| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Solve quadratic inequality |
| Difficulty | Moderate -0.3 This is a straightforward C1 question testing standard techniques: quadratic formula with surds, interpreting roots for an inequality, and sketching a cubic from factored form. All parts follow routine procedures with no problem-solving insight required, making it slightly easier than average but not trivial due to the surd manipulation and multi-part nature. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02g Inequalities: linear and quadratic in single variable1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = \frac{8\pm\sqrt{(-8)^2-(4\times-1\times5)}}{-2}\) | M1 | Correct method to solve quadratic |
| \(= \frac{8\pm\sqrt{84}}{-2}\) | A1 | \(x=\frac{8\pm\sqrt{84}}{-2}\) |
| \(= -4-\sqrt{21}\) or \(= -4+\sqrt{21}\) | A1 (3) | Both roots correct and simplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x \leq -4-\sqrt{21}\), \(x \geq -4+\sqrt{21}\) | M1 | Identifying \(x\leq\) their lower root, \(x\geq\) their higher root |
| \(x \leq -4-\sqrt{21}\), \(x \geq -4+\sqrt{21}\) | A1 (2) | (not wrapped, no 'and') |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Roughly correct negative cubic with max and min | B1 | |
| \((-4, 0)\) | B1 | |
| \((0, 20)\) | B1 | |
| Cubic with 3 distinct real roots | B1 | |
| Completely correct graph | B1 (5) |
## Question 8:
**Part (i):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = \frac{8\pm\sqrt{(-8)^2-(4\times-1\times5)}}{-2}$ | M1 | Correct method to solve quadratic |
| $= \frac{8\pm\sqrt{84}}{-2}$ | A1 | $x=\frac{8\pm\sqrt{84}}{-2}$ |
| $= -4-\sqrt{21}$ or $= -4+\sqrt{21}$ | A1 (3) | Both roots correct and simplified |
**Part (ii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $x \leq -4-\sqrt{21}$, $x \geq -4+\sqrt{21}$ | M1 | Identifying $x\leq$ their lower root, $x\geq$ their higher root |
| $x \leq -4-\sqrt{21}$, $x \geq -4+\sqrt{21}$ | A1 (2) | (not wrapped, no 'and') |
**Part (iii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Roughly correct negative cubic with max and min | B1 | |
| $(-4, 0)$ | B1 | |
| $(0, 20)$ | B1 | |
| Cubic with 3 distinct real roots | B1 | |
| Completely correct graph | B1 (5) | |
---8 (i) Solve the equation $5 - 8 x - x ^ { 2 } = 0$, giving your answers in simplified surd form.\\
(ii) Solve the inequality $5 - 8 x - x ^ { 2 } \leqslant 0$.\\
(iii) Sketch the curve $y = \left( 5 - 8 x - x ^ { 2 } \right) ( x + 4 )$, giving the coordinates of the points where the curve crosses the coordinate axes.
\hfill \mbox{\textit{OCR C1 2009 Q8 [10]}}