| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Prove/show always positive |
| Difficulty | Moderate -0.3 This is a standard C1 completing-the-square question with routine multi-part applications. Part (i) is textbook completion of the square, parts (ii-iv) are straightforward applications requiring no novel insight. Slightly easier than average due to being highly procedural, though the multiple parts require careful execution. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02f Solve quadratic equations: including in a function of unknown1.02g Inequalities: linear and quadratic in single variable1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \((x - 2.5)^2\) o.e. | M1 | |
| \(-2.5^2 + 8\) | M1 | for clear attempt at \(-2.5^2\) |
| \((x - 2.5)^2 + \frac{7}{4}\) o.e. | A1 | allow M2A0 for \((x - 2.5) + \frac{7}{4}\) o.e. with no \((x - 2.5)^2\) seen |
| min \(y = \frac{7}{4}\) o.e. [so above \(x\) axis] or commenting \((x - 2.5)^2 \geq 0\) | B1 | ft, dep on \((x-a)^2 + b\) with \(b\) positive; condone starting again, showing \(b^2 - 4ac < 0\) or using calculus |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| Correct symmetrical quadratic shape | G1 | |
| 8 marked as intercept on \(y\) axis | G1 | or \((0, 8)\) seen in table |
| tp \((\frac{5}{2}, \frac{7}{4})\) o.e. or ft from (i) | G1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(x^2 - 5x - 6\) seen or used | M1 | or \((x - 2.5)^2\ [> \text{or} =]\ 12.25\) or ft \(14 - b\) |
| \(-1\) and \(6\) obtained | M1 | also implies first M1 |
| \(x < -1\) and \(x > 6\) isw or ft their solns | M1 | if M0, allow B1 for one of \(x < -1\) and \(x > 6\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| min \(= (2.5,\ -8.25)\) or ft from (i) | M1 | or M1 for other clear comment re translated 10 down and A1 for referring to min in (i) or graph in (ii); or M1 for correct method for solving \(x^2 - 5x - 2 = 0\) or using \(b^2 - 4ac\) with this and A1 for showing real solns e.g. \(b^2 - 4ac = 33\); allow M1A0 for valid comment but error in \(-8.25\) ft; allow M1 for showing \(y\) can be neg e.g. \((0, -2)\) found |
| so yes, crosses | A1 | and A1 for correct conclusion |
## Question 15:
### Part (i):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $(x - 2.5)^2$ o.e. | M1 | |
| $-2.5^2 + 8$ | M1 | for clear attempt at $-2.5^2$ |
| $(x - 2.5)^2 + \frac{7}{4}$ o.e. | A1 | allow M2A0 for $(x - 2.5) + \frac{7}{4}$ o.e. with no $(x - 2.5)^2$ seen |
| min $y = \frac{7}{4}$ o.e. [so above $x$ axis] or commenting $(x - 2.5)^2 \geq 0$ | B1 | ft, dep on $(x-a)^2 + b$ with $b$ positive; condone starting again, showing $b^2 - 4ac < 0$ or using calculus | **Total: 4** |
### Part (ii):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Correct symmetrical quadratic shape | G1 | |
| 8 marked as intercept on $y$ axis | G1 | or $(0, 8)$ seen in table |
| tp $(\frac{5}{2}, \frac{7}{4})$ o.e. or ft from (i) | G1 | | **Total: 3** |
### Part (iii):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $x^2 - 5x - 6$ seen or used | M1 | or $(x - 2.5)^2\ [> \text{or} =]\ 12.25$ or ft $14 - b$ |
| $-1$ and $6$ obtained | M1 | also implies first M1 |
| $x < -1$ and $x > 6$ isw or ft their solns | M1 | if M0, allow B1 for one of $x < -1$ and $x > 6$ | **Total: 3** |
### Part (iv):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| min $= (2.5,\ -8.25)$ or ft from (i) | M1 | or M1 for other clear comment re translated 10 down and A1 for referring to min in (i) or graph in (ii); or M1 for correct method for solving $x^2 - 5x - 2 = 0$ or using $b^2 - 4ac$ with this and A1 for showing real solns e.g. $b^2 - 4ac = 33$; allow M1A0 for valid comment but error in $-8.25$ ft; allow M1 for showing $y$ can be neg e.g. $(0, -2)$ found |
| so yes, crosses | A1 | and A1 for correct conclusion | **Total: 2** |
**Question Total: 12**15 (i) Write $x ^ { 2 } - 5 x + 8$ in the form $( x - a ) ^ { 2 } + b$ and hence show that $x ^ { 2 } - 5 x + 8 > 0$ for all values of $x$.\\
(ii) Sketch the graph of $y = x ^ { 2 } - 5 x + 8$, showing the coordinates of the turning point.\\
(iii) Find the set of values of $x$ for which $x ^ { 2 } - 5 x + 8 > 14$.\\
(iv) If $\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8$, does the graph of $y = \mathrm { f } ( x ) - 10$ cross the $x$-axis? Show how you decide.
\hfill \mbox{\textit{OCR MEI C1 Q15 [12]}}