HSC 2013 MX2 Marathon (archive) (1 Viewer)

[RealiseNothing]

Re: HSC 2013 4U Marathon

Editing time!

I think I've angered carrot now, nothing good can come of this.

 

[Sy123]

Re: HSC 2013 4U Marathon

 

[Carrotsticks]

Re: HSC 2013 4U Marathon

No anger at all, just a tired night =)

I may as well post a question whilst I'm here.

Consider an upright cylinder of radius R and height H.

From the centre of the cylinder, a vertical slice is made, leaving a solid with half the volume of the cylinder.

From the top of the flat side of the solid, a diagonal cut is made to the other end of the base.

Find the volume of the 'wedge' remaining.

 

[RealiseNothing]

Re: HSC 2013 4U Marathon

Splitting up the power:



Let





So the integral simply becomes:

 

[Carrotsticks]

Re: HSC 2013 4U Marathon

I'll spam some more friendly problems too.

Let P be a point on the ellipse x^2/a^2+y^2/b^2=1. From P, a vertical line is dropped and it meets the X axis at X. From P, a tangent is constructed and it meets the X axis at T.

Prove that OX . OT = a^2.

 

[Carrotsticks]

Re: HSC 2013 4U Marathon

A series of ellipses are drawn with the same foci S and S'.

Prove that there exists a unique ellipse passing through a general point on the plane.

 

[Carrotsticks]

Re: HSC 2013 4U Marathon

A cone has base radius R. A horizontal slice is made to chop off the top, leaving a flat circular surface of radius r.

Let the height of this solid be H.

Find the volume of this solid.

Describe what happens when r=R, and when r=0.

 

[Carrotsticks]

Re: HSC 2013 4U Marathon

Find the area of the standard ellipse x^2/a^2 + y^2/b^2 = 1.

Find the equation of the circle, centred at the origin, that has the same area as the above ellipse.

Find where it intersects the ellipse.

 

[Carrotsticks]

Re: HSC 2013 4U Marathon

 

[Carrotsticks]

Re: HSC 2013 4U Marathon

 

[Sy123]

Re: HSC 2013 4U Marathon

No anger at all, just a tired night =)

I may as well post a question whilst I'm here.

Consider an upright cylinder of radius R and height H.

From the centre of the cylinder, a vertical slice is made, leaving a solid with half the volume of the cylinder.

From the top of the flat side of the solid, a diagonal cut is made to the other end of the base.

Find the volume of the 'wedge' remaining.

Still testing the waters here, and I'm a bit tired at the moment. We know that each cross section is a segment of the semi-circle. So I decided that I would slice up everything and integrate, now I know it doesn't have a common cross section, but I decided to do something a bit different here. For some arbitrary y*, we find the area of the segment of the semi-circle from R to y*, then we multiply that by dz. Then we add up all the y* that exists, so we should end up with:



Is this correct? (i will evaluate it if its right)

EDIT: One sec, I'm pretty sure that's wrong.

EDIT2: wtf no

 

[Carrotsticks]

Re: HSC 2013 4U Marathon

Each cross section, taken perpendicular to the base, is a rectangle.

 

[Sy123]

Re: HSC 2013 4U Marathon

Each cross section, taken perpendicular to the base, is a rectangle.

Yeah I imagined the whole thing completely wrong, I better go to sleep.

 

[Carrotsticks]

Re: HSC 2013 4U Marathon

 

[RealiseNothing]

Re: HSC 2013 4U Marathon

?

 

[Sy123]

Re: HSC 2013 4U Marathon

?

yep, by AM-GM

 

[study1234]

Re: HSC 2013 4U Marathon

In how many ways can one yellow, two red and four green beads be placed on a bracelet if the beads are identical apart from colour? (Hint: This will require a listing of patterns to see if they are identical when turned over.)

 

[anomalousdecay]

Re: HSC 2013 4U Marathon

In how many ways can one yellow, two red and four green beads be placed on a bracelet if the beads are identical apart from colour? (Hint: This will require a listing of patterns to see if they are identical when turned over.)

 

[HeroicPandas]

Re: HSC 2013 4U Marathon

 

[study1234]

Re: HSC 2013 4U Marathon

Nup. It's a circle, and also I think you have to use cases.