SPS SPS FM Pure 2024 June — Question 10 5 marks

Exam BoardSPS
ModuleSPS FM Pure (SPS FM Pure)
Year2024
SessionJune
Marks5
TopicVectors 3D & Lines
TypePerpendicularity conditions
DifficultyStandard +0.3 This is a straightforward perpendicularity problem requiring students to set the dot product equal to zero and solve a trigonometric equation. While it involves double angle identities and requires careful algebraic manipulation, it's a standard application of perpendicular vectors with no novel insight needed—slightly easier than average for Further Maths.
Spec1.05l Double angle formulae: and compound angle formulae1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication
10. $$\begin{aligned} & \boldsymbol { v } _ { \mathbf { 1 } } = \left( \begin{array} { c } \sqrt { 17 } \\ \cos 2 \theta \\ - 4 \end{array} \right) \\ & \boldsymbol { v } _ { \mathbf { 2 } } = \left( \begin{array} { c } - \sin 2 \theta \\ 2 \sqrt { 2 } \\ 1 \end{array} \right) \end{aligned}$$ Given that \(\boldsymbol { v } _ { \mathbf { 1 } }\) and \(\boldsymbol { v } _ { \mathbf { 2 } }\) are perpendicular and that \(0 \leq \theta \leq \pi\), find all possible values of \(\theta\). Give your answers to 3 significant figures.
[0pt] 
10.

$$\begin{aligned}
& \boldsymbol { v } _ { \mathbf { 1 } } = \left( \begin{array} { c } 
\sqrt { 17 } \\
\cos 2 \theta \\
- 4
\end{array} \right) \\
& \boldsymbol { v } _ { \mathbf { 2 } } = \left( \begin{array} { c } 
- \sin 2 \theta \\
2 \sqrt { 2 } \\
1
\end{array} \right)
\end{aligned}$$

Given that $\boldsymbol { v } _ { \mathbf { 1 } }$ and $\boldsymbol { v } _ { \mathbf { 2 } }$ are perpendicular and that $0 \leq \theta \leq \pi$, find all possible values of $\theta$. Give your answers to 3 significant figures.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q10 [5]}}
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