| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Prove/show always positive |
| Difficulty | Moderate -0.8 This is a straightforward C1 completing-the-square question with standard parts: completing the square (routine), sketching a parabola, solving a quadratic inequality, and checking if a shifted parabola crosses the x-axis. All parts use basic techniques with no novel problem-solving required, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02g Inequalities: linear and quadratic in single variable1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| i \((x - 2.5)^2\) o.e.; \(-2.5^2 + 8\); \((x - 2.5)^2 + 7/4\) o.e.; min \(y = 7/4\) o.e. [so above \(x\) axis] or commenting \((x - 2.5)^2 \geq 0\) | Marks: M1, M1, A1, B1 | Guidance: ft, dep on \((x - a)^2 + b\) with \(b\) positive; condone starting again, showing \(b^2 - 4ac < 0\) or using calculus |
| ii correct symmetrical quadratic shape; 8 marked as intercept on \(y\) axis; tp (5/2, 7/4) o.e. or ft from (i) | Marks: G1, G1, G1 | Guidance: or (0, 8) seen in table |
| iii \(x^2 - 5x - 6\) seen or used; \(-1\) and \(6\) obtained; \(x < -1\) and \(x > 6\) isw or ft their solns | Marks: M1, M1, M1 | Guidance: or \((x - 2.5)^2 [> or =] 12.25\) or \(14 - b\); also implies first M1; if M0, allow B1 for one of \(x < -1\) and \(x > 6\) |
| iv min \(= (2.5, -8.25)\) or ft from (i) so yes, crosses | Marks: M1, A1 | Guidance: 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 eg \(b^2 - 4ac = 33\); allow M1A0 for valid comment but error in -8.25 ft; allow M1 for showing \(y\) can be neg eg (0, \(-2\)) found and A1 for correct conclusion |
**i** $(x - 2.5)^2$ o.e.; $-2.5^2 + 8$; $(x - 2.5)^2 + 7/4$ o.e.; min $y = 7/4$ o.e. [so above $x$ axis] or commenting $(x - 2.5)^2 \geq 0$ | **Marks:** M1, M1, A1, B1 | **Guidance:** ft, dep on $(x - a)^2 + b$ with $b$ positive; condone starting again, showing $b^2 - 4ac < 0$ or using calculus | **Total:** 4
**ii** correct symmetrical quadratic shape; 8 marked as intercept on $y$ axis; tp (5/2, 7/4) o.e. or ft from (i) | **Marks:** G1, G1, G1 | **Guidance:** or (0, 8) seen in table | **Total:** 3
**iii** $x^2 - 5x - 6$ seen or used; $-1$ and $6$ obtained; $x < -1$ and $x > 6$ isw or ft their solns | **Marks:** M1, M1, M1 | **Guidance:** or $(x - 2.5)^2 [> or =] 12.25$ or $14 - b$; also implies first M1; if M0, allow B1 for one of $x < -1$ and $x > 6$ | **Total:** 3
**iv** min $= (2.5, -8.25)$ or ft from (i) so yes, crosses | **Marks:** M1, A1 | **Guidance:** 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 eg $b^2 - 4ac = 33$; allow M1A0 for valid comment but error in -8.25 ft; allow M1 for showing $y$ can be neg eg (0, $-2$) found and A1 for correct conclusion | **Total:** 2
**Total for Question 11:** 1211 (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 2008 Q11 [12]}}