| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Solving quadratics and applications |
| Type | Completing the square, form and properties |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing routine completing-the-square technique, reading minimum point coordinates, and solving a basic quadratic inequality by factorization. All parts are standard textbook exercises requiring only procedural knowledge with no problem-solving insight needed. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02g Inequalities: linear and quadratic in single variable1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2(x^2 + \frac{5}{2}x)\) | B1 | \(\left(x + \frac{5}{4}\right)^2\) |
| \(= 2\left[\left(x+\frac{5}{4}\right)^2 - \frac{25}{16}\right]\) | M1 | \(q = -2p^2\) |
| \(= 2\left(x+\frac{5}{4}\right)^2 - \frac{25}{8}\) | A1 | 3 \(q = -\frac{25}{8}\) c.w.o. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left(-\frac{5}{4}, -\frac{25}{8}\right)\) | B1\(\checkmark\) B1\(\checkmark\) | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = -\frac{5}{4}\) | B1 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x(2x+5) > 0\) | M1 | Correct method to find roots |
| A1 | \(0, -\frac{5}{2}\) seen | |
| \(x < -\frac{5}{2},\ x > 0\) | M1 | Correct method to solve quadratic inequality |
| A1 | 4 (not wrapped, strict inequalities, no 'and') |
# Question 8:
## Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2(x^2 + \frac{5}{2}x)$ | B1 | $\left(x + \frac{5}{4}\right)^2$ |
| $= 2\left[\left(x+\frac{5}{4}\right)^2 - \frac{25}{16}\right]$ | M1 | $q = -2p^2$ |
| $= 2\left(x+\frac{5}{4}\right)^2 - \frac{25}{8}$ | A1 | **3** $q = -\frac{25}{8}$ c.w.o. |
## Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left(-\frac{5}{4}, -\frac{25}{8}\right)$ | B1$\checkmark$ B1$\checkmark$ | **2** |
## Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = -\frac{5}{4}$ | B1 | **1** |
## Part (iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x(2x+5) > 0$ | M1 | Correct method to find roots |
| | A1 | $0, -\frac{5}{2}$ seen |
| $x < -\frac{5}{2},\ x > 0$ | M1 | Correct method to solve quadratic inequality |
| | A1 | **4** (not wrapped, strict inequalities, no 'and') |
---8 (i) Express $2 x ^ { 2 } + 5 x$ in the form $2 ( x + p ) ^ { 2 } + q$.\\
(ii) State the coordinates of the minimum point of the curve $y = 2 x ^ { 2 } + 5 x$.\\
(iii) State the equation of the normal to the curve at its minimum point.\\
(iv) Solve the inequality $2 x ^ { 2 } + 5 x > 0$.
\hfill \mbox{\textit{OCR C1 2010 Q8 [10]}}