Further Mathematics 9231: Roots of polynomial equations Topical Past Paper Questions

(a) Use de Moivre's theorem to show that $$ \sec7\theta=\frac{\sec^{7}\theta}{64-112\sec^{2}\theta+56\sec^{4}\theta-7\sec^{6}\theta}. $$ (b) Hence obtain the roots of the equation $$ x^{7}-14x^{6}+112x^{4}-224x^{2}+128=0 $$ in the form $ \sec q\pi $, where q is rational. [5]

The cubic equation $ x^3 + bx^2 + cx + d = 0 $, where $ d \ne 0 $, has roots $ \alpha, \beta, \gamma $ such that $ \gamma = \frac{1}{\beta} $. (a) Show that $ \beta + \frac{1}{\beta} = d - b $. (b) Show also that $ \beta + \frac{1}{\beta} = \frac{1 - c}{d} $. (c) It is given that…

The matrix A is given by $$ \mathbf{A}=\begin{pmatrix}{{{-11}}}&{{{1}}}&{{{8}}} \\{{{0}}}&{{{-2}}}&{{{0}}} \\{{{-16}}}&{{{1}}}&{{{13}}}\end{pmatrix}. $$ (a) Show that $ \begin{pmatrix}1\\1\\1\end{pmatrix} $ is an eigenvector of A and state the corresponding eigenvalue. [2] (b) Sh…