| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Perpendicularity conditions |
| Difficulty | Standard +0.3 This is a straightforward vectors question requiring standard techniques: dot product for perpendicularity (algebraic manipulation with parameters), vector subtraction and magnitude calculation, and finding a unit vector. All parts are routine applications of basic vector operations with no novel insight required, making it slightly easier than average. |
| 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) \(OA.OC = -4p^2 - q^2 + 4p^2 + q^2 = 0\) | M1 A1 | Attempt scalar product. Allow M1 even for e.g. \(OA.OB = 2pq - 2pq\) etc. |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) \(CA = OA - OC = (\pm i)(1 + 4p^2 + q^2)\) (i) | M1 A1 | Ignore \(CA = OC-OA\) Not \(\sqrt{(1+4p^2+q^2)^2}\) |
| \( | CA | = 1 + 4p^2 + q^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| (iii) \(BA = OA - OB = i + 6j + 2k - (2i - 6k) = (\pm)(i + 4j + 8k)\) | M1 | Allow subtn reversed for both M marks |
| \(\frac{-xi + yj + zk}{\sqrt{x^2 + y^2 + z^2}} \rightarrow \frac{1}{9}(i + 4j + 8k)\) | M1A1 | M1 independent of 1st M1 |
(i) $OA.OC = -4p^2 - q^2 + 4p^2 + q^2 = 0$ | M1 A1 | Attempt scalar product. Allow M1 even for e.g. $OA.OB = 2pq - 2pq$ etc.
[2]
(ii) $CA = OA - OC = (\pm i)(1 + 4p^2 + q^2)$ (i) | M1 A1 | Ignore $CA = OC-OA$ Not $\sqrt{(1+4p^2+q^2)^2}$
$|CA| = 1 + 4p^2 + q^2$ | A1 |
[2]
(iii) $BA = OA - OB = i + 6j + 2k - (2i - 6k) = (\pm)(i + 4j + 8k)$ | M1 | Allow subtn reversed for both M marks
$\frac{-xi + yj + zk}{\sqrt{x^2 + y^2 + z^2}} \rightarrow \frac{1}{9}(i + 4j + 8k)$ | M1A1 | M1 independent of 1st M1
[3]6 Relative to an origin $O$, the position vectors of three points, $A , B$ and $C$, are given by
$$\overrightarrow { O A } = \mathbf { i } + 2 p \mathbf { j } + q \mathbf { k } , \quad \overrightarrow { O B } = q \mathbf { j } - 2 p \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = - \left( 4 p ^ { 2 } + q ^ { 2 } \right) \mathbf { i } + 2 p \mathbf { j } + q \mathbf { k }$$
where $p$ and $q$ are constants.\\
(i) Show that $\overrightarrow { O A }$ is perpendicular to $\overrightarrow { O C }$ for all non-zero values of $p$ and $q$.\\
(ii) Find the magnitude of $\overrightarrow { C A }$ in terms of $p$ and $q$.\\
(iii) For the case where $p = 3$ and $q = 2$, find the unit vector parallel to $\overrightarrow { B A }$.
\hfill \mbox{\textit{CAIE P1 2013 Q6 [7]}}