| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Triangle and parallelogram areas |
| Difficulty | Standard +0.3 This is a straightforward multi-part vectors question requiring standard techniques: vector addition for finding C, magnitude calculations to verify not a rhombus, dot product for angle, and cross product for area. All are routine A-level Further Maths procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(\overrightarrow{AB}\) or \(\overrightarrow{AD}\) in component form | B1 | |
| Use a correct method for finding the position vector of \(C\) | M1 | |
| Obtain answer \(4\mathbf{i}+3\mathbf{j}+4\mathbf{k}\), or equivalent | A1 | |
| Using the correct process for the moduli, compare lengths of a pair of adjacent sides, e.g. \(AB\) and \(AD\) | M1 | |
| Show that \(ABCD\) has a pair of unequal adjacent sides | A1 | |
| Alternative method: | ||
| State or imply \(\overrightarrow{AB}\) or \(\overrightarrow{AD}\) in component form | B1 | |
| Use a correct method for finding the position vector of \(C\) | M1 | |
| Obtain answer \(4\mathbf{i}+3\mathbf{j}+4\mathbf{k}\), or equivalent | A1 | |
| Use the correct process to calculate the scalar product of \(\overrightarrow{AC}\) and \(\overrightarrow{BD}\), or equivalent | M1 | |
| Show that the diagonals of \(ABCD\) are not perpendicular | A1 | |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the correct process to calculate the scalar product of a pair of relevant vectors, e.g. \(\overrightarrow{AB}\) and \(\overrightarrow{AD}\) | M1 | |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli of the two vectors and evaluate the inverse cosine of the result | M1 | |
| Obtain answer \(100.3°\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a correct method to calculate the area, e.g. calculate \(AB \cdot AC \sin BAD\) | M1 | |
| Obtain answer \(11.0\) | A1 FT | FT on angle BAD |
| 2 |
## Question 8:
### Part 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\overrightarrow{AB}$ or $\overrightarrow{AD}$ in component form | B1 | |
| Use a correct method for finding the position vector of $C$ | M1 | |
| Obtain answer $4\mathbf{i}+3\mathbf{j}+4\mathbf{k}$, or equivalent | A1 | |
| Using the correct process for the moduli, compare lengths of a pair of adjacent sides, e.g. $AB$ and $AD$ | M1 | |
| Show that $ABCD$ has a pair of unequal adjacent sides | A1 | |
| **Alternative method:** | | |
| State or imply $\overrightarrow{AB}$ or $\overrightarrow{AD}$ in component form | B1 | |
| Use a correct method for finding the position vector of $C$ | M1 | |
| Obtain answer $4\mathbf{i}+3\mathbf{j}+4\mathbf{k}$, or equivalent | A1 | |
| Use the correct process to calculate the scalar product of $\overrightarrow{AC}$ and $\overrightarrow{BD}$, or equivalent | M1 | |
| Show that the diagonals of $ABCD$ are not perpendicular | A1 | |
| | **5** | |
### Part 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the correct process to calculate the scalar product of a pair of relevant vectors, e.g. $\overrightarrow{AB}$ and $\overrightarrow{AD}$ | M1 | |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli of the two vectors and evaluate the inverse cosine of the result | M1 | |
| Obtain answer $100.3°$ | A1 | |
| | **3** | |
### Part 8(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a correct method to calculate the area, e.g. calculate $AB \cdot AC \sin BAD$ | M1 | |
| Obtain answer $11.0$ | A1 FT | FT on angle BAD |
| | **2** | |
---8 Relative to the origin $O$, the points $A , B$ and $D$ have position vectors given by
$$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { O B } = 2 \mathbf { i } + 5 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O D } = 3 \mathbf { i } + 2 \mathbf { k }$$
A fourth point $C$ is such that $A B C D$ is a parallelogram.
\begin{enumerate}[label=(\alph*)]
\item Find the position vector of $C$ and verify that the parallelogram is not a rhombus.
\item Find angle $B A D$, giving your answer in degrees.
\item Find the area of the parallelogram correct to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q8 [10]}}