16 \\
10 \\
22
\end{array} \right)$$

has the form

$$\mathbf { x } = \left( \begin{array} { r } 
1 \\
- 2 \\
- 3 \\
- 4
\end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$

where $\lambda$ and $\mu$ are real numbers and $\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}$ is a basis for $K$.

7 The square matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$ and state the corresponding eigenvalue.

Find the eigenvalues of the matrix $\mathbf { B }$, where

$$\mathbf { B } = \left( \begin{array} { l l l } 
1 & 3 & 0 \\
2 & 0 & 2 \\
1 & 1 & 2
\end{array} \right)$$

Find the eigenvalues of $\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix.

8 The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)$. Find a cartesian equation of $\Pi _ { 1 }$.

The plane $\Pi _ { 2 }$ has equation $2 x - y + z = 10$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.

Find an equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$.

9 The curve $C$ has parametric equations

$$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$

Find the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis.

Find the coordinates of the centroid of the region bounded by $C$, the $x$-axis and the line $x = 1$.

10 The curve $C$ has equation

$$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 } ,$$

where $p$ is a positive constant and $p \neq 3$.\\
(i) Obtain the equations of the asymptotes of $C$.\\
(ii) Find the value of $p$ for which the $x$-axis is a tangent to $C$, and sketch $C$ in this case.\\
(iii) For the case $p = 1$, show that $C$ has no turning points, and sketch $C$, giving the exact coordinates of the points of intersection of $C$ with the $x$-axis.

11 Answer only one of the following two alternatives.

\section*{EITHER}
State the fifth roots of unity in the form $\cos \theta + \mathrm { i } \sin \theta$, where $- \pi < \theta \leqslant \pi$.

Simplify

$$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right)$$

Hence find the real factors of

$$x ^ { 5 } - 1$$

Express the six roots of the equation

$$x ^ { 6 } - x ^ { 3 } + 1 = 0$$

as three conjugate pairs, in the form $\cos \theta \pm \mathrm { i } \sin \theta$.

Hence find the real factors of

$$x ^ { 6 } - x ^ { 3 } + 1$$

OR

Given that

$$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$

and that $v = y ^ { 3 }$, show that

$$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$

Find the particular solution for $y$ in terms of $x$, given that when $x = 0 , y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$.

\hfill \mbox{\textit{CAIE FP1 2013 Q16}}