| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Solving quadratics and applications |
| Type | Quadratic inequality solving |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard quadratic techniques: completing the square or using the quadratic formula for part (i), basic curve sketching for part (ii), and applying roots to solve an inequality for part (iii). All parts are routine C1 content requiring no problem-solving insight, though the surd manipulation and inequality notation make it slightly less trivial than pure recall. |
| 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 |
|---|---|---|
| \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] | M1 | Correct method to solve quadratic |
| \[x = \frac{-8 \pm \sqrt{64-40}}{2}\] | ||
| \[x = \frac{-8 \pm \sqrt{24}}{2}\] | A1 | |
| \[x = \frac{-8 \pm 2\sqrt{6}}{2}\] | ||
| \[x = -4 \pm \sqrt{6}\] | A1 3 |
| Answer | Marks |
|---|---|
| \((x+4)^2 - 16 + 10 = 0\) | M1 A1 |
| \((x+4)^2 = 6\) | |
| \[x + 4 = \pm\sqrt{6}\] | M1 A1 |
| \[x = \pm\sqrt{6} - 4\] | A1 |
| Answer | Marks |
|---|---|
| Positive parabola | B1 |
| Parabola cutting y-axis at (0, 10) where (0, 10) is not min/max point | B1 |
| Parabola with 2 negative roots | B1 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \[x \leq -\sqrt{6} - 4, x \geq \sqrt{6} - 4\] | M1 A1 ft 2 | Fully correct answer, ft from roots found in (i). (allow \(<\), \(>\)) |
## 6(i)
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ | M1 | Correct method to solve quadratic |
$$x = \frac{-8 \pm \sqrt{64-40}}{2}$$ | | |
$$x = \frac{-8 \pm \sqrt{24}}{2}$$ | A1 | |
$$x = \frac{-8 \pm 2\sqrt{6}}{2}$$ | | |
$$x = -4 \pm \sqrt{6}$$ | A1 3 | |
OR
$(x+4)^2 - 16 + 10 = 0$ | M1 A1 |
$(x+4)^2 = 6$ | |
$$x + 4 = \pm\sqrt{6}$$ | M1 A1 |
$$x = \pm\sqrt{6} - 4$$ | A1 |
## 6(ii)
Positive parabola | B1 |
Parabola cutting y-axis at (0, 10) where (0, 10) is not min/max point | B1 |
Parabola with 2 negative roots | B1 3 |
## 6(iii)
$$x \leq -\sqrt{6} - 4, x \geq \sqrt{6} - 4$$ | M1 A1 ft 2 | Fully correct answer, ft from roots found in (i). (allow $<$, $>$) |
---6 (i) Solve the equation $x ^ { 2 } + 8 x + 10 = 0$, giving your answers in simplified surd form.\\
(ii) Sketch the curve $y = x ^ { 2 } + 8 x + 10$, giving the coordinates of the point where the curve crosses the $y$-axis.\\
(iii) Solve the inequality $x ^ { 2 } + 8 x + 10 \geqslant 0$.
\hfill \mbox{\textit{OCR C1 2008 Q6 [8]}}