| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with plane |
| Difficulty | Standard +0.3 This is a straightforward application of standard vector methods: finding a line equation in parametric form, substituting into a plane equation to find intersection, then using the formula for angle between line and plane normal. All steps are routine with no conceptual challenges beyond direct application of learned techniques. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane4.04f Line-plane intersection: find point |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State general vector for point on line, e.g. \(-5\mathbf{i} + 3\mathbf{j} + 6\mathbf{k} + s(10\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})\) or \(5\mathbf{i} + 8\mathbf{j} + \mathbf{k} + t(10\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})\) or equiv | B1 | |
| Substitute line into equation of plane and solve for parameter | M1 | |
| Obtain correct value, \(s = \frac{2}{5}\) or \(t = -\frac{2}{5}\) or equivalent | A1 | |
| Obtain \((-1, 5, 4)\) o.e. | A1 | [4] |
| (ii) State or imply normal vector to \(p\) is \(2\mathbf{i} - \mathbf{j} + 4\mathbf{k}\) | B1 | |
| Carry out process for evaluating scalar product of two relevant vectors | M1 | |
| Using correct process for moduli, divide scalar product by the product of the moduli and evaluate \(\arcsin(\ldots)\) or \(\arccos(\ldots)\) of the result. | M1 | |
| Obtain \(5.1°\) or \(0.089\) rads | A1 | [4] |
**(i)** State general vector for point on line, e.g. $-5\mathbf{i} + 3\mathbf{j} + 6\mathbf{k} + s(10\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})$ or $5\mathbf{i} + 8\mathbf{j} + \mathbf{k} + t(10\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})$ or equiv | B1 |
Substitute line into equation of plane and solve for parameter | M1 |
Obtain correct value, $s = \frac{2}{5}$ or $t = -\frac{2}{5}$ or equivalent | A1 |
Obtain $(-1, 5, 4)$ o.e. | A1 | [4]
**(ii)** State or imply normal vector to $p$ is $2\mathbf{i} - \mathbf{j} + 4\mathbf{k}$ | B1 |
Carry out process for evaluating scalar product of two relevant vectors | M1 |
Using correct process for moduli, divide scalar product by the product of the moduli and evaluate $\arcsin(\ldots)$ or $\arccos(\ldots)$ of the result. | M1 |
Obtain $5.1°$ or $0.089$ rads | A1 | [4]6 The straight line $l$ passes through the points with coordinates $( - 5,3,6 )$ and $( 5,8,1 )$. The plane $p$ has equation $2 x - y + 4 z = 9$.\\
(i) Find the coordinates of the point of intersection of $l$ and $p$.\\
(ii) Find the acute angle between $l$ and $p$.
\hfill \mbox{\textit{CAIE P3 2010 Q6 [8]}}