# harder 3U Q (1 Viewer)

#### mojako

##### Active Member
Can smeone do this pls..?
Note: I'm not really sure what the RHS in part (b) is.

#### Xayma

##### Lacking creativity
Was there any diagram that came with this? Im intrigued by "lower trapezia"

##### New Member
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
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
Xayma said:
Was there any diagram that came with this? Im intrigued by "lower trapezia"
unfortunately there wasnt

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).

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#### mojako

##### Active Member
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..

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#### mojako

##### Active Member
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!!

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#### withoutaface

by inspection only works if they don't give you the answer

#### mojako

##### Active Member
withoutaface said:
by inspection only works if they don't give you the answer
I dont think I give the answer for the polynomial Q

#### ngai

##### Member
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
oh.. my dearest inspector...
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
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
Thank you

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

#### Archman

##### Member
'eres another soln to the circle geo (by inspection )
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.

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#### CrashOveride

##### Active Member
You mean E is to the right of P on the tangent...then all works well thanks for that Archman ^_^

#### mojako

##### Active Member
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
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
mojako said:
On the serious side, how do these things come to your mind?

You inspect it.

hence, 'by inspection'.