| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Unit vector in given direction |
| Difficulty | Moderate -0.8 This is a straightforward two-part question testing basic vector operations: finding a unit vector (subtract vectors, find magnitude, divide) and using the dot product for perpendicularity (set dot product to zero, solve for p). Both are routine procedures with no problem-solving insight required, making it easier than average but not trivial since it requires correct execution of standard techniques. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\vec{AB} = \begin{pmatrix} 2 \\ 3 \\ -6 \end{pmatrix}\); Modulus \(= \sqrt{(4 + 9 + 36)}\) | B1, M1 | co; Correct method for modulus |
| Unit Vector \(= \frac{1}{7}\begin{pmatrix} 2 \\ 3 \\ -6 \end{pmatrix}\) | A1 | co for his vector \(\vec{AB}\) |
| (ii) \(\vec{OB} \cdot \vec{OC} = 4 + 6 - 2p = 0 \to p = 5\) | M1, A1 | Dot product \(= 0\); co |
$\vec{OA} = \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}$, $\vec{OB} = \begin{pmatrix} 4 \\ 2 \\ -2 \end{pmatrix}$, $\vec{OC} = \begin{pmatrix} 1 \\ 3 \\ p \end{pmatrix}$
**(i)** $\vec{AB} = \begin{pmatrix} 2 \\ 3 \\ -6 \end{pmatrix}$; Modulus $= \sqrt{(4 + 9 + 36)}$ | B1, M1 | co; Correct method for modulus
Unit Vector $= \frac{1}{7}\begin{pmatrix} 2 \\ 3 \\ -6 \end{pmatrix}$ | A1 | co for his vector $\vec{AB}$
**(ii)** $\vec{OB} \cdot \vec{OC} = 4 + 6 - 2p = 0 \to p = 5$ | M1, A1 | Dot product $= 0$; coRelative to an origin $O$, the position vectors of the points $A$, $B$ and $C$ are given by
$$\overrightarrow{OA} = \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 4 \\ 2 \\ -2 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 1 \\ 3 \\ p \end{pmatrix}.$$
Find
\begin{enumerate}[label=(\roman*)]
\item the unit vector in the direction of $\overrightarrow{AB}$, [3]
\item the value of the constant $p$ for which angle $BOC = 90°$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q2 [5]}}