| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Standard +0.3 This is a straightforward vectors question requiring finding midpoints, dividing lines in given ratios, forming line equations, and finding intersection points. All techniques are standard A-level procedures with no novel insight needed. The multi-step nature and calculation care required make it slightly above average difficulty, but it remains a routine textbook-style question. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10e Position vectors: and displacement1.10g Problem solving with vectors: in geometry |
| Answer | Marks |
|---|---|
| State that the position vector of \(M\) is \(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}\), or equivalent | B1 |
| Carry out a correct method for finding the position vector of \(N\) | M1 |
| Obtain answer \(3\mathbf{i} - 2\mathbf{j} + \mathbf{k}\), or equivalent | A1 |
| Obtain vector equation of \(MN\) in any correct form, e.g. \(\mathbf{r} = 2\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \lambda(3\mathbf{j} + 3\mathbf{k})\) | A1 |
| Answer | Marks |
|---|---|
| State that the position vector of \(M\) is \(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}\), or equivalent | B1 |
| Carry out a correct method for finding a direction vector for \(MN\) | M1 |
| Obtain answer, e.g. \(\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}\), or equivalent | A1 |
| Obtain vector equation of \(MN\) in any correct form, e.g. \(\mathbf{r} = 2\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \lambda(\mathbf{i} - 3\mathbf{j} + 3\mathbf{k})\) | A1 |
| [SR: The use of \(\overrightarrow{AN} = \overrightarrow{AC}/3\) can earn M1A0, but \(\overrightarrow{AN} = \overrightarrow{AC}/2\) gets M0A0.] | [4] |
| Answer | Marks |
|---|---|
| State equation of \(BC\) in any correct form, e.g. \(\mathbf{r} = 3\mathbf{i} + 2\mathbf{j} - 3\mathbf{k} + \mu(\mathbf{i} - 5\mathbf{j} + 5\mathbf{k})\) | B1 |
| Solve for \(\lambda\) or for \(\mu\) | M1 |
| Obtain correct value of \(\lambda\), or \(\mu\), e.g. \(\lambda = 3\), or \(\mu = 2\) | A1 |
| Obtain position vector \(5\mathbf{i} - 8\mathbf{j} + 7\mathbf{k}\) | A1 |
| [4] |
**(i) EITHER:**
| State that the position vector of $M$ is $2\mathbf{i} + \mathbf{j} - 2\mathbf{k}$, or equivalent | B1 |
| Carry out a correct method for finding the position vector of $N$ | M1 |
| Obtain answer $3\mathbf{i} - 2\mathbf{j} + \mathbf{k}$, or equivalent | A1 |
| Obtain vector equation of $MN$ in any correct form, e.g. $\mathbf{r} = 2\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \lambda(3\mathbf{j} + 3\mathbf{k})$ | A1 |
**OR:**
| State that the position vector of $M$ is $2\mathbf{i} + \mathbf{j} - 2\mathbf{k}$, or equivalent | B1 |
| Carry out a correct method for finding a direction vector for $MN$ | M1 |
| Obtain answer, e.g. $\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}$, or equivalent | A1 |
| Obtain vector equation of $MN$ in any correct form, e.g. $\mathbf{r} = 2\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \lambda(\mathbf{i} - 3\mathbf{j} + 3\mathbf{k})$ | A1 |
| [SR: The use of $\overrightarrow{AN} = \overrightarrow{AC}/3$ can earn M1A0, but $\overrightarrow{AN} = \overrightarrow{AC}/2$ gets M0A0.] | [4] |
**(ii)**
| State equation of $BC$ in any correct form, e.g. $\mathbf{r} = 3\mathbf{i} + 2\mathbf{j} - 3\mathbf{k} + \mu(\mathbf{i} - 5\mathbf{j} + 5\mathbf{k})$ | B1 |
| Solve for $\lambda$ or for $\mu$ | M1 |
| Obtain correct value of $\lambda$, or $\mu$, e.g. $\lambda = 3$, or $\mu = 2$ | A1 |
| Obtain position vector $5\mathbf{i} - 8\mathbf{j} + 7\mathbf{k}$ | A1 |
| [4] |6 With respect to the origin $O$, the points $A , B$ and $C$ have position vectors given by
$$\overrightarrow { O A } = \mathbf { i } - \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$$
The mid-point of $A B$ is $M$. The point $N$ lies on $A C$ between $A$ and $C$ and is such that $A N = 2 N C$.\\
(i) Find a vector equation of the line $M N$.\\
(ii) It is given that $M N$ intersects $B C$ at the point $P$. Find the position vector of $P$.
\hfill \mbox{\textit{CAIE P3 2009 Q6 [8]}}