Past Exam Questions

The vectors \(\mathbf a\), \(\mathbf b\), \(\mathbf c\) and \(\mathbf d\) in \(\mathbb R^3\) are given by

\(\mathbf a=\begin{pmatrix}1\\2\\1\end{pmatrix},\quad \mathbf b=\begin{pmatrix}2\\9\\0\end{pmatrix},\quad \mathbf c=\begin{pmatrix}3\\3\\4\end{pmatrix},\quad \mathbf d=\begin{pmatrix}0\\-8\\3\end{pmatrix}.\)

(i) Show that \(\{\mathbf a,\mathbf b,\mathbf c\}\) is a basis for \(\mathbb R^3\).

(ii) Express \(\mathbf d\) in terms of \(\mathbf a\), \(\mathbf b\) and \(\mathbf c\).

3 The lines \(l_{1}\) and \(l_{2}\) have equations \(\mathbf{r}=6 \mathbf{i}+2 \mathbf{j}+7 \mathbf{k}+\lambda(\mathbf{i}+\mathbf{j})\) and \(\mathbf{r}=4 \mathbf{i}+4 \mathbf{j}+\mu(-6 \mathbf{j}+\mathbf{k})\) respectively. The point \(P\) on \(l_{1}\) and the point \(Q\) on \(l_{2}\) are such that \(P Q\) is perpendicular to both \(l_{1}\) and \(l_{2}\). Find the position vectors of \(P\) and \(Q\).

The line \(l_{1}\) passes through the point with position vector \(8 \mathbf{i}+8 \mathbf{j}-7 \mathbf{k}\) and is parallel to the vector \(4 \mathbf{i}+3 \mathbf{j}\). The line \(l_{2}\) passes through the point with position vector \(7 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}\) and is parallel to the vector \(4 \mathbf{i}-\mathbf{k}\). The point \(P\) on \(l_{1}\) and the point \(Q\) on \(l_{2}\) are such that \(P Q\) is perpendicular to both \(l_{1}\) and \(l_{2}\). In either order,
(i) show that \(P Q=13\),
(ii) find the position vectors of \(P\) and \(Q\).

Relative to an origin \(O\), the points \(A, B, C\) have position vectors
\(\mathbf{i}, \quad \mathbf{j}+\mathbf{k}, \quad \mathbf{i}+\mathbf{j}+\theta \mathbf{k}\),
respectively. The shortest distance between the lines \(A B\) and \(O C\) is \(\frac{1}{\sqrt{2}}\). Find the value of \(\theta\).

The matrix \(\mathbf{M}\) is given by \(\mathbf{M} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\), where \(0 < \theta < 2\pi\).

(a) The matrix \(\mathbf{M}\) represents a sequence of two geometrical transformations in the \(x-y\) plane. State the type of each transformation, and make clear the order in which they are applied.

(b) Find the value of \(\theta\) for which the transformation represented by \(\mathbf{M}\) has a line of invariant points.

The points A, B, C have position vectors

\(\mathbf{i} - 2\mathbf{k}, \quad \mathbf{i} + 2\mathbf{j} + 2\mathbf{k}, \quad 2\mathbf{i} - \mathbf{j} - \mathbf{k},\)

respectively.

(a) Find the equation of the plane ABC, giving your answer in the form \(ax + by + cz = d\).

A point D has position vector \(\mathbf{i} + t\mathbf{k}\), where \(t \neq -2\).

(b) Find the acute angle between the planes ABC and ABD.

(c) Find the values of \(t\) such that the shortest distance between the lines AB and CD is \(\sqrt{2}\).

The matrix M is given by \(\mathbf{M} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\), where \(0 < \theta < 2\pi\).

(a) The matrix M represents a sequence of two geometrical transformations in the x–y plane. State the type of each transformation, and make clear the order in which they are applied.

(b) Find the value of \(\theta\) for which the transformation represented by M has a line of invariant points.

The points A, B, C have position vectors \(\mathbf{i} - 2\mathbf{k}, \mathbf{i} + 2\mathbf{j} + 2\mathbf{k}, 2\mathbf{i} - \mathbf{j} - \mathbf{k}\), respectively.

(a) Find the equation of the plane ABC, giving your answer in the form \(ax + by + cz = d\).

A point D has position vector \(\mathbf{i} + t\mathbf{k}\), where \(t \neq -2\).

(b) Find the acute angle between the planes ABC and ABD.

(c) Find the values of \(t\) such that the shortest distance between the lines AB and CD is \(\sqrt{2}\).

The matrix M represents the sequence of two transformations in the x-y plane given by a stretch parallel to the x-axis, scale factor 14, followed by a rotation anticlockwise about the origin through angle \(\frac{1}{3} \pi\).

(a) Show that \(2\mathbf{M} = \begin{pmatrix} 14 & -\sqrt{3} \\ 14\sqrt{3} & 1 \end{pmatrix}\).

(b) Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane represented by M.

The unit square S in the x-y plane is transformed by M onto the rectangle P.

(c) Find the matrix which transforms P onto S.

The plane \(\Pi\) has equation \(\mathbf{r} = 2\mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) + \mu (3\mathbf{i} + 2\mathbf{j} - 2\mathbf{k})\).

(a) Find a Cartesian equation of \(\Pi\), giving your answer in the form \(ax + by + cz = d\).

The point \(P\) has position vector \(4\mathbf{i} + 2\mathbf{j} + 9\mathbf{k}\).

(b) Find the position vector of the foot of the perpendicular from \(P\) to \(\Pi\).

The line \(l\) is parallel to the vector \(3\mathbf{i} + 5\mathbf{j} - \mathbf{k}\).

(c) Find the acute angle between \(l\) and \(\Pi\).

The points A, B and C have position vectors

\(2\mathbf{j} + 3\mathbf{k}, \quad -5\mathbf{i} + 3\mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{i} + 2\mathbf{j} + 5\mathbf{k}\)

respectively, relative to the origin O.

(a) Find the equation of the plane ABC, giving your answer in the form \(ax + by + cz = d\).

(b) Find the perpendicular distance from O to the plane ABC.

(c) Find the acute angle between the line OA and the plane ABC.

The matrix M represents a sequence of two transformations in the x-y plane given by a one-way stretch in the x-direction, scale factor 3, followed by a reflection in the line y = x.

(a) Find M.

(b) Give full details of the geometrical transformation in the x-y plane represented by M^-1.

The matrix N is such that MN = \(\begin{pmatrix} 1 & 2 \\ 3 & 2 \end{pmatrix}\).

(c) Find N.

\((d) Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane represented by MN.\)

The matrix M represents the sequence of two transformations in the x-y plane given by a stretch parallel to the x-axis, scale factor k (k ≠ 0), followed by a shear, x-axis fixed, with (0, 1) mapped to (k, 1).

(a) Show that M = \(\begin{pmatrix} k & k \\ 0 & 1 \end{pmatrix}\).

\((b) The transformation represented by M has a line of invariant points. Find, in terms of k, the equation of this line.\)

The unit square S in the x-y plane is transformed by M onto the parallelogram P.

(c) Find, in terms of k, a matrix which transforms P onto S.

(d) Given that the area of P is \(3k^2\) units\(^2\), find the possible values of k.

The lines \(l_1\) and \(l_2\) have equations \(\mathbf{r} = \mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + \mathbf{k})\) and \(\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 9\mathbf{k} + \mu(\mathbf{i} - 4\mathbf{j} + 2\mathbf{k})\) respectively. The plane \(\Pi_1\) contains \(l_1\) and is parallel to \(l_2\).

(a) Find the equation of \(\Pi_1\), giving your answer in the form \(ax + by + cz = d\).

The plane \(\Pi_2\) contains \(l_2\) and the point with coordinates \((2, -1, 7)\).

(b) Find the acute angle between \(\Pi_1\) and \(\Pi_2\).

The point \(P\) on \(l_1\) and the point \(Q\) on \(l_2\) are such that \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).

(c) Find a vector equation for \(PQ\).

The line \(l_1\) has equation \(\mathbf{r} = \mathbf{i} + 3\mathbf{j} - \mathbf{k} + \lambda (\mathbf{i} - \mathbf{j} - 4\mathbf{k})\).

The plane \(\Pi\) contains \(l_1\) and is parallel to the vector \(2\mathbf{i} + 5\mathbf{j} - 4\mathbf{k}\).

(a) Find the equation of \(\Pi\), giving your answer in the form \(ax + by + cz = d\).

The line \(l_2\) is parallel to the vector \(5\mathbf{i} - 5\mathbf{j} - 2\mathbf{k}\).

(b) Find the acute angle between \(l_2\) and \(\Pi\).

The matrices A, B and C are given by

\(A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{pmatrix}, \ B = \begin{pmatrix} 0 & -2 \\ -1 & 3 \\ 0 & 0 \end{pmatrix} \text{ and } C = \begin{pmatrix} -2 & -1 \\ 1 & 1 \end{pmatrix}.\)

(a) Show that \(CAB = \begin{pmatrix} 3 & -7 \\ -9 & 3 \end{pmatrix}.\) [3]

(b) Find the equations of the invariant lines, through the origin, of the transformation in the \(x-y\) plane represented by \(CAB.\) [5]

Let \(M = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}.\)

(c) Give full details of the transformation represented by \(M.\) [2]

(d) Find the matrix \(N\) such that \(NM = CAB.\) [3]

The matrix M represents the sequence of two transformations in the x-y plane given by a stretch parallel to the x-axis, scale factor k (k \neq 0), followed by a shear, x-axis fixed, with (0, 1) mapped to (k, 1).

(a) Show that M = \(\begin{pmatrix} k & k \\ 0 & 1 \end{pmatrix}\).

\((b) The transformation represented by M has a line of invariant points. Find, in terms of k, the equation of this line.\)

The unit square S in the x-y plane is transformed by M onto the parallelogram P.

(c) Find, in terms of k, a matrix which transforms P onto S.

(d) Given that the area of P is \(3k^2\) units\(^2\), find the possible values of k.