If `m`, `n` are positive integers and `m+nsqrt(2)=sqrt(41+24sqrt(2))`, then `(m+n)` is equal to (a) 5 (b) 6 (c) 7 (d) 8

Options

A. `5`
B. `6`
C. `7`
D. `8`

Answer

C

Explanation

`(c )` We have `m+nsqrt(2)=sqrt(41+24sqrt(2))`
Squaring both sides, we get
`m^(2)+2n^(2)+2sqrt(2)mn=4a+24sqrt(2)`
`:.` On comparing we get
`m^(2)+2n^(2)=41`…….`(i)`
and `mn=12`…….`(ii)`
`:.` On solving `(i)` and `(ii)`, we get
`implies m^(2)+(2(144))/(m^(2))=41`
`impliesm^(4)=41m^(2)+288=0`
`implies (m^(2)-32)(m^(2)-9)=0`
`implies m^(2) ne 32`
`:. m^(2) =9 implies m =3` and `n=4`
Hence, `(m+n)=7`