3u Mathematics Marathon V 1.1 (1 Viewer)

[Riviet]

Yup... felt bored so I decided to bump a few of these marathon threads.

 

[SoulSearcher]

Almost forgot to post a question:

If tan A = k tan B, show that
(k-1) sin(A+B) = (k+1) sin(A-B)

 

[Sober]

Ain't nothing like maths at 5am.

Spoiler

k.tan(b) = tan(a)

k.sin(b)/cos(b) = sin(a)/cos(a)

2.k.cos(a).sin(b) = 2.sin(a).cos(b)

k.cos(a).sin(b) - sin(a).cos(b) = sin(a).cos(b) - k.cos(a).sin(b)

k.cos(a).sin(b) + k.sin(a).cos(b) - sin(a).cos(b) - cos(a).sin(b) = k.sin(a).cos(b) - k.cos(a).sin(b) + sin(a).cos(b) - cos(a).sin(b)

(k-1)(cos(a).sin(b) + sin(a).cos(b)) = (k+1)(sin(a).cos(b) - cos(a).sin(b))

(k-1)sin(a+b) = (k+1)sin(a-b)

 

[Sober]

A cone has a base radius of 2 and height of 6. What is the volume of the largest cube that can fit inside it.

(note, one side of the cube shares the same plane as the base of the cone, i.e. you need not rotate the cube).

 

[AntiJap]

That's a Hard question.

 

[Sober]

That's a Hard question.

True, so I will give a hint. Draw the vertical cross section of the cone that cuts diagonally through the cube.

 

[SoulSearcher]

Is the answer:

Spoiler

13.824 cm³?

If so, I'll post my method later, too busy to post it up at the moment.

 

[Sober]

Is the answer:

Spoiler

13.824 cm³?

If so, I'll post my method later, too busy to post it up at the moment.

A cube of that volume would overlap the cone, touching it's surface along the top edges like this:

 

[SoulSearcher]

Sorry, my mistake

 

[mshiraev]

Hey, think ive got it...

let side of cube be x.

(...working out...)

x=12/(3rt2 + 2)

therefore, volume is 7.1m^3


.......??

 

[*slurtznow&4eva]

ive got an assignment due tomorrow for extension 1 please help :

AB & CD are two intersecting chords of a circle and CD is parallel to the tangent to the circle at B. prove that AB bisects angle CAD
wat rule do i use????
please help
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[Riviet]

ive got an assignment due tomorrow for extension 1 please help :

AB & CD are two intersecting chords of a circle and CD is parallel to the tangent to the circle at B. prove that AB bisects angle CAD
wat rule do i use????
please help
<!-- google_ad_section_end -->

Use the "angle in the alternate segment" theorem as well as the "alternate angles are equal on parallel lines" rule and the "angles in the same segment" theorem to show that angle CAB is equal to angle DAB, meaning that AB would bisect angle CAD.

edit: SoulSearcher has kindly provided a worked answer in the "circle geometry" thread.

 

[SoulSearcher]

A cone has a base radius of 2 and height of 6. What is the volume of the largest cube that can fit inside it.

(note, one side of the cube shares the same plane as the base of the cone, i.e. you need not rotate the cube).

mshiraev is also right, he just didn't give his working out, which I will do now.

Spoiler

This would be a lot easier with a diagram =P I also cut all the similar triangle proof and finding out the diagonal length of the face of the cube, so if you need that part, ask away.
We have similar triangles in this cross-section, with the larger triangle having side lengths 6 and 2 units and the smaller triangle having side lengths 6-x and x/rt2 units, where x is the side length of the cube. Therefore, using similar triangles,
6/(6-x) = 2/(x/rt2)
6/(6-x) = 2rt2/x
6x = 2rt2(6-x)
6x + 2x*rt2 = 12rt2
3x + x*rt2 = 6rt2
x(3+rt2) = 6rt2
x = 6rt2/(3+rt2)
Therefore, since we have found out the length of the side of the cube, it is easy now to find the volume of the cube.
x³ = 7.102949424 ...
= 7.10 units³

 

[SoulSearcher]

please help

thanks there was another thats really frustratin me:
show that (x-1)(x-2) is a factor of N M M N N M
p(x)=x (2 -1)+x (1-2)+(2-2)









where M and N are positivePlease help<o></o>

What is the question? I can't understand it at the moment.

 

[shimmerz_777]

have u written out the question correctly, it looks a bit stuffed around, well anyways sub in x=1 and x=2 into P(x) and if it equals 0 than (x-2)(x-1) is a factor

 

[*slurtznow&4eva]

sorry about that:
show that (X-1)(X-2) IS A FACTOR OF
P(x) = xn(2m-1) + xm(1-2n) + (2n-2m) WHERE M AND N ARE POSITIVE INTEGERS AND M AND N ARE POWERS
any help would be grealy appreciated

 

[SoulSearcher]

Yes, as shimmerz said, just sub in x = 1 and x = 2 into P(x), and if P(x) = 0 for both values, then (x-2)(x-1) is a factor of P(x).

 

[shimmerz_777]

for subbin in 1, u get
1(2m-1) + 1(1-2n) + 2n-2m
=2m - 2m + 1 -1 +2n - 2m = 0

for 2 it maight be a bit trickyer

 

[SoulSearcher]

Not really, subbing in 2 gives you:
xn(2m-1) + xm(1-2n) + (2n-2m)
= 2n(2m-1) + 2m(1-2n) + (2n-2m)
= 4mn - 2n + 2m - 4mn + 2n - 2m
= 0

 

[Riviet]

New Question:

The point (10,8) divides the interval joining A(x,y) and B(2,3) externally in the ratio 4:1. Find the co-ordinates of A.