| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Distance between parallel planes or line and parallel plane |
| Difficulty | Standard +0.3 This is a straightforward vectors question requiring standard techniques: showing perpendicularity via dot product, finding a midpoint, and using the distance formula between parallel planes. Part (i) is routine verification, and part (ii) applies the formula for distance between parallel planes with minimal problem-solving required. Slightly easier than average due to its mechanical nature. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04i Shortest distance: between a point and a line |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or obtain coordinates \((1, 2, 1)\) for the mid-point of \(AB\) | B1 | |
| Verify that the midpoint lies on \(m\) | B1 | |
| State or imply a correct normal vector to the plane, e.g. \(2\mathbf{i}+2\mathbf{j}-\mathbf{k}\) | B1 | |
| State or imply a direction vector for segment \(AB\), e.g. \(-4\mathbf{i}-4\mathbf{j}+2\mathbf{k}\) | B1 | |
| Confirm that \(m\) is perpendicular to \(AB\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply that the perpendicular distance of \(m\) from the origin is \(\frac{5}{3}\), or unsimplified equivalent | B1 | |
| State or imply that \(n\) has an equation of the form \(2x + 2y - z = k\) | B1 | |
| Obtain answer \(2x + 2y - z = 2\) | B1 |
## Question 6(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or obtain coordinates $(1, 2, 1)$ for the mid-point of $AB$ | B1 | |
| Verify that the midpoint lies on $m$ | B1 | |
| State or imply a correct normal vector to the plane, e.g. $2\mathbf{i}+2\mathbf{j}-\mathbf{k}$ | B1 | |
| State or imply a direction vector for segment $AB$, e.g. $-4\mathbf{i}-4\mathbf{j}+2\mathbf{k}$ | B1 | |
| Confirm that $m$ is perpendicular to $AB$ | B1 | |
**Total: 5**
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## Question 6(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply that the perpendicular distance of $m$ from the origin is $\frac{5}{3}$, or unsimplified equivalent | B1 | |
| State or imply that $n$ has an equation of the form $2x + 2y - z = k$ | B1 | |
| Obtain answer $2x + 2y - z = 2$ | B1 | |
**Total: 3**6 The plane with equation $2 x + 2 y - z = 5$ is denoted by $m$. Relative to the origin $O$, the points $A$ and $B$ have coordinates $( 3,4,0 )$ and $( - 1,0,2 )$ respectively.\\
(i) Show that the plane $m$ bisects $A B$ at right angles.\\
A second plane $p$ is parallel to $m$ and nearer to $O$. The perpendicular distance between the planes is 1 .\\
(ii) Find the equation of $p$, giving your answer in the form $a x + b y + c z = d$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q6 [8]}}