Discussion of paper

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Discussion of paper

• F.6 Pure Mathematics
• 2009-06-29

JOHN NG
http//johnmayhk.wordpress.com
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Q.1 (b) Satisfactory
?
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(No Transcript)
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should be found by definition.
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Q.1(c) Satisfactory
Prove
is continuous at x 0,
check
It is not enough to check
We need
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Q.2 (a) Good
?
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Q.2 (b) Satisfactory
is valid when n is a POSITIVE INTEGER.
is valid when n is a POSITIVE INTEGER.
Thus, it is WRONG to write
The iterance should stop at
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Q.3 (a) Satisfactory
?
Also, students just used lhôpitals rule without
using so-called very important limit
to simplify the work.
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Alternative method
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Q.3 (b) Not satisfactory
?
?
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Make sure to make indeterminate forms like 0/0,
?/? etc. before using lhôpitals rule.
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Q.4 (b) Satisfactory
Prove f(x) is bounded on R, we need to find a
FIXED number M such that f(x) ?? M for ALL x
in R.
Hence, it is wrong to say
?
Just one step further, we can get rid of the x,
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Q.5 (a) Not satisfactory
Students may know what x is, but
not familiarized with the operations.
?
Some wrote 2x 2x or 2x 2n - x
?
Note
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Q.5 (b) Not satisfactory
The graph of y 2x/x is strange to students.
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Q.5 (b)(ii)
Students tried to prove g(x) is injective.
Just tried some values of x and see that g(x) is
NOT.
showing that is NOT injective.
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Q.6 (a)
?
Mistook that
Taking logarithm is the trick.
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Q.6 (a) Not satisfactory
Not many students could show that f(x) gt 0.
A negative sign is in the expression, and it is
not clear enough that f(x) gt 0. Better
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Q.6 (b)(i) Satisfactory
Alternative method
(By (a))
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Q.6 (b)(ii) Not satisfactory
Alternative method
Hence
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Q.6 (b)(ii) Not satisfactory
There is a misunderstanding.
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Q.7 (a)(i) Good
f(x) 0 has a triple root ?, some mistook that
?
?
Some weak in basic differentiation rules, e.g.
?
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Q.7 (a)(ii) Satisfactory
Some used integration in this part. They claimed
?
Thus, ? is a repeated root of f(x) 0
Note the integration is invalid.
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Q.7 (b)(i) Good
Students tried to solve the repeated root
to prove a2? b.
It may be easier to consider that the quadratic
equation g(x) 0 has a real root, hence ? ? 0.
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Q.7 (b)(ii) Not satisfactory
Many students set up Viètes formulas (????) and
obtained complicated relations.
Not many students could solve the repeated root
by elimination
Eliminate ?3,
Eliminate ?2, yield
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Q.7 (c)(i)(ii) Good
Many students could solve (c) without using the
result in (b)(ii).
Students may find it easy to cope with concrete
numerical problems.
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Q.8 (a)(i) Satisfactory
To sketch the graph of
?
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Q.8 (a)(ii) Satisfactory
Some students obtained the following wrongly
and cannot draw the conclusion where f is
differentiable at x -1 or not.
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Q.8 (b) Satisfactory
If students could obtain the first and second
derivatives correctly, it is likely that they
could perform very well in this part
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Q.8 (c) Satisfactory
Many students ignored that (-1,0) is also a
minimum point.
Some said there was no inflection point.
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Q.9 (a)(ii) Satisfactory
Students should pay attention that g(t) means g
is a function of t ONLY, other indeterminate like
x and c, are constants with respect to t.
Hence g(t) is differentiating g with respect to
t.
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Q.9 (a)(ii) Satisfactory
Also, by applying the mean value theorem, state
clearly the range of the value of the d, i.e.
for some d lying between x and c.
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Q.9 (b) Not satisfactory
Not many students could complete this part.
Summing up and use the f(x) lt 0 on (a,b),
result follows.
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Q.9 (c) Not satisfactory
Many students took
instead of
to obtain
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Q.9 (c)
Note that
does not satisfy the condition that
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Q.10 (a)(ii) Not satisfactory
Many students tried to prove by M.I. that
but not success in most of the case. Solution
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Q.10 (a)(ii)
The easiest way should use the increasing of
an, that is
and solve the quadratic inequality above.
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Q.10 (b)(i) Satisfactory
Students may find question in this type is
not-so-familiarized, many could not use the fact
that
for all n ? N, to derive
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Q.10 (b)(ii) Not satisfactory
Few could complete this part. Many used the
previous result wrongly, they wrote
Instead of the fact that (b)(i) valid only for n
? N, i.e.
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Q.10 (c) Not satisfactory
No one could complete this part. No one could
show the following key step.
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Q.10 (c)
Now check the condition
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Q.10 (c)