| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2020 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vector Product and Surfaces |
| Type | Geometric interpretation of vector product |
| Difficulty | Challenging +1.2 This is a Further Maths question testing standard properties of vector products (anti-commutativity, scalar triple product, perpendicularity) with straightforward geometric interpretation. While it requires knowledge beyond A-level Core, the individual parts are direct applications of known results rather than requiring novel problem-solving, placing it moderately above average difficulty. |
| Spec | 1.10b Vectors in 3D: i,j,k notation4.04g Vector product: a x b perpendicular vector |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (a) | (i) |
| [1] | 1.1 | |
| (ii) | a×(b × c) = a×a | |
| = 0 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 1.1 | |
| 1.2 | Must be a vector zero | |
| (iii) | a.(b × c) = a.a | |
| = | a | 2 = 9 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | (i) | Using b × c= a OA is parallel to BC |
| OR OA is normal to the plane OBC | B1 | |
| [1] | 2.2a | |
| (ii) | Using (a) (iii): |
| Answer | Marks |
|---|---|
| OR (FT from (b) (i)) Prism with base OBC and height OA has volume 9 | B1 |
| [1] | 2.2a |
Question 4:
4 | (a) | (i) | c×b = – a | B1
[1] | 1.1
(ii) | a×(b × c) = a×a
= 0 | M1
A1
[2] | 1.1
1.2 | Must be a vector zero
(iii) | a.(b × c) = a.a
= | a |2 = 9 | M1
A1
[2] | 1.1
2.2a
(b) | (i) | Using b × c= a OA is parallel to BC
OR OA is normal to the plane OBC | B1
[1] | 2.2a
(ii) | Using (a) (iii):
tetrahedron OABC has volume 1.5 (cubic units)
OR noting that O, A, B, C are not co-planar
OR (FT from (b) (i)) Prism with base OBC and height OA has volume 9 | B1
[1] | 2.2a4 Points $A , B$ and $C$ have position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ respectively, relative to origin $O$. It is given that $\mathbf { b } \times \mathbf { c } = \mathbf { a }$ and that $| \mathbf { a } | = 3$.
\begin{enumerate}[label=(\alph*)]
\item Determine each of the following as either a single vector or a scalar quantity.
\begin{enumerate}[label=(\roman*)]
\item $\mathbf { c } \times \mathbf { b }$
\item $\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } )$
\item $\mathbf { a } \cdot ( \mathbf { b } \times \mathbf { c } )$
\end{enumerate}\item Describe a geometrical relationship between the points $O , A , B$ and $C$ which can be deduced from
\begin{enumerate}[label=(\roman*)]
\item the statement $\mathbf { b } \times \mathbf { c } = \mathbf { a }$,
\item the result of (a)(iii).
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2020 Q4 [7]}}