| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Plane containing line and point/vector |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question requiring finding intersection conditions and plane equations. Part (i) involves setting up parametric equations and solving simultaneously (routine but multi-step). Part (ii) requires finding a plane through two points parallel to a given vector—a textbook application of cross products and normal vectors. Slightly above average difficulty due to the algebraic manipulation required, but follows standard procedures without requiring novel insight. |
| Spec | 1.10b Vectors in 3D: i,j,k notation4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out a correct method for finding a vector equation for \(AB\) | M1 | |
| Obtain \(\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k} + \lambda(2\mathbf{i} + 3\mathbf{j} - \mathbf{k})\), or equivalent | A1 | |
| Equate two pairs of components of general points on \(AB\) and \(l\) and solve for \(\lambda\) or \(\mu\) | M1 | |
| Obtain correct answer for \(\lambda\) or \(\mu\), e.g. \(\lambda = \frac{5}{7}\) or \(\mu = \frac{3}{7}\) | A1 | |
| Obtain \(m = 3\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| EITHER: Use scalar product to obtain an equation in \(a\), \(b\) and \(c\), e.g. \(a - 2b - 4c = 0\) | B1 | |
| Form a second relevant equation, e.g. \(2a + 3b - c = 0\) and solve for one ratio, e.g. \(a:b\) | M1 | |
| Obtain final answer \(a:b:c = 14:-7:7\) | A1 | |
| Use coordinates of a relevant point and values of \(a\), \(b\) and \(c\) and find \(d\) | M1 | |
| Obtain answer \(14x - 7y + 7z = 42\), or equivalent | A1 | |
| OR 1: Attempt to calculate the vector product of relevant vectors, e.g. \((\mathbf{i} - 2\mathbf{j} - 4\mathbf{k}) \times (2\mathbf{i} + 3\mathbf{j} - \mathbf{k})\) | M1 | |
| Obtain two correct components | A1 | |
| Obtain correct answer, e.g. \(14\mathbf{i} - 7\mathbf{j} + 7\mathbf{k}\) | A1 | |
| Substitute coordinates of a relevant point in \(14x - 7y + 7z = d\), or equivalent, and find \(d\) | M1 | |
| Obtain answer \(14x - 7y + 7z = 42\), or equivalent | A1 | |
| OR 2: Using a relevant point and relevant vectors, form a 2-parameter equation for the plane | M1 | |
| State a correct equation, e.g. \(\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k} + s(\mathbf{i} - 2\mathbf{j} - 4\mathbf{k}) + t(2\mathbf{i} + 3\mathbf{j} - \mathbf{k})\) | A1 | |
| State 3 correct equations in \(x\), \(y\), \(z\), \(s\) and \(t\) | A1 | |
| Eliminate \(s\) and \(t\) | M1 | |
| Obtain answer \(2x - y + z = 6\), or equivalent | A1 | |
| OR 3: Using a relevant point and relevant vectors, form a determinant equation for the plane | M1 | |
| State a correct equation, e.g. \(\begin{vmatrix} x-1 & y+2 & z-1 \\ 1 & -2 & -4 \\ 2 & 3 & -1 \end{vmatrix} = 0\) | A1 | |
| Attempt to expand the determinant | M1 | |
| Obtain or imply two correct cofactors | A1 | |
| Obtain answer \(14x - 7y + 7z = 42\), or equivalent | A1 |
## Question 10(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out a correct method for finding a vector equation for $AB$ | M1 | |
| Obtain $\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k} + \lambda(2\mathbf{i} + 3\mathbf{j} - \mathbf{k})$, or equivalent | A1 | |
| Equate two pairs of components of general points on $AB$ and $l$ and solve for $\lambda$ or $\mu$ | M1 | |
| Obtain correct answer for $\lambda$ or $\mu$, e.g. $\lambda = \frac{5}{7}$ or $\mu = \frac{3}{7}$ | A1 | |
| Obtain $m = 3$ | A1 | |
**Total: 5**
---
## Question 10(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| **EITHER:** Use scalar product to obtain an equation in $a$, $b$ and $c$, e.g. $a - 2b - 4c = 0$ | B1 | |
| Form a second relevant equation, e.g. $2a + 3b - c = 0$ and solve for one ratio, e.g. $a:b$ | M1 | |
| Obtain final answer $a:b:c = 14:-7:7$ | A1 | |
| Use coordinates of a relevant point and values of $a$, $b$ and $c$ and find $d$ | M1 | |
| Obtain answer $14x - 7y + 7z = 42$, or equivalent | A1 | |
| **OR 1:** Attempt to calculate the vector product of relevant vectors, e.g. $(\mathbf{i} - 2\mathbf{j} - 4\mathbf{k}) \times (2\mathbf{i} + 3\mathbf{j} - \mathbf{k})$ | M1 | |
| Obtain two correct components | A1 | |
| Obtain correct answer, e.g. $14\mathbf{i} - 7\mathbf{j} + 7\mathbf{k}$ | A1 | |
| Substitute coordinates of a relevant point in $14x - 7y + 7z = d$, or equivalent, and find $d$ | M1 | |
| Obtain answer $14x - 7y + 7z = 42$, or equivalent | A1 | |
| **OR 2:** Using a relevant point and relevant vectors, form a 2-parameter equation for the plane | M1 | |
| State a correct equation, e.g. $\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k} + s(\mathbf{i} - 2\mathbf{j} - 4\mathbf{k}) + t(2\mathbf{i} + 3\mathbf{j} - \mathbf{k})$ | A1 | |
| State 3 correct equations in $x$, $y$, $z$, $s$ and $t$ | A1 | |
| Eliminate $s$ and $t$ | M1 | |
| Obtain answer $2x - y + z = 6$, or equivalent | A1 | |
| **OR 3:** Using a relevant point and relevant vectors, form a determinant equation for the plane | M1 | |
| State a correct equation, e.g. $\begin{vmatrix} x-1 & y+2 & z-1 \\ 1 & -2 & -4 \\ 2 & 3 & -1 \end{vmatrix} = 0$ | A1 | |
| Attempt to expand the determinant | M1 | |
| Obtain or imply two correct cofactors | A1 | |
| Obtain answer $14x - 7y + 7z = 42$, or equivalent | A1 | |
**Total: 5**
---10 The points $A$ and $B$ have position vectors given by $\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }$ and $\overrightarrow { O B } = 3 \mathbf { i } + \mathbf { j } + \mathbf { k }$. The line $l$ has equation $\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + m \mathbf { k } + \mu ( \mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k } )$, where $m$ is a constant.\\
(i) Given that the line $l$ intersects the line passing through $A$ and $B$, find the value of $m$.\\
(ii) Find the equation of the plane which is parallel to $\mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k }$ and contains the points $A$ and $B$. Give your answer in the form $a x + b y + c z = d$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q10 [10]}}