## Question 13:

### Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_{10} b = 0.0054 \Rightarrow b = 10^{0.0054}$ **or** $\log_{10} a = 0.81 \Rightarrow a = 10^{0.81}$ | M1 | Must use base 10; may be implied by $a \approx$ awrt 6.46 or $b \approx$ awrt 1.01 |
| $b = 1.01$ **or** $a = 6.46$ | A1 | Allow 3 sf |
| $\log_{10} b = 0.0054 \Rightarrow b = 10^{0.0054}$ **and** $\log_{10} a = 0.81 \Rightarrow a = 10^{0.81}$ | M1 | Both values; may be implied if no incorrect work seen |
| $b = 1.013$ **and** $a = 6.457$ | A1 | Requires $a =$ awrt 6.457 and $b =$ awrt 1.013; isw once correct answers seen |

**Special case - constants wrong way round:** $a = 1.013$ **and** $b = 6.457$ scores M1A1M1A0 unless equation formed correctly.

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### Part (b)(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| e.g. The world population **in billions** in 2004 | B1ft | Follow through their $a$; must reference "billions"; allow "original/initial population in billions" |

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### Part (b)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $b = 1.013$ represents the scale factor of the yearly increase in the world population | B1ft | Follow through their $b$; must reference "each year" or "yearly"; allow "proportional increase/change in each year", "population rises by 1.3% each year", "multiplier representing year on year increase" |

Do **not** accept: "the amount it is rising", "how much it is rising", "the rate the population increases", "the percentage increase each year", "the rate of increase in billions annually"

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### Part (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $P = 6.457...(1.013...)^{26}$ or $\log P = 0.81 + 26\times 0.0054 \Rightarrow P = ...$ | M1 | Substitutes $t = 25, 26$ or $27$; must use **their** $a$ and $b$ correctly in $P = ab^t$ |
| awrt 9 billion | A1 | From correct model; condone incorrect/premature rounding leading to awrt 9 billion; allow $9\,000\,000\,000$ or $9\times10^9$; just "9" without "billions" is A0 |

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### Part (d):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Not reliable since data used for model covered 2004–2007 and it would not be sensible to assume the model still holds in 2030 | B1 | Must refer to unreliability **and** reference data being a long way from 2030 |

Accept: "Not good as 2030 is a long way from 2004–2007", "unreliable as based on old data", "questionable as extrapolated over a long time", "not reliable due to how far we have extrapolated"

Do **not** accept: "unreliable, extrapolation" (alone), "not good as outside the range", "disease may happen", "reliable as based on old data"

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