| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Standard +0.8 This is a multi-part vector line intersection problem requiring students to prove lines always intersect (by solving a system with parameter a), then use the distance condition to find specific values of a. It combines vector equations, simultaneous equations with parameters, and distance formula application—more demanding than routine intersection problems but still within standard Further Maths techniques. |
| Spec | 1.10f Distance between points: using position vectors4.04a Line equations: 2D and 3D, cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| State at least two of the equations \(1 + \lambda = a + \mu\), \(4 = 2 + 2\mu\), \(-2 + 3\lambda = -2 + 3a\mu\) | B1 | |
| Solve for \(\lambda\) or for \(\mu\) | M1 | |
| Obtain \(\lambda = a\) (or \(\lambda = a + \mu - 1\)) and \(\mu = 1\) | A1 | |
| Confirm values satisfy third equation | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply point of intersection is \((a+1, 4, 3a-2)\) | B1 | |
| Use correct method for the modulus of the position vector and equate to 9, following their point of intersection | M*1 | |
| Solve a three-term quadratic equation in \(a\) \((a^2 - a - 6 = 0)\) | DM*1 | |
| Obtain \(-2\) and \(3\) | A1 | [4] |
**(i)**
State at least two of the equations $1 + \lambda = a + \mu$, $4 = 2 + 2\mu$, $-2 + 3\lambda = -2 + 3a\mu$ | B1 |
Solve for $\lambda$ or for $\mu$ | M1 |
Obtain $\lambda = a$ (or $\lambda = a + \mu - 1$) and $\mu = 1$ | A1 |
Confirm values satisfy third equation | A1 | [4]
**(ii)**
State or imply point of intersection is $(a+1, 4, 3a-2)$ | B1 |
Use correct method for the modulus of the position vector and equate to 9, following their point of intersection | M*1 |
Solve a three-term quadratic equation in $a$ $(a^2 - a - 6 = 0)$ | DM*1 |
Obtain $-2$ and $3$ | A1 | [4]7 The equations of two straight lines are
$$\mathbf { r } = \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = a \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } + 3 a \mathbf { k } )$$
where $a$ is a constant.\\
(i) Show that the lines intersect for all values of $a$.\\
(ii) Given that the point of intersection is at a distance of 9 units from the origin, find the possible values of $a$.
\hfill \mbox{\textit{CAIE P3 2014 Q7 [8]}}