## Question 1:

### Part (a)

| Answer/Working | Mark | Guidance |
|---|---|---|
| One branch with correct shape in quadrant 1 or quadrant 2 (hyperbola-type curve with asymptotic behaviour to both axes) | **M1** | One branch correct. For one branch with the correct shape in Q1 or Q2. Give tolerance on curves bending away from the axis where it is clear the correct shape is meant, provided there is no clear intention to draw a minimum. Ignore any dashed lines/scale and just look for the shape. The branch must not clearly intentionally meet or cross either axis. Condone gaps between the branches and the axes as long as asymptotic behaviour is intended. |
| Fully correct sketch with both branches showing asymptotic behaviour to both axes | **A1** | Fully correct sketch. Branches should show asymptotic behaviour to the axes but condone gaps as above for this mark; clear bending away is A0. Ignore dotted lines at or near either axis if they are clearly intended to indicate asymptotes. |
| | **(2)** | |

# Question 1(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $3-2x^2 = \frac{1}{x^2} \Rightarrow 2x^4-3x^2+1=0 \Rightarrow x^2 = \ldots$ | M1 | Algebraic method required: multiply both sides by $x^2$ and collect terms, or collect terms and use common denominator |
| $x^2 = 1, \quad \frac{1}{2}$ e.g. $2x^4-3x^2+1=0 \Rightarrow (2x^2-1)(x^2-1)=0$ | A1 | Ignore any reference to $x=0$; if using factors may proceed directly to $x$ |
| $x = \pm 1, \pm\frac{\sqrt{2}}{2}$ | B1 | All four correct exact critical values, no others apart from $x=0$ |
| $-1 < x < -\frac{\sqrt{2}}{2}, \quad \frac{\sqrt{2}}{2} < x < 1$ | M1 | Forms regions using four critical values in correct form, two "inside" inequalities in ascending order |
| (correct regions) | A1 | Accept $\frac{1}{\sqrt{2}}$ or $\sqrt{\frac{1}{2}}$; accept set notation; no other regions |

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