# Question 3:

## Part (i)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\det(\mathbf{A}) = k(4+9) + 7(-4-3) + 4(-6+2)$ | M1, A1 | Obtaining $\det(\mathbf{A})$ in terms of $k$; allow one error; isw unsimplified $65-13k$ M1A0 and allows B1 below |
| $= 13k - 65$ | | |
| $\Rightarrow$ no inverse if $k = 5$ | B1(ag) | May be verified separately |
| At least 4 cofactors correct (including one involving $k$) | M1 | M0 if more than 1 is multiplied by the corresponding element |
| Six signed cofactors correct | A1 | |
| $\mathbf{A}^{-1} = \frac{1}{13k-65}\begin{pmatrix}13 & -26 & -13 \\ 7 & -2k-4 & -3k+8 \\ -4 & 3k-7 & -2k+14\end{pmatrix}$ | M1 | Transposing and $\div$ by $\det(\mathbf{A})$; dependent on previous M1M1 |
| | A1 [7] | Mark final answer |

## Part (ii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| When $k=4$: | M1 | Substituting $k=4$ |
| $\begin{pmatrix}x\\y\\z\end{pmatrix} = \frac{1}{-13}\begin{pmatrix}13 & -26 & -13\\ 7 & -12 & -4\\ -4 & 5 & 6\end{pmatrix}\begin{pmatrix}p\\1\\2\end{pmatrix}$ | M2 | Correct use of inverse; one correct element; condone missing determinant; M0 if wrong order |
| **OR** e.g. $6x - 13y = p+4 \Rightarrow x = -p+4$; $4x - 13y = 3p-4$ | M2, M1 | Eliminating one unknown in two different ways and reaching one unknown in terms of $p$; finding the other two unknowns |
| $\begin{pmatrix}x\\y\\z\end{pmatrix} = \frac{1}{-13}\begin{pmatrix}13p-52\\ 7p-20\\ -4p+17\end{pmatrix}$ | A2 [5] | Dependent on all M marks; terms must be collected; give A1 for one correct; $x=-p+4,\ y=-\frac{7}{13}p+\frac{20}{13},\ z=\frac{4}{13}p-\frac{17}{13}$; $\lambda \times$ correct vector $(\lambda\neq 0)$ A1 |

## Question 3(iii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. $7x - 13y = p + 4$, $7x - 13y = 3p - 4$ | | Or $7x - 13y = 8$ |
| (or $4x + 13z = 7 - 2p$, $4x + 13z = -1$) | | Or $8x + 26z = 3p - 14$ |
| (or $8y + 14z = p - 10$, $4y + 7z = -3$) | | Or $4y + 7z = 5 - 2p$ |
| For solutions, $p + 4 = 3p - 4$ | M2 | Eliminating one unknown in two different ways & obtaining a value of $p$ |
| $\Rightarrow p = 4$ | A1 | |
| **OR** | M2 | A method leading to an equation from which $p$ could be found; e.g. setting $z=0$, augmented matrix, adjoint matrix, etc. |
| $p = 4$ | A1 | |
| $x = \lambda,\ y = \frac{7}{13}\lambda - \frac{8}{13},\ z = -\frac{4}{13}\lambda - \frac{1}{13}$ | M1, A1 | Obtaining general soln. by e.g. setting one unknown $= \lambda$ and finding equations involving the other two and $\lambda$; Accept unknown instead of $\lambda$; e.g. $x = \frac{13}{7}\lambda + \frac{8}{7},\ y = \lambda,\ z = -\frac{4}{7}\lambda - \frac{3}{7}$ |
| Straight line | B1 | Accept "sheaf", "pages of a book", etc. Independent of all previous marks. Ignore other comments. |

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