## Question 10:

### Part (a)(i):

| Answer | Mark | Guidance |
|--------|------|----------|
| Centre $(-3k, k)$ | B1 | May be written as $x = -3k$, $y = k$. Accept without brackets. |

### Part (a)(ii):

| Answer | Mark | Guidance |
|--------|------|----------|
| $(x+3k)^2 - 9k^2 + (y-k)^2 - k^2 + 7 = 0 \Rightarrow (x+3k)^2 + (y-k)^2 = \ldots$ | M1 | Attempts to find $r^2$ by completing the square and collecting terms. Alternatively may use $x^2 + y^2 + 2fx + 2gy + c = 0 \Rightarrow r^2 = f^2 + g^2 - c$ |
| Radius $\sqrt{10k^2 - 7}$ | A1ft | Condone unsimplified equivalents such as $\sqrt{9k^2 + k^2 - 7}$. Must be extracted and explicitly written. Only follow through on centre of form $(\pm 3k, \pm k)$. Do not allow $\pm\sqrt{10k^2-7}$. |

### Part (b):

| Answer | Mark | Guidance |
|--------|------|----------|
| $x^2 + (2x-1)^2 + 6kx - 2k(2x-1) + 7 = 0 \Rightarrow \ldots x^2 + (pk+q)x + rk + s(=0)$ | M1 | Substitutes $y = 2x - 1$ into equation of circle and attempts to collect terms |
| $5x^2 + (2k-4)x + 2k + 8\ (= 0)$ | A1 | Check carefully signs of $2k-4$ since $4-2k$ will lead to same answers and should score maximum M1A0dM1A0ddM1A0 |
| $(2k-4)^2 - 4 \times 5 \times (2k+8) = 0 \Rightarrow k = \ldots$ | dM1 | Attempts to find $b^2 - 4ac$ for their 3TQ and attempts to find at least one critical value |
| Critical values $= 7 \pm \sqrt{85}$ | A1 | $7 \pm \sqrt{85}$ |
| $k <$ "$7 - \sqrt{85}$" or $k >$ "$7 + \sqrt{85}$" o.e. | ddM1 | Attempts to find outside region for their critical values. Must have two values. Condone $k \geq$ "$7+\sqrt{85}$", $k \leq$ "$7-\sqrt{85}$". Dependent on previous two method marks. |
| $k < 7 - \sqrt{85}$ or $k > 7 + \sqrt{85}$ o.e. | A1 | Allow equivalent expressions including set notation for both outside regions. Allow ",", "or", "$\cup$" or space between answers. Do not accept "and", "$\cap$". If a variable is used it must be in terms of $k$. |

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