$(x+1)\log 3 = 2x\log 5$ oe | M1 | or $x + 1 = 2x\log_5 5$ or $(x+1)\log_3 = 2x$; allow recovery from omission of brackets in later working
$\log 3 = x(2\log 5 - \log 3)$ oe | A1 | $x(1-2\log_s5) = -1$ oe; NB $0.4771212254 = 0.920817754x - 1.929947041x - 1$ or $1.317393806x = 0.682606194\ldots$
$\frac{\log 3}{2\log 5 - \log 3}$ oe | A1 | or $\frac{\log_s 3}{2 - \log_s 3}$ oe
$0.518$ cao | A1 | answer only does not score
**Total: [4]**

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# Question 11(i)

$y' = 1 + 8x^{-3}$ | M2 | M1 for just $8x^{-3}$ or $1 - 8x^{-3}$
$y'' = -24x^{-4}$ oe | A1 | but not just $\frac{-24}{x^4}$ as AG
**Total: [3]**

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# Question 11(ii)

their $y' = 0$ soi | M1 |
$x = -2$ | A1 | A0 if more than one $x$-value; $x = -2$ must have been correctly obtained for all marks after first M1
$y = -3$ | A1 | A0 if more than one $y$-value
substitution of $x = -2$: $\frac{-24}{(-2)^4}$ | M1 | or considering signs of gradient either side of $-2$ with negative $x$-values; condone any bracket error
$< 0$ or $-1.5$ oe correctly obtained isw | A1 | signs for gradients identified to verify maximum; must follow from M1 A1 A0 M1 or better
**Total: [5]**

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# Question 11(iii)

$y = -5$ soi | B1 |
substitution of $x = -1$ in their $y'$ | M1 | may be implied by $-7$
grad normal $= ^{1}/_{7}$ | M1* | may be implied by eg $^{1}/_{7}$
$y - \text{their}(-5) = \text{their}^{1}/_{7}(x - (-1))$ | M1dep* | or their $(-5) = $ their $^{1}/_{7} \times (-1) + c$; allow eg $y - \frac{1}{7}x + \frac{34}{7} = 0$
$-x + 7y + 34 = 0$ oe | A1 | allow eg $y - \frac{1}{7}x + \frac{34}{7} = 0$; must see $c = 0$; do not allow eg $y = \frac{x}{7} - \frac{34}{7}$
**Total: [5]**

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# Question 12(i)

$h = 3$ soi | B1 | allow if used with 6 separate trapezia
$\frac{3}{2}(9 + 9.1 + 2(10.7 + 11.7 + 11.9 + 11.0))$ | M1 | basic shape of formula correct with their 3; omission of brackets may be recovered later; M0 if any $x$-values used (NB $y0 = 9$ and $x3 = 9$, so check position)
all $y$-values correctly placed in formula | B1 | condone omission of outer brackets; with 3, 4 or 5 $y$-values in middle bracket, eg $\frac{3}{2}(9 + 2(10.7 + 11.7 + 11.9) + 11.0)$
$163.05$ or $163.1$ or $163$ isw | A1 | answer only does not score; or B1 + B3 if 5 separate trapezia calculated to give correct answer; NB $29.55 + 33.6 + 35.4 + 34.35 + 30.15$
**Total: [4]**

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# Question 12(ii)(A)

$-0.001 \times 13^3 - 0.025 \times 12^2 + 0.6 \times 12 + 9$ soi | M1 | may be implied by $10.872, 10.87$ or $10.9$; NB allow misread if minus sign omitted in first term if consistent in (A) and (B). Lose A1 in this part only
$\pm0.128$[m] or $\pm12.8$cm or $\pm128$mm isw | A1 | B2 if unsupported; appropriate units must be stated if answer not given in metres
**Total: [2]**

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# Question 12(ii)(B)

$F[x] = \frac{-0.001x^4}{4} - \frac{0.025x^3}{3} + \frac{0.6x^2}{2} + 9x$ | M2 | M1 if three terms correct; ignore $+ c$
$F(15) [-F(0)]$ soi | M1 | dependent on at least two terms correct in $F[x]$; condone $F(15) + 0$
$161.7$ to $162$ | A1 | A0 if a numerical value is assigned to $c$; answer only does not score; NB allow misread if minus sign omitted in first term if consistent in (A) and (B). $187.03\ldots$
**Total: [4]**

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# Question 13(i)

$\log_{10}h = \log_{10}ad + bt$ www | B1 | condone omission of base
$m = b, c = \log_{10}a$ | B1 | must be clearly stated: linking equations is insufficient
**Total: [2]**

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# Question 13(ii)

$-0.15, [0.00], 0.23, 0.36, 0.56, 0.67, 0.78, 0.91, 1.08, 1.2[0]$ | B2 | B1 if 1 error
plots correct (tolerance half square) | B1 | condone 1 error – see overlay
single ruled line of best fit for values of $x$ from 5 to 50 inclusive | B1 | line must not go outside overlay between $x = 5$ and $x = 50$
**Total: [4]**

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# Question 13(iii)

$-0.3 \leq y\text{-intercept} \leq -0.22$ | B1 | may be implied by $0.5 \leq a \leq 0.603$
valid method to find gradient of line | M1 | may be embedded in equation; may be implied by eg $m$ between $0.025$ and $0.035$; condone values from table; condone slips eg in reading from graph
$h = \text{their} \times 10^{\text{their}a + \text{their}bt}$ | M1 | if B1M1M0, then SC1 for $\log h = \log a + \text{their}bt$ isw
$0.028 \leq b \leq 0.032$ and $0.5 \leq a \leq 0.603$ or $-0.3 \leq \log a \leq -0.22$ | A1 | if both values in the acceptable range for A1
**Total: [4]**

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# Question 13(iv)

$a10^{60b} - a10^{50b}$ | M1 | or $10^{\log a + b \cdot 60} - 10^{\log a + b \cdot 50}$; condone $15.9$ as second term may follow starting with $\log h = \log a + \text{their}bt$; or their values for $\log a$ and $b$
their values for $a$ and $b$ | M1 |
$8.0$ to $26.1$ inclusive | A1 | NB A0 for estimate without clear valid method using model; both marks available even if $a$ or $b$ or both are outside range in (iii)
**Total: [2]**

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# Question 13(v)

comment on the **continuing reduction in thickness** and its consequences | B1 | eg in long term, it predicts that reduction in thickness will continue to increase, even when the glacier has completely melted
**Total: [1]**