| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Sketch quadratic curve |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing completing the square, a standard C1 technique. All parts follow directly from part (i) with minimal problem-solving: reading off the minimum point, solving by taking square roots, and sketching using these features. This is easier than average as it's purely procedural with no conceptual challenges. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02f Solve quadratic equations: including in a function of unknown1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| \(4x^2 - 24x + 27 = 4(x^2 - 6x) + 27\) | M1 | Factor out 4 |
| \(= 4(x-3)^2 - 36 + 27\) | A1 | Complete the square |
| \(= 4(x-3)^2 - 9\) | A1 A1 | \(a=4\), \(b=3\), \(c=-9\) |
| Answer | Marks | Guidance |
|---|---|---|
| Minimum point: \((3, -9)\) | B1 B1 | Both coordinates correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(4(x-3)^2 = 9\) | M1 | Using completed square form |
| \((x-3)^2 = \frac{9}{4}\) | A1 | |
| \(x - 3 = \pm\frac{3}{2}\) | M1 | |
| \(x = \frac{9}{2}\) or \(x = \frac{3}{2}\) | A1 | Both correct |
| Answer | Marks | Guidance |
|---|---|---|
| Sketch: U-shaped parabola | B1 | Correct shape |
| Minimum at \((3,-9)\) | B1 | Minimum labelled |
| Crosses \(y\)-axis at \((0, 27)\) | B1 | \(y\)-intercept correct |
## Question 12:
**(i)**
$4x^2 - 24x + 27 = 4(x^2 - 6x) + 27$ | M1 | Factor out 4
$= 4(x-3)^2 - 36 + 27$ | A1 | Complete the square
$= 4(x-3)^2 - 9$ | A1 A1 | $a=4$, $b=3$, $c=-9$
**(ii)**
Minimum point: $(3, -9)$ | B1 B1 | Both coordinates correct
**(iii)**
$4(x-3)^2 = 9$ | M1 | Using completed square form
$(x-3)^2 = \frac{9}{4}$ | A1 |
$x - 3 = \pm\frac{3}{2}$ | M1 |
$x = \frac{9}{2}$ or $x = \frac{3}{2}$ | A1 | Both correct
**(iv)**
Sketch: U-shaped parabola | B1 | Correct shape
Minimum at $(3,-9)$ | B1 | Minimum labelled
Crosses $y$-axis at $(0, 27)$ | B1 | $y$-intercept correct
---12 (i) Write $4 x ^ { 2 } - 24 x + 27$ in the form $a ( x - b ) ^ { 2 } + c$.\\
(ii) State the coordinates of the minimum point on the curve $y = 4 x ^ { 2 } - 24 x + 27$.\\
(iii) Solve the equation $4 x ^ { 2 } - 24 x + 27 = 0$.\\
(iv) Sketch the graph of the curve $y = 4 x ^ { 2 } - 24 x + 27$.
\hfill \mbox{\textit{OCR MEI C1 2007 Q12 [12]}}