| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Line intersecting quadratic curve |
| Difficulty | Moderate -0.3 This is a standard C1 question covering routine techniques: solving simultaneous equations to find intersection points, differentiation to find tangent equations, and solving linear equations. All methods are straightforward applications of basic calculus and algebra with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x^2 - 3x + 5 = 2x + 1\) | M1 | |
| \(x^2 - 5x + 4 = 0\) | M1 | |
| \((x-1)(x-4) = 0\) | ||
| \(x = 1, 4\) | A1 | |
| When \(x = 1, y = 2(1) + 1 = 3\) | A1 | |
| \(\therefore P(1,3), Q(4,9)\) | A1 | |
| (ii) \(\frac{dy}{dx} = 2x - 3\) | M1 | |
| grad \(= -1\) | A1 | |
| \(\therefore y - 3 = -(x-1)\) | M1 A1 | |
| \([y = 4 - x]\) | ||
| (iii) grad \(= 5\) | ||
| \(\therefore y - 9 = 5(x - 4)\) | M1 | |
| \(y - 9 = 5x - 20\) | ||
| \(y = 5x - 11\) | A1 | |
| (iv) \(4 - x = 5x - 11\) | M1 | |
| \(x = \frac{5}{2}\) | A1 | |
| \(\therefore \left(\frac{5}{2}, \frac{3}{2}\right)\) | A1 | (13) |
| Answer | Marks |
|---|---|
| Total | (72) |
**(i)** $x^2 - 3x + 5 = 2x + 1$ | M1 |
$x^2 - 5x + 4 = 0$ | M1 |
$(x-1)(x-4) = 0$ | |
$x = 1, 4$ | A1 |
When $x = 1, y = 2(1) + 1 = 3$ | A1 |
$\therefore P(1,3), Q(4,9)$ | A1 |
**(ii)** $\frac{dy}{dx} = 2x - 3$ | M1 |
grad $= -1$ | A1 |
$\therefore y - 3 = -(x-1)$ | M1 A1 |
$[y = 4 - x]$ | |
**(iii)** grad $= 5$ | |
$\therefore y - 9 = 5(x - 4)$ | M1 |
$y - 9 = 5x - 20$ | |
$y = 5x - 11$ | A1 |
**(iv)** $4 - x = 5x - 11$ | M1 |
$x = \frac{5}{2}$ | A1 |
$\therefore \left(\frac{5}{2}, \frac{3}{2}\right)$ | A1 | (13)
---
**Total** | **(72)** |10.\\
\includegraphics[max width=\textwidth, alt={}, center]{76efaf91-a6f3-4493-88d4-3654b023441d-3_646_773_986_477}
The diagram shows the curve $y = x ^ { 2 } - 3 x + 5$ and the straight line $y = 2 x + 1$. The curve and line intersect at the points $P$ and $Q$.\\
(i) Using algebra, show that $P$ has coordinates $( 1,3 )$ and find the coordinates of $Q$.\\
(ii) Find an equation for the tangent to the curve at $P$.\\
(iii) Show that the tangent to the curve at $Q$ has the equation $y = 5 x - 11$.\\
(iv) Find the coordinates of the point where the tangent to the curve at $P$ intersects the tangent to the curve at $Q$.
\hfill \mbox{\textit{OCR C1 Q10 [13]}}