The equation `ax^(4)-2x^(2)-(a-1)=0` will have real and unequal roots if

Options

A. `o lt a lt 1`
B. `a gt 0`, `a ne 1`
C. `a lt 0` , `a ne 1`
D. none of these

Answer

A

Explanation

`(a)` Putting `x^(2)=y`, the given equation in `x` reduces to
`ay^(2)-2y-(a-1)=0` ……….`(i)`
The given biquardratic equation will have four real and distinct roots, if the quadratic equation `(i)` has two distinct and positive roots.
For that, we must have
`D gt 0impliesa^(2)-a+1 gt 0`, which is true `AA a in R`
Product of roots `gt 0implies0 lt a lt 1`
Sum of roots `gt 0 implies a gt 0`
Hence, the acceptable values of `a` are `0 lt a lt 1`.