| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Show line is tangent, verify |
| Difficulty | Moderate -0.8 This is a straightforward line-curve intersection problem requiring substitution to form a quadratic equation, followed by verification that one intersection point is the vertex. Both parts involve routine algebraic manipulation with no conceptual challenges—easier than average A-level questions. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02d Quadratic functions: graphs and discriminant conditions1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Eliminates \(y\) to get \(2x^2 - 7x + 6 = 0\) or \(2y^2 - 5y + 3 = 0\) | M1 A1 | \(y\) (or \(x\)) must be eliminated completely. Setting 3 term quadratic to 0 + soln |
| \(\rightarrow (2x - 3)(x - 2) = 0\) | DM1 | Both correct. |
| \(\rightarrow x = 2\) or \(1\frac{1}{2}\) | A1 | [4] |
| (ii) \(\frac{dy}{dx} = 4x - 8 = 0\) | M1 DM1 | Attempt to differentiate. Setting differential to 0. |
| \(x = 2\) | A1 | co |
| or completes the square and states stationary at \(x = 2\). | [3] |
**(i)** Eliminates $y$ to get $2x^2 - 7x + 6 = 0$ or $2y^2 - 5y + 3 = 0$ | M1 A1 | $y$ (or $x$) must be eliminated completely. Setting 3 term quadratic to 0 + soln
$\rightarrow (2x - 3)(x - 2) = 0$ | DM1 | Both correct.
$\rightarrow x = 2$ or $1\frac{1}{2}$ | A1 | [4]
**(ii)** $\frac{dy}{dx} = 4x - 8 = 0$ | M1 DM1 | Attempt to differentiate. Setting differential to 0.
$x = 2$ | A1 | co
or completes the square and states stationary at $x = 2$. | [3]4 The equation of a curve $C$ is $y = 2 x ^ { 2 } - 8 x + 9$ and the equation of a line $L$ is $x + y = 3$.\\
(i) Find the $x$-coordinates of the points of intersection of $L$ and $C$.\\
(ii) Show that one of these points is also the stationary point of $C$.
\hfill \mbox{\textit{CAIE P1 2008 Q4 [7]}}