| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| 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 straightforward multi-part vectors question requiring standard techniques: finding a fourth vertex using vector addition, calculating magnitudes to verify a rhombus, and finding a plane equation using cross product of two direction vectors. All steps are routine for Further Maths students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.10g Problem solving with vectors: in geometry4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER: State or imply \(\overrightarrow{AB}\) or \(\overrightarrow{BC}\) in component form, or state position vector of midpoint of \(\overrightarrow{AC}\) | B1 | |
| Use a correct method for finding the position vector of \(D\) | M1 | |
| Obtain answer \(3\mathbf{i} + 3\mathbf{j} + \mathbf{k}\), or equivalent | A1 | |
| EITHER: Using the correct process for the moduli, compare lengths of a pair of adjacent sides, e.g. \(AB\) and \(BC\) | M1 | |
| Show that \(ABCD\) has a pair of adjacent sides that are equal | A1 | |
| OR: Calculate scalar product \(\overrightarrow{AC} \cdot \overrightarrow{BD}\), or equivalent | M1 | |
| Show that \(ABCD\) has perpendicular diagonals | A1 | [5] |
| (ii) EITHER: State \(a + 2b + 3c = 0\) or \(2a + b - 2c = 0\) | B1 | |
| Obtain two relevant equations and solve for one ratio, e.g. \(a : b = -7 : 8 : -3\) | M1 | |
| Obtain answer \(a : b : c = -7 : 8 : -3\), or equivalent | A1 | |
| Substitute coordinates of a relevant point in \(-7x + 8y - 3z = d\), and evaluate | M1 | |
| Obtain answer \(-7x + 8y - 3z = 29\), or equivalent | A1 | |
| OR1: Attempt to calculate vector product of relevant vectors, e.g. \((\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) \times (2\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) | M1 | |
| Obtain two correct components of the product | A1 | |
| Obtain correct product, e.g. \(-7\mathbf{i} + 8\mathbf{j} - 3\mathbf{k}\) | A1 | |
| Substitute coordinates of a relevant point in \(-7x + 8y - 3z = d\) and evaluate \(d\) | M1 | |
| Obtain answer \(-7x + 8y - 3z = 29\), or equivalent | A1 | |
| OR2: Attempt to form a 2-parameter equation with relevant vectors | M1 | |
| State a correct equation, e.g. \(\mathbf{r} = 2\mathbf{i} + 5\mathbf{j} - \mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}) + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) | A1 | |
| State 3 equations in \(x, y, z, \lambda\) and \(\mu\) | A1 | |
| Eliminate \(\lambda\) and \(\mu\) | M1 | |
| Obtain answer \(-7x + 8y - 3z = 29\), or equivalent | A1 | |
| OR3: Using a relevant point and relevant direction vectors, form a determinant equation for the plane | M1 | |
| State a correct equation, e.g. \(\begin{vmatrix} x-2 & y-5 & z+1 \\ 2 & 1 & -2 \end{vmatrix} = 0\) | A1 | |
| Attempt to expand the determinant | M1 | |
| Obtain correct values of two cofactors | A1 | |
| Obtain answer \(-7x + 8y - 3z = 29\), or equivalent | A1 |
**(i)** **EITHER:** State or imply $\overrightarrow{AB}$ or $\overrightarrow{BC}$ in component form, or state position vector of midpoint of $\overrightarrow{AC}$ | B1 |
Use a correct method for finding the position vector of $D$ | M1 |
Obtain answer $3\mathbf{i} + 3\mathbf{j} + \mathbf{k}$, or equivalent | A1 |
**EITHER:** Using the correct process for the moduli, compare lengths of a pair of adjacent sides, e.g. $AB$ and $BC$ | M1 |
Show that $ABCD$ has a pair of adjacent sides that are equal | A1 |
**OR:** Calculate scalar product $\overrightarrow{AC} \cdot \overrightarrow{BD}$, or equivalent | M1 |
Show that $ABCD$ has perpendicular diagonals | A1 | [5]
**(ii)** **EITHER:** State $a + 2b + 3c = 0$ or $2a + b - 2c = 0$ | B1 |
Obtain two relevant equations and solve for one ratio, e.g. $a : b = -7 : 8 : -3$ | M1 |
Obtain answer $a : b : c = -7 : 8 : -3$, or equivalent | A1 |
Substitute coordinates of a relevant point in $-7x + 8y - 3z = d$, and evaluate | M1 |
Obtain answer $-7x + 8y - 3z = 29$, or equivalent | A1 |
**OR1:** Attempt to calculate vector product of relevant vectors, e.g. $(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) \times (2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$ | M1 |
Obtain two correct components of the product | A1 |
Obtain correct product, e.g. $-7\mathbf{i} + 8\mathbf{j} - 3\mathbf{k}$ | A1 |
Substitute coordinates of a relevant point in $-7x + 8y - 3z = d$ and evaluate $d$ | M1 |
Obtain answer $-7x + 8y - 3z = 29$, or equivalent | A1 |
**OR2:** Attempt to form a 2-parameter equation with relevant vectors | M1 |
State a correct equation, e.g. $\mathbf{r} = 2\mathbf{i} + 5\mathbf{j} - \mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}) + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$ | A1 |
State 3 equations in $x, y, z, \lambda$ and $\mu$ | A1 |
Eliminate $\lambda$ and $\mu$ | M1 |
Obtain answer $-7x + 8y - 3z = 29$, or equivalent | A1 |
**OR3:** Using a relevant point and relevant direction vectors, form a determinant equation for the plane | M1 |
State a correct equation, e.g. $\begin{vmatrix} x-2 & y-5 & z+1 \\ 2 & 1 & -2 \end{vmatrix} = 0$ | A1 |
Attempt to expand the determinant | M1 |
Obtain correct values of two cofactors | A1 |
Obtain answer $-7x + 8y - 3z = 29$, or equivalent | A1 |9 The points $A , B$ and $C$ have position vectors, relative to the origin $O$, given by $\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$, $\overrightarrow { O B } = 4 \mathbf { j } + \mathbf { k }$ and $\overrightarrow { O C } = 2 \mathbf { i } + 5 \mathbf { j } - \mathbf { k }$. A fourth point $D$ is such that the quadrilateral $A B C D$ is a parallelogram.\\
(i) Find the position vector of $D$ and verify that the parallelogram is a rhombus.\\
(ii) The plane $p$ is parallel to $O A$ and the line $B C$ lies in $p$. Find the equation of $p$, giving your answer in the form $a x + b y + c z = d$.
\hfill \mbox{\textit{CAIE P3 2016 Q9 [10]}}