Here are some moderately difficult problems for your revision:
1)
Consider dD(t)/dt=k1*L(t)-k2*D(t). Also, L(t)=Lo*e^(-k1*t). Find an expression for the critical time Tc: the time when D(t) is maximum. Take D(0)=Do. Note that k1,k2, Lo and Do are constants and k2>k1.
--3 marks
What is this time (Tc) when k1*Lo is less than Do*(k2-k1)?
--3 marks
2)What is the magnetic moment of a disc sector (angle=pi/3) of radius R and charge Q (uniformly spread) rotated with angular velocity w about its natural axis??
--2 marks
3)A point charge is kept at the center of a cylinder of length L and radius R. What is the ratio of electric flux thru the curved surface to the flux thru the end caps?
--2 marks
4)The volume charge density varies with the distance from origin 'r' as rho=a*r+b*r*r.
What is the variation of electric field with r?
--2 marks
5)A 3:1 (by moles) mixture of oxygen and nitrogen effuses through a tiny orifice. What is the molecular mass of the mixture effusing out? (Do NOT consult your class notes!).
--1 mark
Note: Part 2 of Q1 is my favorite. Please try it! It looks (and is) difficult but it is solvable by you!
Note: You're through with JEE if you can solve 4 problems!
Expectations
A random variable is a variable whose value results from a measurement on some type of random process. Expectation E(X) of a random variable X is given by :
Suppose that n balls are tossed into n bins. Each toss is independent and each ball is equally likely to end up in any bin.
Using (or maybe not) the fact that E(X+Y) = E(X) + E(Y) even if X&Y are dependent variables, give the expected value of:
- Number of balls in a bin?
- Number of tosses till a given bin contains a ball?
- Number of tosses till every bin has a ball?
- Number of tosses till a given bin contains two balls?
- Number of tosses till at least a bin contains two balls?
- Number of empty bins?
- Number of bins with exactly one ball?
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