r -l · 2019. 4. 3. · r l-29 - a committee of 4 persons is to be chosen from 8 persons, including...
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r % TEST CODE 02234020-lMAY/JUNE 2018FORM TP 2018301
CARIBBEAN EXAMINATIONS COUNCIL
C ARIB B EAI\ AD VA NC E D PROF'IC IE N C Y E XA1VIINATIO N@
PTJRE MATHEMATICS
UNIT 2 -Paper 02
ANALYSTS, MATRTCES AI\D COMPLEX NUMBERS
2 hours 30 minutes
Examination Materials Permitted
Mathematical formulae and tables (provided) - Revised 2012
Mathematical instrumentsSilent, non-programmable, electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1. This examination paper consists of THREE sections
2.
3.
4.
5.
6.
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 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
onthe original page, you must use the extra page(s) provided at the back ofthis booklet. Remember to draw a line through your original answer.
8. If you use the extra page(s) you MUST write the question numberclearly in the box provided at the top of the extra page(s) and, where
relevant, include the question part beside the answer.
Copyright @ 2Ol7 Caribbean Examinations CouncilAll rights reserved.
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-lr -4-
SECTION A
Module I
Answer BOTH questions.
A curve P is defined parametrically as x = #r, y: *,l. (a) (i)
Determine the gradient of the curve at the point ( + , +)
[5 marksl
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022340201CAPE 2018
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(iD Hence, or otherwise, determine the x and y intercepts of the tangent that touches
the curve at the Point
[4 marksl
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(+,+)
022340201CAP8 2018
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0223402005
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r -6-
(b) Let the function/(x, y) = sin (Ix) sin (alcy). Determine
02234020/CAPE 2018
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0x 0y
[3 marksl
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_tL 0223402006
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r -7 -
(c) Use DeMoivre'stheoremto showthatsin 50: sin5 0-lO sin3 0 cos2 0 * 5 cosa 0sin0
[5 marks]
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022340201CAP8 2018
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0223402007
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r -8-
(i) Write the complexnumber z=(l -r) in theformreia where r:lz I and0: arg(z).
[3 marksl
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-l(d)
02234020/CAPE 2018
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-lr 9-
(ii) Hence, show that (1 - i)': 16(1 - i)
[5 marksl
Total25 marks
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022340201CAP8 2018
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-lr -10-
2. (a) Determine
I(i) xt cos (x3)dx
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-lr - 11-
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[8 marksl
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022340201CAP8 2018
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02234020/CAPE 2018
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[5 marksl
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2re
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dx.
JLY.!F
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(b) (i)
022340201CAPE 2018
-13-
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-lUse partial fractions to showthat #+ = + -#,
lQY.,"4
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[8 marksl
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-15- -l(ii) Hence, determine
f x4+ Il;641y ax
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[4 marksl
Total25 marks
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-lr -16-
SECTION B
Module 2
Answer BOTH questions.
Asequence {a,} is suchthat or= 0 dfrddn*r: Jr+q
(a) (i) State the third term, ar, of the sequence.
02234020/CAPE 2018
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[2 marksl
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3
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-lr -t7 -
(ii) Use mathematical induction to prove that x, is increasing and that it is bounded
above by 3, that is,4, ( a*, and o, ' 3 for all z € N'
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-lr -18-
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[8 marksl
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-ts' -l(b) (i) Letf (x): "-". By calculating the first three non-zero terms and assuming that
the pattem continues, show that the Maclaurin series expansion of f (x) may be
@
expressed as/(x) :Ik=O
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02234020/CAPE 2018
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[8 marksl
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-21 -
(iD Hence, or otherwise, determine the values of x for which the expansion is valid.
[3 marksl
-l
(c) Determine the sum of the ,".i", ,;, (.,"( ; )
.i,, ( * )
[4 marks]
Total25 marks
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4. (a) Determine the coefficient of the term in x7 in the expression
022340201CAPE, 2018
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[4 marksl
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-23 -
(b) By expressing nC, and 'C,-, in terms of factorials, show that"C, * 'C,-r- '*'C,
-l
[6 marksl
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02234020/CAPE 2018
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r(c) (i)
-24 -
Use the intermediate value theorem to show that the equation 4 cos r - x3 + 2:0has a root in the interval (1, 1.5).
[3 marksl
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-lr -25 -
(ii) Use linear interpolation to approximate the value of the root of the equation4 cos x - f + 2 = 0 in the interval (1, 1.5), correctto two decimal places.
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[8 marksl
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-l
022340201CAP8 2018
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-27 - -l(d)
022340201CAPE 2018
The equation 3d = | - 2 ln.r is known to have a root in the interval (0, 1).
Taking xr= 0.2 as the first approximation of the root, use the NeMon-Raphson methodto find a second approximation, rr, of the root in the interval (0, 1).
[4 marksl
Total25 marks
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r -28 -
SECTION C
Module 3
Answer BOTH questions.
5. (a) Events A and.B are such that P(A): 0.4, P(B) :0.45 and P(A w B) = 0'68
(i) Calculate P(A a B).
-l
[3 marksl
(ii) Determine whether the events A and B are independent. Justiff your response.
[3 marksl
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022340201CAPE 2018
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r
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-29 -
A committee of 4 persons is to be chosen from 8 persons, including Mr Smith and his
wife. Mr Smith will not join the committee without his wife, but his wife will join the
committee without him.
Calcutate the number of possible ways the committee of 4 persons can be formed
[5 marksl
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-l(b)
02234020/CAPE 2018
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r -30-
How rnany odd numbers greater than 500 000 can be made from the digits 2, 3, 4, 5 , 6, 7 ,
without repetitions?
[5 marksl
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-l(c)
02234020/CAPE 20r8
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r - 31 - -l(d) Two matrices, are given as
(D
-2 3-)3 4 | n:0 6)
5
02
A
1
By finding AB, deduce that A-t : gg B.
[5 marksl
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-lr -32-
(ii) Hence, or otherwise, solve the system of equations given as
7
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-2 3l341o 6)
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[4 marksl
Total25 marks
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02234020/CAPE 2018
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r -33-
A differential equation is given as y' cos x = y sin x + sin 2x.
(i) Show that the general solution of the differential equation is
!r..r-cosrt.Csecx.2
-l6. (a)
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-lr -34-
(ii) Hence, or otherwise, solve the initial value problemy'cos x:y sin x + sin 2x, Y (0):0.
[9 marks]
[2 marks]
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-35-
A differential equation is given as yo + 2y' + y: xe-'.
(D Determine the solution of the complementary equation yn + 2y' 't y: 0
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-l(b)
02234020/CAPE 2018
[4 marksl
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-lr -36-
(ii) Given that the particular solution of a differential equation has the form
ye= (Af + Bx2)e-', or otherwise, determine the general solution of the differential
equation.
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-lr -37 -
[10 marksl
Total25 marks
END OF TEST
IF YOU FINISII BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
022340201CAPE 2018
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