| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Line-curve intersection conditions |
| Difficulty | Standard +0.3 This is a standard discriminant problem requiring students to form a quadratic equation from the intersection condition and apply Δ > 0 for two distinct points, then find tangent conditions (Δ = 0) and verify their intersection. While it involves multiple steps and algebraic manipulation, the technique is routine for P1 level with no novel insight required—slightly easier than average due to its predictable structure. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3kx - 2k = x^2 - kx + 2 \rightarrow x^2 - 4kx + 2k + 2\ (=0)\) | B1 | \(kx\) terms combined correctly — implied by correct \(b^2 - 4ac\) |
| Attempt to find \(b^2 - 4ac\) | M1 | Form a quadratic equation in \(k\) |
| \(1\) and \(-\frac{1}{2}\) | A1 | SOI |
| \(k > 1,\ k < -\frac{1}{2}\) | A1 | Allow \(x > 1,\ x < -1/2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = 3x - 2\), \(\quad y = -\frac{3}{2}x + 1\) | M1 | Use of their \(k\) values (twice) in \(y = 3kx - 2k\) |
| \(3x - 2 = -\frac{3}{2}x + 1\) OR \(y + 2 = 2 - 2y\) | M1 | Equate their tangent equations OR substitute \(y = 0\) into both lines |
| \(x = \frac{2}{3}\), \(\rightarrow y = 0\) in one or both lines | A1 | Substitute \(x = \frac{2}{3}\) in one or both lines |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3kx - 2k = x^2 - kx + 2 \rightarrow x^2 - 4kx + 2k + 2\ (=0)$ | B1 | $kx$ terms combined correctly — implied by correct $b^2 - 4ac$ |
| Attempt to find $b^2 - 4ac$ | M1 | Form a quadratic equation in $k$ |
| $1$ and $-\frac{1}{2}$ | A1 | SOI |
| $k > 1,\ k < -\frac{1}{2}$ | A1 | Allow $x > 1,\ x < -1/2$ |
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = 3x - 2$, $\quad y = -\frac{3}{2}x + 1$ | M1 | Use of their $k$ values (twice) in $y = 3kx - 2k$ |
| $3x - 2 = -\frac{3}{2}x + 1$ OR $y + 2 = 2 - 2y$ | M1 | Equate their tangent equations OR substitute $y = 0$ into both lines |
| $x = \frac{2}{3}$, $\rightarrow y = 0$ in one or both lines | A1 | Substitute $x = \frac{2}{3}$ in one or both lines |
---6 A line has equation $y = 3 k x - 2 k$ and a curve has equation $y = x ^ { 2 } - k x + 2$, where $k$ is a constant.\\
(i) Find the set of values of $k$ for which the line and curve meet at two distinct points.\\
(ii) For each of two particular values of $k$, the line is a tangent to the curve. Show that these two tangents meet on the $x$-axis.\\
\hfill \mbox{\textit{CAIE P1 2019 Q6 [7]}}