given that
f3
J
,_1
3
[3f (x) + g (x)]
.3
t
l
,'{')
dx: 5 and !
[5f ("r) -2e @)l
dx: l, determine
*
-3
a
!, eot a.'
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02134020/CAPE 2017
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0213402030
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$
i-:*
i-:ls
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:ll*
'.-.8
'
[5 marks]
Total25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORI( ON THIS TEST.
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