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Yep, I know very little about it haha.

They do in projective geometry :).

The product law for limits works even if infinity is allowed though. And we could try to define: x^inf=0 if 0=<x<1 x^inf=inf if x>1 x^inf=? if x=1. However, no choice of '?' would make our desired exponentiation limit law true. Many times infinity can be adjoined to the real numbers...

:P. I figured it would get my point across. Mathematics is fun.

Its not THAT obvious... If x->a and y-> b, then xy->ab. This is true for real a,b. It is even true for infinite a,b given suitable definitions. From this it is natural to 'guess' that: If x->a and y->b then x^y-> a^b might hold for positive/infinite a,b given suitable definitions. This is not...

Why wouldnt 1 to the power of a chair equal 1? Infinity is not a real number, the current definition of exponentiation does not apply to this situation and we cannot extrapolate any information from the fact that 1 to the power of a real number is 1. In any case the OP meant something...

Whenever you do something (which you suspect may be illegal) in mathematics you should be asking yourself why you CAN do it. Steps in your working are invalid until proven valid, not the other way around! (So if you want a more specific answer on why we cannot say (1+1/n)^n->1 or something...

At least thats how I interpret what you are asking, if you are instead referring to some limiting process you might need to be a bit more specific. Eg If x->1 and y->inf does x^y->1? The answer is an emphatic no here, as can be seen from the identity (1+1/n)^n->e.

No, 1^inf is undefined...you cannot just extrapolate something like that and claim it to be true of the 'object' infinity. I stress the use of the word object here as you have gone from considering a real number with a real exponent to a real number with infinity (not a real number) as a...

Let f(x) be your integrand and let I be the sought definite integral. It is easy to show f(x)+f(pi/2-x)=1. Integrate this expression between the given limits and make the substitution u=pi/2-x to arrive at I=pi. (Note that the integral of a T-periodic function over any interval of length T is...

Hey yep, the circle of Apollonius (or line if the fixed ratio is 1:1) is how I proved D=D'. That injectivity argument is clever :). Probably wouldn't be received well in the HSC but then again, most HSC geometry questions are much easier.

Pretty good. Not entirely happy with the "work backwards to complete the proof" though...I think some justification is needed for why e=a+b, f=pi-b+c is the only possible solution.

For ii) I should clarify something. A point having rational coordinates means BOTH of its coordinates are rational. This is NOT equivalent to having a rational x-coordinate.

Let ABCD be a cyclic quadrilateral and let X be the point of intersection of the tangents at A and C. (We assume that AC is not a diameter of the circumcircle of ABCD.) Prove that X lies on BD if and only if AB.CD=AD.BC.

Anyone want to have a crack at it? It isn't beyond an interested MX2 student. I will post a solution tomorrow otherwise.

We define a primitive Pythagorean triad to be a triple of positive integers (a,b,c) with greatest common divisor 1 such that: a^2+b^2=c^2. Examples of Pythagorean triads include (3,4,5) and (5,12,13). In this question we find ALL primitive Pythagorean triads using high-school level methods...

That said, most high school teachers wouldn't really know the difference, so its not a particularly important point.

Meh, so many people use different notation...but in most calculus books I've seen upper case delta is the standard when looking at difference quotients. Lower case delta is used more in analysis-type arguments and at a higher level, in calculus of variations.

sorry :P.

The symbol \delta should not be used here. The 'd' is standard and \delta is connected to things like the variation of a functional. It is more common to write \frac{\Delta y}{\Delta x} to denote the difference quotient of a function though, and in this case we have: \frac{dy}{dx}=\lim_{\Delta...