## Question 11(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $(0,-3)$ | B1 | Condone $y=-3$, isw; if not coordinates, must be clear which is $x$ and which is $y$ |
| $(-\frac{1}{2},0)$ and $(3,0)$ www | B2 | Condone $x=-\frac{1}{2}$ and $3$; B1 for one correct www; or M1 for $(2x+1)(x-3)$ or correct use of formula or reversed coordinates |

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## Question 11(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $2x^2-6x-6\ [=0]$ isw or $x^2-3x-3\ [=0]$ or $2y^2-18y+30\ [=0]$ | M1 | For equating curve and line, and rearrangement to zero, condoning one error; allow rearranging to constant if they go on to attempt completing the square |
| Use of formula or completing the square, with at most one error | M1 | No ft from $2x^2-6x=0$ or other factorisable equations; if completing the square must get to the stage of complete square only on lhs as in 9(ii) |
| $\left(\frac{6\pm\sqrt{84}}{4},\frac{18\pm\sqrt{84}}{4}\right)$ or $\left(\frac{3\pm\sqrt{21}}{2},\frac{9\pm\sqrt{21}}{2}\right)$ oe isw | A2 | A1 for one set of coords or for $x$ values correct (or $y$s from quadratic in $y$); need not be written as coordinates; A0 for unsimplified $y$ coords e.g. $\frac{3+\sqrt{21}}{2}+3$ |

## Question 11(iii):

| Answer | Mark | Guidance |
|--------|------|----------|
| $2x^2 - 5x - 3 = x + k$ | M1 | for equating curve and line |
| $2x^2 - 6x - 3 - k\ [= 0]$ | M1 | for rearrangement to zero, condoning one error, but must include $k$; this second M1 implies the first |
| $b^2 - 4ac < 0$ for non-intersecting lines | M1 | allow for just quoting this condition; allow 'discriminant is negative' if further work implies $b^2 - 4ac$ |
| $36 - 8 \times -(3 + k)\ [\leq 0]$ oe | A1 | for correct substitution into $b^2 - 4ac$; no ft from wrong equation; if brackets missing must be followed by correct simplified version |
| $k < -\dfrac{15}{2}$ oe | A1 | isw; if 3rd M1 not earned, allow **B1** for $-\dfrac{15}{2}$ obtained for $k$ with any symbol |

**Alternative (tangent condition with trials):**

| Answer | Mark | Guidance |
|--------|------|----------|
| Rearrangement with correct boundary value, e.g. $2x^2 - 6x + 4.5\ [=0]$ or $2x^2 - 6x - (3-7.5)\ [=0]$ | M2 | **M1** for $2x^2 - 5x - 3 = x - 7.5$ |
| Showing $36 - 8 \times -(3-7.5) = 0$ or $36 - 8 \times 4.5 = 0$ oe | M1 | may be in formula; implies previous M2 |
| $k < -\dfrac{15}{2}$ oe | A2 | **B1** for $-\dfrac{15}{2}$ obtained for $k$ as final answer with any symbol |

**Alternative (tangent with differentiation):**

| Answer | Mark | Guidance |
|--------|------|----------|
| $y' = 4x - 5$ | M1 | |
| [when $y = x + k$ is tgt] $4x - 5 = 1$ | M1 | |
| $x = 1.5,\ y = -6$ | A1 | |
| $-6 = 1.5 + k$ or $k = -7.5$ oe | A1 | |
| $k < -7.5$ oe | A1 | |
| **[5]** | | |