| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Write inequalities from graph |
| Difficulty | Moderate -0.8 Part (a) requires reading inequalities directly from a graph with clear boundaries (a routine skill), while part (b) involves solving a quadratic inequality—a standard technique requiring rearrangement, factoring/quadratic formula, and sign analysis. Both are textbook exercises with no novel problem-solving required, making this easier than average. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation1.02i Represent inequalities: graphically on coordinate plane |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y > x + 5\) | B1 | Allow interchange of \(>\) and \(\geq\) or \(<\) and \(\leq\) for one inequality as long as direction is correct |
| \(y \leq 8 - 2x - x^2\) | B1 [2] | Both inequalities fully correct. Allow \(x + 5 < y \leq 8 - 2x - x^2\) oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Boundary values when \(8 - 2x - x^2 = x + 5\), giving \(x^2 + 3x - 3 = 0\) | M1 | A correct three term quadratic equation (or inequality) must be seen. Any method including BC acceptable for solving the quadratic clearly seen in form \(ax^2 + bx + c = 0\) |
| Giving \(x = \dfrac{-3 \pm \sqrt{21}}{2}\) | A1 | Correct roots of the equation, must be exact |
| \(\dfrac{-3-\sqrt{21}}{2} < x < \dfrac{-3+\sqrt{21}}{2}\) or \(\left\{x : x > \dfrac{-3-\sqrt{21}}{2}\right\} \cap \left\{x : x < \dfrac{-3+\sqrt{21}}{2}\right\}\) | B1 [3] | Shows correct inequality FT their roots. Allow B1 for \(-3.79 < x < 0.79\) www |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y > x + 5$ | B1 | Allow interchange of $>$ and $\geq$ or $<$ and $\leq$ for one inequality as long as direction is correct |
| $y \leq 8 - 2x - x^2$ | B1 [2] | Both inequalities fully correct. Allow $x + 5 < y \leq 8 - 2x - x^2$ oe |
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## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Boundary values when $8 - 2x - x^2 = x + 5$, giving $x^2 + 3x - 3 = 0$ | M1 | A correct three term quadratic equation (or inequality) must be seen. Any method including BC acceptable for solving the quadratic clearly seen in form $ax^2 + bx + c = 0$ |
| Giving $x = \dfrac{-3 \pm \sqrt{21}}{2}$ | A1 | Correct roots of the equation, must be exact |
| $\dfrac{-3-\sqrt{21}}{2} < x < \dfrac{-3+\sqrt{21}}{2}$ or $\left\{x : x > \dfrac{-3-\sqrt{21}}{2}\right\} \cap \left\{x : x < \dfrac{-3+\sqrt{21}}{2}\right\}$ | B1 [3] | Shows correct inequality FT their roots. Allow B1 for $-3.79 < x < 0.79$ www |
---3 (a) The diagram shows the line $y = x + 5$ and the curve $y = 8 - 2 x - x ^ { 2 }$. The shaded region is the finite region between the line and the curve. The curved part of the boundary is included in the region but the straight part is not included.
Write down the inequalities that define the shaded region.\\
\includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-04_846_716_1379_322}
\section*{(b) In this question you must show detailed reasoning.}
Solve the inequality $8 - 2 x - x ^ { 2 } > x + 5$ giving your answer in exact form.
\hfill \mbox{\textit{OCR MEI Paper 1 2021 Q3 [5]}}