Easy -1.2 This is a straightforward discriminant question requiring rearrangement to standard form (2x² - x + 7 = 0), then showing b² - 4ac < 0. It's simpler than average A-level questions as it only requires one standard technique with no problem-solving insight, though the 'show detailed reasoning' requirement adds minimal complexity.
Must be clearly argued from a correct discriminant which need not be evaluated if clearly negative. Expression for discriminant must be precise if not evaluated.
Stationary point is minimum so \(y \geq 6.875\) so is never zero
A1
Must be clearly argued from correct working
[3]
# Question 1:
**DR** (Detailed Reasoning required)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Rearrange as $2x^2 - x + 7 = 0$ | M1 (AO 1.1a) | |
| Discriminant is $(-1)^2 - 4 \times 2 \times 7$ | M1 dep (AO 1.1a) | |
| $= -55 < 0$ so no real roots | A1 (AO 2.2a) | Must be clearly argued from a correct discriminant which need not be evaluated if clearly negative. Expression for discriminant must be precise if not evaluated. |
| **[3]** | | |
**Alternative method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Rearrange as $2x^2 - x + 7 = 0$ | M1 | |
| Attempt to complete the square | M1 dep | Allow for $2(x-0.25)^2 + \ldots$ soi |
| $2(x - 0.25)^2 + 6.875 = 0$ | | |
| $x - 0.25 = \pm\sqrt{-3.4375}$ so no real roots | A1 | Must be clearly argued from correct working |
| **[3]** | | |
**Second alternative method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Rearrange as $[y =]\, 2x^2 - x + 7 = 0$ | M1 | Must equate to zero |
| Differentiate $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 4x - 1 = 0$ | M1 dep | |
| Stationary point at $(0.25,\ 6.875)$ | | |
| Stationary point is minimum so $y \geq 6.875$ so is never zero | A1 | Must be clearly argued from correct working |
| **[3]** | | |
1 In this question you must show detailed reasoning.
Show that the equation $x = 7 + 2 x ^ { 2 }$ has no real roots.
\hfill \mbox{\textit{OCR MEI AS Paper 1 2019 Q1 [3]}}