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1. Exponentials Problem

I have ln((V+v)/(V-v)) = 2ctV, and I've checked this is right. But how do I go from here to v = V((e^Vct - e^-Vct)/(e^Vct+e^-Vct))? The solutions just give them as successive lines. (I appreciate that's not the easiest thing to read, it's taken from the 2004 MEI specimen paper for A-level...
2. Normal Subgroups

I'm working on introductory group theory and am stuck on this proof. I don't even know where to start, so I'd appreciate any help at all! "A subgroup H of a finite group G is said to be a normal subgroup if, for each element h ∈ H and each element g ∈ G, the element g^1hg ∈ H Prove that if...
3. Direction vector of a line

How would I find the direction vector of a line given in the form ax + by + cz +d =0? Thankyou.
4. Finding the speed of a bullet

I'm doing this without a teacher, so this isn't really a homework question, but this still seemed the most appropriate forum. I have the following question: A square board, of side 2a and mass M, is to be used to estimate the speed of bullets. It is freely hinged about one horizontal edge...
5. Infinite series of complex numbers.

I have the two series: C = 1 + (1/3)cosx + (1/9)cos2x + (1/27)cos3x ... (1/3^n)cosnx S = (1/3) sinx + (1/9)sin2x ... (1/3^n)sinnx I have to express, in terms of x, the sum to infinity of these two series. Here's what I've done so far: Let z represent cosx + jsinx C + jS = z^0 +...