(a) $\frac{3}{2} = -2x^2 + 4x$ | M1
$4x^2 - 8x + 3(=0)$ | A1
$(2x-1)(2x-3) = 0$ | M1
$x = \frac{1}{2}, \frac{3}{2}$ | A1 | (4 marks)

(b) Area of $R = \int_{\frac{1}{2}}^{\frac{3}{2}} (-2x^2+4x)\,dx - \frac{3}{2}$ | B1 (for $-\frac{3}{2}$)
$\int(-2x^2+4x)\,dx = \left[-\frac{2}{3}x^3+2x^2\right]$ | M1 [A1] (Allow $\pm\left[\,\,\right]$, accept $-\frac{4}{2}x^2$)
$\int_{\frac{1}{2}}^{\frac{3}{2}}(-2x^2+4x)\,dx = \left(-\frac{2}{3}\cdot\frac{3^3}{2^3}+2\times\frac{3^2}{2^2}\right) - \left(-\frac{2}{3}\cdot\frac{1}{2^3}+2\times\frac{1}{2^2}\right)$ | M1 M1
$\left(= \frac{11}{6}\right)$
Area of $R = \frac{11}{6} - \frac{3}{2} - \frac{1}{3}$ | A1 cao (Accept exact equivalent but not 0.33...) | (6 marks)

**Guidance:**
- (a) 1st M1 for forming a correct equation; 1st A1 for a correct 3TQ (condone missing $=0$ but must have all terms on one side); 2nd M1 for attempting to solve appropriate 3TQ
- (b) B1 for subtraction of $\frac{3}{2}$. Either "curve $-$ line" or "integral $-$ rectangle"
  - 1st M1 for some correct attempt at integration ($x^n \rightarrow x^{n+1}$)
  - 1st A1 for $-\frac{2}{3}x^3 + 2x^2$ only i.e. can ignore $-\frac{2}{3}x$
  - 2nd M1 for some correct use of their $\frac{3}{2}$ as a limit in integral
  - 3rd M1 for some correct use of their $\frac{1}{2}$ as a limit in integral and subtraction either way round

**Special Case:** Line $-$ curve gets B0 but can have the other A marks provided final answer is $+\frac{1}{3}$.

**Total: 10 marks**

---

# General Principles for C1 & C2 Marking:

## Method mark for solving 3 term quadratic:

1. **Factorisation:** $(x^2+bx+c) = (x+p)(x+q)$, where $|pq|=|c|$, leading to $x=\ldots$
   $(ax^2+bx+c) = (mx+p)(nx+q)$, where $|pq|=|c|$ and $|mn|=|a|$, leading to $x=\ldots$

2. **Formula:** Attempt to use correct formula (with values for $a, b$ and $c$).

3. **Completing the square:** Solving $x^2+bx+c=0$: $(x \pm p)^2 \pm q \pm c, p \neq 0, q \neq 0$, leading to $x=\ldots$

## Method marks for differentiation and integration:

1. **Differentiation:** Power of at least one term decreased by 1. ($x^n \rightarrow x^{n-1}$)

2. **Integration:** Power of at least one term increased by 1. ($x^n \rightarrow x^{n+1}$)

## Use of a formula:

Where a method involves using a formula that has been learnt, the advice given in recent examiners' reports is that the formula should be quoted first.

**Normal marking procedure is as follows:**

**Method mark for quoting a correct formula and attempting to use it, even if there are mistakes in the substitution of values.**

Where the formula is not quoted, the method mark can be gained by implication from correct working with values, but will be lost if there is any mistake in the working.

## Exact answers:

Examiners' reports have emphasised that where, for example, an exact answer is asked for, or working with surds is clearly required, marks will normally be lost if the candidate resorts to using rounded decimals.

## Answers without working:

The rubric says that these may gain no credit. Individual mark schemes will give details of what happens in particular cases. General policy is that if it could be done "in your head", detailed working would not be required. Most candidates do show working, but there are occasional awkward cases and if the mark scheme does not cover this, please contact your team leader for advice.

## Misreads:

A misread must be consistent for the whole question to be interpreted as such.

These are not common. In clear cases, please deduct the first 2 A (or B) marks which would have been lost by following the scheme. (Note that 2 marks is the maximum misread penalty, but that misreads which alter the nature or difficulty of the question cannot be treated so generously and it will usually be necessary here to follow the scheme as written).

Sometimes following the scheme is written is more generous to the candidate than applying the misread rule, so in this case use the scheme as written.

If in doubt please send to review or refer to Team Leader.