diksha and punnima, there is a limit to how much patience we have...

For god's sake, stop posting rubbish. If you are so sure you can integrate it, post the solution here. If you notice, so far there have been 132 replies to this post and no person has found a solution yet. I hope you have enough dignity to stop this discussion immeditately or post the solution.

 

 

 

 

So the square:  forms the sample space. The quantity  will be the the vertical height of the needle. Now, we know that the needle will intersect any one of the lines if and only if . So the event space(favourable) is given by the area of region that lies between the curves and . Thus the required probability is equal to :





In our question , and

 

Of course effective resistance cant be negative. So we reject the negative root.

Hence X =

Let .

This problem can done by using the method of symmetry.

(If A and B are connected to battery and AB or it's perpendicular bisector is a line of symmetry, then all points lying on the perpendiculars drawn to AB are at same potential).

 

In the question, the perpendicular to AB is a line of symmetry. Hence the current through AF is equal to the current throguh FB which means no corrent flows from F to C or D. This implies that even if we break the connection at F, the currents in the circuit will not be affected. : Note that I have drawn the circuit only for the topmost layer of triangles.

 

 

Now calculate the resistance between CD.

 

 

 

Now simplifying the circuit, it consists of AB parallel to another branch (of resistance( ) .

Hence,

 

Now for the next layer of triangles, repeat the above procedure to see that



Now we see a pattern. For any particular layer is the resistance is , the next layer the resistance is .

 

So in effect the circuit is this:

 

Hence,

 

   ohms

 

where .

Is the answer

for third question,

 

 

 

  Solve



correspondingly,

deedee, there is no need to find the inverse or make any substitutions.

 

   
and   obviosully has no solutions.

Hence C