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Q1
Answer
Three planes have equations $$\begin{align*}x - y + 2z &= 4 \\ 2x + y - z &= 2 \\ 3x + py + qz &= r\end{align*}$$ where $p$, $q$ and $r$ are constants.
  1. Show that the planes never meet at a unique point when $p = 2$ and $q = -1$, regardless of the value of $r$.
    1. For $p = 2$, $q = -1$, find the value of $r$ such that the planes form a sheaf.
      1. Find the point of intersection of the planes when $p = 0$, $q = 1$ and $r = 10$.
      1. Determinant is zero
      2. $r = 6$
      3. $(1, -1, 2)$
      Q2
      Answer
      Three planes have equations $$\begin{align*}x + 3y - 2z &= a \\ 2x - y + z &= 3 \\ 5x + y + \lambda z &= 7\end{align*}$$ where $\lambda$ and $a$ are constants.
      1. Find the value of $\lambda$ for which the system does not have a unique solution.
        1. For this value of $\lambda$, determine the value of $a$ for which the planes form a sheaf.
          1. Use a matrix method to solve the system when $\lambda = 0$ and $a = 1$.
          1. $\lambda = -4$
          2. $a = 1$. Line: $\mathbf{r} = \begin{pmatrix}2\\0\\\frac{1}{2}\end{pmatrix} + t\begin{pmatrix}1\\-5\\-7\end{pmatrix}$
          3. $\left(\dfrac{2}{3},\, \dfrac{1}{3},\, 1\right)$
          Q3
          Answer
          A transformation of the plane is represented by the matrix $\mathbf{M} = \begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}$.
          1. Find the invariant lines of the transformation through the origin.
            1. Determine whether either invariant line is a line of invariant points.
              1. $y = -2x$ and $y = x$
              2. Neither.
              Q4
              Answer
              A transformation of the plane is represented by the matrix $\mathbf{M} = \begin{pmatrix} 4 & -3 \\ 2 & -1 \end{pmatrix}$.
              1. Find the line of invariant points of the transformation.
                1. Find all other invariant lines.
                  1. $y = x$
                  2. $y = \dfrac{2}{3}x + c$
                  Q5
                  Answer
                  A transformation $T$ of the plane has associated matrix $\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$. Show that the line $y = x + c$ is invariant under $T$ regardless of the value of $c$.
                    Any value of $c$ works
                    Q6
                    Answer
                    The matrix $\mathbf{M} = \begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix}$ represents a transformation.
                    1. Find the line of invariant points.
                      1. Show that every line parallel to the $y$-axis is an invariant line.
                        1. Describe the transformation represented by $M$.
                        1. $x = 0$
                        2. See hint
                        Q7
                        Answer
                        $$\mathbf{M} = \begin{pmatrix} k & 2 \\ 3 & k-1 \end{pmatrix}$$ Find the values of $k$ for which $\mathbf{M}$ has a line of invariant points.
                          $k = 4$ or $k = -1$
                          Q8
                          Answer
                          $$\mathbf{M} = \begin{pmatrix} 2k & k+1 \\ 1 & k \end{pmatrix}$$ Find the values of $k$ for which $\mathbf{M}$ has a line of invariant points, and for each value find the equation of the line.
                            $k = 1$ $y = x$, or $k = -\dfrac{1}{2}$ $y = -2x$