| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vector Product and Surfaces |
| Type | Conditions for vector product to be zero |
| Difficulty | Challenging +1.8 This Further Maths question requires understanding that a×b=0 with non-parallel vectors is impossible (leading to a contradiction in part a), then solving a system where b=λa with integer constraints. The conceptual insight about vector products and the algebraic manipulation with parametric constraints elevate this above standard exercises, though the techniques themselves are accessible to FM students. |
| Spec | 1.10b Vectors in 3D: i,j,k notation4.04g Vector product: a x b perpendicular vector |
4 The vectors $\mathbf { a }$ and $\mathbf { b }$ are given by $\mathbf { a } = \left( \begin{array} { c } p - 1 \\ q + 2 \\ 2 r - 3 \end{array} \right)$ and $\mathbf { b } = \left( \begin{array} { c } 2 p + 4 \\ 2 q - 5 \\ r + 3 \end{array} \right)$, where $p , q$ and $r$ are real numbers.
\begin{enumerate}[label=(\alph*)]
\item Given that $\mathbf { b }$ is not a multiple of $\mathbf { a }$ and that $\mathbf { a } \times \mathbf { b } = \mathbf { 0 }$, determine all possible sets of values of $p , q$ and $r$.
\item You are given instead that $\mathbf { b } = \lambda \mathbf { a }$, where $\lambda$ is an integer with $| \lambda | > 1$.
By writing each of $p , q$ and $r$ in terms of $\lambda$, show that there is a unique value of $\lambda$ for which $p , q$ and $r$ are all integers, stating this set of values of $p , q$ and $r$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2024 Q4 [10]}}