Preliminary mathematics marathon (1 Viewer)

[hscishard]

What is your final answer?

 

[hscishard]

Sorry exploit, I got a trig question. How can you show in b i) without using the double angle forumale?

 

[ohexploitable]

This marathon ain't so good cos liek most ppl haven't done as much as us, liek my school has only done basic algebra, and geometry.

 

[ohexploitable]

Liek the only people who have posted is me, you, 2010ers and ppl who have already done the HSC.

 

[mitchy_boy]

New question?

LM and FG are two chords of the circle at right angles to each other, and they intersect at P. From P, a line perpendicular to GM is drawn, meeting it at S. SP is produced to meet LF at T.

a) Show that ∆ LTP is isosceles.
b) Show that T is the midpoint of LF.

 

[hscishard]

Liek the only people who have posted is me, you, 2010ers and ppl who have already done the HSC.

You're right.

 

[party]

This marathon ain't so good cos liek most ppl haven't done as much as us, liek my school has only done basic algebra, and geometry.

yeah your right

 

[Drongoski]

New question?

LM and FG are two chords of the circle at right angles to each other, and they intersect at P. From P, a line perpendicular to GM is drawn, meeting it at S. SP is produced to meet LF at T.

a) Show that ∆ LTP is isosceles.
b) Show that T is the midpoint of LF.

Let angle LFG = A
and angle FLM = B

,: A + B = 90 deg (since in triangle LPF angle LPF = 90 deg)

.: angle LMG = angle LFG = A (angles on same chord LG)
& angle FGM = angle FLM = B (angles on same chord MF)

.: angle SPM = B (angle PSM = 90 deg)

.: angle LPT = angle SPM = B (vert opp angles)

.: angle TLP = B = angle LPT in triangle LTP

.: triangle LTP is isosceles

Similarly in triangle GPS, angle GPS = A (angle PSG = 90 deg)
.: angle TPF = angle GPS = A (vert opp angles)
.: in triangle TFP, angle TPF = A = angle TFP
i.e TPF is an isosceles triangle
.: PT = TF
but TP = LT
.: PT = TF

i.e. T is mid-point of LF

QED


Edit

Didn't realise this is a marathon question; can someone pl post a question for me.

 

[ohexploitable]

Didn't realise this is a marathon question; can someone pl post a question for me.

Sorry exploit, I got a trig question. How can you show in b i) without using the double angle forumale?

.

 

[frenzal_dude]

Then make it into a general quadratic expression and find the minimum value of the parabola.

What's the final answer? I got 34.45 and 35.46.

 

[hscishard]

What's the final answer? I got 34.45 and 35.46.

Hi 5!

 

[mitchy_boy]

Let angle LFG = A
and angle FLM = B

,: A + B = 90 deg (since in triangle LPF angle LPF = 90 deg)

.: angle LMG = angle LFG = A (angles on same chord LG)
& angle FGM = angle FLM = B (angles on same chord MF)

.: angle SPM = B since angle PSM = 90 deg

.: angle LPT = angle SPM = B (vert opp angles)

.: angle TLP = B = angle LPT in triangle LTP

.: trangle LTP is isosceles

Similarly in triangle GPS, angle GPS = A since angle PSG = 90 deg
.: angle TPF = angle GPS = A (vert opp angles)
.: in triangle TFP, angle TPF = A = angle TFP
i.e TPF is an isosceles triangle
.: PT = TF
but TP = LT
.: PT = TF

i.e. T is mid-point of LF

QED


Edit

Didn't realise this is a marathon question; can someone pl post a question for me.

Woah that was ext2 harder ext1 question. I don't have time to go through now but I think you might have hit it on the head. Nice work man.

 

[hscishard]

Why would you post that on a prelim? Lol

That is pretty hard

 

[ohexploitable]

New Question:

Differentiate:

 

[Drongoski]

 

[Trebla]

Very tempting to do that one above using logarithmic differentiation....

Have you Prelimers done series? I have a nice series question here:



Otherwise, another one from differential calculus:

The tangent to the hyperbola xy = k at the point P(x₁, y₁) cuts the x-axis at X and the y-axis at Y. Show that the area of triangle OXY is constant.

 

[hscishard]

Very tempting to do that one above using logarithmic differentiation....

Have you Prelimers done series? I have a nice series question here:



Otherwise, another one from differential calculus:

The tangent to the hyperbola xy = k at the point P(x₁, y₁) cuts the x-axis at X and the y-axis at Y. Show that the area of triangle OXY is constant.

I'll try the second one.
f'(x) = -k/x^2

Therefore tangent is

y-intercept:

x intercept


Area of triangle = 1/2AB
B is perp height.


k is a constant and points of P(x1,y1) are also constants.
Therefore area is constant?

Edit: Oh For the first one, I think S = 7/4 (calculator LOL)

 

[ohexploitable]

I think P is any point on the curve, so not constant.

 

[nikkifc]

I'll try the second one.
f'(x) = -k/x^2

Therefore tangent is

y-intercept:

x intercept


Area of triangle = 1/2AB
B is perp height.


k is a constant and points of P(x1,y1) are also constants.
Therefore area is constant?

P is not a constant lol. You need to use the condition (ie. xy=k) since you are not using parametrics.

And Trebla, this is a typical 4U conics question isn't it? It's a bit mean asking it to the prelimers

 

[ohexploitable]

Otherwise, another one from differential calculus:

The tangent to the hyperbola xy = k at the point P(x₁, y₁) cuts the x-axis at X and the y-axis at Y. Show that the area of triangle OXY is constant.

I was thinking of like a logical approach...

A = 1/2 ab

height = y
length = x

xy=k

therefore A = 1/2 k

since k is constant, area is constant

amirite?