| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Find discriminant, state roots |
| Difficulty | Moderate -0.8 Part (i) is direct application of the discriminant formula b²-4ac with straightforward arithmetic, requiring only recall of the relationship between discriminant and number of roots. Part (ii) uses the equal roots condition (discriminant = 0) leading to a simple equation to solve for p. Both parts are routine textbook exercises with no problem-solving insight required, making this easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | Marks | Guidance |
| \(49 - 4 \times (-2) \times 3 = 73\) | M1 | Uses \(b^2 - 4ac\) |
| 2 real roots | A1 | |
| B1 \(\sqrt{3}\) | 2 real roots (ft from their value) | |
| Attempts \(b^2 - 4ac = 0\) (involving p) or attempts to complete square (involving p) | ||
| (ii) | Marks | Guidance |
| \((p+1)^2 - 64 = 0\) or \(2[(x + \frac{p+1}{4})^2 - \frac{(p+1)^2}{16} + 4] = 0\) | M1 | |
| A1 | \((p+1)^2 - 64 = 0\) aef | |
| \(p = -9, 7\) | B1 | \(p = -9\) |
| B1 4 | \(p = 7\) | |
| 7 |
**(i)** | **Marks** | **Guidance**
---|---|---
$49 - 4 \times (-2) \times 3 = 73$ | M1 | Uses $b^2 - 4ac$
2 real roots | A1 |
| B1 $\sqrt{3}$ | 2 real roots (ft from their value)
| | Attempts $b^2 - 4ac = 0$ (involving p) or attempts to complete square (involving p)
**(ii)** | **Marks** | **Guidance**
---|---|---
$(p+1)^2 - 64 = 0$ or $2[(x + \frac{p+1}{4})^2 - \frac{(p+1)^2}{16} + 4] = 0$ | M1 |
| A1 | $(p+1)^2 - 64 = 0$ aef
$p = -9, 7$ | B1 | $p = -9$
| B1 4 | $p = 7$
| **7** |6 (i) Calculate the discriminant of $- 2 x ^ { 2 } + 7 x + 3$ and hence state the number of real roots of the equation $- 2 x ^ { 2 } + 7 x + 3 = 0$.\\
(ii) The quadratic equation $2 x ^ { 2 } + ( p + 1 ) x + 8 = 0$ has equal roots. Find the possible values of $p$.
\hfill \mbox{\textit{OCR C1 2005 Q6 [7]}}