# Question 3:

## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2k \times 3 = 9$, $k = 1.5$ | M1 | Attempt to find $k$. Substitute $x=2$ and $\frac{dy}{dx}=9$ into given differential equation and attempt to find $k$ |
| | A1 | Obtain $k=1.5$. Allow any exact equiv. including $\frac{9}{6}$ |
| | [2] | |

## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = x^3 - 0.75x^2 + c$ | M1 | Expand bracket and attempt integration. M0 if bracket not expanded first. M1 can still be gained for integrating an incorrect expansion as long as there are two terms. For an 'integration attempt' there must be an increase in power by 1 for both terms |
| | A1ft | Obtain at least one correct term (allow still in terms of $k$). Follow through on their value of $k$ (but not on an incorrect expansion at start of part (ii)). Can also get A1 if still in terms of $k$. Allow unsimplified coefficients |
| | A1 | Obtain $x^3 - 0.75x^2$ (condone no $+c$). Must now be numerical, and no f-t. Allow unsimplified coefficients. A0 if integral sign or $dx$ still present, unless it later disappears |
| $7 = 8 - 3 + c$ hence $c = 2$, $y = x^3 - 0.75x^2 + 2$ | M1 | Attempt to find $c$ using $(2, 7)$. There must have been an attempt at integration, but can follow M0 e.g. if the bracket was not expanded first. Need to get as far as actually attempting $c$. M1 could be implied by e.g. $7=8-3$ followed by an attempt to include a constant to balance the equation. M0 if no $+c$ seen or implied. M0 if using $x=7$, $y=2$ |
| | A1 | Obtain $y = x^3 - 0.75x^2 + 2$. Coefficients now need to be simplified ($0.75$ or $\frac{3}{4}$). Must be an equation ie $y=...$, so A0 for '$f(x)=...$' or 'equation $=...$'. Allow aef, such as $4y = 4x^3 - 3x^2 + 8$ |
| | [5] | |

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