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1.) use derivatives, remember d/dx arccos(x) = - 1/sqrt(1-x^2) or you could use pythagoras' theorem 2.) that means we only define F(x) for x in [1,2] F(3) for example is undefined, intuitively, imagine erasing the graph of f(x) for all x>2 and all x<1

let the capacity of the tank be C let the rate water inflow from the 2 taps be L and S respectively we have (L+S)*80 = C C/L = C/S - 120 80(L+S)/C = 1 C(L - S) = 120 LS 80(L^2 -S^2) = 120 LS solving, L = 2S so the smaller pipe takes 240 mins and the larger pipe than 120 mins...

intuitively, think monotonic increasing means the graph doesn't go down and monotonically decreasing means it doesn't go up. the formal definitions are: if f(x) is monotonically increasing, then f(a) <= f(b) for any a <= b. if f(x) is monotonically decreasing, then f(a) >= f(b) for any a...

quanta to quarks : traditional, easy, overlaps with chem astrophysics: bit more to remember than QtQ, still good Age of silicon: If you are into these things.. and only if so. Medical physics: don't recommend this Geophysics: who's doing this?

briefly(this is not enough): Eddy currents are currents induced in a piece of metal due to the change of magnetic field through it. Eddy currents induced in the piece of metal would it self produce a field which opposes the change in magnetic field according to lenz's law. The induced eddy...

anhydrous -> 'without water' polar -> the molecues have it's positive and negative charges distributed unevenly non-polar is the opposite of polar

below circa 77

hmm.. draw the point 3+4i in the argand diagram and find the argument... naturally it involves arctan. for other values for z hmmm.. same drill for general z=x+iy |z| = sqrt(x^2 + y^2) arg(z) = arctan(y/x) if x is positive and = arctan(y/x) +Pi if x is negative

somedude: notice w^4 = w and w^8 = w^2 maniacguy: 'trivial' hmm....

Yr 11!!! get a life :D curve sketching is hardest for me personally, but I guess it's generally considered easy. Mechanics and conics are where the maths gets yucky

it's macroeconomics.. I don't think we have many economists here.. try maniacguy

(i) notice how it says 'a condition'? so a perfectly legitimate answer would be z,w,v are distinct cube roots of unit. the general condition would be (z-w),(w-v),(v-z) are k, xk,x^2k in any order,where x is a complex cube root of unity ii) rotate 1+3i clockwise or anticlockwise 60 degrees...

ranks => higher the better

it's because moving electrons need to reach a certain speed before they can pass energy electrons in the gas atoms, and they must pass energy in chunks

Hmm I would say: Nitrate, Sulphate, Phosphate, Carbonate, Acetate (ethanoate), Permanganate, Ammonium, Hydroxide, Hydroxonium(or hydronium as they call it here) should be enough

1) cA is a scalar multiple of A, so geometrically, it's parallel to A (except in the degenerate case where c = 0), cA's magnitude (length) is equal to |c|*length of A to find cA algebraically, just multiply each element of A by c eg A= (w,x,y,z) then cA = (cw,cx,cy,cz) 2.) there may be...

80s

hey CM what did you think I mean by "a=b (true statement)"? it means you start from something that is true.. and not the meaning you suggested And it's completely valid..

xyama: actually, you can raise expressions to even powers when working forward (ie, not the 'backward' method), the operations you need to becareful of are those such as squareroots by the way, q1 is from 2003 q7, where did the other 2 questions come from?

not fair, my profile gives too much away :S