# Question 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $h = 0.2$ | B1 | o.e. including $\frac{5-4}{5}$; seen or implied by sight of e.g. $\frac{0.2}{2}$ or $\frac{1}{10}$ in front of bracket. May be implied by correct answer if no incorrect working. $h = -0.2$ is B0 |
| $\frac{1}{2} \times \text{"0.2"} \times [9.2 + 8.6 + 2(8.4556 + 3.8512 + 5.0342 + 7.8297)]$ | M1 | Correct application of trapezium rule with their $h$. Correct inner bracket structure $9.2 + 8.6 + 2(8.4556 + 3.8512 + 5.0342 + 7.8297)$, condoning slips copying values or omission of final brackets on rhs |
| $6.814$ | A1 | awrt 6.814 isw once correct answer seen. Correct answer with no working scores B1M1A1. If $h = -0.2$ used, maximum is B0M1A0. A mark cannot be awarded without both B1M1 |

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# Question 2a:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $f\!\left(\frac{3}{2}\right) = 4\!\left(\frac{3}{2}\right)^3 - 8\!\left(\frac{3}{2}\right)^2 + 5\!\left(\frac{3}{2}\right) + a = 0 \Rightarrow a = \ldots$ | M1 | Substitutes $x = \frac{3}{2}$ into expression, sets equal to 0 and rearranges to find $a$. Substituting $x = -\frac{3}{2}$ is M0. Attempting via long division scores M0 |
| $\Rightarrow \frac{27}{2} - 18 + \frac{15}{2} + a = 0 \Rightarrow a = -3$ | A1* | $a = -3$ achieved with no errors seen including omission of brackets. Must see $x = \frac{3}{2}$, at least one intermediate step, and "$= 0$" somewhere before the answer |

# Question 2b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to find quadratic factor of $4x^3 - 8x^2 + 5x - 3$ using $2x-3$ as factor, obtaining $2x^2 \pm x \pm \ldots$ | M1 | Score for $2x^2 \pm x \pm \ldots$ if algebraic division used, or $2x^2 \pm \ldots x \pm 1$ if equating coefficients/inspection used |
| $2x^2 - x + 1$ (for $2x-3$ as linear factor) | A1 | Or $4x^2 - 2x + 2$ for $x - \frac{3}{2}$ as linear factor |
| $(-1)^2 - 4 \times 2 \times 1 = -7 < 0$ | dM1 | Dependent on previous M. Attempts to show quadratic factor has no real roots via $b^2 - 4ac$, solving quadratic formula, or completing the square |
| $\Rightarrow$ no real roots so $x = \frac{3}{2}$ is the only real root | A1* | Requires correct factorisation of cubic, correct calculation for quadratic, reason and conclusion. Must reference root $x = \frac{3}{2}$ |

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# Question 3a(i) and (ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $r = \sqrt{(8-2)^2 + (-3-5)^2} = 10$ | M1 | Attempts to find radius; values embedded in expression finding difference in $x$ and $y$ coordinates |
| $(r =)\ 10$ | A1 | Ignore units. $\pm 10$ is A0. Do not accept $\sqrt{100}$. isw once correct answer seen |
| $(x-2)^2 + (y-5)^2 = 100$ | A1ft | o.e. e.g. $(x-2)^2+(y-5)^2 = \text{"10}^2\text{"}$. Follow through on their positive single value of $r$ |

# Question 3b:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Gradient between centre and $P = -\frac{4}{3}$ | B1 | May be implied by their gradient of the tangent |
| Perpendicular gradient $= \frac{3}{4}$ | M1 | Attempts negative reciprocal of their gradient; may be embedded in equation of tangent |
| $y + 3 = \frac{3}{4}(x - 8)$ | M1 | Attempts equation of tangent at $P$ using changed gradient and point $(8, -3)$. If $y=mx+c$ used must proceed to $c = \ldots$ |
| $3x - 4y - 36 = 0$ | A1 | o.e. provided all coefficients are integers and all terms on one side. isw once correct equation seen |