# Question 5(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{vmatrix}a&a&1\\-a&4&0\\4&a&5\end{vmatrix} = a(4\times5-0)-a(-5a-0)+1(-a^2-(4\times4))$ → $20a+5a^2-a^2-16=0$ → $a^2+5a-4=0$ → $a=\frac{-5+\sqrt{41}}{2}$ | M1 | Uses correct method for det $\mathbf{A}$ (implied by two correct parts) to obtain expression in $a$ |
| $a = \frac{-5+\sqrt{41}}{2}$ | A1 | Correct exact value; condone $\frac{-5\pm\sqrt{41}}{2}$ |

**(2)**

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# Question 5(b)(i) Way 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $|\mathbf{A}-\lambda\mathbf{I}| = \begin{vmatrix}a-\lambda&a&1\\-a&4-\lambda&0\\4&a&5-\lambda\end{vmatrix}$ $=(a-\lambda)(4-\lambda)(5-\lambda)-a\cdot a(5-\lambda)+(-a^2-4(4-\lambda))$ | M1 | Obtains expression for $|\mathbf{A}-\lambda\mathbf{I}|$ in terms of $a$ and $\lambda$, or just $a$ if $\lambda$ replaced by 2; if method unclear insist on 2 out of 3 correct parts |
| $\lambda=2 \Rightarrow (a-2)\times2\times3+3a^2-a^2-8=0$ → $2a^2+6a-20=0 \Rightarrow a^2+3a-10=0$ → $(a-2)(a+5)=0$ | dM1 | Following use of $\lambda=2$, forms and solves 3TQ in $a$; requires previous M mark |
| $(a>0 \Rightarrow)\, a=2$ | A1 | Correct value of $a$; if $-5$ offered, imply rejection if 2 alone used in (ii) |

**(3)**

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# Question 5(b)(i) Way 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{Ax}=2\mathbf{x} \Rightarrow ax+ay+z=2x,\; -ax+4y=2y,\; 4x+ay+5z=2z$ | M1 | Uses $\mathbf{Ax}=2\mathbf{x}$ or $(\mathbf{A}-2\mathbf{I})\mathbf{x}=\mathbf{0}$ to obtain three simultaneous equations |
| $\Rightarrow a^2+3a-10=0 \Rightarrow (a-2)(a+5)=0$ | dM1 | Forms and solves 3TQ in $a$; requires previous M mark |
| $(a>0\Rightarrow)\, a=2$ | A1 | Correct value of $a$ |

**(3)**

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# Question 5(b)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(2-\lambda)(4-\lambda)(5-\lambda)+4(5-\lambda)+(-4-16+4\lambda)=0$ → $(5-\lambda)[(2-\lambda)(4-\lambda)+4-4]=0$ → $(5-\lambda)(2-\lambda)(4-\lambda)=0 \Rightarrow \lambda=\ldots$ | M1 | Uses $a=2$ in recognisable characteristic equation; achieves real non-zero eigenvalue $\neq 2$; some algebra required |
| $\lambda = 4$ and $5$ | A1 | Both correct, no extra, from correct work |

**(2)**

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# Question 5(c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{A}\mathbf{x}=4\mathbf{x}$ or $(\mathbf{A}-4\mathbf{I})\mathbf{x}=\mathbf{0}$: $-2x+2y+z=0$, $-2x=0$, $4x+2y+z=0$ | M1 | Uses $\mathbf{Ax}=\lambda\mathbf{x}$ or $(\mathbf{A}-\lambda\mathbf{I})\mathbf{x}=\mathbf{0}$ with $a=2$ and real non-zero $\lambda\neq2$; or attempts vector product of two rows of $\mathbf{A}-4\mathbf{I}$ |
| $\pm\begin{pmatrix}0\\-1\\2\end{pmatrix}$ or $\pm\begin{pmatrix}1\\-2\\7\end{pmatrix}$ (one correct) | A1 | One correct eigenvector; accept multiples |
| $\pm\begin{pmatrix}0\\-1\\2\end{pmatrix}$ and $\pm\begin{pmatrix}1\\-2\\7\end{pmatrix}$ (both correct) | A1 | Both correct eigenvectors; accept multiples |
| $\pm\frac{1}{\sqrt{5}}\begin{pmatrix}0\\-1\\2\end{pmatrix}$, $\pm\frac{1}{\sqrt{54}}\begin{pmatrix}1\\-2\\7\end{pmatrix}$ oe | M1 A1 | M1: Correct method to normalise at least one eigenvector; A1: Both correct, any exact equivalents |

**(5) Total: 12**

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