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$f(x) = \phi (\sin x ) + \phi(\cos x) \\ \\<br/>and \ \\ \\<br/>\phi$

Find the interval of monotonicity of f(x)

hi pitchu aka upgrade , ssup ?

not as straightforward as it seems at first glance

Lets consider the case of one turning point only (all other cases can be shown equivalent to it )->

In this also , lets make 2 cases

Case 1 ->

In this case, lets consider the most symmetric case , ie when it turns at t=1/2 sec and it has same acceleration throughout

At t=1/2

s1=at^2 /2 =a/8

v1=at=a/2

t=1/2 to t=1

v2= 0 = u -At =a/2 - At =a/2- A/2

implies a = A

s2=ut - at^2/2 =a/4 - a/8 =a/8

but s1 + s2 = 1 = a/4

implies a = 4 m/s^2

the velocity time graph in this case is as shown in graph 1

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Case 2- >

Now , suppose instead of t=1/2 , the turning point comes at some other time , then it is obvious that if acceleration is less than 4 for interval before turning point , acceleration would be greater than 4 for for interval after turning point and vice versa

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Now , we have to proove that instead of 1 turning point suppose there are infinitely many turning points yet the above condition holds true .

For it , consider the graph 2 in which there are many turns .

Now we can consider any point P , and then we can see the the part before P will correspond to a certain average acceleration and part after P will correspond to a certain average acceleration , and hence the corresponding graph would come to be of either case 1 or case 2.

Now suppose , before P ,the average acceleration is greater than 4 , then since there are many turning points its obvious that to maintain average acceleration of more than 4 , at some points instantaneous accelerations would have to be greater than 4 .Same would be true for interval after P , if it has average acceleration greater than 4.

Hence , option a and c are correct

A polynomial f(x) leaves remainder 15 when divided by (x-3) and 2x+1 when divided by (x-1)^2.

Find the remainder when it is divided by (x-1)^2 * (x-3).

I do have a method , but still i think more than one approach maybe a distinct possibility :)

http://www.ephysics.org/simulations/bec/background.html

$P = e^{-2 G L}$

where $G = \sqrt{\dfrac{2 m \left(V_0 - E \right)}{\hbar^2}}$

$P$ is the probability of the object tunneling through the barrier, where 1 is 100% chance, and 0 is no chance.

$L$ is the length of the barrier (in meters)

$m$ is the mass of the object (in kilograms)

$V_0$ is the energy of the barrier

$E$ is the energy of the object before tunneling

$\hbar$ is Dirac constant

Now the probability of tunneling is certainly increased in case of electron cloud as the electrons are of very small mass , however the breadth of the wall should be very less . This was based on quantum mechanics.

Actually classical mechanics says only 2 things ->

if energy of electron cloud is higher than the wall , then it will pass from above it , else it will collide and reverse , but since classical mechanics ignores wave nature , tunneling concept cannot be applied there.

@reddevil ->
how have u kept the plates , in the plane of incline(ie sideways to the incline) or in the plane perpendicular to incline (ie above and below the incline )

@conjurer->
whats the condition for going up ?

maybe the forces are like this

how ?

A very , very easy approach to this would be nodal analysis

Considering the top leftmost point at 0 potential , then the distribution is as shown

However its given that 120 - X =100

hence , X =20

Also , since the right 2 resistors are in parallel the sum of currents that flows through them individually = current flowing through 10 ohm resistor

or

[120-20]/R + [120-20]/100=[20-0]/10

which gives R =100

its difficult to comment without exact info , do u have the answer to this one ?

My answer is coming to be 3C/8 , and i feel the method which has been given is correct.

acha hai , benzyne is out of course LOL :D

Dunno , but what is the advantage of posting problems (and soln ) from net ??

http://www.qbyte.org/puzzles/

excellent !!!

well , these are the opinions of me and one of my good friends ,

One thing that stands out is that hs bhatt sir has worked selflessly for this site and students. We all must respect that , secondly , he is humble to the core , and knows effective use of english :D , ie he really uses the softest of words in the best of ways (even if he is targeted ) . The thing that happened in this thread before was an exception , and we all do know that there is no reason for bhatt sir to not be angry , after all he is also a human being and has all the rights to pour out what is in his mind .Moreover i can assure you , he'll always take these things on a humourous note as far as possible

(For instance , when feynmann said that he had filched the soln , he said that it was a compliment for him !!!! Cos his soln was so brilliant that it seemed it was filched LOL :D )

One thing for which i really appreciate sir, respect him, etc etc kind words :D , is that he still has interest in solving problems , i mean at this age i think i would like to stay as far away from problems as possible !!!

Just talking about that thing , sir said about ramyani , i have seen some other instances as well of that and i just wanted to say to those people as well , WELL DONE !!!

Now , coming to mr feynmann ->

You are plain rude . I had a lot of respect for you initially .After you said easy before each qn , i did get to learn a thing or two about your pride, but still it was ok (If you know it you flaunt it :D ) ,but now it seems there is something wrong (you don't know it yet u flaunt it !!!! )

(Mind you i am not saying knowing and flaunting is right , but is was acceptable to me )