| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2018 |
| Session | March |
| Marks | 7 |
| Topic | 3x3 Matrices |
| Type | Matrix properties verification |
| Difficulty | Moderate -0.8 This is a straightforward Further Pure 1 question testing basic properties of transformation matrices. Parts (i) and (iv) require simple recall that reflections have determinant -1 and rotations have determinant +1. Parts (ii) and (v) involve elementary equation solving (a² = -1 and 0.36 + b² = 1). Part (iii) requires recognizing eigenvectors, which is standard FP1 content. All parts are routine applications of well-known results with minimal problem-solving required. |
| Spec | 4.03g Invariant points and lines4.03i Determinant: area scale factor and orientation4.03j Determinant 3x3: calculation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(-1\) | B1 | |
| [1] | ||
| (ii) \(a^2 = -1\) | M1 | |
| \(a = \pm i\) | A1 | |
| [2] | ||
| (iii) \(R\) lies in the \(x\)-\(z\) plane. | E1 | Or plane \(y = 0\) |
| [1] | ||
| (iv) \(1\) | B1 | |
| [1] | ||
| (v) \(\det \mathbf{B} = 0.6^2 + b^2 = 1\) | M1 | |
| \(b = \pm 0.8\) | A1 | |
| [2] |
**(i)** $-1$ | B1 |
| [1] |
**(ii)** $a^2 = -1$ | M1 |
$a = \pm i$ | A1 |
| [2] |
**(iii)** $R$ lies in the $x$-$z$ plane. | E1 | Or plane $y = 0$
| [1] |
**(iv)** $1$ | B1 |
| [1] |
**(v)** $\det \mathbf{B} = 0.6^2 + b^2 = 1$ | M1 |
$b = \pm 0.8$ | A1 |
| [2] |
---5 The matrix $\mathbf { A }$ is given by $\left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & a ^ { 2 } & 0 \\ 0 & 0 & 1 \end{array} \right)$ and the matrix $\mathbf { B }$ is given by $\left( \begin{array} { c c c } 0.6 & b & 0 \\ - b & 0.6 & 0 \\ 0 & 0 & 1 \end{array} \right)$.\\
(i) $\mathbf { A }$ represents a reflection. Write down the value of $\operatorname { det } \mathbf { A }$.\\
(ii) Hence find the possible values of $a$.\\
(iii) $\mathbf { r }$ is the position vector of a point $R$. Given that $\mathbf { A r } = \mathbf { r }$ describe the location of $R$.\\
(iv) $\mathbf { B }$ represents a rotation. Write down the value of $\operatorname { det } \mathbf { B }$.\\
(v) Hence find the possible values of $b$.
\hfill \mbox{\textit{OCR FP1 AS 2018 Q5 [7]}}