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Re: HSC 2013 2U Marathon Well done, though I was intending that you didn't refer to the actual trigonometry and just used 't' only, but its all good. ======= $The graph$ \ \ f(x)=\frac{Ax}{1+x^2} \ \ $is sketched below$ $A rectangle$ \ \ ABCD \ \ $is drawn, with$ \ \ B(a,0) $i) Prove$...

Re: HSC 2013 4U Marathon $Let$ \ \ I_n = \int_0^{\pi /4} \tan^{2n+1} x \ dx $Find a recurrence formula for this integral, and use it to prove$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}

Re: MX2 Integration Marathon Good idea x^3\sqrt{1-x^2} = -x\sqrt{1-x^2} (1-x^2-1) = x\sqrt{1-x^2} - x(1-x^2)^{\frac{3}{2}} Which can be done using reverse chain rule What makes integration one of my favorite topics is the vast amount of ways to get to the same answer, I think we've all...

Re: HSC 2013 2U Marathon $Consider the right angle triangles$ \ \ \triangle ABC \ , \ \triangle ACD \ $and$ \ \triangle AFE $Let$ \ \angle BAC = \angle CAD = x AC = 1 $Let$ \ t=\cos x $i) Prove that$ \ \ DE = \frac{1}{t^2} \sqrt{1-t^2} \ \ \ \fbox{2} $ii) Prove that$ \ \ \triangle...

Re: HSC 2013 3U Marathon Thread $A projectile is fired from the origin with initial velocity$ \ \ U $to pass through the point$ \ (a,b) $Prove that there are 2 possible trajectories if$ (U^2-gb)^2 > g^2(a^2+b^2) \ \ \ \ \fbox{5} Difficulty: Q14

Re: HSC 2013 3U Marathon Thread I'll try Most if not all questions I post will be at the higher end of difficulty in terms of HSC, for both the 2U and 4U marathons

Re: MX2 Integration Marathon Alternatively we do IBP with u=x^2, dv = xsqrt(1-x^2) dx == \int \cot^{-1} (x^2+x+1) \ dx y=\cot^{-1} x \ \ $is the inverse function of$ \ y=\cot x \ \ $for$ \ 0\leq x \leq \pi

Re: HSC 2013 2U Marathon Yes well done, it was the long way however. \cos \alpha = \tan \alpha \therefore \frac{\sin \alpha}{\cos^2 \alpha} = 1 \ \ \fbox{1} Now, differentiate the curves, and find the gradients of each curve at P -\sin \alpha = m_1 m_2 = \sec^2 \alpha m_1 m_2 =...

Re: MX2 Integration Marathon \int_0^1 x^3 \sqrt{1-x^2} \ dx

Re: HSC 2013 4U Marathon $The complex numberss$ \ \ a_1, a_2, a_3, \dots, a_n $satsify$ \ \ |a_i| \leq A \ \ $for$ \ i=1,2, \dots, n \ \ $for positive real$ \ A \geq 1 $For the complex equation$ a_n z^n + a_{n-1}z^{n-1} + \dots + a_1 z = B \ \ $for positive real$ \ B \leq 1 $Show that...

Re: HSC 2013 3U Marathon Thread $i) For the polynomial$ \ \ P(x) = (x-a)^2 Q(x) $Show that$ \ \ x=a \ \ $is a root of$ \ P'(a) $ii) The polynomial$ \ \ S(y) = ay^3 + by + c \ \ $has a double zero at$ \ y= 1 $and has a remainder of$ \ 4 \ $when divided by$ \ \ y+1 $Find$ \ a,b,c...

Re: HSC 2013 2U Marathon $The curves$ \ y=\cos x \ \ $and$ \ y=\tan x \ \ $intersect at a point$ \ P \ $with$ \ x$-coordinate of$ \ \alpha 0 \leq \alpha < \frac{\pi}{2} $i) Show that the curves intersect at right angles to each other at$ \ P $ii) Show that$ \sec^2 \alpha =...

Lol no way, I worked backwards, I decided to get a relatively alright integral, which was quite short, I plugged in numerous values into wolfram alpha, and settled with 83.9 and 4. Then I substituted in something to the 'simple' integral to make it much more complex, then some manipulation. The...

Re: HSC 2013 2U Marathon Yep well done. $Consider the functions$ \ \ C(x) \ \ $and$ \ \ S(x) $Which are such that$ C(\cos x) = x S(\sin x) = x $i) By differentiating, show that$ S'(z) + C'(z) = 0 $For$ \ -1 \leq z \leq 1 $Consider another function$ \ \ f(z) = S(z) + C(z)...

$Calculate to the nearest 10th$ \int_{\frac{15}{4}}^{\frac{703821}{8390}} \frac{(x^2+1)(x^8-4x^6 + 6x^4-4x^2+1)}{x^{10} - 4x^8 + 7x^6 - 4x^4 + x^2} \ dx (are we allowed to do only approximations?) Make sure that when doing this integral, if you do a substitution, make sure you substitute in...

Re: MX2 Integration Marathon $Using squeeze theorem$ $Prove$ \ \ \lim_{n \to \infty} \frac{1}{n} \int_1^2 \left(1 - \frac{1}{x} \right)^n \ dx = 0

Re: HSC 2013 2U Marathon That is essentially what I was aiming for, well done. $By multiplying the numerator and denominator by the same function (essentially multiplying by 1 in a clever way), Find$ \int \frac{e^{2x} - 1}{e^{2x} + 1} \ dx

Re: HSC 2013 4U Marathon Yeah that sounds more correct, thank you

Re: HSC 2013 2U Marathon Yes that is true but you haven't explained why (a=0) gives maximum, I've asked for the sketch in the first part for a reason. $Prove that$ x^{\frac{1}{x}} \leq e^{\frac{1}{e}} $For$ \ x > 0

Re: HSC 2013 4U Marathon $A number from the sequence$ \ \ F_n = F_{n-1} + F_{n-2} \ , \ F_0 = 0 \ F_1=1 \ \ $is called a Fibonacci number$ $A formula for the$ \ n^{th} \ $Fibonacci number is$ F_n = \frac{1}{\sqrt{5}} \left(\left(\frac{1+\sqrt{5}}{2} \right )^n - \left( \frac{1-\sqrt{5}}{2}...