| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Midpoint of line segment |
| Difficulty | Standard +0.3 Part (i) requires finding a stationary point by differentiation, calculating a midpoint using the standard formula, then substituting into a linear equation—all routine procedures. Part (ii) involves using the discriminant condition for a quadratic with no real roots, which is a standard technique. Both parts are straightforward applications of well-practiced methods with no novel problem-solving required. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = 2x - 4 = 0\) | Can use completing the square. | |
| \(\rightarrow x = 2,\ y = 3\) | B1 B1 | |
| Midpoint of \(AB\) is \((3, 5)\) | B1 FT | FT on (*their* 2, *their* 3) with (4,7) |
| \(\rightarrow m = \frac{7}{3}\) (or 2.33) | B1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Simultaneous equations \(\rightarrow x^2 - 4x - mx + 9 (= 0)\) | *M1 | Equates and sets to 0 must contain \(m\) |
| Use of \(b^2 - 4ac \rightarrow (m+4)^2 - 36\) | DM1 | Any use of \(b^2 - 4ac\) on equation set to 0 must contain \(m\) |
| Solves \(= 0 \rightarrow -10\) or \(2\) | A1 | Correct end-points. |
| \(-10 < m < 2\) | A1 | Don't condone \(\leqslant\) at either or both end(s). Accept \(-10 < m,\ m < 2\). |
| Total: 4 |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 2x - 4 = 0$ | | Can use completing the square. |
| $\rightarrow x = 2,\ y = 3$ | B1 B1 | |
| Midpoint of $AB$ is $(3, 5)$ | B1 FT | FT on (*their* 2, *their* 3) with (4,7) |
| $\rightarrow m = \frac{7}{3}$ (or 2.33) | B1 | |
| **Total: 4** | | |
## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Simultaneous equations $\rightarrow x^2 - 4x - mx + 9 (= 0)$ | *M1 | Equates and sets to 0 must contain $m$ |
| Use of $b^2 - 4ac \rightarrow (m+4)^2 - 36$ | DM1 | Any use of $b^2 - 4ac$ on equation set to 0 must contain $m$ |
| Solves $= 0 \rightarrow -10$ or $2$ | A1 | Correct end-points. |
| $-10 < m < 2$ | A1 | Don't condone $\leqslant$ at either or both end(s). Accept $-10 < m,\ m < 2$. |
| **Total: 4** | | |7 Points $A$ and $B$ lie on the curve $y = x ^ { 2 } - 4 x + 7$. Point $A$ has coordinates $( 4,7 )$ and $B$ is the stationary point of the curve. The equation of a line $L$ is $y = m x - 2$, where $m$ is a constant.\\
(i) In the case where $L$ passes through the mid-point of $A B$, find the value of $m$.\\
(ii) Find the set of values of $m$ for which $L$ does not meet the curve.\\
\hfill \mbox{\textit{CAIE P1 2017 Q7 [8]}}