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HSC 2016 MX2 Marathon (archive) (5 Viewers)

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integral95

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Re: HSC 2016 4U Marathon

Consider the excess solutions that do not satisfy the given equation and eliminate them.
Oh wait I was right, is the answer




EDIT: This is because from the above equation the denominator can't be 0 which is why theta can't be 0, dammit how did I not see that?
 
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Paradoxica

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Re: HSC 2016 4U Marathon

I'm bored so here's how you do itttt
Well technically one could also do a triple and double angle expansion, then proceed to solve both polynomials simultaneously...

But that's only if you're feeling masochistic.
 

integral95

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Re: HSC 2016 4U Marathon

Well technically one could also do a triple and double angle expansion, then proceed to solve both polynomials simultaneously...

But that's only if you're feeling masochistic.
ewwww no.
 

Paradoxica

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Re: HSC 2016 4U Marathon

ewwww no.
Being the masochistic person I am, here is the alternative bashy solution



The solutions to the cosine equation can be found very easily. These can be substituted into the second equation and then the second equation can be solved for the corresponding solutions.
 

seanieg89

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Re: HSC 2016 4U Marathon

(In this question you need to know how tennis scoring works.)

In a basic model of a tennis match, every point is won by the server with probability p and the receiver with probability 1-p.

a) What is the probability of winning a service game in terms of p?

b) Suppose it is known that both sides win equally often from the game score of 0-30. Approximate the value of p that this condition forces.

c) Approximate the probability of winning a service game in terms of the p found in part b.


(Use Wolfram Alpha if you like for the approximations, unless you are feeling particularly algebraically macho and would like to do it using something like Newton's.)
 
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seanieg89

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Re: HSC 2016 4U Marathon

Suppose you have never heard of exponentials or logarithms before.

Define for positive reals x, the function

Prove:

a) L(1)=0
L(xy)=L(x)+L(y)
L(x^r)=rL(x) for rational r
L is differentiable in its domain and L'(1)=1.

b) Prove that L is the unique function on the positive reals with the properties in a.

c) Prove that L is invertible and denote its inverse function by E(x), what is the domain of E?

d) Prove that E(r)=E(1)^r for rational r. Prove that E'(0)=1. (You may assume without proof that E is differentiable).

e) Suppose another differentiable function A(x) satisfies A(r)=A(1)^r for all rationals r (with A(1) not equal to E(1)). Show that A'(0) is not equal to 1.
 

hedgehog_7

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Re: HSC 2016 4U Marathon

The area between y = 2x +1 and y = (x+1)^2 is rotated about y = -1 with x=0 as the boundary axis. Find the volume
 

Paradoxica

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Re: HSC 2016 4U Marathon

a) For any natural n, find and classify the solutions to the following equation:




b) Repeat the above for all non-natural integers n.
 

KingOfActing

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Re: HSC 2016 4U Marathon

a) For any natural n, find and classify the solutions to the following equation:




b) Repeat the above for all non-natural integers n.
For natural , by "cross multiplying":



In the case



Notice also that the solution for some negative n is the same as the solution for -n + 2.
E.g. n = -2 has the solutions to z^3 = 1 which is the same as n = 4.
 

Paradoxica

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Re: HSC 2016 4U Marathon

For natural , by "cross multiplying":



In the case



Notice also that the solution for some negative n is the same as the solution for -n + 2.
E.g. n = -2 has the solutions to z^3 = 1 which is the same as n = 4.
-1 is not necessarily a solution for some values of n anyway, but I think I can accept that non-intersection.
 

Katebate

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Re: HSC 2016 4U Marathon

In mechanics, banked circular tracks, if there is not lateral force stated, do we just assume it isn't there?
 

leehuan

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Re: HSC 2016 4U Marathon

In mechanics, banked circular tracks, if there is not lateral force stated, do we just assume it isn't there?
What do you mean by lateral force?

You are meant to consider friction, normal reaction and gravity when analysing banked tracks, to which the overall effect is uniform circular motion
 

InteGrand

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Re: HSC 2016 4U Marathon

What do you mean by lateral force?

You are meant to consider friction, normal reaction and gravity when analysing banked tracks, to which the overall effect is uniform circular motion
By lateral force, Katebate most likely meant the friction force (which is a lateral force, i.e. up or down the bend).
 
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