Just took the exam, compiling all the mistakes and providing the correct (I think) solutions here:

2a) The answer was correct, but in the notes it says 2sec

^{2}(x+pi/4) is an equivalent expression, when it's actually (1/2)sec

^{2}(x+pi/4).

2b) As people have said, you have to take the positive AND negative root, which gives you a smaller positive value of x.

sin(x) = +-1/sqrt(2) --> x+pi/4 = 3pi/4 --> x=pi/2

4a) I'm just going to give the answers for question 4 since all errors in the question stem from saying E(P) = 1/3.

E(P) = 1/4

4b) Pr(P≥1/3) = Pr(P=0) = 1-(3/4)^3 = 1-27/64 = 37/64

4c) sqrt( (1/4)(3/4) / (27) ) = sqrt( 3/(16*27) ) = (1/4)*sqrt( 3/27 ) = (1/4)*sqrt(1/9) = 1/(4*3) = 1/12

4d) E(P) = 1/4, SD(P) = 1/12. Pr( P > 1/3 ) = Pr( Z > ((1/3)-(1/4))/(1/12) ) = Pr( Z > (1/12)/(1/12) ) = Pr ( Z > 1 ). By 68-95-99.7 rule, the tail one standard deviation away from the mean has area (1-0.68)/2 = 0.16. Therefore Pr ( P > 1/3 ) = 0.16.

7a) (1-1/2)(1-2/5)(1/6) = 1/20, not 1/30. Substituting this value correctly gives 3/4.

7b) Incorrect usage of the conditional probability formula, should be Pr(Not rejected | Not rejected first), not the other way around.

Pr(Not rejected | Not rejected first) = Pr(Not rejected ∩ Not rejected first) / Pr(Not rejected first).

Pr(Not rejected ∩ Not rejected first) = Pr(Not rejected) = 1 - 3/4 = 1/4, as ALL non-rejected toasters are not rejected first.

Pr(Not rejected first) = 1/2

Therefore Pr(Not rejected | Not rejected first) = (1/4) / (1/2) = 1/2

9b) As Matthew said, the answer you should be getting to is (1/2pi)*(2pi-cos(k)). The incorrect part was the factoring in the very final step.

c) Maximum value of the mean occurs when cos(k) = -1, hence k = pi. Subbing into the correct equation;

(1/2pi)*(2pi-cos(pi))=(1/2pi)*(2pi+1) = 1 + 1/2pi.

I know this isn't very useful to compile now since I doubt anyone's going to be doing any early morning practice exams, but maybe next year

Also, I tallied up the answer key's mark and it got 27/40 = 67.5% lol. Don't mean to be

*that* guy but you should probably have a couple more people review this next time haha.