harder 3U Q (1 Viewer)

mojako

Active Member
Joined
Mar 27, 2004
Messages
1,333
Gender
Male
HSC
2004
Can smeone do this pls..? ;)
Note: I'm not really sure what the RHS in part (b) is.
 

Xayma

Lacking creativity
Joined
Sep 6, 2003
Messages
5,955
Gender
Undisclosed
HSC
N/A
Was there any diagram that came with this? Im intrigued by "lower trapezia"
 

googleplex

New Member
Joined
Mar 30, 2004
Messages
29
Gender
Male
HSC
2009
just drawing trapeziums under the graph getting all those areas < integrating from 1 to n
?

yeh was there any pics or lead ups to this q
 

nike55

New Member
Joined
Aug 13, 2004
Messages
6
hehe usually i dont post anymore but im in a good mood from my umat scores :)
(btw it would be good if someone got latex or some math program for this board like other maths forums do)

anyway

area under y = lnx (from 1 to n) > area under sum of trapeziums from 1 to n
nlnn - n + 1 > 1/2(0+ln2) + 1/2(ln2+ln3) + .... + 1/2 (ln(n-1) + lnn)
1/2lnn + nlnn - n + 1 > lnn! (adding 1/2lnn to each side)
e(1/2lnn + nlnn - n + 1) > n!
e(1/2lnn) * e(nlnn) * e(-n) * e > n!
(n^1/2 * n^n * e)/e(n) > n!
(e * n^(n+1/2)) / e(n) > n! as required
 

mojako

Active Member
Joined
Mar 27, 2004
Messages
1,333
Gender
Male
HSC
2004
Xayma said:
Was there any diagram that came with this? Im intrigued by "lower trapezia"
unfortunately there wasnt :(

I'll read nike's answer.
Thanks!

When I tried to do this question I used the wrong formula for trapezoid rule.... and also I tried to find the area from 1 to (n+1).
 
Last edited:

mojako

Active Member
Joined
Mar 27, 2004
Messages
1,333
Gender
Male
HSC
2004
Is there a term to say if I make "x" becomes "2^x" or "e^x"?
If you dont get what I mean... making "x" becomes "x^2" is called.. raising x to the power of two.

EDIT: I know now... somebody told me.. it's "raising 2 to the x" or "raising e to the x", hehe..
 
Last edited:

mojako

Active Member
Joined
Mar 27, 2004
Messages
1,333
Gender
Male
HSC
2004
another Q:
solve 6x^4 - 7x^3 - 28x^2 +35x -10 = 0

I'm sure many, many of you are much, much smarter than me ;)
Just do the circle and polynomial questions by "inspection" like in <a href=http://www.boredofstudies.org/community/showthread.php?t=41677>here</a> ;)

Inspection inspection.. beware.. be careful.. the inspectors are coming.. hide hide hide... cover up any mess.
Now, where are our inspectors??
I need them!!
 
Last edited:

withoutaface

Premium Member
Joined
Jul 14, 2004
Messages
15,098
Gender
Male
HSC
2004
by inspection only works if they don't give you the answer:p
 

mojako

Active Member
Joined
Mar 27, 2004
Messages
1,333
Gender
Male
HSC
2004
withoutaface said:
by inspection only works if they don't give you the answer:p
I dont think I give the answer for the polynomial Q
 

ngai

Member
Joined
Mar 24, 2004
Messages
223
Gender
Male
HSC
2004
mojako said:
solve 6x^4 - 7x^3 - 28x^2 +35x -10 = 0
lol u asked, so here it is
clearly, x = 1/2, 2/3, +- sqrt(5) (by inspection)
 

mojako

Active Member
Joined
Mar 27, 2004
Messages
1,333
Gender
Male
HSC
2004
oh.. my dearest inspector...
anyway I sneaked into your office and found your inspection method :p
now I'm gonna make the whole community knows

since the coefficients of the polynomial are all integers, the numerator of any rational roots is a factor of the constant term, and the denominator is a factor of the leading coefficient.
BTW do we have to know this?? It's not mentioned in the syllabus, which however has a section (section 7.1) which specifically talks about integer roots being factors of the constant term. I've seen a question like this but I thought it was a bad question.

Now that you secret inspection procedure has been broken, would be very friendly to do the circle question not by inspection?
 

ngai

Member
Joined
Mar 24, 2004
Messages
223
Gender
Male
HSC
2004
mojako said:
since the coefficients of the polynomial are all integers, the numerator of any rational roots is a factor of the constant term, and the denominator is a factor of the leading coefficient.
BTW do we have to know this?? It's not mentioned in the syllabus, which however has a section (section 7.1) which specifically talks about integer roots being factors of the constant term. I've seen a question like this but I thought it was a bad question.

Now that you secret inspection procedure has been broken, would be very friendly to do the circle question not by inspection?
ROFL i didnt even bother with that
i used maple

hahaha alrite i'll do the circle geo
extend PO to meet circle again at E
BD & OP intersection is F
now, let angle FDP = a
so CDB = a
and so CAB = a
and so PEC = a
and so triangles ECP, FDP similar (equal angles a and P)
so angle DFP = angle ECP = 90
 

mojako

Active Member
Joined
Mar 27, 2004
Messages
1,333
Gender
Male
HSC
2004
Thank you :)

How do you know how to do this though??
How long did it take you?
 

Archman

Member
Joined
Jul 29, 2003
Messages
337
Gender
Undisclosed
HSC
N/A
'eres another soln to the circle geo (by inspection :p)
draw in tangent at P, call a point to the right of P on the tangent E
let PO meet BD at F
EPA = PCA = DBP so PE and DB are parallel.
now since PE is a tangent, so PE is perpendicular to OP, so DB is perpendicular to OP.
 
Last edited:

CrashOveride

Active Member
Joined
Feb 18, 2004
Messages
1,488
Location
Havana
Gender
Undisclosed
HSC
2006
You mean E is to the right of P on the tangent...then all works well thanks for that Archman ^_^
 

mojako

Active Member
Joined
Mar 27, 2004
Messages
1,333
Gender
Male
HSC
2004
omg.. inspection..
its such a wonderful and powerful technique..
I wonder why we weren't taught that technique at school ;)

On the serious side, how do these things come to your mind?
 

Archman

Member
Joined
Jul 29, 2003
Messages
337
Gender
Undisclosed
HSC
N/A
CrashOveride said:
You mean E is to the right of P on the tangent...then all works well thanks for that Archman ^_^
thx, its corrected.
 

Rorix

Active Member
Joined
Jun 29, 2003
Messages
1,818
Gender
Male
HSC
2005
mojako said:
On the serious side, how do these things come to your mind?

You inspect it.

hence, 'by inspection'.
 

withoutaface

Premium Member
Joined
Jul 14, 2004
Messages
15,098
Gender
Male
HSC
2004
mojako said:
omg.. inspection..
its such a wonderful and powerful technique..
I wonder why we weren't taught that technique at school ;)

On the serious side, how do these things come to your mind?
You fuck around with a few things in your head until you find one that works.
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top