(1) The no. of positive integral solution of the equation

7x²-2xy+3y² = 27 

ans = 1, 2 ,0 ,None of these.

(2) find the no. of positive integral solution

x³-y² = 2

I think the answer for the 1st question is 2

for the 2^nd question

x³=y²+2

x=(y²+2)^1/3

now the only solution i could think of was x=3,y=5

Yes, Rahul.. you have guess it right. The only positive integral solution for the equation x³ - y² = 2 is (3 , 5). This equation is an example of Mordell's equation which is an elliptic curve:

                                                          y² = x³  + n

for an integer n. Till 2006, the general soultion for the above equation had not been found. Follow this link to know a bit more about (if you are interseted):

http://www.lrz-muenchen.de/~hr/numb/mordell.html

i am not sure of the answer. so please correct if wrong

The first one looks like case-work to me.

Since only positive integral solutions are asked for, the pairs (0,3) and (0,-3) are not counted.

We can rewrite the equation as

x=y gives no integer solutions.

Since the second term is even and is atleast 14 (since x and y are atleast 1 and x, y are unequal), (x-y)² must be odd and less than or equal to 9

Hence the only possibilites are

Case 2 gets straightaway ruled out as y must be divisible by 3 and the least such +ve integer is 3 which means x = 0 which is not possible as x is also positive.

Case 1: y can be 1, 2 or 3.

 y =1, x =2 is the only integral solution obtained

Hence (2,1) is the only solution among positive integers

another way of solving