यदि `x=1+sqrt(2)+sqrt(3)` और `y=1+sqrt(2)-sqrt(3)` तो `(x^(2)+4xy+y^(2))/(x+y)` का मान क्या होगा?

Options

A. `2sqrt(2)`
B. `2(2+sqrt(2))`
C. `1`
D. `6`

Answer

D

Explanation

According to the question
`x=1+sqrt(2)+sqrt(3)`………….i
`y=1+sqrt(2)-sqrt(3)`……………..ii
`implies (x^(2)+4xy+y^(2))/(x+y)`
`implies((x+y)^(2)+2xy)/(x+y)`
From equation i + ii
`x+y=2+2sqrt(2)`
`xy=(1+sqrt(2))^(2)-(sqrt(3))^(2)`
`=3+2sqrt(2)-3`
`=2sqrt(2)`
`implies ((2+2sqrt(2))^(2)+4sqrt(2))/(2+2sqrt(2))`
`implies (4+8+8sqrt(2)+4sqrt(2))/(2+2sqrt(2))`
`implies (12+12sqrt(2))/(2+2sqrt(2))`
`=(12(1+sqrt(2)))/(2(1+sqrt(2)))=6`