HSC 2013 Maths Marathon (archive) (3 Viewers)

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.

HSC 2013 Maths Marathon (archive) (3 Viewers)

This too shall pass

Member

what is that?It is Cowpea

what is that?It is Cowpea

Member

what is that?It is Cowpea

Active Member

Heroic!

Attachments

Member

Retired

Member

Heroic!

This too shall pass

Member

This too shall pass

Member

This too shall pass

Member

what is that?It is Cowpea

Member

Users Who Are Viewing This Thread (Users: 0, Guests: 3)