| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Perpendicularity conditions |
| Difficulty | Moderate -0.3 Part (a) is a straightforward application of the midpoint formula in vector form requiring simple algebraic manipulation. Part (b) involves two simultaneous equations from magnitude and perpendicularity conditions, but these are standard techniques with routine algebraic solving. Both parts test fundamental vector concepts without requiring novel insight or extended reasoning. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement |
| Answer | Marks |
|---|---|
| \(\overrightarrow{PR} = 2\overrightarrow{PQ} = 2(\mathbf{q} - \mathbf{p})\) | (B1 |
| \(\overrightarrow{OR} = \mathbf{p} + 2\mathbf{q} - 2\mathbf{p} = 2\mathbf{q} - \mathbf{p}\) | M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\overrightarrow{QR} = \overrightarrow{PQ} = \mathbf{q} - \mathbf{p}\) | (B1 | |
| \(\overrightarrow{OR} = \overrightarrow{OQ} + \overrightarrow{QR} = \mathbf{q} + \mathbf{q} - \mathbf{p} = 2\mathbf{q} - \mathbf{p}\) | M1A1 | Or other valid method |
| Answer | Marks | Guidance |
|---|---|---|
| \(6^2 + a^2 + b^2 = 21^2\) SOI | B1 | |
| \(18 + 2a + 2b = 0\) | B1 | |
| \(a^2 + (-a-9)^2 = 405\) | M1 | Correct method for elimination of a variable |
| \((2)(a^2 + 9a - 162)(= 0)\) | A1 | Or same equation in \(b\) |
| \(a = 9\) or \(-18\) | A1 | |
| \(b = -18\) or \(9\) | A1 |
## Question 8(a):
**EITHER:**
$\overrightarrow{PR} = 2\overrightarrow{PQ} = 2(\mathbf{q} - \mathbf{p})$ | (B1 |
$\overrightarrow{OR} = \mathbf{p} + 2\mathbf{q} - 2\mathbf{p} = 2\mathbf{q} - \mathbf{p}$ | M1A1 |
**OR:**
$\overrightarrow{QR} = \overrightarrow{PQ} = \mathbf{q} - \mathbf{p}$ | (B1 |
$\overrightarrow{OR} = \overrightarrow{OQ} + \overrightarrow{QR} = \mathbf{q} + \mathbf{q} - \mathbf{p} = 2\mathbf{q} - \mathbf{p}$ | M1A1 | Or other valid method
---
## Question 8(b):
$6^2 + a^2 + b^2 = 21^2$ SOI | B1 |
$18 + 2a + 2b = 0$ | B1 |
$a^2 + (-a-9)^2 = 405$ | M1 | Correct method for elimination of a variable
$(2)(a^2 + 9a - 162)(= 0)$ | A1 | Or same equation in $b$
$a = 9$ or $-18$ | A1 |
$b = -18$ or $9$ | A1 |
---8
\begin{enumerate}[label=(\alph*)]
\item Relative to an origin $O$, the position vectors of two points $P$ and $Q$ are $\mathbf { p }$ and $\mathbf { q }$ respectively. The point $R$ is such that $P Q R$ is a straight line with $Q$ the mid-point of $P R$. Find the position vector of $R$ in terms of $\mathbf { p }$ and $\mathbf { q }$, simplifying your answer.
\item The vector $6 \mathbf { i } + a \mathbf { j } + b \mathbf { k }$ has magnitude 21 and is perpendicular to $3 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$. Find the possible values of $a$ and $b$, showing all necessary working.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2017 Q8 [9]}}