| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Parallel or perpendicular vectors condition |
| Difficulty | Moderate -0.8 This is a straightforward vectors question testing basic concepts: (i) uses the dot product condition for perpendicularity (routine calculation), and (ii) requires finding the magnitude of AB and solving a quadratic equation. Both parts are standard textbook exercises with no problem-solving insight required, making it easier than average but not trivial due to the algebraic manipulation needed in part (ii). |
| 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 |
|---|---|---|
| (i) \(-8 + 3 + p = 0 \rightarrow p = 5\) | M1, A1 [2] | Must be scalar. co. |
| (ii) Vector \(\overrightarrow{AB} = \mathbf{b} - \mathbf{a} = 6\mathbf{i} - 2\mathbf{j} + (p-1)\mathbf{k}\) | M1 | Must be \(\mathbf{b} - \mathbf{a}\) or \(\mathbf{a} - \mathbf{b}\). |
| \(36 + 4 + (p-1)^2 = 49 \rightarrow p = 4\) or \(p = -2\) | M1 A1, A1 [4] | Must be sum of 3 squares. A1 √ lost. co. |
(i) $-8 + 3 + p = 0 \rightarrow p = 5$ | M1, A1 [2] | Must be scalar. co.
(ii) Vector $\overrightarrow{AB} = \mathbf{b} - \mathbf{a} = 6\mathbf{i} - 2\mathbf{j} + (p-1)\mathbf{k}$ | M1 | Must be $\mathbf{b} - \mathbf{a}$ or $\mathbf{a} - \mathbf{b}$.
$36 + 4 + (p-1)^2 = 49 \rightarrow p = 4$ or $p = -2$ | M1 A1, A1 [4] | Must be sum of 3 squares. A1 √ lost. co.Relative to an origin $O$, the position vectors of the points $A$ and $B$ are given by
$$\overrightarrow{OA} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix} \text{ and } \overrightarrow{OB} = \begin{pmatrix} 4 \\ 1 \\ p \end{pmatrix}$$
\begin{enumerate}[label=(\roman*)]
\item Find the value of $p$ for which $\overrightarrow{OA}$ is perpendicular to $\overrightarrow{OB}$. [2]
\item Find the values of $p$ for which the magnitude of $\overrightarrow{AB}$ is 7. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2010 Q5 [6]}}