| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing standard completing-the-square technique, basic quadratic properties (line of symmetry, discriminant), and interpreting the discriminant. All parts are routine recall and application of formulas with no problem-solving required, making it easier than average but not trivial since it requires multiple techniques. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(5(x^2+4x)-8\), \(p=5\) | B1 | \(p=5\) |
| \(= 5[(x+2)^2-4]-8\), \((x+2)^2\) seen | B1 | \((x+2)^2\) seen or \(q=2\) |
| \(= 5(x+2)^2 - 20 - 8\) | M1 | \(-8-5q^2\) or \(-\frac{8}{5}-q^2\) |
| \(= 5(x+2)^2 - 28\), \(r=-28\) | A1 (4) | \(r=-28\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = -2\) | B1 ft (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(20^2 - 4\times5\times-8 = 560\) | M1 | Uses \(b^2-4ac\) |
| \(560\) | A1 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 2 real roots | B1 (1) | 2 real roots |
## Question 6:
**Part (i):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $5(x^2+4x)-8$, $p=5$ | B1 | $p=5$ |
| $= 5[(x+2)^2-4]-8$, $(x+2)^2$ seen | B1 | $(x+2)^2$ seen or $q=2$ |
| $= 5(x+2)^2 - 20 - 8$ | M1 | $-8-5q^2$ or $-\frac{8}{5}-q^2$ |
| $= 5(x+2)^2 - 28$, $r=-28$ | A1 (4) | $r=-28$ |
**Part (ii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = -2$ | B1 ft (1) | |
**Part (iii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $20^2 - 4\times5\times-8 = 560$ | M1 | Uses $b^2-4ac$ |
| $560$ | A1 (2) | |
**Part (iv):**
| Answer | Mark | Guidance |
|--------|------|----------|
| 2 real roots | B1 (1) | 2 real roots |
---6 (i) Express $5 x ^ { 2 } + 20 x - 8$ in the form $p ( x + q ) ^ { 2 } + r$.\\
(ii) State the equation of the line of symmetry of the curve $y = 5 x ^ { 2 } + 20 x - 8$.\\
(iii) Calculate the discriminant of $5 x ^ { 2 } + 20 x - 8$.\\
(iv) State the number of real roots of the equation $5 x ^ { 2 } + 20 x - 8 = 0$.
\hfill \mbox{\textit{OCR C1 2009 Q6 [8]}}