**(a)**

| $\int f(x)\,dx = \int x^2 + 10x^{\frac{1}{2}} + 25x^{-\frac{1}{2}}\,dx = \frac{2}{5}x^{\frac{5}{2}} + \frac{20}{3}x^{\frac{3}{2}} + 50x^{\frac{1}{2}} + c$ | M1 A1 A1 A1 (4) | For attempting to divide by $x^2$ and increasing a correct power by 1 (ie must have expanded). Must be seen in part (a) |

**(b)(i)**

| $f'(x) = \frac{3}{2}x^{\frac{1}{2}} + 5x^{-\frac{1}{2}} - \frac{25}{2}x^{-\frac{3}{2}}$ | M1 A1 | For attempting to divide by $x^2$ and decreasing a correct power by one. Must be seen or used in part (b). For attempts at the quotient or product rule look for correct denominator and one correct numerator term in $\frac{A(x+5)\sqrt{x}-B(x+5)^2x^{-\frac{1}{2}}}{\left(\sqrt{x}\right)^2}$ (oe for product rule) |
| $f'(x) = 0 \Rightarrow \frac{3}{2}x^{\frac{1}{2}} + 5x^{-\frac{1}{2}} - \frac{25}{2}x^{-\frac{3}{2}} = 0 \text{ AND } \times x^{\frac{3}{2}}$ | dM1 | Sets $\alpha x^{\frac{1}{2}} + \beta x^{-\frac{1}{2}} + \gamma x^{-\frac{3}{2}} = 0$ and attempts to multiply by $x^{\frac{3}{2}}$ seen in at least two terms or explicitly shown in a step before the final answer. |
| $3x^2 + 10x - 25 = 0 \Rightarrow 3x^2 + 10x - 25 = 0 \,*$ | A1* | Reaches $3x^2 + 10x - 25 = 0$ showing all steps with no errors. Note that they must work correctly with the equation, if they state $f'(x) = 3x^2 + 10x - 25$ before setting equal to 0 then allow M (if suitable method shown) but it is A0. |
| $\left(x = \frac{5}{3}\text{ only}\right)$ | B1 (5) | $\left(x = \frac{5}{3}\right)$ only. The other root need not be seen, but if shown it must be subsequently rejected in some way. Note that 1.6 (1.6 recurring) is acceptable as the answer for B1, but 1.67 is B0. |

**Total: 9 marks**

**Guidance for (a):**
- M1: For attempting to divide by $x^2$ and increasing a correct power by 1 (ie must have expanded). Must be seen in part (a)
- A1: For one correct term (allow un simplified)
- A1: For two correct terms simplified.
- A1: For $\frac{2}{5}x^{\frac{5}{2}} + \frac{20}{3}x^{\frac{3}{2}} + 50x^{\frac{1}{2}} + c$ or exact equivalent. Must include the constant of integration. Condone if they write "y =" and ignore any spurious integral signs or extra dx

**Guidance for (b)(i):**
- M1: For attempting to divide by $x^2$ and decreasing a correct power by one. Must be seen or used in part (b). For attempts at the quotient or product rule look for correct denominator and one correct numerator term in $\frac{A(x+5)\sqrt{x}-B(x+5)^2x^{-\frac{1}{2}}}{\left(\sqrt{x}\right)^2}$ (oe for product rule)
- A1: For $f'(x) = \frac{3}{2}x^{\frac{1}{2}} + 5x^{-\frac{1}{2}} - \frac{25}{2}x^{-\frac{3}{2}}$ (oe e.g from quotient rule) which may be left un simplified.
- dM1: Sets $\alpha x^{\frac{1}{2}} + \beta x^{-\frac{1}{2}} + \gamma x^{-\frac{3}{2}} = 0$ and attempts to multiply by $x^{\frac{3}{2}}$ seen in at least two terms or explicitly shown in a step before the final answer. This may be derived in stages, first multiplying by $x^{\frac{1}{2}}$ and then by x. This is permissible for the M (and A) as evidence of the correct method has been shown by the stages. The dM can be scored if the index work is correct in at least two indices, and A1 if all is correct.

So $\frac{3}{2}x^{\frac{1}{2}} + 5x^{-\frac{1}{2}} - \frac{25}{2}x^{-\frac{3}{2}} = 0 \Rightarrow \frac{3}{2}x + 5 - \frac{25}{2}x^{-1} = 0 \Rightarrow \frac{3}{2}x^2 + 5x - \frac{25}{2} = 0 \Rightarrow 3x^2 + 10x - 25 = 0$ is sufficient for the dM1 as long as the middle step is seen with two correct powers (and the A1 if they then multiply through be 2).

**Guidance for (b)(ii):**
- B1: $\left(x = \frac{5}{3}\right)$ only. The other root need not be seen, but if shown it must be subsequently rejected in some way. Note that 1.6 (1.6 recurring) is acceptable as the answer for B1, but 1.67 is B0.

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