Post on 15-Sep-2019
Marking Scheme
Module 2 (Algebra and Calculus)
This document was prepared for markers'reference. It should not be regarded as a set of model answers. Candidates and teachers who were not involved in the marking process are advised to interpret its contents with care.
General Marking Instructions
1. It is very important that all markers should adhere as closely as possible to the marking scheme. In manycases, however, candidates will have obtained a correct answer by an alternative method not specified in themarking scheme. In general, a correct answer merits all the marks allocated to that part, unless a particularmethod has been specified in the question. Markers should be patient in marking alternative solutions notspecified in the marking scheme.
2. In the marking scheme, marks are classified into the following three categories:'M'marks awarded for correct methods being used; 'A'marks awarded for the accuracy of the answers; Marks without'M'or'A' awarded for correctly completmg a proof or arriving
at an answer given in a question. In a question consisting of several parts each depending on the previous parts,'M'marks should be awarded to steps or methods correctly deduced from previous answers, even if these answers are erroneous. However, 'A'marks for the corresponding answers should NOT be awarded恤less otherwise specified).
3. For the convenience of markers, the marking scheme was written as detailed as possible. However, it is stilllikely that candidates would not present their solution in the same explicit manner, e.g. some steps wouldeither be omitted or stated implicitly. In such cases, markers should exercise their discretion in markingcandidates'work. In general, marks for a certain step should be awarded if candidates'solution indicated thatthe relevant concept/technique had been used.
4. In marking candidates'work, the benefit of doubt should be given in the candidates'favour.
5. In the marking scheme,'r.t.'stands for'accepting answers which can be ro皿ded off to'and'f.t.'stands for'follow through'. Steps which can be skipped are璽璽璽whereas alternative answers are enclosed with転ctangle�. All fractional answers must be simplified.
6. Unless otherwise specified in the question, numerical answers not given in exact values should not beaccepted.
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Solution I Marks I Remarks
I. (5 + x)4
= 54 +4(53 )x+6(5勺x2 + 4(5)x3 + x4 lM = 625 + 500x + 150x2 + 20x3 + x4 lA
曰］三正訂訂 lM
6 12 8 =1--+—-— X X 2 X 3
The constant term = (625)(1) + (500)(-6) + (150)(12) + (20)(-8) IM I withhold 1 M if the step is skipped =-735 IA
----------(5)
�') .1 1
了 一了亡- ✓x✓x(x+h)
严汀戸鬥- .Jx(x+h) .[;+元」lM
x+h-x ＝
(x+h)」;+x✓x+hh
(x+h)」；+x✓x+h 1
昷=lim一1二✓3 －可✓3
h今Oh lM
-Jim 1( -語〕h丑 h (x+h)✓x+x✓x+h
= lim -✓3曰 (x+ h)乒+x✓x-i歹 lM I withhold lM if the step is skipped
-✓3
2x」 I lA
---(5)
64
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3.
Solution
(a) Note that the coordinates of Q are (0, 2e勺，where u >0·.The area of LlOPQ
u(2e勺2
=ue u
(b) Let v be they-coo「dinate of P.
Since v = 2e" , we have -dv = 2eu 一
du－ ．
dt dt
Therefore we have 一 ＝＿＿ ．du 1囯， dt 2eu
Let A square units be the area of AOPQ .
By (a), we have A= ue" . dA dt
du = (eu + ueu ) 一
dt
=〔eu +ue"
)�
2e" dt
｛言So, we have 扭-I = 〔与汀(6) = 15 .
dt u .. 4Thus, the required rate of change is 15 square units per second.
Let v be the y-coordinate of P .
Since v =2才，we have u = In囯·Let A square units be the area of t:..OPQ .
By (a), we have A =蝨2 ln〔汀2
·
dA dt
＝（（司〔f〕〔½)+½ln長〕〕窋
丑刊譚When u = 4 , we have v = 2e4 .
閂dt u=4
侶們）(6)
=15
Thus, the required rate of change is 15 square units per second.
65
I Marks I Remarks
lA
IM
lM+lM
IA
IM
I lM+lM
lA
一一一一(5)
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