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TRANSCRIPT
r %. TEST CODE O2I34O2OMAY/JTINE 2016
_lFORM TP 2016279
CARIBBEAN EXAMINATIONS COUNCIL
CARIBBEAN ADVANICED PROFICIENCY EXAMINATION@
PURE MATHEMATICS
UNITl-Paper02
ALGEBRA, GEOMETRY AND CALCULUS
2 hours 30 minutes
Examination Materials Permitted
Mathematical formulae and tables (provided) - Revised 2012Mathematical instrumentsSilent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
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READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
This examination paper consists of THREE sections.
Each section consists of TWO questions.
Answer ALL questions from the THREE sections.
Write your answers in the spaces provided in this booklet.
Do NOT write in the margins.
Unless otherwise stated in the question, any numerical answer that is notexact MUST be written correct to three signiflcant figures.
If you need to rewrite any answer and there is not enough space to do soon the original page, you must use the extra lined page(s) provided at theback of this booklet. Remember to draw a line through your originalanswer.
rf you use the extra page(s) you MUsr write the question numberclearly in the box provided at the top of the extra page(s) and, whererelevant, include the question part beside the answer.
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SECTION A
Module 1
Answer BOTH questions.
l. (a) Let f(x) :2x3 - f + Px + q.
(i) Given that x + 3 is a factor of f(x) and that there is a remainder of 10, when (x) isdivided by x + I show that p: )5 and q: -12.
[7 marks]
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02t34020lC.APE 2016
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(ii) Hence, solve the equation (r):0.
[6 marks]
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(b) Use mathematical induction to prove that 6'- 1 is divisible by 5 for all naturalnumbers n.
[6 marks]
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02t34020lcAPE 2016
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IL 0213402006
t- -7 - -l(c) (i) Given that p and q are two propositions, complete the truth table below:
p q p ---+ q pvq p^q (p v q)--- (p n q)
T T
T F
F T
F F
[4 marks]
(ii) State, giving a reason for your response, whether the following statements arelogically equivalent:
p--+q
(pvq)---(pnq)
[2 marks]
Total25 marks
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-lr2. (a)
02r34020lcAPE 2016
Solve the following equation for x
log, (10 --r) + logrx: 4
-8-
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[6 marks]
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-lrA tunction f is defined by f(x) : ++ , x + |
-9-
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(b)
02134020/CAPE 20t6
Determine whether f is bijective, that is, both one-to-one and onto
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JL 0213402009
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[8 marks]
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JL 0213402010
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(c)
o2t34020tcAPE 2016
Let the roots ofthe equation 2x3 - 5l + 4x + 6 : 0 be o, B and y
(D State the values of a + I + y, o0 * oy + By and opy.
[3 marks]
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(ii) Hence, or otherwise, determine an equation with integer coefficients which has
I)
Y-andI
e1)
o-roots
Note: (og)' + (oy)' + (Fy)' : (ap + ay + Ey)' - 2o-Ft(cr+ B + y;
a, +9, + l, :(o + p + y), -2 (ap + oy + 0y)
02r34020/cAPE 2016
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JL 0213402012
r -13-
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[8 marks]
Total25 marks
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.l
02t34020lcAPE 2016
L 0213402013 I
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3. (a) (i) Show that sec2 0 :
-t4-
SECTION B
Module 2
Answer BOTH questions.
cosec 0cosecd-sin0
[4 marks]
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02t34020lcAPE 2016
r ilil til ilil ilil| ililr ilil llfl llil illl lllil lil lil JL 0213402014
Itr_-.:ftf.Et.--t+-:
..t4.,f,*.j*i--:.'-l{-:,
El\:.Ir;t---\r--.€{.i-I* -:
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:,-,=*- -
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Iilil liltil!] lill lillllilllllllllllllllll lllll lil llll
[5 marks]
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-l(ii) Hence, or otherwise, solve the equatio, tot-t" d, ,cosecA-sind
for 0 < 0<2n.
4J
02t34020/cAPE 2016
JL 0213402015
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t- -t6-
(b) (D Express the function f@) : sin 0 + cos d in the form r sin (0 * a), where
r>Oand}<e<TE-2
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[5 marks]
GO ON TO THE NEXT PAGE02t34020/cAPE 2016
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,-t+-...G.-Q..
::$:.-ao
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(ii) Hence, find the maximum value of f and the smallest non-negative value of 0 atwhich it occurs.
[5 marks]
GO ON TO THE NEXT PAGE02t34020/cAPE 20t6
I tiltil illl ilt] ilil tilil ilil illl illll ult tilt ll] il _lL 0213402017
t-(c) Prove that
tan(A+B+C):
-18-
tan A -r tan B * tan C - tan A tan B tan C
-l
1 - tan A tan B - tan A tan C - tala B tan C
GO ON TO THE NEXT PAGE02r34020/cAPE 2016
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[6 marksl
Total25 marks
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02t34020lcAPE 2016
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4. (a) (i) Given that sin d: x, show that tan e: ___L, where 0 < e < a^tt -7 2
[3 marks]
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(ii) Hence, or otherwise, determine the Cartesian equation of the curve deflnedIE
parametrically by y : tan 2t and x : sin r for 0 ., . T.
[5 marks]
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021340201CAPE 2016
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(b) Let u : and v: be two position vectors in RP
(i)
(ii) Find cos d where d is the angle between u and v in R3
-l2
1
5
Calculate the lengths of u and v respectively.
[3 marks]
[4 marks]
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(c) A point, P (x, y), moves such that its distance from the x-axis is half its distance from the
oflgln.
Determine the Cartesian equation of the locus of P
[5 marksl
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02r34020lCAPE 2016
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t- -24- -l(d)
02134020/CAPF. 2016
The line Z has the equation 2x + y + 3 : 0 and the circle Chas the equation x2 + 1? : 9Determine the points of intersection of the circle C and the line L.
[5 marksl
Total25 marks
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SECTION C
Module 3
Answer BOTH questions.
I
[4 marksl
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5. (a) Use an appropriate substitution to find J f,
* f f dx
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02t34020lcAPE 2016 IL 0213402025
t- -26-
The diagram below represents the finite region R which is enclosed by the curve/ : x3 - |and the lines x:0 andy:6.
y:# -r
-l(b)
02134020/CAPE 2016
v
x0 I
Calculate the volume of the solid that results from rotating R about the y-axis
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,.E-!i+i
[5 marks]
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r .\1 -l(c) Given that Io (r) dx : J, n" - x) dx a ) O,show that
dI Id-x
A +7- 2
[6 marksl
GO ON TO THE NEXT PAGE02t34020lcAPE 2016
L il|]il ililililI ilil tilil til[llfl rill l]ll lill lill llll0213402027 I
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An initial population of 10 000 bacteria grow exponentially at a rate of 2%o per hour,where y : f (t) is the number of bacteria present / hours later.
(i) Solve an appropriate differential equation to show that the number of bacteriapresent at any time can be modelled by the equation y : l0 000 e0.o2t
[7 marks]
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-l(d)
02t34020/cAPE 2016
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t- -29 -
(iD Determine the time required for the bacteria population to double in size
-l
[3 marksl
Total25 marks
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02t34020lcAPE 2016
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6. (a) Find the equation of the tangent to the curve f(x) :2x3 + 5f - x + 12 at the point wherex:3
[4 marks]
GO ON TO THE NEXT PAGE02t34020lcAPE 2016
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i+2x+3 -x<0f(r) :
ax+b x>0
lim f(x)x --- 0-
o2t34020lcAPE 2016
r iltil ffi ilil ilil rilil lflr lllll llll lll! lllll lil llll
(i) Calculate the and
[4 marksl
(iD Hence, determine the values of a and D such that (r) is continuous at x : 0.
[5 marks]
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lim f(x)r --- 0*
IL 0213402031
r 32-
(iii) If the value of b :3, determine a such that f '(0) :
-llim
I ---+ 0
(0+r-f(0)t
[6 marks]
GO ON TO THE NEXT PAGE02t34020/cAPE 2016
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-lr -33-
(c) Use first principles to differentiate f(x) : {Twith respect to x
[6 marks]
Total25 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORI( ON THIS TEST.
02134020|CAPE 2016
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Question No.
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