I think everyone got Ashwin...

 

Let me put down a very simple example :

Initially it is given that, a and b are two positive integers such that :

             

                                                       a = b

                                                     a² = ab

                                               a² - b² = ab - b²

                                       (a + b)(a - b) = b(a - b)

                                                (a + b) = b

                                                        a = 0

 

But is not 'a', a positive integer ??!! Now the hidden mistake made over here is that in the 4th step, you cancelled (a - b) on both sides which is not possible as a = b and that (a - b) = 0. These types of tricky proofs can be brought through very complex words that you may not notice the mistakes made....   

Hi Vivazcool,

 

As per your question, A = {1, 2, 3, ...., n}

Now,

 

The no. of subsets of this set with 1 as one of the elements among the three = ^{(n - 1)}C₂ -------------------- 1 is the least term in each subset. 

The no. of subsets with 2 as one of the elements(where 1 is not included) = ^{(n - 2)}C₂ ----------------------- 2 is the least term.

The no. of subsets with 3 as one of the elements(where 1 and 2 are not included) = ^{(n - 3)}C₂ ------------------------ 3 is the least term.

.

.

.

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The no. of subsets  with (n - 2) as one of the element where the other 2 elements are (n - 1) and n  = ²C₂ = 1 ---------------------- (n - 2) is the least term.

 

ie, Arithmetic mean of all the least terms in each of these subsets :

 

{ 1*^{(n -1)}C₂ + 2*^{(n - 2)}C₂ + 3*^{(n - 3)}C₂ + ............ + (n - 3)*³C₂ + (n - 2)*²C₂ }  /  { ^{(n - 1)}C₂ + ^{(n - 2)}C₂ + ^{(n - 3)}C₂ +............+ ³C₂ + ²C₂ }

 

I haven't done my thorough studies on these " Combinations " so, simplification ???!!

 

Please let me know whether I'm right or wrong...

 

                                                                      Thathwamasi

 

Hey, Jaysun.. It's (5.4 / 5)mm as : pitch = distance moved / no. of rotations given..

 

And.. Soumi,

For all those quantities which are not fundamental, directly or indirectly they will have a relation with those fundamentals qts.

 

Eg : Velocity = Displacement / Time

                    = [L] / [T]

 

And like this, all those simple quantities may be dimensionalised from which we can find those quantities like 'G' as our Bipin Sir has done...                       

Hey, put down some more questions kya ?? 

Given : 

Initial velocity,u = 10 m/s

Angle of projection,  = 30⁰  

So, Max. height = u²sin² / 2g = 25 / 2g

 

ie, When the projectile is at a height of 25 / 4g,

Horizontal velocity,               v_x  = ucos  = 53

Vertical velocity,                   v_y  =  { u²sin² - 2g(25 / 4g) }

                                                = 5 / 2

 

ie, Velocity at that height, v =  { v_x²  + v_y² }

                                         = 5 ( 7 / 2)

 

SO : Change in velocity : 10 - 5 ( 7 / 2) = 5( 2 - ( 7 / 2)) 

 

If my answer is wrong, please let me know my mistakes..

If right, rate me yaar....

 

                                          " Thathwamasi "

Hey someone, do this one :

 

Prove that :

 

[ (cos - cos) + i(sin  - sin) ]ⁿ + [ (cos - cos) - i(sin - sin) ]ⁿ

                                                                 =

              2^{n +1}sinⁿ( (  -  ) / 2 ) cos n( ( + +  ) / 2 )

                                               

                                       

 

1)

 

   (sin x + cosec x) + (cos x + sec x)²

 

  = sin²x +cosec²x + 2 + cos²x + sec²x + 2

 

  = 5 + cosec²x + sec²x

 

  = 5 + tan²x + cot²x + 2

 

  >= 9            (minimum value of tan²x + cot²x =2) 

 

2)

 

  cosec - sin =m  ie, (1 - sin²) / sin = cos² / sin =m

  

  Similarily:  sin² / cos = n

 

  Now m²n = cos³  and  mn² = sin³

 

  ie, sin² + cos² = (mn²)^2/3 + (m²n)^2/3 =1  

 

  SO : m ^2/3n^2/3 (m^2/3 + n^2/3 ) = 1

 

I will be posting the last one later.....           

Oh!!!!!!!!!!!!!!!!

Hey VMC, I really forgot about these old ones. Sorry yaar.... Now let me put down the new ones...and Joy.... what you said is correct. There is no need of mugging up these, instead solve the qustions directly........

 

But if you are very strong in your fundamentals and at the sametime you know all these, accuracy comes along with time management...  

 

 

I agree with Remya...take a thumbsup.. 

 

Now let me just go direct to the answer :

 

Given : |x|, |x - 1|, |x + 1| are in A.P. then,  

 

 

 

 

 

|x| = 1/4

|x - 1| = 3/4

|x + 1| = 5/4

 

hence common difference = 1/2

 

 

 

  

 

                                              Thathwamasi