**(a)**

| States or implies that $f(x) = kx(x-4)$ | M1 | States or implies that $f(x) = kx(x-4)$ . Allow with $k = 1$. The f(x) need not be seen – assume they are working with f(x) in part (a) |
| Attempts to find k. E.g. $-4.8 = k \times 2 \times (2-4) \Rightarrow k = ...$ | dM1 | Attempts to find all the constants in their equation for f(x) using $(2, -4.8)$ . E.g. $-4.8 = k \times 2 \times (2-4) \Rightarrow k = ...$ |
| $f(x) = 1.2x(x-4)$ | A1 (3) | $\left[f(x)\right] = 1.2x(x-4)$ o.e. such as $\left[f(x)\right] = 1.2x^2 - 4.8x$ or $f(x) = 1.2(x-2)^2 - 4.8$ and isw. Accept with $(x-0)$. The f(x) may be missing, it can be assumed (but use of g(x) is A0). Allow if the full expression is given in latter parts if all the work was correct in (a) to find the constants. |

**(b)**

| States or implies that $g(x) = \lambda x(x-4)^2$ | M1 | States or implies that $g(x) = \lambda x(x-4)^2$ . Allow with $\lambda = 1$. The g(x) need not be seen – assume they are working with g(x) in part (b). |
| Attempts to find $\lambda$ . E.g. $7.2 = \lambda \times 6 \times (6-4)^2 \Rightarrow \lambda = ...$ | dM1 | Attempts a full equation for g(x) using $(6, 7.2)$ |
| $g(x) = 0.3x(x-4)^2$ | A1 (3) | $\left[g(x)\right] = 0.3x(x-4)^2$ o.e. such as $\left[g(x)\right] = 0.3x^3 - 2.4x^2 + 4.8x$ and isw. Accept with $(x-0)$. The g(x) may be missing, it can be assumed (but use of f(x) is A0). Allow if the full expression in used in (c) if all the work was correct in (b) to find the constants. |

**(c)**

| Sets their $1.2x(x-4) = 0.3x(x-4)^2$ | B1ft | Sets their $1.2x(x-4) = 0.3x(x-4)^2$ . It is for a "correct" ft equation on their constants, but the form (quadratic = cubic) must be correct. |
| Valid attempt to solve $1.2x(x-4) = 0.3x(x-4)^2 \Rightarrow x = 4 + \frac{1.2}{0.3}$ | M1 | Valid attempt to solve their equations to find at least one non-zero solution as long as they have a cubic and quadratic equated. E.g. $1.2x(x-4) = 0.3x(x-4)^2 \Rightarrow x = 4 + \frac{1.2}{0.3}$ . Alternatively, they may expand, factor out the x and solve the quadratic (allowing for slips), or even expand to a cubic (allowing slips) and solve via calculator (may need to check, and you may ignore "incorrect factorisation" if the answers are correct for their equation). |
| $x = 8$ | A1 | $x = 8$ identified as the x coordinate of $P$. A0 if they give 0.4 and 8 and do not select the correct answer. |
| $\left(8, 38.4\right)$ | A1 A1 (4) | $\left(8, 38.4\right)$ or exact equivalent for 38.4 |

**Total: 10 marks**

**Guidance for (a):**
- M1: States or implies that $f(x) = kx(x-4)$ . Allow with $k = 1$. The f(x) need not be seen – assume they are working with f(x) in part (a)

Alternatives include $f(x) = k(x-2)^2 - 4.8$ . Allow with $k = 1$ and $f(x) = ax^2 + bx$ with at least one of $4a + b = 0$ and/or $4a + 2b = -4.8$.

- dM1: Attempts to find all the constants in their equation for f(x) using $(2, -4.8)$ . E.g. $-4.8 = k \times 2 \times (2-4) \Rightarrow k = ...$

- A1: $\left[f(x)\right] = 1.2x(x-4)$ o.e. such as $\left[f(x)\right] = 1.2x^2 - 4.8x$ or $f(x) = 1.2(x-2)^2 - 4.8$ and isw. Accept with $(x-0)$. The f(x) may be missing, it can be assumed (but use of g(x) is A0). Allow if the full expression is given in latter parts if all the work was correct in (a) to find the constants.

**Guidance for (b):**
- M1: States or implies that $g(x) = \lambda x(x-4)^2$ . Allow with $\lambda = 1$. The g(x) need not be seen – assume they are working with g(x) in part (b).

Alternatives include $g(x) = ax^3 + bx^2 + cx$ with at least two of $64a + 16b + 4c = 0$, $48a + 8b + c = 0$, and/or $216a + 36b + 6c = 7.2$ (allow minor slips).

- dM1: Attempts a full equation for g(x) using $(6, 7.2)$

E.g. $7.2 = \lambda \times 6 \times (6-4)^2 \Rightarrow \lambda = ...$

- A1: $\left[g(x)\right] = 0.3x(x-4)^2$ o.e. such as $\left[g(x)\right] = 0.3x^3 - 2.4x^2 + 4.8x$ and isw. Accept with $(x-0)$. The g(x) may be missing, it can be assumed (but use of f(x) is A0). Allow if the full expression in used in (c) if all the work was correct in (b) to find the constants.

**Guidance for (c):**
- B1ft: Sets their $1.2x(x-4) = 0.3x(x-4)^2$ . It is for a "correct" ft equation on their constants, but the form (quadratic = cubic) must be correct.

- M1: Valid attempt to solve their equations to find at least one non-zero solution as long as they have a cubic and quadratic equated. E.g. $1.2x(x-4) = 0.3x(x-4)^2 \Rightarrow x = 4 + \frac{1.2}{0.3}$ . Alternatively, they may expand, factor out the x and solve the quadratic (allowing for slips), or even expand to a cubic (allowing slips) and solve via calculator (may need to check, and you may ignore "incorrect factorisation" if the answers are correct for their equation).

- A1: $x = 8$ identified as the x coordinate of $P$. A0 if they give 0.4 and 8 and do not select the correct answer.

- A1: $\left(8, 38.4\right)$ or exact equivalent for 38.4

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