# Question 9:

## Part (i)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{(3x+2)^2}{4\sqrt{x}} = \frac{9x^2+12x+4}{4\sqrt{x}} = \frac{9}{4}x^{\frac{3}{2}}+3x^{\frac{1}{2}}+x^{-\frac{1}{2}}$ | M1 | Attempts to multiply out numerator and divide (any term) by $4\sqrt{x}$. Award for one correct index from correct working. Do not award for $\frac{\cdots}{x^{\frac{1}{2}}}$ unless implied by further work. |
| $\int \frac{(3x+2)^2}{4\sqrt{x}}\,dx = \frac{2}{5}\times\frac{9}{4}x^{\frac{5}{2}}+\frac{2}{3}\times3x^{\frac{3}{2}}+2x^{\frac{1}{2}}(+c)$ | dM1 | Raises the power of any correct index by one: $x^{\frac{3}{2}}\to x^{\frac{5}{2}}$, $x^{\frac{1}{2}}\to x^{\frac{3}{2}}$, $x^{-\frac{1}{2}}\to x^{\frac{1}{2}}$. Indices must be processed. |
| $= \frac{9}{10}x^{\frac{5}{2}}+2x^{\frac{3}{2}}+2x^{\frac{1}{2}}+c$ | A1 | One correct simplified term. Allow $0.9$ instead of $\frac{9}{10}$ |
| | A1 | Two correct simplified terms. Allow $0.9$ instead of $\frac{9}{10}$ |
| | A1 | All three terms correct with $+c$ on one line. Allow $0.9$ instead of $\frac{9}{10}$. Accept e.g. $\frac{9}{10}(\sqrt{x})^5+2(\sqrt{x})^3+2\sqrt{x}+c$ |

**(5 marks)**

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## Part (ii)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $f'(x)=x^2+ax+b$ | — | Given |
| $f'(3)=2 \Rightarrow 2=9+3a+b$ | M1 | Attempts to use $f'(3)=2$. Expression need not be simplified. Condone slips when squaring but do not allow $f'(3)=-2$ |
| $f(x)=\frac{1}{3}x^3+\frac{1}{2}ax^2+bx+c$ | M1 | Attempts to integrate and achieves $f(x)=\ldots x^3+\ldots ax^2+bx+(c)$ with or without $+c$, where $a,b\neq 0$ |
| Uses $y$-intercept $=-8$ and $(3,-2)$ in $f(x)=\frac{1}{3}x^3+\frac{1}{2}ax^2+bx+c$ | dM1 | Dependent on previous M mark. Alternatively uses $(3,-2)$ in $f(x)=\ldots x^3+\ldots ax^2+bx-8$ |
| $-2=9+\frac{9}{2}a+3b-8$ | A1 | Correct unsimplified equation in $a$ and $b$. Note: simplified form is $9a+6b=-6$. Alternatively correct equation in $a$ only: $-2=9+\frac{9}{2}a+3(-7-3a)-8$, e.g. $4.5a=-18$ |
| Solves simultaneously to get values for $a$ and $b$ | ddM1 | Dependent on all previous M marks. Solve two equations with $c=-8$ to find $a$ and $b$, or solve equation in $a$ then substitute into $b=-7-3a$ |
| $a=-4,\ b=5 \Rightarrow f(x)=\frac{1}{3}x^3-2x^2+5x-8$ | A1 | $\frac{1}{3}x^3-2x^2+5x-8$ as final answer. Do not allow e.g. multiplied through to give $x^3-6x^2+15x-24$. Do not allow $5x^1$. |

**(6 marks)**

**(11 marks total)**