| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Perpendicularity conditions |
| Difficulty | Moderate -0.3 This is a straightforward multi-part vectors question testing standard techniques: dot product for perpendicularity, magnitude formula for equal lengths, and unit vector scaling. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\overrightarrow{OA} = \begin{pmatrix}2\\1\\-2\end{pmatrix}\), \(\overrightarrow{OB} = \begin{pmatrix}5\\-1\\k\end{pmatrix}\), \(\overrightarrow{OC} = \begin{pmatrix}2\\6\\-3\end{pmatrix}\) | ||
| \(10 - 1 - 2k = 0 \rightarrow k = 4\frac{1}{2}\) | M1 A1 | Use of scalar product \(= 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\overrightarrow{AB} = \begin{pmatrix} 3 \\ -2 \\ k+2 \end{pmatrix}\) | B1 | |
| \( | \overrightarrow{OC} | = 7\) (seen or implied) |
| \(3^2 + (-2)^2 + (k+2)^2 = 49 \rightarrow k = 4\) or \(-8\) | M1 A1 [4] | Correct method. Both correct. Condone sign error in \(\overrightarrow{AB}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \( | \overrightarrow{OA} | = 3\) |
| \(\overrightarrow{OD} = 3\overrightarrow{OA} = \begin{pmatrix} 6 \\ 3 \\ -6 \end{pmatrix}\) and \( | \overrightarrow{OE} | = 2\) |
| \(\overrightarrow{OC} = \begin{pmatrix} 4 \\ 12 \\ -6 \end{pmatrix}\) | ||
| \(\overrightarrow{DE} = \overrightarrow{OE} - \overrightarrow{OD} = \begin{pmatrix} -2 \\ 9 \\ 0 \end{pmatrix}\) | M1 | Correct vector subtraction. |
| \(\rightarrow\) Magnitude of \(\sqrt{85}\) | A1 [4] |
## Question 10(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\overrightarrow{OA} = \begin{pmatrix}2\\1\\-2\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}5\\-1\\k\end{pmatrix}$, $\overrightarrow{OC} = \begin{pmatrix}2\\6\\-3\end{pmatrix}$ | | |
| $10 - 1 - 2k = 0 \rightarrow k = 4\frac{1}{2}$ | **M1 A1** | Use of scalar product $= 0$ |
# Question 10 (continued):
## Part (ii):
$\overrightarrow{AB} = \begin{pmatrix} 3 \\ -2 \\ k+2 \end{pmatrix}$ | **B1** | |
$|\overrightarrow{OC}| = 7$ (seen or implied) | **B1** | |
$3^2 + (-2)^2 + (k+2)^2 = 49 \rightarrow k = 4$ or $-8$ | **M1 A1** [4] | Correct method. Both correct. Condone sign error in $\overrightarrow{AB}$ |
## Part (iii):
$|\overrightarrow{OA}| = 3$ | | |
$\overrightarrow{OD} = 3\overrightarrow{OA} = \begin{pmatrix} 6 \\ 3 \\ -6 \end{pmatrix}$ and $|\overrightarrow{OE}| = 2$ | **M1 A1** | Scaling from magnitudes/unit vector – oe. |
$\overrightarrow{OC} = \begin{pmatrix} 4 \\ 12 \\ -6 \end{pmatrix}$ | | |
$\overrightarrow{DE} = \overrightarrow{OE} - \overrightarrow{OD} = \begin{pmatrix} -2 \\ 9 \\ 0 \end{pmatrix}$ | **M1** | Correct vector subtraction. |
$\rightarrow$ Magnitude of $\sqrt{85}$ | **A1** [4] | |
---10 Relative to an origin $O$, the position vectors of points $A , B$ and $C$ are given by
$$\overrightarrow { O A } = \left( \begin{array} { r }
2 \\
1 \\
- 2
\end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r }
5 \\
- 1 \\
k
\end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r }
2 \\
6 \\
- 3
\end{array} \right)$$
respectively, where $k$ is a constant.\\
(i) Find the value of $k$ in the case where angle $A O B = 90 ^ { \circ }$.\\
(ii) Find the possible values of $k$ for which the lengths of $A B$ and $O C$ are equal.
The point $D$ is such that $\overrightarrow { O D }$ is in the same direction as $\overrightarrow { O A }$ and has magnitude 9 units. The point $E$ is such that $\overrightarrow { O E }$ is in the same direction as $\overrightarrow { O C }$ and has magnitude 14 units.\\
(iii) Find the magnitude of $\overrightarrow { D E }$ in the form $\sqrt { } n$ where $n$ is an integer.
\hfill \mbox{\textit{CAIE P1 2016 Q10 [10]}}