HSC 2013-14 MX1 Marathon (archive) (1 Viewer)

obliviousninja · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

RealiseNothing said:
like this?

$\frac{sin4x}{sinx} = \frac{sin(x+x+x+x)}{sinx} = \frac{sinx+sinx+sinx+sinx}{sinx} = \frac{1+1+1+1}{1}$

$=4$

I'm still laughing

Sy123 · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

$a) How many ways can you arrange the word CARROT, if the letters R and A must be together$

$b) By considering the scenario of separating$ \ n \ $people into 2 groups, in 2 different ways show that$

$\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n} =2^n$

$c) A particle is moving in Simple Harmonic Motion, show that the function for displacement over time for this motion is trigonometric$

$$d)$ \ \ \int_{-1}^{1} \sin^{-1} x + \cos^{-1} x + \tan^{-1}x \ dx$

$$e) A projectile is fired onto an inclined plane from the start of the plane$

$the plane is inclined at angle$ \ \ \alpha \ \ $show that the angle of projection must be$

$\frac{\pi}{4} + \frac{\alpha}{2} \ \ $in order to yield maximum range up the plane$$

$f) Consider a parabola, a light beam is shone above the parabola that is parallel$

$to the parabola's axis, it hits the parabola is reflected off, show that this beam travels through the focus$

$g) Prove the Binomial theorem by Mathematical Induction$

$h) By using the Binomial Theorem, prove$

$\binom{n}{r-1} \binom{n}{r+1} - \binom{n}{r-2} \binom{n}{r+2} + \dots +(-1)^r \binom{n}{1} \binom{n}{2r-1} - (-1)^r \binom{n}{2r}$

$= \frac{1}{2} \binom{n}{r} \left( \binom{n}{r} - 1 \right)$

$i) Ten equal circular discs can be placed tightly within a frame in the form of an equilateral triangle. Nine of them touch the frame; the tenth, in the centre touches six of the other discs. If the internal sides of the frame are each of equal length measure R, express the radius of each circular discs in terms of R, and show that the outside perimeter of the figure which remains when the frame is removed is$

$\frac{11}{12} (3- \sqrt{3}) \pi R$

$j) Find the general solutions to$

$\sin 2x + \sin 4x = \sin 3x$

$k) Find the values of$ \ x \ $for which the inequalities are simultaneously satisfied$

$x + \frac{1}{|x|} > 0$

$x^2-x-2 < 0$

$l) Prove that the acute angle$ \ \theta \ \ $between two lines of gradient$ \ \ m_1 \ \ $and$ \ \ m_2 \ \ $is given by$

$\tan \theta = \left|\frac{m_1 - m_2}{1+ m_1 m_2} \right|$

$$m) Approximate$ \ \ \sqrt[3]{2}$

$$n) Consider the fixed point$ \ \ A(1,1) \ \ $and the line$ \ \ y=-x$

$The variable point$ \ \ B \ \ $lies on the line, find the locus of the point$

$P \ \ $which divides$ \ \ AB \ \ $in ratio$ \ \ 1:2$

$o) Using the substitution$ \ \ u=1-x^3 \ \ $Evaluate$

$\int_0^1 x^5 \sqrt{1-x^3} \ dx$

-----

I think I've covered all topics?

obliviousninja · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

Sy123 said:
a) How many ways can you arrange the word CARROT, if the letters R and A must be together

billyp245 · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

Carrotsticks said:

Is part i) -(√2)t = ∫1/(bx-1) ?

dontfuckmypostbox · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

Can someone please post solutions to Carrot's and Sy's questions its getting late and I am lost for some of these questions

mahoo_mk · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

dontfuckmypostbox said:
Can someone please post solutions to Carrot's and Sy's questions its getting late and I am lost for some of these questions

im lost for all lol

dontfuckmypostbox · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

mahoo_mk said:
im lost for all lol

Yea same I just said "some" so i don't seem too stupid on this forum but seriously solutions will be appreciated

Carrotsticks · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

billyp245 said:
Is part i) -(√2)t = ∫1/(bx-1) ?

That's not the displacement time equation.

andybandy · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

is it this carrot sticks?

<a href="http://www.codecogs.com/eqnedit.php?latex=t=&space;\frac{\sqrt{2(bx-1)^{2}}}{2b^{2}x-2}&space;+&space;\frac{1}{\sqrt2b}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?t=&space;\frac{\sqrt{2(bx-1)^{2}}}{2b^{2}x-2}&space;+&space;\frac{1}{\sqrt2b}" title="t= \frac{\sqrt{2(bx-1)^{2}}}{2b^{2}x-2} + \frac{1}{\sqrt2b}" /></a>

Carrotsticks · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

Displacement time equation doesn't need to be t with respect to x. It can be x with respect to t, which is often easier to find.

andybandy · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

Carrotsticks said:
Displacement time equation doesn't need to be t with respect to x. It can be x with respect to t, which is often easier to find.

yeah i know, but the velocity was in terms of displacement, so i had to find t in terms of x

Carrotsticks · Oct 24, 2013

Re: HSC 2013 3U Marathon Thread

I don't remember getting such an expression.

Anyway, good luck for tomorrow!

andybandy · Oct 24, 2013

Re: HSC 2013 3U Marathon Thread

haha its okay,
and thankyou!

omgiloverice · Oct 26, 2013

Re: HSC 2013 3U Marathon Thread

$Temperature and the period of a pendulum For oscillations of small amplitude (short swings), we may safely model the relationship between the period $ T $ and the length $ L $ of a simple pendulum with the equation:$

$T=2\pi \sqrt{\frac{L}{g}}$

$$where g is the constant acceleration of gravity at the pendulum's location. If we measure $g$ in centimeters per second squared, we measure $L$ in centimeters and $T$ in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to $L$. In symbols, with $u$ being temperature and $k$ the proportionality constant,$

$\frac{dL}{du}=kL.$

$$Assuming this to be the case, show that the rate at which the period changes with respect to temperature is $kT/2.$

Ikki · Nov 17, 2013

Re: HSC 2013 3U Marathon Thread

$\fbox{1} \int \frac{secxtanx}{\sqrt{1-sec^2x} } \mathrm{d}x $ between b and a, where $ b>a$

$$Hint: Use the substitution $ u=secx$

asianese · Jan 13, 2014

Re: HSC 2013 3U Marathon Thread

$$The parabola $ P(x) = Ax^2+Bx+C $ is used to approximate the area under the curve of $ y = f(x), $ which is difficult to integrate, from $ x=-h $ to $ x=h.\\ $i) Find the area under the curve $ y=P(x) $ from $ -h $ to $ h $ in terms of $ A, B $ and $ C. \\ $ii) Show, with all working, that this area can be expressed as $ \dfrac{h}{3}\left( y_0 +4y_1+y_2\right). \\ $iii) Compare this method of approximating areas to the trapezoidal method and comment on when either would be more useful than the other. Give examples to support your answers.$

asianese · Jan 13, 2014

Re: HSC 2013 3U Marathon Thread

Timely revival.

asianese · Jan 15, 2014

Re: HSC 2013 3U Marathon Thread

$Consider the curve $ y=\dfrac{\ln x}{x} $ which is shown. Suppose a certain tangent of the curve goes through the origin. $\\ $a) By considering the equation of the tangent, find the co-ordinates of the point $P. $(3 marks)$ \\ $b) Write down the $ x-$intercept of the curve. (1 mark) $\\ $c) Show that the curve bisects the area given by the triangle spanned by the origin, the point $ P $ and the point $ (x_0,0) $(3 marks).$

Sy123 · Jan 31, 2014

Re: HSC 2013 3U Marathon Thread

$\\ $Let$ \\ \\ x = \frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)} \\ \\ $Find$ \ \ \ (x+1)^{48}$

nightweaver066 · Mar 1, 2014

Re: HSC 2013 3U Marathon Thread

$$Show that $ (x+1)(x^2+1)(x^4+1)(x^8+1)(x^{16}+1) = 1 + x + x^2 + x^3 + ... + x^{31}$

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