| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Tangency condition for line and curve |
| Difficulty | Moderate -0.3 This is a standard C1 simultaneous equations question involving a quadratic and linear equation. The substitution method is straightforward, leading to a quadratic with no real solutions. While it requires careful algebraic manipulation (5 marks suggests multiple steps), it's a routine textbook exercise with no novel insight needed beyond recognizing the discriminant interpretation. Slightly easier than average due to its mechanical nature. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution |
| Answer | Marks | Guidance |
|---|---|---|
| *M1 | Substitute for \(y\) or attempt to get an equation in 1 variable only | or \(10x + 2(2x^2 - 3x - 5) + 11 = 0\). If \(x\) is eliminated, expect \(k(8y)^2 + 48y + 72) = 0\) |
| A1 | Obtain correct 3 term quadratic – could be a multiple e.g. \(2x^2 + 2x + 0.5 = 0\) | |
| DM1 | Correct method to solve resulting 3 term quadratic | |
| A1 | SC If DM0 and \(x = -\frac{1}{2}\) spotted. B1 for \(x\) value, B1 for \(y\) value. B1 justifying only one root | |
| A1 | ||
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| B1√ | Must be consistent with their answers to their quadratic in (i). 1 repeated root – indicates one point. Accept tangent, meet at, intersect, touch etc. but do not accept cross. 2 roots – indicates meet at two points. 0 roots – indicates do not meet. Do not accept "do not cross" | Follow through from their solution to (i) |
| [1] |
### (i)
$2x^2 - 3x - 5 = \frac{-10x - 11}{2}$, $4x^2 + 4x + 1 = 0$, $(2x + 1)(2x + 1) = 0$, $x = -\frac{1}{2}$, $y = -3$
| *M1 | Substitute for $y$ or attempt to get an equation in 1 variable only | or $10x + 2(2x^2 - 3x - 5) + 11 = 0$. If $x$ is eliminated, expect $k(8y)^2 + 48y + 72) = 0$ |
| A1 | Obtain correct 3 term quadratic – could be a multiple e.g. $2x^2 + 2x + 0.5 = 0$ | |
| DM1 | Correct method to solve resulting 3 term quadratic | |
| A1 | | SC If DM0 and $x = -\frac{1}{2}$ spotted. B1 for $x$ value, B1 for $y$ value. B1 justifying only one root |
| A1 | | |
| [5] | | |
### (ii)
Line is a tangent to the curve
| B1√ | Must be consistent with their answers to their quadratic in (i). 1 repeated root – indicates one point. Accept tangent, meet at, intersect, touch etc. but do not accept cross. 2 roots – indicates meet at two points. 0 roots – indicates do not meet. Do not accept "do not cross" | Follow through from their solution to (i) |
| [1] | | |\begin{enumerate}[label=(\roman*)]
\item Solve the simultaneous equations
$$y = 2x^2 - 3x - 5, \quad 10x + 2y + 11 = 0.$$ [5]
\item What can you deduce from the answer to part (i) about the curve $y = 2x^2 - 3x - 5$ and the line $10x + 2y + 11 = 0$? [1]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 2013 Q4 [6]}}