## Question 11(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $ar = 6$ oe | B1 | must be in $a$ and $r$ |
| $\frac{a}{1-r} = 25$ oe | B1 | must be in $a$ and $r$ |
| $25 = \frac{a}{1 - \frac{6}{a}}$ | M1 | or $\frac{6}{r} = 25(1-r)$; NB assuming $a = 10$ earns M0 |
| $a^2 - 25a + 150\ [= 0]$ | A1 | or $25r^2 - 25r + 6\ [= 0]$; all signs may be reversed |
| $a = 10$ obtained from formula, factorising, Factor theorem or completing the square | A1 | if M0, B1 for $r = 0.4$ **and** $0.6$ and B1 for $a = 15$ by trial and improvement; mark to benefit of candidate |
| $a = 15$ | A1 | $a = 15$ |
| $r = 0.4$ and $0.6$ | A1 | $a = \frac{6}{0.6} = 10$ oe |

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## Question 11(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $10 \times (3/5)^{n-1}$ and $15 \times (2/5)^{n-1}$ seen | M1 | |
| $15 \times 2^{n-1} : 10 \times 3^{n-1}$ or $3 \times \frac{2^{n-1}}{5^{n-1}} : 2 \times \frac{3^{n-1}}{5^{n-1}}$ | M1 | may be implied by $3 \times 2^{n-1} : 2 \times 3^{n-1}$; condone ratio reversed |
| $3 \times 2^{n-1} : 2 \times 3^{n-1}$ | A1 | and completion to given answer www; condone ratio reversed |

# Mark Scheme Examples (June 2012, Paper 4752)

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## Example 7:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Graph showing curve with points $(-1.3, 4.3)$ and $(1.3, -4.3)$ marked, crossing x-axis at $-\sqrt{5}$ and $\sqrt{5}$ | B1 | $3^{\text{rd}}$ B1 condone extreme ends ruled |

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## Example 8:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Graph crossing x-axis at $-2$, $\sqrt{c}$, and $\sqrt{5}$ with correct cubic shape | B0 | $3^{\text{rd}}$ B0 — doesn't meet x-axis 3 times |

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## Example 9:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Graph crossing x-axis at $-\sqrt{5}$, $0$, and $\sqrt{5}$ with point $(0, 10)$ marked | B1 | $4^{\text{th}}$ B1 is earned in spite of the curve not being a cubic |

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## Example 10:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0 = x(x^2 - 5)$ | | |
| $x^2 = 5$ | | |
| $x = \sqrt{5} = 2.23$ or $-2.23$ | | |
| Points $(0, -2.23)$, $(0, 0)$, $(0, 2.23)$ marked on graph | B1 | $4^{\text{th}}$ B1 — x-intercepts: co-ordinates reversed but condone this as candidates who write $-2.23$, $2.23$ only would not be penalised |

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## Example 11:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Graph with $x$-intercepts at $-\sqrt{5}$, $0$, $\sqrt{5}$ and local maximum $(1.3, 7.8)$ marked, value $7.8$ on y-axis | B1 | $3^{\text{rd}}$ B0: incorrect orientation — $4^{\text{th}}$ B1 earned |

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## Example 12:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = 0,\ y = 0$ | | |
| $x = 2.13,\ y = -4.3$ | | |
| $x_2 = -1.3,\ y = 4.3$ | | |
| Points $(-\sqrt{5}, 0)$, $(0, 0)$, $(\sqrt{5}, 0)$ on x-axis; $(-1.3, 4.3)$ and $(1.3, -4.3)$ as turning points | B1 | $4^{\text{th}}$ B1 earned |