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\lhead{Geometry of Plane Intersections}
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\rhead{\href{https://danieljsmith.org}{danieljsmith.org}}
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\begin{document}
\vspace*{20pt}
\tableofcontents
\newpage
\begin{enumerate}[
  label=\textbf{\arabic*.},
  leftmargin=*,
  topsep=0pt,
  partopsep=0pt,
  parsep=0pt
]

% ---- Question 1 ----
\questionitem
% [Source: _QBT__Plane_Intersections, Q1]

Three planes have equations
\[
\begin{aligned}
x-y+2z&=3\\
2x+y+z&=4\\
3x+(m+1)z&=7
\end{aligned}
\] \\
where $m$ is a constant. \\
\begin{enumerate}[label=(\roman*)]
  \item Investigate the arrangement of the planes:

  when $m=2$;

  when $m\ne 2$. \marks{6} \\
  \item A student claims that the position vectors $\mathbf{i}-\mathbf{j}+2\mathbf{k}$, $2\mathbf{i}+\mathbf{j}+\mathbf{k}$ and $3\mathbf{i}+4\mathbf{k}$ all lie in a plane through the origin. Determine whether or not the student is correct. \marks{2} \\
\end{enumerate}

\newpage

% ---- Question 2 ----
\questionitem
% [Source: _QBT__Plane_Intersections, Q2]

\begin{enumerate}[label=\textbf{(\alph*)}]
  \item Consider the linear system $A\mathbf{x}=\mathbf{b}$, where
  \[
  A=\begin{pmatrix}
  3&-2&1\\
  2&1&-4\\
  7&0&-7
  \end{pmatrix},
  \qquad
  \mathbf{x}=\begin{pmatrix}x\\y\\z\end{pmatrix},
  \qquad
  \mathbf{b}=\begin{pmatrix}5\\1\\7\end{pmatrix}
  \]
  Show that $A$ is singular, and deduce that the system does not have a unique solution. \marks{2}\\
  
  \item Show that the system is consistent. Interpret the solution set geometrically. \marks{3} \\
\end{enumerate}

\newpage

% ---- Question 3 ----
\questionitem
% [Source: _QBT__Plane_Intersections, Q3]

\begin{enumerate}[label=\textbf{(\alph*)}]
  \item Show that the system of equations
  \[
  \begin{aligned}
  x+2y-z&=3\\
  2x-y+4z&=1\\
  3x+y+3z&=a
  \end{aligned}
  \]
  where $a$ is a constant, does not have a unique solution. \marks{2}\\
  \item Given that $a=4$, show that the system of equations in part (a) is consistent. Interpret this situation geometrically. \marks{3} \\
  \item Given instead that $a\ne 4$, show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically. \marks{2} \\
\end{enumerate}

\newpage

% ---- Question 4 ----
\questionitem
% [Source: _QBT__Plane_Intersections, Q4]

The equations
\[
\begin{aligned}
x-y+2z&=4\\
(k+1)x+ky+(2-k)z&=4-k\\
(5-2k)x+(2-2k)y+(k+2)z&=8-k
\end{aligned}
\] \\
represent three planes, where $k$ is a real constant. \\
The planes do not intersect in exactly one point. \\
\begin{enumerate}[label=\textbf{(\alph*)}]
  \item Find the possible values of $k$. \marks{3} \\
  \item For each value of $k$ found in part (a), describe the configuration of the planes. \\
  Fully justify your answer, stating in each case whether the corresponding system is consistent or inconsistent. \marks{5} \\
\end{enumerate}

\newpage

% ---- Question 5 ----
\questionitem
% [Source: _QBT__Plane_Intersections, Q5]

Three planes have equations
\[
\begin{aligned}
x+ay&=-1\\
2x+y+z&=3\\
3x+by+2z&=c
\end{aligned}
\] \\
where $a$, $b$ and $c$ are constants. \\
\begin{enumerate}[label=\textbf{(\alph*)}]
  \item In the case where the planes \textbf{do not} intersect at a unique point,
  \begin{enumerate}[label=(\roman*)]
    \item find $b$ in terms of $a$ \marks{4}
    \item find the value of $c$ for which the planes form a sheaf. \marks{3} \\
  \end{enumerate}
  \item In the case where $b=1-a$ and $c=5$, find the coordinates of the point of intersection of the planes in terms of $a$. \marks{6} \\
\end{enumerate}

\newpage

% ---- Question 6 ----
\questionitem
% [Source: _QBT__Plane_Intersections, Q6]

\[
\mathbf{M}=\begin{pmatrix}
1&1&0\\
3&k&1\\
2&-1&1
\end{pmatrix}
\] \\
where $k$ is a constant. \\
\begin{enumerate}[label=\textbf{(\alph*)}]
  \item Find the values of $k$ for which the matrix $\mathbf{M}$ has an inverse. \marks{2} \\
  \item Find, in terms of $p$, the coordinates of the point where the following planes intersect:
  \[
  \begin{aligned}
  x+y&=1\\
  3x+2y+z&=p\\
  2x-y+z&=4
  \end{aligned}
  \]
  \marks[-20pt]{5}
  \item
  \begin{enumerate}[label=(\roman*)]
    \item Find the value of $q$ for which the set of simultaneous equations
    \[
    \begin{aligned}
    x+y&=1\\
    3x+z&=q\\
    2x-y+z&=4
    \end{aligned}
    \]
    has at least one solution. \marks[-20pt]{2}
    \item For this value of $q$, interpret the solution set geometrically. \marks{2} \\
  \end{enumerate}
\end{enumerate}

\newpage

% ---- Question 7 ----
\questionitem
% [Source: _QBT__Plane_Intersections, Q7]

You are given that $a$ is a parameter which can take only real values. \\
The matrix $A$ is given by
\[
A=\begin{pmatrix}
3&2&a\\
1&3&0\\
2&-1&1
\end{pmatrix}
\] \\
\begin{enumerate}[label=\textbf{(\alph*)}]
  \item Find an expression for the determinant of $A$ in terms of $a$. \marks{2} \\
\end{enumerate}
You are given the following system of equations in $x$, $y$ and $z$.
\[
\begin{aligned}
3x+2y+az&=8\\
x+3y&=5\\
2x-y+z&=3
\end{aligned}
\] \\
The system can be written in the form
\[
A\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}8\\5\\3\end{pmatrix}
\] \\
\begin{enumerate}[label=\textbf{(\alph*)}, resume]
  \item
  \begin{enumerate}[label=(\roman*)]
    \item In the case where $A$ is not singular, solve the given system by using $A^{-1}$. \marks{5}
    \item In the case where $A$ is singular, describe the configuration of the planes whose equations are the three equations of the system. \marks{3} \\
  \end{enumerate}
\end{enumerate}
The transformation represented by $A$ is denoted by $T$. \\
A 3-D solid of volume $a^2+3$ is transformed by $T$ to an image. \\
\begin{enumerate}[label=\textbf{(\alph*)}, resume]
  \item
  \begin{enumerate}[label=(\roman*)]
    \item Determine the range of values of $a$ for which the orientation of the image is the reverse of the orientation of the object. \marks{1}
    \item Determine the range of values of $a$ for which the volume of the image is less than the volume of the object. \marks{2} \\
  \end{enumerate}
\end{enumerate}
\end{enumerate}
\end{document}
