| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Line intersection: show lines are skew |
| Difficulty | Standard +0.3 Part (a) requires checking if lines intersect by solving a system of equations and verifying they're not parallel—a standard multi-step procedure. Part (b) is routine application of the dot product formula for angle between direction vectors. Both are textbook exercises with well-established methods, slightly above average difficulty only due to the 3D coordinate work and multiple steps involved. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks |
|---|---|
| 7(a) | Express general point of a line in component form, e.g. |
| (1 + 2s, 3 – s, 2 + 3s) or (2 + t, 1 – t, 4 + 4t) | B1 |
| Equate at least two pairs of components and solve for s or for t | M1 |
| Answer | Marks |
|---|---|
| 5 | A1 |
| Verify that all three component equations are not satisfied | A1 |
| Show that the lines are not parallel and are thus skew | A1 |
| Answer | Marks |
|---|---|
| 7(b) | Carry out correct process for evaluating the scalar product of the direction |
| vectors | M1 |
| Answer | Marks |
|---|---|
| product of the moduli and evaluate the inverse cosine of the result | M1 |
| Obtain answer 19.1° or 0.333 radians | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 7:
--- 7(a) ---
7(a) | Express general point of a line in component form, e.g.
(1 + 2s, 3 – s, 2 + 3s) or (2 + t, 1 – t, 4 + 4t) | B1
Equate at least two pairs of components and solve for s or for t | M1
2
Obtain correct answer for s or for t (possible answers are –1, 6, for s and
5
1
–3, 4, − for t )
5 | A1
Verify that all three component equations are not satisfied | A1
Show that the lines are not parallel and are thus skew | A1
5
--- 7(b) ---
7(b) | Carry out correct process for evaluating the scalar product of the direction
vectors | M1
Using the correct process for the moduli, divide the scalar product by the
product of the moduli and evaluate the inverse cosine of the result | M1
Obtain answer 19.1° or 0.333 radians | A1
3
Question | Answer | Marks | GuidanceTwo lines have equations $\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}$.
\begin{enumerate}[label=(\alph*)]
\item Show that the lines are skew. [5]
\item Find the acute angle between the directions of the two lines. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q7 [8]}}