| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | September |
| Marks | 5 |
| Topic | Vectors 3D & Lines |
| Type | Linear combinations of vectors |
| Difficulty | Standard +0.3 This is a standard vector geometry problem requiring students to find a position vector using section formula and intersection of lines. It involves routine techniques (finding P using ratio, expressing OQ two ways, equating coefficients) with straightforward algebra. While multi-step, it follows a well-practiced method with no novel insight required, making it slightly easier than average. |
| Spec | 1.10d Vector operations: addition and scalar multiplication1.10g Problem solving with vectors: in geometry |
12.
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\caption{Diagram NOT accurately drawn}
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The diagram shows a quadrilateral $O A C B$ in which
$$\overrightarrow { O A } = 4 \mathbf { a } \quad \overrightarrow { O B } = 3 \mathbf { b } \quad \overrightarrow { B C } = 2 \mathbf { a } + \mathbf { b }$$
The point $P$ lies on $A C$ such that $A P : P C = 3 : 2$\\
The point $Q$ is such that $O P Q$ and $B C Q$ are straight lines.\\
Using a vector method, find $\overrightarrow { O Q }$ in terms of $\mathbf { a }$ and $\mathbf { b }$\\
Give your answer in its simplest form.\\
Show your working clearly.\\
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q12 [5]}}