# Question 1:

## Part (a)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_1^2 k\left(\frac{1}{2}x^3 - 3x^2 + ax + 1\right)dx [=1]$ | M1 | Attempting to integrate f(x), at least one term $x^n \to x^{n+1}$. Ignore limits. No need to equate to 1 |
| $k\left[\frac{1}{8}x^4 - x^3 + \frac{1}{2}ax^2 + x\right]_1^2 [=1]$ | A1 | Fully correct integration. Allow unsimplified. Ignore limits, accept any letters. Allow $+C$. No need to equate to 1 |
| $k(2-8+2a+2) - k\left(\frac{1}{8}-1+\frac{1}{2}a+1\right)=1$ **or** $k(2a-4)-k\left(\frac{1}{8}+\frac{1}{2}a\right)=1$ | dM1 | Dep on 1st M1. Subst correct limits, subtracting results and equate to 1. Allow use of F(2)=1 and F(1)=0 |
| $-\frac{33}{8}k + \frac{3}{2}ka = 1 \therefore k(12a-33)=8$ ✳ | A1* | Given answer. Correct solution only. At least one correct line of working required between $k(2a-4)-k\left(\frac{1}{8}+\frac{1}{2}a\right)=1$ and final answer |

## Part (b)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{df(x)}{dx} = k\left(\frac{3}{2}x^2 - 6x + a\right)$ | M1 | Attempting to differentiate f(x), at least one term $x^n \to x^{n-1}$. Condone missing $k$ or incorrect value for $k$ |
| $\frac{3}{2}x^2 - 6x + 5 = 0$ **or** $\frac{4}{9}x^2 - \frac{16}{9}x + \frac{40}{27}=0$ | dM1 | Dep on first M1. Setting differential (or multiple of) $= 0$. May be implied by awrt 1.18 or awrt 2.82 |
| $x = \frac{6 \pm \sqrt{6^2 - 4\times1.5\times5}}{3}$ | M1 | Correct method for solving their 3-term quadratic. May be implied by awrt 1.18 or awrt 2.82. Minimum for method if final answer incorrect: form $\frac{6\pm\sqrt{6}}{3}$ |
| $x = 2 - \frac{\sqrt{6}}{3}$ oe or 1.183... (awrt 1.18) | A1 | Allow equivalent exact answer. awrt 1.18. Must eliminate the 2.816... or clearly indicate which of the 2 solutions is their answer |

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