| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Show line is tangent, verify |
| Difficulty | Moderate -0.3 This is a structured multi-part question covering completing the square, curve sketching, and simultaneous equations to verify a tangent. While it requires multiple techniques, each part is routine C1 material with clear guidance. The 'show that' in part (iii) is straightforward once the simultaneous equations are solved, making this slightly easier than average but not trivial. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(x - \frac{5}{2}\right)^2 - \frac{1}{4}\) | B3 | B1 for \(a = \frac{5}{2}\); M1 for \(6 - \text{their } a^2\) soi; condone \(\left(x-\frac{5}{2}\right)^2 - \frac{1}{4} = 0\); condone omission of index |
| \(\left(\frac{5}{2}, -\frac{1}{4}\right)\) | B1 | Accept \(x = 2.5, y = -0.25\) |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((2, 0)\) and \((3, 0)\) | B2 | B1 each; or B1 for both correct plus extra; or M1 for \((x-2)(x-3)\) or correct use of formula or for their \(a \pm \sqrt{\text{their } b}\) from (i) |
| \((0, 6)\) | B1 | |
| Graph of quadratic the correct way up and crossing both axes | B1 | Ignore label of turning point; condone stopping at \(y\)-axis; condone 'U' shape or slight curving back; must not be ruled |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^2 - 5x + 6 = 2 - x\) | M1 | For attempt to equate or subtract equations or attempt at rearrangement and elimination of \(x\); accept calculus approach: \(y' = 2x - 5\) |
| \(x^2 - 4x + 4\ [= 0]\) | M1 | For rearrangement to zero ft and collection of terms; condone one error; if using completing the square, need to get as far as \((x-k)^2 = c\); \([(x-2)^2 = 0\) if correct]; use of \(y' = -1\) M1 |
## Question 11(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(x - \frac{5}{2}\right)^2 - \frac{1}{4}$ | B3 | B1 for $a = \frac{5}{2}$; M1 for $6 - \text{their } a^2$ soi; condone $\left(x-\frac{5}{2}\right)^2 - \frac{1}{4} = 0$; condone omission of index |
| $\left(\frac{5}{2}, -\frac{1}{4}\right)$ | B1 | Accept $x = 2.5, y = -0.25$ |
| **[4]** | | |
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## Question 11(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(2, 0)$ and $(3, 0)$ | B2 | B1 each; or B1 for both correct plus extra; or M1 for $(x-2)(x-3)$ or correct use of formula or for their $a \pm \sqrt{\text{their } b}$ from (i) |
| $(0, 6)$ | B1 | |
| Graph of quadratic the correct way up and crossing both axes | B1 | Ignore label of turning point; condone stopping at $y$-axis; condone 'U' shape or slight curving back; must not be ruled |
| **[4]** | | |
---
## Question 11(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 - 5x + 6 = 2 - x$ | M1 | For attempt to equate or subtract equations or attempt at rearrangement and elimination of $x$; accept calculus approach: $y' = 2x - 5$ |
| $x^2 - 4x + 4\ [= 0]$ | M1 | For rearrangement to zero ft and collection of terms; condone one error; if using completing the square, need to get as far as $(x-k)^2 = c$; $[(x-2)^2 = 0$ if correct]; use of $y' = -1$ M1 |11 (i) Express $x ^ { 2 } - 5 x + 6$ in the form $( x - a ) ^ { 2 } - b$. Hence state the coordinates of the turning point of the curve $y = x ^ { 2 } - 5 x + 6$.\\
(ii) Find the coordinates of the intersections of the curve $y = x ^ { 2 } - 5 x + 6$ with the axes and sketch this curve.\\
(iii) Solve the simultaneous equations $y = x ^ { 2 } - 5 x + 6$ and $x + y = 2$. Hence show that the line $x + y = 2$ is a tangent to the curve $y = x ^ { 2 } - 5 x + 6$ at one of the points where the curve intersects the axes.
\hfill \mbox{\textit{OCR MEI C1 2013 Q11 [12]}}