Past Exam Questions

The linear transformation \(\mathrm{T}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is represented by the matrix \(\mathbf{M}\), where
\(\mathbf{M}=\left(\begin{array}{rrrr} 1 & -3 & -1 & 2 \\ 4 & -10 & 0 & 2 \\ 1 & -1 & 3 & -4 \\ 5 & -12 & 1 & 1 \end{array}\right) .\)

Find, in either order, the rank of \(\mathbf{M}\) and a basis for the null space \(K\) of T .

Evaluate
\(\mathbf{M}\left(\begin{array}{r} 1 \\ -2 \\ -3 \\ -4 \end{array}\right),\)
and hence show that every solution of
\(\mathbf{M} \mathbf{x}=\left(\begin{array}{r} 2 \\ 16 \\ 10 \\ \end{array}\right)\)
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 \(K\).

The linear transformation \(\mathrm{T}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is represented by the matrix \(\mathbf{M}=\left(\begin{array}{rrrr}1 & 3 & -2 & 4 \\ 5 & 15 & -9 & 19 \\ -2 & -6 & 3 & -7 \\ 3 & 9 & -5 & 11\end{array}\right)\).
(i) Find the rank of \(\mathbf{M}\).

(ii) Obtain a basis for the null space of T .

The linear transformation \(\mathrm{T}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is represented by the matrix \(\mathbf{M}\), where
\(\mathbf{M}=\left(\begin{array}{rrrr} 2 & -1 & 1 & 3 \\ 2 & 0 & 0 & 5 \\ 6 & -2 & 2 & 11 \\ 10 & -3 & 3 & 19 \end{array}\right) .\)
(i) Find the rank of \(\mathbf{M}\) and state a basis for the range space of T .

(ii) Obtain a basis for the null space of T .

The linear transformation \(\mathrm{T}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is represented by the matrix \(\mathbf{M}\), where
\(\mathbf{M}=\left(\begin{array}{rrrr} 3 & 4 & 2 & 5 \\ 6 & 7 & 5 & 8 \\ 9 & 9 & 9 & 9 \\ 15 & 16 & 14 & 17 \end{array}\right) .\)

Find
(i) the rank of \(\mathbf{M}\) and a basis for the range space of T ,

(ii) a basis for the null space of T .

Answer only one of the following two alternatives.

EITHER
By considering \(\sum_{k=0}^{n-1}(1+\mathrm{i} \tan \theta)^{k}\), show that
\(\sum_{k=0}^{n-1} \cos k \theta \sec ^{k} \theta=\cot \theta \sin n \theta \sec ^{n} \theta,\)
provided \(\theta\) is not an integer multiple of \(\frac{1}{2} \pi\).

Hence or otherwise show that
\(\sum_{k=0}^{n-1} 2^{k} \cos \left(\frac{1}{3} k \pi\right)=\frac{2^{n}}{\sqrt{ } 3} \sin \left(\frac{1}{3} n \pi\right) .\)

Given that \(0\lt x\lt 1\), show that
\(\sum_{k=0}^{n-1} \frac{\cos \left(k \cos ^{-1} x\right)}{x^{k}}=\frac{\sin \left(n \cos ^{-1} x\right)}{x^{n-1} \sqrt{ }\left(1-x^{2}\right)} .\)

OR
The linear transformations \(\mathrm{T}_{1}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) and \(\mathrm{T}_{2}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) are represented by the matrices \(\mathbf{M}_{1}\) and \(\mathbf{M}_{2}\), respectively, where
\(\mathbf{M}_{1}=\left(\begin{array}{rrrr} 1 & 1 & 1 & 2 \\ 1 & 4 & 7 & 8 \\ 1 & 7 & 11 & 13 \\ 1 & 2 & 5 & 5 \end{array}\right), \quad \mathbf{M}_{2}=\left(\begin{array}{rrrr} 2 & 0 & -1 & -1 \\ 5 & 1 & -3 & -3 \\ 3 & -1 & -1 & -1 \\ 13 & -1 & -6 & -6 \end{array}\right) .\)
(i) Find a basis for \(R_{1}\), the range space of \(\mathrm{T}_{1}\).

(ii) Find a basis for \(K_{2}\), the null space of \(\mathrm{T}_{2}\), and hence show that \(K_{2}\) is a subspace of \(R_{1}\).

The set of vectors which belong to \(R_{1}\) but do not belong to \(K_{2}\) is denoted by \(W\).
(iii) State whether \(W\) is a vector space, justifying your answer.

The linear transformation \(\mathrm{T}_{3}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is the result of applying \(\mathrm{T}_{1}\) and then \(\mathrm{T}_{2}\), in that order.
(iv) Find the dimension of the null space of \(\mathrm{T}_{3}\).

Answer only one of the following two alternatives.

EITHER
The line \(l_{1}\) passes through the point \(A\) whose position vector is \(3 \mathbf{i}+\mathbf{j}+2 \mathbf{k}\) and is parallel to the vector \(\mathbf{i}+\mathbf{j}\). The line \(l_{2}\) passes through the point \(B\) whose position vector is \(-\mathbf{i}-\mathbf{k}\) and is parallel to the vector \(\mathbf{j}+2 \mathbf{k}\). The point \(P\) is on \(l_{1}\) and the point \(Q\) is on \(l_{2}\) and \(P Q\) is perpendicular to both \(l_{1}\) and \(l_{2}\).
(i) Find the length of \(P Q\).

(ii) Find the position vector of \(Q\).

(iii) Show that the perpendicular distance from \(Q\) to the plane containing \(A B\) and the line \(l_{1}\) is \(\sqrt{ } 3\).

OR
The linear transformation \(\mathrm{T}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is represented by the matrix \(\mathbf{M}=\left(\begin{array}{rrrr}1 & 1 & 5 & 7 \\ 3 & 9 & 17 & 25 \\ 1 & 7 & 7 & 11 \\ 3 & 6 & 16 & 23\end{array}\right)\).
(i) In either order,
(a) show that the dimension of \(R\), the range space of T , is equal to 2 ,
(b) obtain a basis for \(R\).

(ii) Show that the vector \(\left(\begin{array}{r}1 \\ -15 \\ -17 \\ -6\end{array}\right)\) belongs to \(R\).

(iii) It is given that \(\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\}\) is a basis for the null space of T , where \(\mathbf{e}_{1}=\left(\begin{array}{r}14 \\ 1 \\ -3 \\ 0\end{array}\right)\) and \(\mathbf{e}_{2}=\left(\begin{array}{r}19 \\ 2 \\ 0 \\ -3\end{array}\right)\). Show that, for all \(\lambda\) and \(\mu\),
\(\mathbf{x}=\left(\begin{array}{r} 4 \\ -3 \\ 0 \\ \end{array}\right)+\lambda \mathbf{e}_{1}+\mu \mathbf{e}_{2}\)
is a solution of
\(\mathbf{M x}=\left(\begin{array}{r} 1 \\ -15 \\ -17 \\ -6 \end{array}\right) .\)
(iv) Hence find a solution of \((*)\) of the form \(\left(\begin{array}{l}\alpha \\ 0 \\ \gamma \\ \delta\end{array}\right)\).

(a) Solve the equation \(5|2x-1|+8=23\).

(b) On the axes, sketch the graph of \(y=5\sin x-2\) for \(0^\circ\leqslant x\leqslant360^\circ\).

Solve the equation \(4|7x-3|-5=9\).

Solve the equation

\(|4x+9|=|6-5x|.\)

Solve the equation

\(|3x-2|=4+x.\)

Solve

\(|3x+2|=x+4.\)

Solve the equation

\(|5x-3|=-3x+13.\)

Solve \(\lvert 5x+3\rvert=\lvert 1-3x\rvert\).

Solve the equation \(|3x-1|=|5+x|\).

The diagram shows the graph of \(y=|3x+3|\).

Use a graphical method to solve the inequality \(|3x+3|\geqslant |x-2|\).

Solve the following inequalities.

(a) \(x^2-x-6\geqslant0\)

(b) \(|3x-4|\lt x+2\)

Solve the inequality \(|5 x+2| \geqslant 3\).

(a) On the axes, sketch the graphs of \(y=4|x-1|\) and \(y=|3x+2|\), stating the intercepts with the axes.

(b) Solve the inequality \(4|x-1|\leqslant |3x+2|\).

Use a graphical method to solve the inequality \(|2 x-8|\gt 4\).

(a) Solve the equation \(2|8-4 x|+5=25\).

(b) Solve the inequality \(16 x-5 x^{2}-3\lt \frac{57-9 x}{6}\).