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#1 2007-03-21 10:08:36

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Several Questions

These questions come from the 1st pre-test of College Entrance in China.

   1.


,
.
   (1) Find Sequence

   (2) compare
where m and k are integers but not equal.
   The above two are trivial .
  Here is the last question.
  (3)When q>1  , compare

Cleaned up latex from over running the page - Ricky


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#2 2007-03-21 10:18:09

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Several Questions

2. A circle

has intersections with line
at point P and Q,
There's a point
and

  (1)when b=1 , find the value of k
  (2)when
find the domain of k .


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#3 2007-03-21 10:26:24

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Several Questions

Two functions

have intersections Point A and O .
A line
has intersection with the two functions Point B and D .
(1) Find the relationship of the Area of AOBD and t , that is to find

(2) Find the maximum value of A while 
.


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#4 2007-03-21 10:27:45

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Several Questions

I think the Chinese math exam is a little harsh , Since everyone has to deal with problem like these , no matter what your major is.

Last edited by Stanley_Marsh (2007-03-21 10:28:10)


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#5 2007-03-21 11:43:06

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Several Questions

#2

(1) C is the circle of radius 1 with centre at (1,1). It touches (is tangent to) the y-axis at (0,1). So if M is this point and MP is perpendicular to MQ, PQ must be the diameter of the circle. Thus, the line ℓ passes through the centre of the circle – so k = 1.

EDIT:
(2) Still working on the second part. The relationship I’ve derived between b and k is

Last edited by JaneFairfax (2007-03-21 12:30:07)

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#6 2007-03-21 12:45:27

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Several Questions

These are the 3  last questions,  the entrance exam just like sat , but only got 120 minute to finish 21 questions.

Yeah , the second is kind of tricky.


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#7 2007-03-22 16:31:01

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Several Questions

Okay, here’s I(1).

From which we can see that


(otherwise we would have 0 = 1).

If we carry on, we find

From the emerging pattern, we can guess that

It remains to prove our guess by induction. That’s the easy part – the hard part is finding the pattern in the first place. Well, not so much hard as laborious. dizzy

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#8 2007-03-22 16:49:46

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Several Questions

Actually there's a shortcut ~


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