person able to solve this will qualify for IIT(part 3)

since I was not able to get a good response frm goiitians in my previous posts ,so this time I  am posting some easy ones,which even a tenther can do.

1.   cuberoot(20+14(square root 2)) +cuberoot(20-14(squareroot2))=  ?

2. if x and y are natural numbers and  (x+y+90)/xy =1 ,x and y >2

  then( x+y+2)/4 is equal to  ?

3.  If x and y are postive real numbers which satisfy following three conditions

a) x+y+1>xy

b)(x²+1)(y²+1)=10

c)(x+y)(xy-1)=3

THen find the value of x²+y²+4xy-x²y²

cuberoot(20+14(square root 2)) +cuberoot(20-14(squareroot2))

let it be t ..

cubing both the sides ..

t^3 = 20 + 14root2 + 20 - 14root2 + 3 (cuberoot8)(t)

t^3 = 40 + 6t

cubic equation qith only one real root ...t = 4.

the end.

 leaving the remaining two for a tenther to come and solve ...wats the fun in solving everything ?.

am i rite guys ?

NUGO ,try the other one if u want to get rates.BY the way this is just a hit and trial method and will not work for all the equations

1. Put the given equation equal to t. Cubing both sides, u'll get

    40 + 6t = t³. So, t³ - 6t - 40 = 0. Now solve the equation to get t=4.

2. x + y + 90 / xy = 1

or, x - xy + y = -90

or, 1 + (x-1) (1-y) = -90

or, (1-x) (y-1) = -91 = (-13) . ( 7)

So, x -1 = -13 and y - 1 = 7 and therefore, x = 14 and y = 8.

So, x + y + 2 / 4 = 14 + 8 + 2 / 4 = 6

3. I am not sure about the third question. I tried it but sry. Yeah, but i solved it in 2 sec using hit and trial method, i.e. x and y = 1 and 2.

And so ur answer would cum out to be 9.

Here goes no.3

x²y²+x²+y²=9(according to second eq)

(x+y)²-xy(2-xy)=9

again replace (x+y) by 3/(xy-1)

thus

9/(xy-1)² =9+xy(2-xy)

9(1/(xy-1)² -1)=xy(2-xy)

9(xy)(xy-2)/(xy-1)²=xy(2-xy).

from this we infer that either xy=2 or (xy-1)²=-9 which is impossible .

x^2+y^2+2xy+9-x^2y^2(=x^2+y^2)-x^2y^2+2xy=k(reqd value )+9-x^2y^2

we have 2(x+y)^2=K+9

=>2(3/xy-1)^2=K+9(xy=2)

=>k=9

feelin sleepy..mum mum

finally I would like to tell the answers

1.}   4

2.}   6

3) 9

everybody has done a good job.

for question 2 ,DEEPAK's method is the shortest , for question 3, Shankh is correct ,  however for question 1 ,there is an easier way

:---if u see carefully  the equation of first question reduces to ( 2+ sqrooot2) +(2-sqroot2)  =4