STEP I 1997 question 6 solution - The Student Room
The Student Room

STEP I 1997 question 6 solution

TSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP I 1997 question 6 solution


\displaystyle x^4(1-x)^4 = x^8-4x^7+6x^6-4x^5+x^4. The easiest way of doing the next bit would probably be long division, but that will be painful to layout in TeX. So, let's compare coefficients...


Coeff of \displaystyle x^8:  a_6 = 1

Coeff of \displaystyle x^7:  a_5 = -4

Coeff of \displaystyle x^6:  a_6+a_4 = 6 \implies a_4 = 5

Coeff of \displaystyle x^5:  a_5+a_3 = -4 \implies a_3 = 0

Coeff of \displaystyle x^4:  a_4+a_2 = 1 \implies a_2 = -4

Coeff of \displaystyle x^3:  a_3+a_1 = 0 \implies a_1 = 0

Coeff of \displaystyle x^2:  a_2+a_0 = 0 \implies a_0 = 4

Coeff of \displaystyle x^1:  a_1 = 0 (Which we knew it should be, but it never hurts to check).

Coeff of \displaystyle x^0: a_0+b = 0 \implies b=-4


Using this in the integral we get:

\displaystyle \int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx = \int_0^1 x^6-4x^5+5x^4-4x^2+4 - \frac{4}{1+x^2} dx

RHS = \displaystyle \frac 17 - \frac 46 + 1 - \frac 43 + 4 - \pi = \frac {22}{7} - \pi as required.

\displaystyle \int_0^1 x^4(1-x)^4 dx = \int_0^1 x^8-4x^7+6x^6-4x^5+x^4 dx \\ = \frac 19 - \frac 12 + \frac 67 - \frac 46 + \frac 15 \\ = \frac{70 - 315 + 540 - 420 + 126}{630} = \frac{1}{630}

But \displaystyle 1+x^2 > 1 \text{ for } x > 0 and so \displaystyle 0 < \int_0^1 \frac{x^4(1-x)^4}{1+x^2} < \int_0^1 x^4(1-x)^4 = 1 / 630.

So since we know \displaystyle \pi = \frac{22}{7} - \frac{x^4(1-x)^4}{1+x^2} we deduce \displaystyle \frac {22}{7} > \pi > \frac{22}{7} - \frac{1}{630}.

Solution by DFranklin

Discussions Toggle
Tsr Pakistani Society VI... Saag and Toast for all members!!
started by: killa78
forum: International Lounge
replies: 3166
last post: 2 Minutes Ago
So does Manchester uni really have a semen problem??!!
started by: SkyBreeze x
forum: General University Discussion
replies: 5
last post: 2 Minutes Ago
Do you want an identical twin?
started by: AkDo
forum: Advice on Everyday Issues
replies: 58
last post: 2 Minutes Ago
God cannot know everything. Carnal love, reproduction or sex.
started by: Greatest I am
forum: Religion
replies: 21
last post: 3 Minutes Ago
AS Physics help - Mechanics
started by: JoeUtd
forum: Physics
replies: 1
last post: 3 Minutes Ago
Second racist gang attacks on white man by "asians"
started by: Zeffy
forum: News and Current Affairs
replies: 114
last post: 3 Minutes Ago
Easiest A Levels.
started by: Fool In The Rain
forum: A-Levels, ASs, A2s, VCEs
replies: 27
last post: 3 Minutes Ago
Any Interviews for Epsom UCA??
started by: naomijayde`
forum: Creative Arts
replies: 19
last post: 3 Minutes Ago
Higher Maths Questions
started by: JaggySnake95
forum: Scottish Qualifications
replies: 4
last post: 3 Minutes Ago
Didn't sit GCSEs now 18, completely stuck
started by: Troglet
forum: Apprenticeships and alternatives to University
replies: 93
last post: 3 Minutes Ago
Official Imperial Applicants Thread (2012 Entry)
started by: Beth1234
forum: Imperial College
replies: 2783
last post: 4 Minutes Ago
OCR Jan 2012 c3
started by: Classical Liberal
forum: Maths Exams
replies: 1
last post: 5 Minutes Ago
The "I'm applying to Edinburgh for 2012 entry" applicant chat thread
started by: oxymoronic
forum: University of Edinburgh
replies: 1231
last post: 5 Minutes Ago
What do you think will happen in Syria?
started by: Roberto-MOr
forum: International
replies: 18
last post: 5 Minutes Ago
Southampton Uni chat thread
started by: marcusmerehay
forum: Southampton Unis
replies: 6817
last post: 6 Minutes Ago
Jews want Arabs to pay 300 Billion for oppression in 1948
started by: Nayred
forum: International
replies: 119
last post: 6 Minutes Ago
Favourite Tekken Character?
started by: Salmaa!
forum: Gaming
replies: 58
last post: 6 Minutes Ago
Travelling in London
started by: mespannerhanz
forum: Travel
replies: 8
last post: 6 Minutes Ago
Offer holders open day
started by: Cats150
forum: University of Exeter
replies: 5
last post: 6 Minutes Ago
Akala is a prophet.
started by: Bonged.
forum: Society
replies: 39
last post: 6 Minutes Ago
Article Updates Toggle
Law Offers 2012
edited by: tiffsays, 21 Minutes Ago
Music Technology Degree
edited by: Potally_Tissed, 49 Minutes Ago
Durham 2012 Applicants Stalking Page
edited by: zebra281, 2 Hours Ago
Medicine Applicants Stalking Page 2012 Entry
edited by: Pembilton, 2 Hours Ago
Biomedical Sciences Offer 2012
edited by: sammythealleycat, 2 Hours Ago
Veterinary Medicine Stalking Page 2012
edited by: sezza33155, 3 Hours Ago
English Offers 2012
edited by: HannahtheFauves, 3 Hours Ago
Model House of Commons Parliament
edited by: Metrobeans, 3 Hours Ago
UCL 2012 Applicants Stalking Page
edited by: Fleximetrics, 3 Hours Ago
Pharmacy Offers 2012
edited by: charb16, 3 Hours Ago
Contact Us | Site Rules | Staying Safe on TSR | Advertising | Staff Blog | Essays & Coursework | Terms & Conditions | Top
Customise your TSR | Life Advice | Hobbies and Interests | Debate and Current Affairs | Study Help | University and University courses
Universities and HE Colleges | Careers, Employment and Gap Years | General Discussion

Customise your TSR