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HSC 2015 MX2 Marathon (archive) (5 Viewers)

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InteGrand

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Re: HSC 2015 4U Marathon

Unless the question says that it's a vertical motion question right?
Well it doesn't have to be completely vertical, is long as there is gravity influencing it somehow (e.g. for conical pendulums, the motion is in a horizontal circle, but gravity still needs to be taken into account, or for a ball rolling down an inclined plane, there is a vertical component to motion, so gravity needs to be considered).

In the case with the boat in that question, gravity didn't affect its motion in the horizontal direction, so we didn't need to consider it. And I think the HSC will usually tell you all the relevant forces acting on the particle in question.
 

Drsoccerball

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Re: HSC 2015 4U Marathon

My test for :
integration
Volumes
Mechanics
is on friday can we get some hard volumes and mechanics question :)?
 

InteGrand

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Re: HSC 2015 4U Marathon

Hi if a particle is moving in a circular path of radius 'r' and VARIABLE angular velocity (θ') what would be the tangential and normal components of the acceleration in terms of θ' and θ"
Thanks!


 
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seanieg89

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Re: HSC 2015 4U Marathon

Given that every degree n polynomial with real coefficients has exactly n complex roots (counting multiplicity), show that the same is true of every degree n polynomial with complex coefficients.

(Obviously, you may not assume the fundamental theorem of algebra for this question.)
 

seanieg89

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Re: HSC 2015 4U Marathon

Two more polynomial questions:

Let p be a degree n polynomial with complex coefficients.

1. If the roots of p form an equilateral n-gon, show that p' has a single root of multiplicity (n-1), located at the centroid of this n-gon.

2. Given that p(ix) is real for every real x, show that p(-x) is the conjugate of p(x) for every real x.

(1 and 2 are completely unrelated to each other.)
 

Sy123

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Re: HSC 2015 4U Marathon

Given that every degree n polynomial with real coefficients has exactly n complex roots (counting multiplicity), show that the same is true of every degree n polynomial with complex coefficients.

(Obviously, you may not assume the fundamental theorem of algebra for this question.)
I've proven it for when the polynomial's domain is restricted among the reals














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InteGrand

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Re: HSC 2015 4U Marathon

Two more polynomial questions:

Let p be a degree n polynomial with complex coefficients.

1. If the roots of p form an equilateral n-gon, show that p' has a single root of multiplicity (n-1), located at the centroid of this n-gon.

2. Given that p(ix) is real for every real x, show that p(-x) is the conjugate of p(x) for every real x.

(1 and 2 are completely unrelated to each other.)










 

seanieg89

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Re: HSC 2015 4U Marathon

I've proven it for when the polynomial's domain is restricted among the reals

What do you mean by this? Are you then only saying something about the real roots of an n'th degree polynomial? You definitely shouldn't be able to obtain an "exactly n" statement in this case!



This identity holds for all real x at least.




Real roots or complex roots? Given that you use the previous identity I can only assume real?
--------









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Should "complex" be replaced by real here? This is fair enough then, and says that any complex deg n poly has at most n real roots. (The "at most" statement in solving polynomials is the easy part though, we can get it straight from the factor theorem and the notion of degree. In fact this would tell us that any complex deg n poly has at most n COMPLEX roots, which is closer to what we need overall in this problem.)



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Remember that R is a subset of C! By C I will assume you mean C-R.





Why should the square of a nonreal number be nonreal? P(\beta)=i is a counterexample to your claim. (2) certainly shouldn't be a contradiction given that you are just talking about real numbers...it is very possible for a complex deg n poly to have < n real roots.




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complex or real? I don't see how you have proven the complex statement.
.
 
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seanieg89

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Re: HSC 2015 4U Marathon

Given that every degree n polynomial with real coefficients has exactly n complex roots (counting multiplicity), show that the same is true of every degree n polynomial with complex coefficients.

(Obviously, you may not assume the fundamental theorem of algebra for this question.)
So an approach similar to Sy's in spirit works here.

Recall from my previous post that a complex polynomial p such that p(x) is real for all real x must in fact be a real polynomial.



As a real degree 2n polynomial, q has exactly 2n roots counting multiplicity by the problem assumption.

But the roots of are just the conjugates of those of , counting multiplicity.

So p(z) must have n roots, counting multiplicity.
 
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simpleetal

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Re: HSC 2015 4U Marathon

Hey Seanieg, or Sy or whoever sees this, my friend claims that if sin(2a)*sin(2b)*cosa + sin(2a)sin(a)cos(2b)= sin(2b)sin(2a)cos(b)+sin(2b)sin(b)cos(2a) because the lhs is basically the rhs except with the a and b switched, this automatically means that a must equal b. That was his reasoning, given that a and b are actue angles of a triangle. Is his reasoning valid?
 

seanieg89

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Re: HSC 2015 4U Marathon

Hey Seanieg, or Sy or whoever sees this, my friend claims that if sin(2a)*sin(2b)*cosa + sin(2a)sin(a)cos(2b)= sin(2b)sin(2a)cos(b)+sin(2b)sin(b)cos(2a) because the lhs is basically the rhs except with the a and b switched, this automatically means that a must equal b. That was his reasoning, given that a and b are actue angles of a triangle. Is his reasoning valid?
a might be forced to equal b (not going to check this myself right now), but the reasoning is not.

Eg x+y = y+x for all x and y whatsoever, but this does not tell us that x=y.
 

simpleetal

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Re: HSC 2015 4U Marathon

a might be forced to equal b (not going to check this myself right now), but the reasoning is not.

Eg x+y = y+x for all x and y whatsoever, but this does not tell us that x=y.
Thank you, for some reason he isn't convinced though, apparently his tutor said otherwise
 

seanieg89

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Re: HSC 2015 4U Marathon

Thank you, for some reason he isn't convinced though, apparently his tutor said otherwise
Lol, his tutor really shouldn't be tutoring then.

Is he really not convinced? Ask him how the situation in his question differs from that in my example.

It seems his claim is that for an arbitrary function f(x,y), that if f(x,y)=f(y,x) then x=y. Ask him if he thinks this is true.

Nearly any function you can think of violates this. (Any symmetric function violates this really badly!)
 

InteGrand

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Re: HSC 2015 4U Marathon

I'm guessing the tutor was including the fact that a and b were acute angles, although it still doesn't seem obvious that a must equal b.
 

seanieg89

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Re: HSC 2015 4U Marathon

I'm guessing the tutor was including the fact that a and b were acute angles, although it still doesn't seem obvious that a must equal b.
Of course. I suspect it could well be true for this function with that condition, but you certainly can't phrase it as an injectivity type argument as he/the tutor did. Injectivity is precisely what you need to prove to draw that conclusion!
 
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