HSC 2013 Maths Marathon (archive) (1 Viewer)

Sy123 · Oct 9, 2013

Re: HSC 2013 2U Marathon

Yea, you don't need to write full solutions, just the important steps will do.

$Consider the function$ \ \ f(x) = \frac{A+x}{2\sqrt{Ax}} \ \ \ $where$ \ \ A>0 \ $is constant$

$$i) Show that the minimum value of the function occurs at$ \ \ x=A \ \ \ \fbox{2}$

$$ii) Find the minimum value of the function$ \ \ \ \fbox{1}$

$iii) Hence show that$

$A+ B \geq 2\sqrt{AB} \ \ \ \fbox{1}$

$$for positive$ \ \ A, B$

mahmoudali · Oct 9, 2013

Re: HSC 2013 2U Marathon

Sy123 said:
Yea, you don't need to write full solutions, just the important steps will do.

$Consider the function$ \ \ f(x) = \frac{A+x}{2\sqrt{Ax}} \ \ \ $where$ \ \ A>0 \ $is constant$

$$i) Show that the minimum value of the function occurs at$ \ \ x=A \ \ \ \fbox{2}$

$$ii) Find the minimum value of the function$ \ \ \ \fbox{1}$

$iii) Hence show that$

$A+ B \geq 2\sqrt{AB} \ \ \ \fbox{1}$

$$for positive$ \ \ A, B$

$($how about you fuck yourself with that question till i go fucking eat $)$

RealiseNothing · Oct 9, 2013

Re: HSC 2013 2U Marathon

Sy123 said:
Fixed latex, that is the right idea anyway, if you didn't make any mistakes its right

EDIT: Second integral is wrong
EDIT2: First integral is wrong as well

hahahah lovin' the edits

RealiseNothing · Oct 9, 2013

Re: HSC 2013 2U Marathon

Sy123 said:
Yea, you don't need to write full solutions, just the important steps will do.

$Consider the function$ \ \ f(x) = \frac{A+x}{2\sqrt{Ax}} \ \ \ $where$ \ \ A>0 \ $is constant$

$$i) Show that the minimum value of the function occurs at$ \ \ x=A \ \ \ \fbox{2}$

$$ii) Find the minimum value of the function$ \ \ \ \fbox{1}$

$iii) Hence show that$

$A+ B \geq 2\sqrt{AB} \ \ \ \fbox{1}$

$$for positive$ \ \ A, B$

I was going to say an easy way to do this was using the AM-GM inequality, then I saw part (iii) lol.

superSAIyan2 · Oct 9, 2013

Re: HSC 2013 2U Marathon

^ can the AM-GM inequality be stated in 4U to show/prove things like (x+y+z) >= 3 x cube root of x

RealiseNothing · Oct 9, 2013

Re: HSC 2013 2U Marathon

superSAIyan2 said:
^ can the AM-GM inequality be stated in 4U to show/prove things like (x+y+z) >= 3 x cube root of x

You'll have to prove the AM-GM inequality, it's not hard though:

$(\sqrt{a} - \sqrt{b})^2 \geq 0$ (squares are always positive)

$a - 2\sqrt{ab} + b \geq 0$

$a + b \geq 2\sqrt{ab}$

User5555555 · Oct 10, 2013

Re: HSC 2013 2U Marathon

HeroicPandas said:
$(a)$ If $y = x^x,$ find $\frac{dy}{dx} \\ \\ (b)$If z = x^{x^{x}}, $ $ hence using part a or otherwise find $\frac{dz}{dx}$

a) x^x(x-1)

HeroicPandas · Oct 16, 2013

Re: HSC 2013 2U Marathon

$$Show that \tan^2 \gamma = \tan^2 \alpha + \tan^2 \beta$

Shazer2 · Oct 16, 2013

Re: HSC 2013 2U Marathon

HeroicPandas said:
$$Show that \tan^2 \gamma = \tan^2 \alpha + \tan^2 \beta$

Uhhhh, there's no 3D trig in 2U? :/

Carrotsticks · Oct 16, 2013

Re: HSC 2013 2U Marathon

Shazer2 said:
Uhhhh, there's no 3D trig in 2U? :/

This.

Shazer2 · Oct 16, 2013

Re: HSC 2013 2U Marathon

That scared me so much, haha.

HeroicPandas · Oct 16, 2013

Re: HSC 2013 2U Marathon

Shazer2 said:
Uhhhh, there's no 3D trig in 2U? :/

i am very sorry

Sy123 · Oct 16, 2013

Re: HSC 2013 2U Marathon

$i) Show that$

$\frac{1-\cos^4 x}{\sin x \cos x + \cot x \cos^4 x} = \frac{\tan x \sec^2 x - \sin x \cos x}{\tan^2 x + \cos^2 x}$

$ii) Hence find$

$\int_0^{\pi/4} \frac{1-\cos^4x}{\sin x \cos x + \cot x \cos^4 x } \ dx$

Shazer2 · Oct 16, 2013

Re: HSC 2013 2U Marathon

How come there are never any questions that are around the difficulty of the average 2U student?

As much as I wish I could answer these, I just can't.

Sy123 · Oct 16, 2013

Re: HSC 2013 2U Marathon

Shazer2 said:
How come there are never any questions that are around the difficulty of the average 2U student? As much as I wish I could answer these, I just can't.

Did you try part (i) yet?

-----

$Differentiate$

$(x + \sin^2(e^x))^{4}$

Shazer2 · Oct 16, 2013

Re: HSC 2013 2U Marathon

Yeah all I got to was changing $sec^{2}x$ to $tanx+1$ .

-----

$\indent \frac {d}{dx} (x+sin^{2}(e^{x}))^{4} \\ \\ \indent = 4 \times (x+sin^{2}(e^{x}))^{3} \times (1+2e^{x}cos(e^{x})sin(e^{x})) \\ \\ \indent = (4+8e^{x}cos(e^{x})sin(e^{x})) \times (x+sin^{2}(e^{x}))^{3}$

Sy123 · Oct 16, 2013

Re: HSC 2013 2U Marathon

Shazer2 said:
Yeah all I got to was changing $sec^{2}x$ to $tanx+1$ .

-----

$\indent \frac {d}{dx} (x+sin^{2}(e^{x}))^{4} \\ \\ \indent = 4 \times (x+sin^{2}(e^{x}))^{3} \times (1+2e^{x}cos(e^{x})sin(e^{x})) \\ \\ \indent = (4+8e^{x}cos(e^{x})sin(e^{x})) \times (x+sin^{2}(e^{x}))^{3}$

You don't need to use that identity

And your answer is correct, wel done

Shazer2 · Oct 16, 2013

Re: HSC 2013 2U Marathon

Thanks! Another question that's not too hard?

RealiseNothing · Oct 16, 2013

Re: HSC 2013 2U Marathon

Shazer2 said:
Thanks! Another question that's not too hard?

For part (i) of Sy's trig question, think about multiplying everything on the LHS by something.

Shazer2 · Oct 16, 2013

Re: HSC 2013 2U Marathon

RealiseNothing said:
For part (i) of Sy's trig question, think about multiplying everything on the LHS by something.

I've got no idea.

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