Moderate -0.8 This is a straightforward Further Maths question testing basic recall of transformation matrices. Part (i) requires knowing the standard shear matrix form and simple substitution. Part (ii) is immediate recognition of an enlargement. Part (iii) involves sketching a transformation and stating that determinant gives area scale factor—all standard textbook exercises with no problem-solving or novel insight required.
Write down the matrix, \(\mathbf { A }\), that represents a shear with \(x\)-axis invariant in which the image of the point \(( 1,1 )\) is \(( 4,1 )\).
The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { c c } \sqrt { 3 } & 0 \\ 0 & \sqrt { 3 } \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { B }\).
The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 2 & 6 \\ 0 & 2 \end{array} \right)\).
Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\).
Write down the determinant of \(\mathbf { C }\) and explain briefly how this value relates to the transformation represented by \(\mathbf { C }\).
8 The quadratic equation \(2 x ^ { 2 } - x + 3 = 0\) has roots \(\alpha\) and \(\beta\), and the quadratic equation \(x ^ { 2 } - p x + q = 0\) has roots \(\alpha + \frac { 1 } { \alpha }\) and \(\beta + \frac { 1 } { \beta }\).
Show that \(p = \frac { 5 } { 6 }\).
Find the value of \(q\).
9 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1 \\ 3 & a & 1 \\ 4 & 2 & 1 \end{array} \right)\).
Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
Hence find the values of \(a\) for which \(\mathbf { M } ^ { - 1 }\) does not exist.
Determine whether the simultaneous equations
$$\begin{aligned}
& 6 x - 6 y + z = 3 k \\
& 3 x + 6 y + z = 0 \\
& 4 x + 2 y + z = k
\end{aligned}$$
where \(k\) is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
Show that \(\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }\).
Hence find an expression, in terms of \(n\), for
$$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
Show that \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }\).
(i) Write down the matrix, $\mathbf { A }$, that represents a shear with $x$-axis invariant in which the image of the point $( 1,1 )$ is $( 4,1 )$.\\
(ii) The matrix $\mathbf { B }$ is given by $\mathbf { B } = \left( \begin{array} { c c } \sqrt { 3 } & 0 \\ 0 & \sqrt { 3 } \end{array} \right)$. Describe fully the geometrical transformation represented by $\mathbf { B }$.\\
(iii) The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { l l } 2 & 6 \\ 0 & 2 \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { C }$.
\item Write down the determinant of $\mathbf { C }$ and explain briefly how this value relates to the transformation represented by $\mathbf { C }$.
8 The quadratic equation $2 x ^ { 2 } - x + 3 = 0$ has roots $\alpha$ and $\beta$, and the quadratic equation $x ^ { 2 } - p x + q = 0$ has roots $\alpha + \frac { 1 } { \alpha }$ and $\beta + \frac { 1 } { \beta }$.
\begin{enumerate}[label=(\roman*)]
\item Show that $p = \frac { 5 } { 6 }$.
\item Find the value of $q$.
9 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1 \\ 3 & a & 1 \\ 4 & 2 & 1 \end{array} \right)$.
\begin{enumerate}[label=(\roman*)]
\item Find, in terms of $a$, the determinant of $\mathbf { M }$.
\item Hence find the values of $a$ for which $\mathbf { M } ^ { - 1 }$ does not exist.
\item Determine whether the simultaneous equations
$$\begin{aligned}
& 6 x - 6 y + z = 3 k \\
& 3 x + 6 y + z = 0 \\
& 4 x + 2 y + z = k
\end{aligned}$$
where $k$ is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.\\
(i) Show that $\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$.\\
(ii) Hence find an expression, in terms of $n$, for
$$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
\item Show that $\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }$.
\end{enumerate}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2011 Q7 [9]}}