Today's exam was horrendous. I think I had a maths mental block. From getting a spread of marks between 83% on P4 and 100% on P1, this pure module is by far the lowest...every question I struggled on. Even 1 b - not a good confidence boost. My calculator didnt agree with my answer, but i checked my answers over and over again...and didnt see an error so stuck with my answers, as I had the answer as ln2, and the calc showed -ln2. I was a bit worried when it said leave it as klnb or similar and I only came out with ln 2.
Here goes:
cosechx - 2coth x = 2
1/sinhx - 2coshx/sinhx = 2
1-2coshx = 2sinhx
At this point, my maths colleagues turned it into the e^x form, which is what I did all the time in revising...but not in the exam! I noticed a potential cosh²x-sinh²x = 1 coming up - so I squared everything.
1-4coshx+4cosh²x = 4sinh²x
So 4(cosh²x-sinh²x)=4coshx-1
4coshx-1=4
So 4coshx = 5
so cosh x = 5/4
So x = arcosh (5/4)
Using the form given in the formula booklet for arcosh, i.e. arcosh x = ln (x + root (x²-1)) I got x=ln2.
Subbing this back into the original equation (three times) gave me -2 instead of the required 2. Panic and confusion set in as I could not spot my error....can any of you? Surely squaring both sides is a valid move?! I never took any square roots either, so I haven't got to worry about negatives or positives?
Reduction formulae are easy! Fantastic! This question, YUK! I couldn't show the first bit, nor could I get the next bit with root 2 involved, but I managed to work out I4 I think!
The *only* thing I could do fully was the radius of curvature worth 9 marks, woohoo! 9/75, I'm well chuffed. No, seriously, about 40/75 is what I am expecting...a dire performance on my part and I actually felt a tear in my eye at the end of my exam as I am fully aware my university place could have already been lost thanks to one bad exam.
Kirk