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congratulations to Vinh :)

Hi .ben, hope this isn't too late of a reply if you actually need(ed) solutions to any of these questions. QUESTION 1: Find ∫dx/[x(4x2 -3).Sqrt(1-x2)] Make the substitution Sqrt(1-x2) = u ; -xdx/Sqrt(1-x2) = du I = -∫du/(1-u2)(1-4u2) then use the method of partial fractions to obtain...

If this appeared in your Calculus course under Improper Integrals this semester I'm sure you would have been taught the integral (rather than having to derive it empirically yourself): si(0) = Int{infinity to 0}[Sinc(x).dx] = -pi/2 ; where the Sinc function is defined as the continuous...

btw, if you are not already familiar with the equation W = Int[F.dx], where 'Int' is the integral sign, it's derived from the simple Newtonian relationship W = Fd = Fx, where 'd' or 'x' denotes the one dimensional displacement (not distance) of the particle. So in the case where F is a...

indeed, there are many ways to doing this problem... so much so that this simple result appears an an exercise problem in the Year 11 3unit (red) Cambridge text.

lol, Iruka's simple proof of the collinearity of the inverse of all points on the circle really renders your restricted and unnecessarily complex method quite amusing Buchanan. But on the issue of other elegant proofs of this result no_arg, you can think about [hint] rotating the circle about...

but just remember that induction on its own is not sufficient for a proof of the generalised DeMoivre's Theorem. (in other words, at the HSC level, you can only restrict the use of DMT to {k: k is a non-negative integer}) although the case for all rationals is simple, the case for all reals...

take the tan of both sides of 2arctan(2) = pi - arccos(3/5) then use tan identities to show LHS = RHS is also an expedient method.

this really should be in the Physics forum... but anyways, hope i didn't see this too late. W(ork) = Int[F.dx] = Int[F.(dx/dt).dt] = Int[F.v.dt] Given a graph of Force vs. Time, and knowing the equation of that graph (ie. Force as a function of time), then the area under the graph gives...

^ exactly, and that's why the word 'practical' was put in inverted commas in my quote Templar ;) Oh and one more thing - only dumb experimenters would ever even think of doing it manually. the smart ones usually use a little thing we call 'computer' ;)

the moral of this thread should be: do NOT use the 3rd, or nth, derivative test in HSC exams! NEVER. the testing-of-values with the second derivative is a flawless test for POI 100% of times! use that always. the use of buchanan's results along with his overtly trivial examples really...

the famous Buffon's Needle problem is one of the "extension" questions in the Probability chapter of the 3u Cambridge (either the Yr11 or Yr12 book, i forgot which one exactly) book. interestingly, there have been real life experiments of the Buffon Needle scenario used to provide a...

wow, good to see such refereshing mathematics being discussed during school breaks... and a nice surprise for me after a 3-month sabbatical from maths in an exceptional problem here :p before reading my take on the problem, here's an explanation of most of the notations i've used in the proof...

^ is someone else using your account Zedmeister? (he's probably at your house now isn't he :p ?) or are you just bluffing? cause you don't even do Latin!!! :p

yes, precisely what i thought too this time last year - except i've since learnt, and you will probably too, that assignments are so much nicer than cramming for exams (unless you cram for assignments too :p)... believe me, getting to do things at your own pace, combined with the freedom of...

^ ah ic... pretty much the same as i did, except i spent a whole first page just getting to the equivalent of your use of the secant theorem :p

^ hmmm.... nope, our school never had one :( we just got our "portfolio" (containing the 'certificate') from the deputy principal...

graduation? don't you go to one of those school that teach yr 11 and yr 12 after yr 10???

I too would be grateful should you post the solution that employs the elegance of the theory of circle geometry :)

yes, a good question Estel :) but that wasn't much of a hint you gave. i thought hints were supposed to help ppl solve the problem?:rolleyes: {i won't use spoiler tags since solution is quite long, it would be inconvenient.} Question 3: new question, Question 4: hmm ... can't...