## Question 9(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int(2x - 5 + 4x^{-2})\,dx = x^2 - 5x - 4x^{-1}$ | M1 | Attempt to rewrite integrand in a suitable form. Attempt to divide all 3 terms by $x^2$, or attempt to multiply all 3 terms by $x^{-2}$ |
| Obtain $2x - 5 + 4x^{-2}$ | A1 | Allow if third term is written in fractional form |
| Attempt integration of their integrand | M1 | Integrand must be written as a polynomial ie with all terms of the form $kx^n$, and no brackets. At least two terms must increase in power by 1. Allow if the $-5$ disappears |
| Obtain $x^2 - 5x - 4x^{-1}$ | A1 | Allow unsimplified (eg $\frac{4}{-1}x^{-1}$) |
| $(4a^2 - 10a - \frac{2}{a}) - (a^2 - 5a - \frac{4}{a}) = 0$, $3a^2 - 5a + \frac{2}{a} = 0$, $3a^3 - 5a^2 + 2 = 0$ **AG** | M1 | Attempt use of limits. Must be $F(2a) - F(a)$ ie subtraction with limits in the correct order. Must be in integration attempt |
| Equate to 0 and rearrange to obtain $3a^3 - 5a^2 + 2 = 0$ | A1 | Must be equated to 0 before multiplying through by $a$. At least one extra line of working required between $(4a^2-10a-\frac{2}{a})-(a^2-5a-\frac{4}{a})=0$ and final answer. **AG** so look carefully at working |

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## Question 9(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $f(1) = 3 - 5 + 2 = 0$ **AG** | B1 | Confirm $f(1)=0$ — detail required. $3(1)^3 - 5(1)^2 + 2 = 0$ is enough. B0 for just $f(1)=0$. If using division must show '0' on last line. If using coefficient matching must show '$R=0$'. If using inspection there must be some indication of no remainder |
| $f(a) = (a-1)(3a^2 - 2a - 2)$ | M1 | Attempt full division by $(a-1)$, or equiv method. Must be complete method — ie all 3 terms attempted |
| Obtain $3a^2$ and one other correct term | A1 | Could be middle or final term depending on method. Must be correctly obtained |
| Obtain fully correct quotient | A1 | For coeff matching it must now be explicit: $A=3$, $B=-2$, $C=-2$ |
| $a = \frac{2 \pm \sqrt{4+24}}{6} = \frac{2 \pm 2\sqrt{7}}{6} = \frac{1\pm\sqrt{7}}{3}$ | M1 | Attempt to solve quadratic. Using quadratic formula or completing the square. M0 if factorising attempt as expected root is a surd. Quadratic must come from division attempt |
| hence $a = \frac{1}{3}(1+\sqrt{7})$ only | A1 | Must give the positive root only, so A0 if negative root still present (but condone $a=1$ also given). Allow aef but must be a simplified surd as per request on question paper (ie simplify $\sqrt{28}$) |

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