1. Re: HSC 2018 MX2 Marathon

Originally Posted by CapitalSwine
Cant seem to figure out how to do this question:
https://imgur.com/a/jRaWrmq
BSQ = PQB (alternate angle theorem)
PQB = 180-RPQ (co-interior angles in a parallelogram are supplementary)
BSQ + RPQ = 180

Hence opposite angles are supplementary so RSQP is a cyclic quad

Hope this reasoning is legit, forgot maths

EDIT: Nvm too slow

2. Re: HSC 2018 MX2 Marathon

It is easy to prove RSQP is cyclic; so that this can be easily shown in many ways.

e.g.

Let angle AQS = @; .: angle QBS = @; .: angle PRS = @ (PR // QB)

.: angle AQS = angle PRS

.: external angle AQS of quadrilateral RSQP = int opp angle PRS

.: RSQP must be cyclic

QED

3. Re: HSC 2018 MX2 Marathon

$\noindent For n = 0, 1, 2... let I_n = \int_{0}^{\pi/4} tan^{n}\theta d\theta$
$\noindent a) Show that I_1 = \frac{1}{2}ln2$
$\noindent b) Show that, for n \geq 2, I_n + I_{n-2} = \frac{1}{n-1}$

$\noindent c) For n \geq 2, explain why I_n < I_{n-2} and deduce that$

$\noindent \frac{1}{2(n+1)} < I_n < \frac{1}{2(n-1)}$

$\noindent d) Use the reduction formula in part b) to find I_5, and hence deduce that$

$\frac{2}{3} < ln2 < \frac{3}{4}$

4. Re: HSC 2018 MX2 Marathon

a)
\begin{aligned} I_1 &= \int^{\frac\pi4}_0 \tan \theta \,d\theta \\ &= \left[\phantom{\frac\,\,}-\log \cos\theta\right]^{\frac\pi4}_0\\&=\log 1 - \log \frac{\sqrt2}{2} \\ &= \log \frac{2}{\sqrt 2} \\ &= \frac12\log 2 \end{aligned}

b)

\begin{aligned} I_n + I_{n+2} &= \int_0^{\frac\pi4} \tan^n\theta + \tan^{n-2}\theta\,d\theta \\ &=\int_0^{\frac\pi4} \tan^{n-2}\theta(1+\tan^2\theta)\,d\theta \\ &=\int^{\frac\pi4}_{0} \left(\tan\theta\right)^{n-2} \sec^2\theta\,d\theta \\ &= \left[\frac{1}{n-1} \left(\tan \theta\right)^{n-1}\right]^{\frac\pi4}_0\\&= \frac{1}{n-1}\end{aligned}

c)

For $0 \leq x \leq \pi/4$, $0\leq \tan \theta\leq 1$.

If $0\leq \tan \theta\leq 1$, then clearly $\left(\tan\theta\right)^n >\left(\tan\theta\right)^{n+1}$.

Therefore, $\left(\tan\theta\right)^n <\left(\tan\theta\right)^{n-2}\implies I_n < I_{n-2}$.

From b) $I_{n-2} = \frac{1}{n-1} - I_n$. Since $I_n < I_{n-2}$,

\begin{aligned} I_{n} &< \frac{1}{n-1} - I_n \\ I_n&< \frac{1}{2(n-1)} \end{aligned}

Replacing $n$ with $n +2$, we get $I_{n} = \frac{1}{n+1} - I_{n+2} \implies I_{n+2} = \frac{1}{n+1} - I_{n}$ with $I_{n+2} < I_{n}$ and so

\begin{aligned} \frac{1}{n+1}-I_{n} &< I_n \\\frac{1}{2(n+1)} &< I_n \end{aligned}

Giving, as required,

$\frac{1}{2(n+1)} < I_n < \frac{1}{2(n-1)}$

d)

From b),

$I_n = \frac{1}{n-1} -I_{n-2}$.

$I_1 = \frac 12 \log 2$

$I_3 = \frac12 -\frac 12 \log 2$

$I_5 = \frac 14 - I_3 = \frac 12 \log 2 - \frac 14$

From c),

\begin{aligned} \frac{1}{12} < \,&I_5 < \frac18 \\\phantom{0}\\ \frac{1}{3} <\frac12&\log2 < \frac38 \\\phantom{0}\\ \frac23 <\,&\log2<\frac34\end{aligned}

5. Re: HSC 2018 MX2 Marathon

$Let\, I_n = \int_0^{\frac12} \frac{(\tan^{-1}2x)^n}{4x^2+1}\,dx\,for\,n\in\mathbb{Z}^+.$

$i) Show that\, I_n =\frac{1}{2(n+1)}\left(\frac\pi4\right)^{n+1}$

$ii) Hence show that\, I_0 \times I_1 \times I_2 \times ... \times I_n = \frac{1}{2^{2n}(2n)!}\left(\frac\pi4\right)^{2n^2+ n}$

6. Re: HSC 2018 MX2 Marathon

yoyoyo
there was a nice question i found a while back and thought i would post it here:

given that e^x can be written as a sum of an odd and even function, find the two functions.

7. Re: HSC 2018 MX2 Marathon

Originally Posted by mrbunton
yoyoyo
there was a nice question i found a while back and thought i would post it here:

given that e^x can be written as a sum of an odd and even function, find the two functions.
Is the intended answer the Taylor series for $e^x$?

Edit: nevermind, I found some others

8. Re: HSC 2018 MX2 Marathon

Originally Posted by fan96
Is the intended answer the Taylor series for $e^x$?

Edit: nevermind, I found some others

its Hyperbolic function of sin and cos
so e^x = sinhx+coshx
Its legit like deriving cosx and sinx in complex field

The taylor expansion is just an approximate for e^x for values less than 1 or close to 0 but can get quite close. You would just factorise the even and odd parts of the function. However, that does not mean you cant use it, I tried it with Maclaurin series and it seems to look solid.

9. Re: HSC 2018 MX2 Marathon

$\noindent By the way, you can do this with any function, not just e^x (provided that it makes sense to input both x and -x into your function). We can write a given f(x) as a sum of an \color{red}{even }\color{black} function and an \color{blue}{odd }\color{black} function as \boxed{f(x) = \color{red}\frac{1}{2}(f(x) + f(-x))\color{black} + \color{blue}\frac{1}{2}(f(x) - f(-x))\color{black}}, and in fact this is the \emph{only} way to make such a decomposition.$

10. Re: HSC 2018 MX2 Marathon

Yeah it clicked in when i was thinking of expressing these interms of Complex version of cos and sin and basically simultaneous

11. Re: HSC 2018 MX2 Marathon

Originally Posted by InteGrand
$\noindent By the way, you can do this with any function, not just e^x (provided that it makes sense to input both x and -x into your function). We can write a given f(x) as a sum of an \color{red}{even }\color{black} function and an \color{blue}{odd }\color{black} function as \boxed{f(x) = \color{red}\frac{1}{2}(f(x) + f(-x))\color{black} + \color{blue}\frac{1}{2}(f(x) - f(-x))\color{black}}, and in fact this is the \emph{only} way to make such a decomposition.$
ya. You can generate that through utilising the definition of even and odd functions which was the intent of the question.

More qtns

13. Re: HSC 2018 MX2 Marathon

$\\ i) A point P moves on the rectangular hyperbola x^2 - y^2 = a^2. Let M be the point on the tangent to the hyperbola at P be the point closest to the origin. \\\\Show that this curve is given by the locus \ (x^2 + y^2)^2 = a^2(x^2 - y^2). \\\\ This curve is called the \textit{Lemniscate of Bernoulli}$

$\\ ii) Show that if F_1\left(\frac{a}{\sqrt{2}}, 0 \right) and F_2\left(\frac{-a}{\sqrt{2}}, 0\right) are fixed points in the plane. Then for any point Q on the Lemniscate of Bernoulli, then |QF_1| \cdot |QF_2| = 1$

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