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Some complex number problems could be done quiet elegantly using a geometric approach. How about the other way around? Let ABC be a triangle, D be the midpoint of AB and E a point on the side AC for which AE = 2EC. Prove that BE bisects the segment CD. a) by geometry b) by complex numbers

We had a thread last year on elegance -Spice Girl and ND were the main correspondents. To do well in Ext. 2, students need to develop their EQ elegance quotient. Here are three problems to practise on, the first two could easily be done by a Year 11 doing the preliminary, the third - well...

Who says OLDMAN can't attract attention?:D Played basketball with the young son this morning, and the court was filled with hoppers- yes the locusts have finally reached Mudgee. Even without a blade of grass, a grasshopper's orientation after it lands seem pretty random - so its next hopping...

An interesting binomial probability question with a twist. A die is thrown n times. What is the probability of getting an odd number of sixes. Here's the twist : use (p-q)^n rather than the usual binomial distribution of (p+q)^n.

Here's a retake of an otherwise useful question. If w, w^2, ...,w^n are the n complex roots of 1, with w being of smallest positive argument, what is the condition that need to be satisfied for W=w^k, so that W, W^2, W^3,...,W^n are also the n roots of unity; and why?

Some of the following have been discussed already. It might be useful to bring them all together to show that there usually is physical meaning behind the coefficient. Use a combinatorial argument to show : a) nCr = nC n-r b) nCr = n-1 C r-1 + n-1 Cr c) (nCo)^2 + (nC1)^2 +...

By now all ext 2 students should be familiar with de Moivre's^*, here's an interesting exercise on it. Prove that cos(1^o), cos(2^o),cos(3^o),cos(4^o),cos(5^o) are all irrationals. de Moivre's^* : is your teacher a...

Got a new online student courtesy of bored, must not offend God(laz) and contribute some. The idea for this question is from a discussion in a 3-unit thread. Prove without using formulas : nCk = n-1Ck-1 + n-1Ck

edit: sorry, wrong forum!

Sorry to post this in this forum- where else do I belong! Is it possible with the Adobe/Acrobat software (the purchased version) to do the following : say I want to print the 10 exponential growth and decay problems from 2-unit HSC 1990-1999 (assuming there was one a year), can I cut and...

Wishing you all the very best. Remember : write fast; don't look back; don't be too careful and lose time; better off investing your time on a new question with fresh points than checking an old one that in all probability has been answered sufficiently; it is not the olympics (meaning...

This question is from Moriah College 2003 Trial Q8. P_1,P_2,P_3,...,P_n represent the complex numbers z1,z2,z3,...,zn (zn=1) and are the vertices of a regular polygon on a unit circle. Prove that (z1-z2)^2+(z2-z3)^2+(z3-z4)^2+...+(zn-z1)^2=0

Consider the eqn. 2^x=1+x^2 i) find two obvious solutions. ii) show that there is another solution between 4 and 5. iii) show that these are the only solutions.

____________________________________________________ quote freaking_out: actually, the fact that the question is posted by "OLDMAN", turns me off straight away. ____________________________________________________ Criticism noted. An examiner's dilemma : pose a challenge, but be fair. Only...

Explain why the polynomial b_0+b_1x+...+b_nx^n has at least one real root if, b_0/1+b_1/2+...+b_n/(n+1)=0.

Show that the coeff. of x^k in the expression, (1+x+x^2+x^3)^n is SUM(j:0--->int(k/2))[nCj.nC(k-2j)] where int(k/2) is the highest integer <= to k/2.

A clock's minute and hour hands are lengths 4 and 3 respectively. At the moment when the distance between the two tips is increasing most rapidly: i) what is the distance? ii) what is the speed? iii) what is the time after 3 o'clock does this first happen?

A test on recursive probability tree. Di and Dot bet on the total roll of two standard dice. Di bets that a 12 will be rolled first. Dot bets that two consecutive 7's will be rolled first. They keep rolling until one wins. What is the probability that Di will win? Sorry about the silly...

Bob tosses 11 coins, Penny tosses 10. (i) What is the probability that Bob has more heads than Penny. (ii) Generalize to n+1,n respectively.

Two players bet on the outcome of a toss of two coins. Bob bets that double heads will be tossed first. Penny bets that two consecutive single head will be tossed first (that is, exactly one head and one tail, Penny wins if this happens twice, one after the other). They keep tossing until one...