Q45
If the equation x3 – ax2 + bx – a = 0 has three real roots, then it must be the case that,
Answer: B
x3 – ax2 + bx – a = x3 + bx – ax2 – a = x(x2 + b) – a(x2 + 1)
∴ x(x2 + b) – a(x2 + 1) = 0
If b = 1 then x(x2 + 1) – a(x2 + 1) = (x – a)(x2 + 1) = 0
∴ (x – a) = 0 or (x2 + 1) = 0
But then (x2 + 1) = 0 does not have real roots.
∴ b ≠ 1
Hence, option (b).