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Re: 2012 HSC MX2 Marathon Did you mean 'Conjugate' instead of 'Congruent'? A congruent number is an integer that is the area of the triangle with 3 rational sides. A cool theorem related to this is "There does NOT exist a square number that is congruent". It is called Fermat's Right Triangle...

(3) Sketch y=1/ln(x) and observe that it is an injective function in (2,3) so we can construct an upper/lower bound rectangle. I is between the area of them. (2) a and b are just the points of intersections because if we go beyond it, we get 'negative area'.

What other subjects are you doing? If Mathematics is not your forte and you think diverting energy into it is not worth it, then perhaps best not to choose it...

Excellent ranks!

Alternatively, you can construct the vector z-w (using vector addition and subtraction), and then use basic Euclidean Geometry to prove that Arg(z-w) is 15 degrees (or pi/12) 1. Draw the usual parallelogram, including z-w. 2. Notice that Arg(w) = 2pi/3, which means that the bigger angle of the...

If you expand the cis into cosx + isinx, then z = 2i, and w = -1 + sqrt(3).i z - w = 2i - (-1 + sqrt(3).i ) = 1 + i ( 2 - sqrt(3) ) Arg (z-w) = Arg (1 + i ( 2 - sqrt(3) ) ) = arctan (2 - sqrt(3) ) Note: arctan is a short hand way of saying inverse tan. Edit: Which is equal to pi/12, I...

wut.

Says the person who named themselves the 'Mathmaster'