**(a)**

$\frac{2}{x+2} + \frac{4}{x^2+5} - \frac{18}{(x+2)(x^2+5)} = \frac{2(x^2+5) + 4(x+2) - 18}{(x+2)(x^2+5)}$ | M1A1 | Combines the three fractions to form a single fraction with a common denominator. Allow errors on the numerator but at least one must have been adapted. Condone 'invisible' brackets for this mark. Accept three separate fractions with the same denominator. Amongst possible options allowed for this method are: **Eg 1** An example of 'invisible' brackets: $\frac{2x^2 + 5 + 4x + 2 - 18}{(x+2)(x^2+5)}$; **Eg 2** An example of an error (on middle term), $1^{\text{st}}$ term has been adapted: $\frac{2(x^2+5) + 4(x+2) - 18}{(x+2)(x^2+5)}$; **Eg 3** An example of a correct fraction with a different denominator: $\frac{2(x^2+5)^2(x+2) + 4(x+2)^2(x^2+5) - 18x^2 + 5(x+2)}{(x+2)^2(x^2+5)^2}$ |

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$= \frac{2x(x+2)}{(x+2)(x^2+5)}$ | M1 | There must be a single denominator. Terms must be collected on the numerator. A factor of $(x+2)$ must be taken out of the numerator and then cancelled with one in the denominator. The cancelling may be assumed if the term 'disappears'. |

$= \frac{2x}{(x^2+5)}$ | A1* | Cso. $\frac{2x}{(x^2+5)}$. This is a given solution and this mark should be withheld if there are any errors. | (4 marks)

**(b)**

$h'(x) = \frac{(x^2+5) \times 2 - 2x \times 2x}{(x^2+5)^2}$ | M1A1 | Applies the quotient rule to $\frac{2x}{(x^2+5)}$, a form of which appears in the formula book. If the rule is quoted it must be correct. There must have been some attempt to differentiate both terms. If the rule is not quoted (nor implied by their working, meaning terms are written out $u=...., u' =....., v=...., v' =...., \text{followed by their } \frac{vu'-uv'}{v^2}$) then only accept answers of the form $\frac{(x^2+5) \times A - 2x \times Bx}{(x^2+5)^2}$ where $A, B > 0$ |

$h'(x) = \frac{10-2x^2}{(x^2+5)^2}$ | cso A1 | Correct unsimplified answer $h'(x) = \frac{(x^2+5) \times 2 - 2x \times 2x}{(x^2+5)^2}$. **A1** $h'(x) = \frac{10-2x^2}{(x^2+5)^2}$. The correct simplified answer. Accept $\frac{2(5-x^2)}{(x^2+5)^2}$, $\frac{-2(x^2-5)}{(x^2+5)^2}$, $\frac{10-2x^2}{(x^4+10x^2+25)}$ | (3 marks)

**(c)**

Maximum occurs when $h'(x) = 0 \Rightarrow 10 - 2x^2 = 0 \Rightarrow x = ..$ | M1 | Sets their $h'(x)=0$ and proceeds with a correct method to find $x$. There must have been an attempt to differentiate. Allow numerical errors but do not allow solutions from 'unsolvable' equations. |

$\Rightarrow x = \sqrt{5}$ | A1 | Finds the correct $x$ value of the maximum point $x=\sqrt{5}$. Ignore the solution $x= -\sqrt{5}$ but withhold this mark if other positive values found. |

When $x = \sqrt{5} \Rightarrow h(x) = \frac{\sqrt{5}}{5}$ | M1, A1 | Substitutes their answer into their $h'(x)=0$ in $h(x)$ to determine the maximum value. $\text{Cso} -$ the maximum value of $h(x) = \frac{\sqrt{5}}{5}$. Accept equivalents such as $\frac{2\sqrt{5}}{10}$ but not $0.447$. |

Range of $h(x)$ is $0 \leq h(x) \leq \frac{\sqrt{5}}{5}$ | A1ft | Range of $h(x)$ is $0 \leq h(x) \leq \frac{\sqrt{5}}{5}$. Follow through on their maximum value if the M's have been scored. Allow $0 \leq y \leq \frac{\sqrt{5}}{5}$, $0 \leq$ Range $\leq \frac{\sqrt{5}}{5}$, $\left[0, \frac{\sqrt{5}}{5}\right]$ but not $0 \leq x \leq \frac{\sqrt{5}}{5}$, $\left(0, \frac{\sqrt{