| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| 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 vector arithmetic (finding vectors from position vectors) and standard scalar product application to find an angle. The geometric setup is clearly described, and both parts follow routine procedures taught in P3 with no novel problem-solving required. Slightly easier than average due to the step-by-step structure and standard techniques. |
| 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 |
|---|---|
| 5(i) | dy |
| Answer | Marks |
|---|---|
| dx | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| x2 ( x+3y ) | B1 | dy dy |
| Answer | Marks | Guidance |
|---|---|---|
| dx | M1 | Given answer so check working carefully |
| Obtain the given answer | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 5(ii) | Equate derivative to – 1 and solve for y | M1* |
| Answer | Marks | Guidance |
|---|---|---|
| in x or y | M1(dep*) | |
| Obtain answer (1, – 2) | A1 | |
| Obtain answer (33, 0) | B1 | 1ln3 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 5:
--- 5(i) ---
5(i) | dy
State or imply 3y2 as derivative of y3
dx | B1
dy
State or imply 6xy+3x2 as derivative of 3x2y
dx
OR State or imply 2x ( x+3y )+x2 1+3 dy as derivative of
dx
x2 ( x+3y ) | B1 | dy dy
3x2 +6xy+3x2 −3y2 =0
dx dx
dy
Equate derivative of the LHS to zero and solve for
dx | M1 | Given answer so check working carefully
Obtain the given answer | A1
4
--- 5(ii) ---
5(ii) | Equate derivative to – 1 and solve for y | M1*
Use their y = – 2x or equivalent to obtain an equation
in x or y | M1(dep*)
Obtain answer (1, – 2) | A1
Obtain answer (33, 0) | B1 | 1ln3
Must be exact e.g. e3 but ISW if decimals after exact value
seen
4
Question | Answer | Marks | Guidance\includegraphics{figure_5}
The diagram shows a three-dimensional shape. The base $OAB$ is a horizontal triangle in which angle $AOB$ is 90°. The side $OBCD$ is a rectangle and the side $OAD$ lies in a vertical plane. Unit vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $OA$ and $OB$ respectively and the unit vector $\mathbf{k}$ is vertical. The position vectors of $A$, $B$ and $D$ are given by $\overrightarrow{OA} = 8\mathbf{i}$, $\overrightarrow{OB} = 5\mathbf{j}$ and $\overrightarrow{OD} = 2\mathbf{i} + 4\mathbf{k}$.
\begin{enumerate}[label=(\roman*)]
\item Express each of the vectors $\overrightarrow{DA}$ and $\overrightarrow{CA}$ in terms of $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$. [2]
\item Use a scalar product to find angle $CAD$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q5 [6]}}