| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Acute angle between two planes |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question testing routine techniques: showing a line is parallel to a plane (dot product of direction vector with normal equals zero), finding angle between planes (using normal vectors), and finding points at a given distance from a plane. All three parts use well-practiced methods with straightforward algebra, making it slightly easier than average for Further Maths content. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane4.04j Shortest distance: between a point and a plane |
| Answer | Marks | Guidance |
|---|---|---|
| 10(i) | EITHER: Expand scalar product of a normal to m and a direction vector of l | M1 |
| Verify scalar product is zero | A1 | |
| Verify that one point of l does not lie in the plane | A1 | |
| OR: Substitute coordinates of a general point of l in the equation of the plane m | M1 | |
| Obtain correct equation in λ in any form | A1 | |
| Verify that the equation is not satisfied for any value of λ | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 10(ii) | Use correct method to evaluate a scalar product of normal vectors to m and n | M1 |
| Answer | Marks |
|---|---|
| moduli and evaluate the inverse cosine of the result | M1 |
| Obtain answer 74.5° or 1.30 radians | A1 |
| Answer | Marks |
|---|---|
| 10(iii) | EITHER: Using the components of a general point P of l form an equation in λ by |
| equating the perpendicular distance from n to 2 | M1 |
| Answer | Marks |
|---|---|
| length of the projection of QP onto a normal to plane n to 2 | M1 |
| Obtain a correct modular or non-modular equation in any form | A1 |
| Solve for λ and obtain a position vector for P, e.g. 7i + 5j + 7j from λ = 3 | A1 |
| Obtain position vector of the second point, e.g. 3i + j – k from λ = – 1 | A1 |
Question 10:
--- 10(i) ---
10(i) | EITHER: Expand scalar product of a normal to m and a direction vector of l | M1
Verify scalar product is zero | A1
Verify that one point of l does not lie in the plane | A1
OR: Substitute coordinates of a general point of l in the equation of the plane m | M1
Obtain correct equation in λ in any form | A1
Verify that the equation is not satisfied for any value of λ | A1
3
--- 10(ii) ---
10(ii) | Use correct method to evaluate a scalar product of normal vectors to m and n | 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 74.5° or 1.30 radians | A1
3
--- 10(iii) ---
10(iii) | EITHER: Using the components of a general point P of l form an equation in λ by
equating the perpendicular distance from n to 2 | M1
OR: Take a point Q on l, e.g. (5, 3, 3) and form an equation in λ by equating the
length of the projection of QP onto a normal to plane n to 2 | M1
Obtain a correct modular or non-modular equation in any form | A1
Solve for λ and obtain a position vector for P, e.g. 7i + 5j + 7j from λ = 3 | A1
Obtain position vector of the second point, e.g. 3i + j – k from λ = – 1 | A1
4The planes $m$ and $n$ have equations $3x + y - 2z = 10$ and $x - 2y + 2z = 5$ respectively. The line $l$ has equation $\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} + 2\mathbf{k})$.
\begin{enumerate}[label=(\roman*)]
\item Show that $l$ is parallel to $m$. [3]
\item Calculate the acute angle between the planes $m$ and $n$. [3]
\item A point $P$ lies on the line $l$. The perpendicular distance of $P$ from the plane $n$ is equal to 2. Find the position vectors of the two possible positions of $P$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q10 [10]}}