HSC Stuff > HSC Mathematics Extension 2
Mathematics Extension 2 Challenge Marathon
RuiAce:
Mahan:
--- Quote from: RuiAce on November 08, 2016, 10:56:34 pm ---
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Yes, using the hint makes the question pretty simple.Just for the fun of it, I just presented a different proof. :)
RuiAce:
--- Quote from: Paradoxica on October 05, 2016, 07:28:48 pm ---
Hint: Divide.
--- End quote ---
RuiAce:
--- Quote from: Paradoxica on October 05, 2016, 06:32:06 pm ---Consider an alphabet with n different available letters.
Let P(k) be the number of ways you can use exactly k different letters in an n letter word.
i) Explain why
ii) Show that
Let the Score of a word, X, be defined as 1/(1+ρ(X)), where ρ(X) is the number of letters that were not used by the word X.
iii) Show that the sum of all the Scores, S, over all possible n letter words, is given by:
iv) Hence, evaluate S in closed form.
--- End quote ---
To help explain part (i) (because I took a while understanding what was going on)
ExplanationConsider a three letter alphabet: {a, b, c}
Then, P(1) is the number of ways we can make three letter words, out of just one letter of the alphabet. If that letter was a, then aaa is the only word.
P(2) is the number of ways we can make three letter words, out of any two letters of the alphabet. If the letters are a and b, then the words are:
aab, aba, baa, bba, bab, abb
P(3) is the number of ways we can make three letter words using all the letters. So that's the easy 3!
The point of introducing the nCk is to quantify the fact we could've chosen any k of the 3 letters. In P(1), we could've chosen a, b or c to be our letter. (And indeed, 3Ck=3 possible letters.)
And now for the question
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If a 4U student has not seen that Greek letter before, that is rho.
ExplanationGoing back to the case n=3
If the words satisfied k=1 (e.g. aaa), then ρ(X) = 2 (Note: 1+2=3)
If the words satisfied k=2 (e.g. aab), then ρ(X) = 1 (Note: 2+1=3)
If the words satisfied k=3 (e.g. abc), then ρ(X) = 0 (Note: 3+0=3)
This can be seen by just realising how man letters we did not use, in each case.
The important thing to realise is that for each value of k, ρ(X) differs. As a matter of fact, ρ(X) always goes down by 1.
In fact, ρ(X) = 3 - k
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I don't fully trust what I say from here due to how P(k) has been defined.
Paradoxica:
--- Quote from: Ali_Abbas on August 22, 2016, 11:55:14 am ---Here's a question for people to try:
Suppose we have the hyperbola y = 1/x defined over the positive real numbers (first quadrant). Let P(p, 1/p) and Q(q, 1/q) be two arbitrarily fixed points along the curve, with p < q. Define M as the midpoint of the chord PQ. The line segment OM intersects the hyperbola at R(r, 1/r), where O is the origin (0,0).
Without expressing the coordinates of R in terms of p and q, i.e. without deriving the equation of the line OM, prove that the tangent to the hyperbola at R is parallel to the chord PQ.
Mod edit: Altered the language to make it a bit "easier" to comprehend with respect to the HSC 4U course
--- End quote ---
Consider an arbitrary chord AB parallel to PQ using the following parameters: A(p/r,r/p), B(qr,1/qr) where r is a positive scaling factor. Construct tangents at A and B and denote the point of intersection I.
With some algebra, it is verified that OI and OM share the same gradient. Thus, I always lies on OM for any and every positive value of r.
Taking the limit as , the chord becomes a tangent.
The tangent at point P is unique to P, so conversely, the point I degenerates and becomes P.
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