Alan

I suspect the 6 should be \(\theta\), in which case we have: \(f(\theta)=10(\cos\theta+i\sin\theta)\)

This is already in rectangular form!  In polar form it is: \(f(\theta)=10e^{i\theta}\)

The area of the whole rectangle is (5 - 1)*0.25 = 1

You want the value of x=a such that (a - 1)*0.25 = 0.75

The period is the time for one complete cycle (2).

The amplitude is half the distance between peak and trough (10)

Frequency is 1/period cycles/second or 2pi/period radians per second.

You have used 5 N instead of 50 N!  Apart from that you have done the calculation correctly.  (Your torque, T, should be 10 times larger than your 1.4 Nm).

Yes, that is correct!

Add up all the numbers to get 236.  In row a1, column b2 is 21, so Probability is 21/236.

I get the same as you Will.  Looks like whoever did the marking scheme added an extra 35 into the total!

X is simply a vector whose first component is x and second component is y. The answer given means 

x = c1*e^3t - c2*e^-t and y = -c1*e^3t + 3c2*e^-t

(Incidentally, it is the wrong answer!)

.

Express \(x^2-x-2 \text{ as }(x+1)(x-2)\)

Multiply every term by \((x+1)(x-2) \text{ to get } 3x(x-2)+5(x+1)=15\)

Collect terms:  \(3x^2-x-10=0\)

Solve the quadratic to get \(x=-\frac{5}{3}\text{ and }x=2\)

2 is an extraneous solution (try substituting 2 into the original equation and see what you get!).

Fermi process:  Typical brick volume vb ≈ 60 in³

                          Typical bath volume vt ≈ 30000 in³

                          Number of bricks n = vt/vb = 30000/60 = 500