| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2018 |
| Session | March |
| Marks | 10 |
| Topic | Linear transformations |
| Type | Area scale factor from determinant |
| Difficulty | Standard +0.3 This is a straightforward Further Pure 1 question involving matrix multiplication, determinant conditions for singularity, and applying the standard result that |det(M)| gives area scale factor while sign indicates orientation. All parts are routine applications of well-practiced techniques with no novel problem-solving required, making it slightly easier than average even for FP1. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03i Determinant: area scale factor and orientation4.03l Singular/non-singular matrices |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\begin{pmatrix} 1 & 2 \\ 1 & a \end{pmatrix}\begin{pmatrix} 2 & 1 \\ -1 & b \end{pmatrix} = \begin{pmatrix} 1 \times 2 + 2 \times -1 & 1 \times 1 + 2 \times b \\ 1 \times 2 + a \times -1 & 1 \times 1 + a \times b \end{pmatrix}\) | M1 | Must be right way round. Must be correct matrix multiplication (ie rows into columns) |
| \(\begin{pmatrix} 0 & 1 + 2b \\ 2 - a & 1 + ab \end{pmatrix}\) | A1 | Must be in exactly the correct form. |
| [2] | ||
| (ii) \(\begin{vmatrix} 0 & 1 + 2b \\ 2 - a & 1 + ab \end{vmatrix} = -(1 + 2b)(2 - a)\) | B1ft | Could be from detA × detB probably as \((a - 2)(2b + 1)\) |
| \(0 = -(1 + 2b)(2 - a)\) | M1ft | Knowledge that singular means zero determinant (could be stated) |
| (Either) \(a = 2\) | A1 | If no "or" then A1 for 1 or 2 of \(a\) and \(b\) correct. |
| or \(b = -\frac{1}{2}\) | A1 | |
| [4] | ||
| (iii)(a) Area is multiplied by 3 | E1 | Not \(-3\) |
| P'Q'R'S' run anticlockwise oe (eg orientation is reversed) | E1 | Must mention orientation or handedness or clockwise or anticlockwise |
| [2] | ||
| (iii)(b) New area is 0 | E1 | Allow informal explanation (eg the shape is flattened) |
| The orientation is lost or neither clockwise nor anticlockwise oe | E1 | Allow "points now all lie on a straight line" |
| [2] |
**(i)** $\begin{pmatrix} 1 & 2 \\ 1 & a \end{pmatrix}\begin{pmatrix} 2 & 1 \\ -1 & b \end{pmatrix} = \begin{pmatrix} 1 \times 2 + 2 \times -1 & 1 \times 1 + 2 \times b \\ 1 \times 2 + a \times -1 & 1 \times 1 + a \times b \end{pmatrix}$ | M1 | Must be right way round. Must be correct matrix multiplication (ie rows into columns)
$\begin{pmatrix} 0 & 1 + 2b \\ 2 - a & 1 + ab \end{pmatrix}$ | A1 | Must be in exactly the correct form.
| [2] |
**(ii)** $\begin{vmatrix} 0 & 1 + 2b \\ 2 - a & 1 + ab \end{vmatrix} = -(1 + 2b)(2 - a)$ | B1ft | Could be from detA × detB probably as $(a - 2)(2b + 1)$
$0 = -(1 + 2b)(2 - a)$ | M1ft | Knowledge that singular means zero determinant (could be stated)
(Either) $a = 2$ | A1 | If no "or" then A1 for 1 or 2 of $a$ and $b$ correct.
or $b = -\frac{1}{2}$ | A1 |
| [4] |
**(iii)(a)** Area is multiplied by 3 | E1 | Not $-3$
P'Q'R'S' run anticlockwise oe (eg orientation is reversed) | E1 | Must mention orientation or handedness or clockwise or anticlockwise
| [2] |
**(iii)(b)** New area is 0 | E1 | Allow informal explanation (eg the shape is flattened)
The orientation is lost or neither clockwise nor anticlockwise oe | E1 | Allow "points now all lie on a straight line"
| [2] |
---6 The matrix $\mathbf { A }$ is given by $\left( \begin{array} { l l } 1 & 2 \\ 1 & a \end{array} \right)$ and the matrix $\mathbf { B }$ is given by $\left( \begin{array} { c c } 2 & 1 \\ - 1 & b \end{array} \right)$.\\
(i) Find the matrix $\mathbf { A B }$.\\
(ii) State the conditions on $a$ and $b$ for $\mathbf { A B }$ to be a singular matrix.\\
$P Q R S$ is a quadrilateral and the vertices $P , Q , R$ and $S$ are in clockwise order. A transformation, T , is represented by the matrix $\mathbf { A B }$.\\
(iii) State the effect on both the area and also the orientation of the image of $P Q R S$ under T in each of the following cases.
\begin{enumerate}[label=(\alph*)]
\item $\quad a = 1$ and $b = 1$
\item $\quad a = 2$ and $b = 3$
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2018 Q6 [10]}}