| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Vector geometry in 3D shapes |
| Difficulty | Standard +0.3 This is a straightforward 3D vectors question requiring basic coordinate setup and standard scalar product application. Part (i) involves simple ratio calculation along a diagonal, and part (ii) is a routine angle-between-vectors calculation using the dot product formula. The rectangular base and clear vertical height make visualization easy, and all required techniques are standard A-level procedures with no novel problem-solving required. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication4.04c Scalar product: calculate and use for angles |
| Answer | Marks | Guidance |
|---|---|---|
| 9(a) | Substitute and obtain a correct equation in x and y | B1 |
| Use i2 = −1 and equate real and imaginary parts | M1 |
| Answer | Marks |
|---|---|
| 3x – y = 1 and 3y – x = 5 | A1 |
| Solve and obtain answer z = 1 + 2(i) | A1 |
| Total: | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| 9(b) | Show a circle with radius 3 | B1 |
| Show the line y = 2 extending in both quadrants | B1 | |
| Shade the correct region | B1 | |
| Carry out a complete method for finding the greatest value of arg z | M1 | |
| Obtain answer 2.41 | A1 | |
| Total: | 5 | |
| Question | Answer | Marks |
Question 9:
--- 9(a) ---
9(a) | Substitute and obtain a correct equation in x and y | B1
Use i2 = −1 and equate real and imaginary parts | M1
Obtain two correct equations in x and y, e.g.
3x – y = 1 and 3y – x = 5 | A1
Solve and obtain answer z = 1 + 2(i) | A1
Total: | 4
--- 9(b) ---
9(b) | Show a circle with radius 3 | B1
Show the line y = 2 extending in both quadrants | B1
Shade the correct region | B1
Carry out a complete method for finding the greatest value of arg z | M1
Obtain answer 2.41 | A1
Total: | 5
Question | Answer | Marks\includegraphics{figure_9}
The diagram shows a pyramid $OABCD$ with a horizontal rectangular base $OABC$. The sides $OA$ and $AB$ have lengths of 8 units and 6 units respectively. The point $E$ on $OB$ is such that $OE = 2$ units. The point $D$ of the pyramid is 7 units vertically above $E$. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $ED$ respectively.
\begin{enumerate}[label=(\roman*)]
\item Show that $\overrightarrow{OE} = 1.6\mathbf{i} + 1.2\mathbf{j}$. [2]
\item Use a scalar product to find angle $BDO$. [7]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q9 [9]}}