Ok I needed to refer to the examiners' report but if I was in a real exam I might be able to do it because force myself to do it.
Anyway, the style of Q10(b) of 2002 and 2003 (algebra mess) are characteristics of the new exam committee and is likely to appear in 2004 paper as well.
So I'll give some tips. Hopefully they make sense and don't sound too complicated.
1. Don't get scared by the plethora of algebraic expressions. It's just like how our ext2 english students use big confusing strange words, mathematicians also use long strange algebraic expressions. They're really nothing and are just there to distract you.
For example in 2002 Q10(b)(i), just relax and differentiate it normally.
1 / [ b^2 + (x+8)^2 ] = [ b^2 + (x+8)^2 ]^(-1)
=-1 * [ b^2 + (x+8)^2 ]^(-2) * derivative of the thing in [...]
=-1 * [ b^2 + (x+8)^2 ]^(-2) * [2*(x+8)*derivative of (x+8)]
Do the same with 1 / [ b^2 + (x-8)^2 ]
Then, since the given expression (-2P/Q) contains no + sign, you need to combine the two terms into one, by taking a common denominator.
u/a + v/b = [u(b)+v(a)]/ab
2. For derivative to be zero, the numerator must be zero. For numerator to be zero, one of the factors in its expression must be zero. For the derivative to be negative (when testing for maximum or minimum), the product of signs of the factors in numerator and denominator must be negative. -*-=+, -*+=-, etc. * is for times.
So for 2002 Q10(b)(ii), we have 3 factors for P.
When the first factor=0, x=0
When the second factor=0, x^2=-[64+15^2+16sqrt(..)] which is negative. but x^2 can't be negative so there's no x value for this factor to be zero.
Similar for the third factor.
To confirm that its indeed maximum TP, just put x=1 and x=-1.
Another technique would be to look at the 3 factors. the second and third can never be negative when b=15 [the third factor simplifies to x^2+17]
Hence the first factor determines the sign of numerator.
Also, dI/dx=-2P/Q. From its ugly expression in (i), Q must be positive because it contains squares only.
3. Extract info from the question. In 2003 Q10(b)(iii), you want to find where l is maximum and the determining info to confirm that the stationary point is a max is that x<1 and that r<1. Ideally you should show clearly how you determine it's a max (by actually finding the sign of the second factor when x is slightly less than the value for the stat pt). But you can pretend to be smart and just do like what Geff Jeha did in his solutions in the Resources section. Just mention the key info that you think/guess determines the sign of factors.
You might get the mark by justifying it informally.
Consider a person walking from A to N, N is a point at a fixed distance above the floor, below or exactly at P.
Code:
X_____A
| /
P| /
| /
N|/
|
|
M|
He follows the route "AP then PN". There exists an angle XAP at which he can do the least walking, with walking at 45 degrees giving the least overal distance.
Least walking means least rope is used to reach N. The remaining of the rope can be used to extend to M (vertically down), so least AN corresponds to largest l.
