| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Line-curve intersection conditions |
| Difficulty | Moderate -0.3 This is a standard discriminant problem requiring students to set equations equal, form a quadratic, and apply b²-4ac < 0 for no intersection. Part (ii) extends to the tangency condition (discriminant = 0). While it involves multiple steps and careful algebraic manipulation, it's a routine textbook exercise on a core AS-level topic with no novel insight required, making it slightly easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02f Solve quadratic equations: including in a function of unknown1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(kx^2 - kx + 1 = 0\) | M1 | \(y\) eliminated |
| \(k^2 - 4k < 0\) | M1 | Applying \(b^2 - 4ac < 0\) or \(=\) or \(\leq\) or \(\geq\) |
| \(0 < k < 4\) | A1 | co |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(k = 4\) only | B1\(\sqrt{}\) | ft from their \(k^2 - 4k = 0\) (Not \(k=0\)) |
| \((2x-1)^2 = 0\) | M1 | ft from their \(k\) |
| \(x = \frac{1}{2},\ y = 2\) or \((\frac{1}{2}, 2)\) | A1, A1 |
## Question 6:
**Part (i)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $kx^2 - kx + 1 = 0$ | M1 | $y$ eliminated |
| $k^2 - 4k < 0$ | M1 | Applying $b^2 - 4ac < 0$ or $=$ or $\leq$ or $\geq$ |
| $0 < k < 4$ | A1 | co |
**Part (ii)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $k = 4$ only | B1$\sqrt{}$ | ft from their $k^2 - 4k = 0$ (Not $k=0$) |
| $(2x-1)^2 = 0$ | M1 | ft from their $k$ |
| $x = \frac{1}{2},\ y = 2$ or $(\frac{1}{2}, 2)$ | A1, A1 | |
---6 A curve has equation $y = k x ^ { 2 } + 1$ and a line has equation $y = k x$, where $k$ is a non-zero constant.\\
(i) Find the set of values of $k$ for which the curve and the line have no common points.\\
(ii) State the value of $k$ for which the line is a tangent to the curve and, for this case, find the coordinates of the point where the line touches the curve.
\hfill \mbox{\textit{CAIE P1 2010 Q6 [7]}}