IAL 2026 Jan FP1 Q2 | 國際數學站

Question

[!problem]

$f(z) = z^4 - \lambda z^3 + 14z^2 - 9\lambda z + 45$

(a) Show that $f(3i) = 0$ .

(2)

(b) Hence determine the exact range of values of $\lambda$ for which the equation $f(z) = 0$ has no real solutions.

(5)

中文翻译

$f(z) = z^4 - \lambda z^3 + 14z^2 - 9\lambda z + 45$

(a) 证明 $f(3i) = 0$ 。

(2)

(b) 由此确定 $\lambda$ 的精确取值范围，使得方程 $f(z) = 0$ 没有实数解。

(5)