1. A detector is used to count the number of \gamma\ rays emitted by a radio active source. If the number of counts recorded in exactly 20 secs. is 10000, the error in the counting rate of per second is what?

AVERAGE counts per second = 10000/20 = 500. The error (standard deviation) in the number of counts is the square root of the number of counts (according to poisson statistics which apply to systems such as this where the average rate is constant and events are independent from one-another) and therefore the uncertainty in 500 counts per second is about 22

2. A point particle is moving in the (X,Y) plane on a trajectory given in polar coordinates by the equation

.

Then the trajectory of the particle is what?

Its not clear to me exactly what you mean by trajectory here, since as far as I am concerned, the equation written above defines the trajectory. If you want the equation which defines the trajectory in terms of X and Y, then you can use this method.

First note how to change variables:

The idea is to just sub these in to replace r and theta by X and Y, but it looks clear that the sine part will be ugly, so split it first into stuff using the sin rule [sin(A+B) =sinA cosB - sinB cosA] which we can use more easily with our transformations between co-ordinate systems:

which is

Now substituting in for X and Y:

In principal you could use the quadratic formula to write this as X = f(Y) for some function f, but I am pretty sure it will be very ugly, so the last equation I wrote is probably what they want as the final solution, assuming i didnt make any mistakes.

Hope that helped