The equation `|((1+x)^2,(1-x)^2,-(2+x^2)),(2x+1,3x,1-5x),(x+1,2x,2-3x)|+|((1+x)^2,2x+1,x+1),((1-x)^2,3x,2x),(1-2x,3x-2,2x-3)|=0` has (a) no real solution (b) 4 real solutions (c) two real and two non-real solutions (d) infinite number of solutions, real or non-real

Options

A. has no real solution
B. has `4` real solutions
C. has two real and two non-real solutions
D. has infinite number of solutions, real or non-real

Answer

D

Explanation

`(d)` `|{:((1+x)^(2),(1-x)^(2),-(2+x^(2))),(2x+1,3x,1-5x),(x+1,2x,2-3x):}|+|{:((1+x)^(2),2x+1,x+1),((1-x)^(2),3x,2x),(1-2x,3x-2,2x-3):}|=0`
`implies|{:((1+x)^(2),2x+1,x+1),((1-x)^(2),3x,2x),(-(2+x^(2)),1-5x,2-3x):}|+|{:((1+x)^(2),2x+1,x+1),((1-x)^(2),3x,2x),(1-2x,3x-2,2x-3):}|=0`
(taking transpose of `1^(st)` determinant)
`implies|{:((1+x)^(2),2x+1,x+1),((1-x)^(2),3x,2x),(-(2+x^(2))+1-2x,1-5x+3x-2,2-3x+2x-3):}|=0`
`implies|{:((1+x)^(2),2x+1,x+1),((1-x)^(2),3x,2x),(-(1+x)^(2),-2x-1,-1-x):}|=0`
Here `1^(st)` row and `3^(rd)` row are the same
`implies` This is an identity
`implies` Infinite roots