## Question 7:

### Part (i):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(x) = \frac{4}{x^3} + 2x - 1 \Rightarrow f(x) = \frac{4}{-2}x^{-2} + x^2 - 1x \ (+c)$ | M1A1 | M1 for attempt to integrate $f'(x)$ |
| Sub $(2,3)$: $3 = -\frac{1}{2} + 4 - 2 + c \Rightarrow c = \frac{3}{2}$ | dM1A1 | dM1 dependent on previous M, for substituting point $(2,3)$ to find $c$ |
| $f(1) = -\frac{2}{1^2} + 1^2 - 1 + \frac{3}{2} = -\frac{1}{2}$ | A1 | cso |

### Part (ii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_1^4 (3\sqrt{x} + A)\, dx = \left[\frac{3x^{1.5}}{1.5} + Ax\right]_1^4 = 21$ | M1A1 | M1 for attempt to integrate; A1 for correct integration |
| $\left(\frac{3 \times 4^{1.5}}{1.5} + 4A\right) - \left(\frac{3 \times 1^{1.5}}{1.5} + 1A\right) = 21$ | M1 | For substituting limits and subtracting |
| $16 + 4A - 2 - A = 21$ | | |
| $3A = 7 \Rightarrow A = \frac{7}{3}$ | dM1A1 | dM1 dependent on previous M, for solving to find $A$ |

# Question (i) [Integration]:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Raise power by 1 in any of the 3 terms: $x^{-3} \to x^{-2}$, $x \to x^2$ or $1 \to x$ | M1 | Accept any one correct power raise |
| $f(x) = 4\frac{x^{-2}}{-2} + 2\frac{x^2}{2} - 1x$ $(+c)$ | A1 | Correct unsimplified form; no requirement for $+c$ |
| Substitutes $(2,3)$ into their $f(x)$, which must have $+c$, finds numerical value of $c$ | dM1 | Dependent on previous M; $f(x)$ must contain $+c$ |
| $c = \frac{3}{2}$ | A1 | Or equivalent |
| $f(1) = -\frac{1}{2}$ | A1 | Accept exact equivalents such as $-0.5$ |

# Question (ii) [Integration with limits]:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $3\sqrt{x} \to \ldots x^{\frac{3}{2}}$ or $A \to Ax$ | M1 | Score for either correct power raise |
| $3\frac{x^{1.5}}{1.5} + Ax$ $(+c)$ | A1 | Fully correct unsimplified integral; no need to set $=21$ or sub limits yet |
| Sub limits 4 and 1 into changed function, subtract and set $=21$: $\left(\frac{3\times4^{1.5}}{1.5}+4A\right)-\left(\frac{3\times1^{1.5}}{1.5}+1A\right)=21$ | M1 | Either subtraction order; with or without brackets |
| Proceed to solve equation in $A$ | dM1 | Dependent on **both** previous M's **and** having $A \to Ax$; must end with numerical value |
| $A = \frac{7}{3}$ | A1 | Accept $2\frac{1}{3}$, $2.\dot{3}$; do not accept $2.3$ or $2.33$ |

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