| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| 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 routine quadratic skills: solving by formula/factoring, finding the vertex by differentiation or completing the square, sketching with intercepts, and identifying increasing intervals. All parts are standard C1 techniques with no problem-solving required, making it easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| \((3x + 7)(3x - 1) = 0\) | M1, A1 | Correct method to find roots; Correct factorisation |
| \(x = -\dfrac{7}{3},\ x = \dfrac{1}{3}\) | A1 [3] | Correct roots |
| Answer | Marks | Guidance |
|---|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 18x + 18\) | M1 | Attempt to differentiate \(y\) |
| \(18x + 18 = 0\) | M1 | Uses \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\) |
| \(x = -1\) | A1 | |
| \(y = -16\) | A1ft [4] |
| Answer | Marks |
|---|---|
| Positive quadratic curve | B1 |
| \(y\) intercept \((0, -7)\) | B1 |
| Good graph with correct roots indicated and minimum point in correct quadrant | B1 [3] |
| Answer | Marks |
|---|---|
| \(x > -1\) | B1 [1] |
## Question 10:
### Part (i):
$(3x + 7)(3x - 1) = 0$ | M1, A1 | Correct method to find roots; Correct factorisation
$x = -\dfrac{7}{3},\ x = \dfrac{1}{3}$ | A1 [3] | Correct roots
### Part (ii):
$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 18x + 18$ | M1 | Attempt to differentiate $y$
$18x + 18 = 0$ | M1 | Uses $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0$
$x = -1$ | A1 |
$y = -16$ | A1ft [4] |
### Part (iii):
Positive quadratic curve | B1 |
$y$ intercept $(0, -7)$ | B1 |
Good graph with correct roots indicated and minimum point in correct quadrant | B1 [3] |
### Part (iv):
$x > -1$ | B1 [1] |
---10 (i) Solve the equation $9 x ^ { 2 } + 18 x - 7 = 0$.\\
(ii) Find the coordinates of the stationary point on the curve $y = 9 x ^ { 2 } + 18 x - 7$.\\
(iii) Sketch the curve $y = 9 x ^ { 2 } + 18 x - 7$, giving the coordinates of all intercepts with the axes.\\
(iv) For what values of $x$ does $9 x ^ { 2 } + 18 x - 7$ increase as $x$ increases?
\hfill \mbox{\textit{OCR C1 2009 Q10 [11]}}