## Question 16:

| Answer | Mark | Guidance |
|--------|------|----------|
| $\int \frac{dy}{y} = \int \frac{9\,dx}{(x-1)(x+2)}$ oe | M1 | Separation of variables; condone omission of integral signs or of $dx$ and/or $dy$; allow 1 slip such as omission of 9 or sign error in bracket |
| $\frac{A}{x-1} + \frac{B}{x+2}$ | M1 | Allow 1 sign error in bracket |
| $\frac{9}{(x-1)(x+2)} = \frac{3}{(x-1)} - \frac{3}{(x+2)}$ oe | A1 | One of two terms correct |
| | A1 | All correct |
| $\ln y = A\ln(x-1) + B\ln(x+2) + c$ | M1* | Any ln integral correct; FT their $A$ and $B$; condone omission of $+ c$ |
| $\ln y = 3\ln(x-1) - 3\ln(x+2) + c$ oe | A1 | All three terms correct including $+ c$; may see $\frac{1}{9}\ln y = \frac{1}{3}\ln(x-1) - \frac{1}{3}\ln(x+2) + c$; NB $\ln y + c$ is equivalent to $\ln Ay$ where $A$ is a constant |
| $\ln 16 = 3\ln(2-1) - 3\ln(2+2) + c$ | M1dep* | Substitution of $(2, 16)$ in their expression; may be implied by e.g. $\ln 16 = 3\ln 1 - 3\ln 4 + c$; must see substitution for incorrect expressions |
| $c = 5\ln 4$ or $\ln 4^5$ or $\ln 1024$ | A1 | May see $c = \frac{5}{9}\ln 4$ oe; allow exact equivalents only |
| $\ln \frac{\text{their } 1024(x-1)^A}{(x+2)^B}$ oe | M1 | Correctly combines their RHS into a single logarithm; their $+ c$ must be correctly incorporated into their $\ln f(y)$ or their $\ln f(x)$ |
| $\ln y = \ln \frac{1024(x-1)^3}{(x+2)^3}$ | A1 | All correct |
| $e^{\ln y} = e^{\ln\frac{(x-1)^3}{(x+2)^3}}$ oe | M1 | Correctly exponentiates their expressions; may be awarded before combination into single logarithm |
| $y = \frac{1024(x-1)^3}{(x+2)^3}$ | A1 | All correct; must see "$y =$" at some stage |
| **Total: [12]** | | |

**Alternative (for last 6 marks):**

| Answer | Mark | Guidance |
|--------|------|----------|
| $\ln\left\{\frac{(x-1)^A}{(x+2)^B} \times e^c\right\}$ oe | M1dep* | Correctly combines their RHS into a single logarithm |
| $\ln y = \ln\left[\frac{(x-1)^3}{(x+2)^3} \times e^c\right]$ oe | A1 | All correct |
| $e^{\ln y} = e^{\ln\frac{(x-1)^A}{(x+2)^B} \times D}$ oe | M1 | Correctly exponentiates their expressions; may be awarded before combining into single logarithm |
| $y = \frac{(x-1)^3}{(x+2)^3} \times D$ | A1 | All correct |
| $16 = \frac{(2-1)^A}{(2+2)^B} \times D$ oe | M1 | Substitution of $(2, 16)$ in their expression; may be implied by e.g. $16 = \frac{1^3}{4^3} \times D$; must see substitution for incorrect expressions; may be awarded before exponentiating |
| $y = \frac{1024(x-1)^3}{(x+2)^3}$ | A1 | All correct; must see "$y =$" at some stage |

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