L.H.S

2 log (a+b)= log(a+b)^2 = log(a^2 + b^2 + 2ab)

but a^2+b^2 = 7ab

=>log(a^2 + b^2 + 2ab) = log9ab = log9 + loga + logb  (since log(a*b) = loga + logb)

=R.H.S Hence proved :)

log(1000/256)= log1000 - log256 = log1000 - 8log2   (since 256 = 2^8 hence log256 = log2^8 = 8log2 (since loga^b = bloga))

= 3 - 8 * .301= 3 - 2.408 =  0.592

velocity of particle at time t = ft

at time 2t = ft + 2ft

at time 3t = ft + 2ft + 3ft

at time nt = ft + 2ft + 3ft  +  ...... + nt

so velocity at time nt = n(n+1)/2 * ft

distance travelled = integral zero to t (n(n+1)/2 * ft) dt = n(n+1)/4 * ft^2 

since the balls & boxes are identical so we are not bothered about which ball goes into which box.We are only going to consider the no. of balls going into each box. & since boxes are identical hence arrangements such as (6,5,1) & (5.6.1) will be same

now suppose no. of balls going into first,second & third boxes are x₁, x₂, x₃ respectively

then x₁ + x₂ +x₃ = 12

& furthermore (x1,x2,x3) this pair of three should not rearrange

so first total no. of integral solns. = 14C2 = 13*7 = 91

no when x1=x2=x3 we have have one soln i.e (4,4,4)

when x1=x2!=x3 we have

(0,0,12)(1,1,10),(2,2,8),(3,3,6),(5,5,2),(6,6,0) i.e 6 solns

when x1!= x2!=x3

[91 - 1 - 6 * (3!/2!)]/3! = (91-19)/6 = 72/6 = 12

so to verify 12* 3! + 6 * 3!/2! + 1 = 91

hence required answer is 12 + 6 + 1 = 19

lamda = h/mv

since v is same

lambda is inversely proportional to mass so

lambda(electron) / (lambda(proton) =  mass(proton) / mass(electron)  = 1833

no. of ways of dividing n identical things to r different people(say)

then divisions will vary only with the quantity given to the persons (since things are identical)

so let x1 , x2 ,x3, .......... xR be the quantity given to person 1 person 2,person 3...... & so on till person r resp.

so we have to find the no. of non negative integral solns of  x1 + x2 + x3 + ....... + xR = n

which is coeff . of x^n in (1+x+x^2....... + x^n)^r

= coeff. of x^n in[ (1-x^(n+1))/(1-x)]^r

= coeff .of x^n in (1-x)^-r       

coeff of x^n in (1 + rC1x +  r+1C2x^2  +........... + n+r-1Cnx^n )

= n+r-1Cn = n+r-1Cr-1 

a) work done = change in kinetic enrgy

work done from x = 1m to x =0.5m     =  integral -k/2x^2dx (x from 1 to .5)

= [k/2 * 1/x] = k/2

=>k/2 = 1/2mv^2  => v^2 = k/m =.01/.01 = 1

since intitially particle is at rest & acceleration is towards left hence final velocity is also left hence v = - 1m/sec

V=6x-8xy^2-8y+6yz-4z^2-4x^2.

E =  - (partial diff of V with x   + partial diff of V with y + partial diff of V with z)

  =    - ( (6 - 8y^2 - 8x) + (-16xy - 8 + 6z) + ( 6y-8z))

now since particle is at origin hence x,y,z =(0,0,0)

E = -(6-8) = -(-2) = 2  

so force acting =qE = 2*2 = 4N

*edited*

1)   

 

= e^(i^2* pi/2) = e ^ (-pi/2) = a purely real no.

 

2  )  1^i = (e^i*0)^i = e^(I^2*0) = e^0 = 1

 

1^i = e^[(i^2)*0)]=e^-0 = 1

e^i = cos1 + isin1 

to reach in shortest time the vertical component of velocity should be maximum so HE SHOULD SWIM DUE NORTH

VAMOS RAFA !!!!!!!

In a 4 hr 48 min epic final, the longest ever in wimbledon history,Rafael Nadal wins Wimbledon after french emulating borg who did the same 28 yrs ago.

I'm really happy , Nadal deserves this victory, he came close to winning last year and finally he has ended federer's 5 times winning streak.

VAMOS RAFA!!!!!!!!!!!!!!!!!!!

Congo!!!!!!!!!!!!!!!!

VAMOS RAFA !!!!!!!

In a 4 hr 48 min epic final, the longest ever in wimbledon history,Rafael Nadal wins Wimbledon after french emulating borg who did the same 28 yrs ago.

I'm really happy , Nadal deserves this victory, he came close to winning last year and finally he has ended federer's 5 times winning streak.

VAMOS RAFA!!!!!!!!!!!!!!!!!!!

Congo!!!!!!!!!!!!!!!!

I wanted to know that preferences filled during off campus counselling is final or it can be changed in on campus as well

Pls reply as fast as u can

amity does not admit students on the basis of merit

it has only one aim to earn money and thats all

All in all i 100% agree with akshaytalwar.

attending ip university counselling as u wont get nething good in aieee

my score is 265 what can i get in 2nd iteration.Should i waitlist myself, is it worth it??????????