k no beatin abt d bush

1) by practice of course

2) draw d grafs to get an idea nd proceed using simple geometeical methods or integration if required....if itz mod u wont need much of integration....can b solved by simpl geometry

white.....coz its north pole.....nd itz a polar bear....as evry wall faces south in north pole

heyya it shud b:

f(x) = 1 +{ sinx / 1 - sin²x}

       = 1 + sinx/cos²x

       range(-inf,inf)---------->>1 is included!!

cheers!

sinx----->>(-1,1)

cos^2 x--->>(0,1]

so we'll have -infinity too

newton leibniz rule fr continuous fns.........plz refer ur texts fr this......

df/dx = log(x + rt(1 + x²))

let df/dx = p(x)

p(x) = log(x + rt(1 + x²))

p(-x) = log[rt(1 + x²) - x]-------------(1)

-p(x) = - log (x + rt(1 + x²))

           =  log [1/(x + rt(1 + x²))]

         = log[rt(1 + x²) - x] / (x² + 1 - x²)-------------(2)

         = p(-x)

so p(x) is an odd function

so its integral.........f(x) is an even function

@sachin gupta.....

dats d mistake we all made d last tym.........

its not mentioned 2 nos.--got it!

accordin to ma soln.....the max pdt is e^271/e ~~~approx....(2.7)¹⁰⁰.....which is way higher than ur 135.5^2

xperts......reply

oops........stupid me!

it s not mentioned 2 + real nos...is it........

let me assume the number of numbers to b 'n'

so

x₁ + x₂ + ....+ x_n   >=  n (x₁x₂.....x_n)^1/n

.......pdt....<= (271/n)ⁿ

so pdt is a function of number of terms

so p(n) = (271/n)ⁿ.........

logp = n log(271) - nlogn

1/p * dp/dn = log271 - logn - 1

dp/dn = p[log (271 / n) - 1]

dp/dn = 0 ------->> log(271/n) = 1 or p=0

n = 271/e or p = 0

but p cant be zero so

n = 271/e

n x₁ = nx₂ = .......= nx_n = 271

so...

x₁ = x₂ = x₃=.......=x_n = e

so the positive real numbers required are:

271/e................n times..........

x₁ = x₂ = x₃=.......=x_n = e.........

uhhhhhh........now anothr prob.....,m gettin not gettin n as an integer.......ne1 help.....

check this out

solved last year:

http://www.goiit.com/posts/list/algebra-number-theory-13315.htm

cheers!

simple....use congruence modular theory:

2222 nd 7 r coprime to each other

so eulers totient function phi(n)= 7(1 - 1/7)= 6

so

2222⁶ = 1(mod 7)

so

(2222⁶)⁹²⁵ = 1⁹²⁵(mod 7) = 1 mod7

so

2222⁵⁵⁵⁰ = 1mod7------------(1)

now

2222 = 3mod7

2222² = 9mod7 = 2mod7------(2)

2222³ = (-1)mod7------(3)

so 2222⁵⁵⁵⁵ = 1*2*(-1)mod7 = (-2)mod7 = 5 mod7

remainder is 5

cheers!

now fer explanation:

consider car A movin in rite with u m/s

and car B movin in left with v m/s

.....

now relative velocity of A wrt B is.....U - V (caps are fer vectors nd lower case fer scalars)

relative displacement is S = 8 m/s

S = (rel.velocity)(time)

  = U -V....coz time is 1s

now U = +u

and V = - v(takin rite to left as negative)

so s = u + v

8 = u + v

similarly

wen both travel in same direction it is

0.8 = u - v.........

solve u get above ans

let velocities b ---u and v

now dis is easy coz it in 1 second......

so sum of velocities mus b 8 and diff mus b 0.8

so ans is obv....4.4 and 3.6-----(b)

hey

u c itz positive real nos......

so d first thing u shud look at is

x + y/2 >= rt(xy)

so itz obviously (135.5)^2........

in maximisin probs the value u luk fer mostly lies in middle

awryt....itz a try....

k m assuming dat he is buildin d cube from bottom up....row by row......

for bottom row---he needs4 vth 3face colored......4 vth 2 face colored nd 1 vth 1 face colored.......

he picks em out in

8C₄ * 12C₄* 1C₁  ............nd aftr selection....the 3color cubes can b placed on black face in 3/6 ways......nd rotated in 1/4 ways.......------>so in 1/8 ways

2 color cubes can b placed in 2/6 ways nd rotated in 1/4 ways.......---->>1/12

1color cube can b placed in 1way and rotated in 1/4----->>1/4.......

proceed lyk dis fer othr rows!

m still not sure

hmmm...

there'll b

12 smaller cubes vth 2 black faces

8 with 3black faces

6 vth 1 black face

1 vth no face black

we'll consider each category separately......

now one 2 face colored cube can occupy the position of anothr 2 face colored cube....nd so on......each can b fitted in2 d place of anothr of itz kind nd not newer else......

so we find the number of ways.....fer each category.....nd divide by total no.of ways.......nd there u have it!

this is basic idea!

vll post complete solution l8r......am i r8 little_genius??

PS: dont 4get the along vth position of cube color of d side mus also b taken in2 account!

whoa-----ive been fooling arnd......got an idea.....

in 12 hrs hands vll meet 11 tyms....

so answer

12/11 hrs