| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Standard +0.3 This is a standard two-part vectors question requiring equating components to find intersection (routine algebraic manipulation), then finding a plane equation using cross product of direction vectors. While it involves multiple steps, all techniques are textbook-standard with no novel insight required, making it slightly easier than average. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting4.04f Line-plane intersection: find point |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply general point of either line has coordinates \((5+s,\;1-s,\;-4+3s)\) or \((p+2t,\;4+5t,\;-2-4t)\) | B1 | |
| Solve simultaneous equations and find \(s\) and \(t\) | M1 | |
| Obtain \(s = 2\) and \(t = -1\) or equivalent in terms of \(p\) | A1 | |
| Substitute in third equation to find \(p = 9\) | A1 | |
| State point of intersection is \((7,\,-1,\,2)\) | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Either: Use scalar product to obtain a relevant equation in \(a, b, c\), e.g. \(a - b + 3c = 0\) or \(2a + 5b - 4c = 0\) | M1 | |
| State two correct equations in \(a, b, c\) | A1 | |
| Solve simultaneous equations to obtain at least one ratio | DM1 | |
| Obtain \(a : b : c = -11 : 10 : 7\) or equivalent | A1 | |
| Obtain equation \(-11x + 10y + 7z = -73\) or equivalent with integer coefficients | A1 | |
| Or 1: Calculate vector product of \(\begin{pmatrix}1\\-1\\3\end{pmatrix}\) and \(\begin{pmatrix}2\\5\\-4\end{pmatrix}\) | M1 | |
| Obtain two correct components of the product | A1 | |
| Obtain correct \(\begin{pmatrix}-11\\10\\7\end{pmatrix}\) or equivalent | A1 | |
| Substitute coordinates of a relevant point in \(\mathbf{r}\cdot\mathbf{n} = d\) to find \(d\) | DM1 | |
| Obtain equation \(-11x + 10y + 7z = -73\) or equivalent with integer coefficients | A1 | |
| Or 2: Using relevant vectors, form correctly a two-parameter equation for the plane | M1 | |
| Obtain \(\mathbf{r} = \begin{pmatrix}5\\1\\-4\end{pmatrix} + \lambda\begin{pmatrix}1\\-1\\3\end{pmatrix} + \mu\begin{pmatrix}2\\5\\-4\end{pmatrix}\) or equivalent | A1 | |
| State three equations in \(x, y, z, \lambda, \mu\) | A1 | |
| Eliminate \(\lambda\) and \(\mu\) | DM1 | |
| Obtain \(11x - 10y - 7z = 73\) or equivalent with integer coefficients | A1 | [5] |
## Question 8:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply general point of either line has coordinates $(5+s,\;1-s,\;-4+3s)$ or $(p+2t,\;4+5t,\;-2-4t)$ | B1 | |
| Solve simultaneous equations and find $s$ and $t$ | M1 | |
| Obtain $s = 2$ and $t = -1$ or equivalent in terms of $p$ | A1 | |
| Substitute in third equation to find $p = 9$ | A1 | |
| State point of intersection is $(7,\,-1,\,2)$ | A1 | [5] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **Either:** Use scalar product to obtain a relevant equation in $a, b, c$, e.g. $a - b + 3c = 0$ or $2a + 5b - 4c = 0$ | M1 | |
| State two correct equations in $a, b, c$ | A1 | |
| Solve simultaneous equations to obtain at least one ratio | DM1 | |
| Obtain $a : b : c = -11 : 10 : 7$ or equivalent | A1 | |
| Obtain equation $-11x + 10y + 7z = -73$ or equivalent with integer coefficients | A1 | |
| **Or 1:** Calculate vector product of $\begin{pmatrix}1\\-1\\3\end{pmatrix}$ and $\begin{pmatrix}2\\5\\-4\end{pmatrix}$ | M1 | |
| Obtain two correct components of the product | A1 | |
| Obtain correct $\begin{pmatrix}-11\\10\\7\end{pmatrix}$ or equivalent | A1 | |
| Substitute coordinates of a relevant point in $\mathbf{r}\cdot\mathbf{n} = d$ to find $d$ | DM1 | |
| Obtain equation $-11x + 10y + 7z = -73$ or equivalent with integer coefficients | A1 | |
| **Or 2:** Using relevant vectors, form correctly a two-parameter equation for the plane | M1 | |
| Obtain $\mathbf{r} = \begin{pmatrix}5\\1\\-4\end{pmatrix} + \lambda\begin{pmatrix}1\\-1\\3\end{pmatrix} + \mu\begin{pmatrix}2\\5\\-4\end{pmatrix}$ or equivalent | A1 | |
| State three equations in $x, y, z, \lambda, \mu$ | A1 | |
| Eliminate $\lambda$ and $\mu$ | DM1 | |
| Obtain $11x - 10y - 7z = 73$ or equivalent with integer coefficients | A1 | [5] |8 Two lines have equations
$$\mathbf { r } = \left( \begin{array} { r }
5 \\
1 \\
- 4
\end{array} \right) + s \left( \begin{array} { r }
1 \\
- 1 \\
3
\end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r }
p \\
4 \\
- 2
\end{array} \right) + t \left( \begin{array} { r }
2 \\
5 \\
- 4
\end{array} \right) ,$$
where $p$ is a constant. It is given that the lines intersect.\\
(i) Find the value of $p$ and determine the coordinates of the point of intersection.\\
(ii) Find the equation of the plane containing the two lines, giving your answer in the form $a x + b y + c z = d$, where $a , b , c$ and $d$ are integers.
\hfill \mbox{\textit{CAIE P3 2012 Q8 [10]}}