## Exam-Style Question on Vectors## A mathematics exam-style question with a worked solution that can be revealed gradually |

Question id: 655. This question is similar to one that appeared on an IB AA Higher paper in 2023. The use of a calculator is allowed.

The following diagram shows parallelogram OABC with \( \overrightarrow{OA}\ = \pmb{a} \), \( \overrightarrow{OC}\ = \pmb{c} \) and \(|\pmb{c}| = 2|\pmb{a}| \)

The angle \(ABC = \theta \), where \(0 \lt \theta \lt \pi\).

Point M is on [AB] such that \( \overrightarrow{AM} = k\overrightarrow{AB} \), where \( 0 \le k \le 1 \).

OM is perpendicular to MC.

(a) Express \( \overrightarrow{OM} \) and \( \overrightarrow{MC} \) in terms of \(\pmb{a}\) and \(\pmb{c}\).

(b) Hence, use a vector method to show that \( |a|^2(1-2k)(2\cos{\theta}-(1-2k))=0 \).

(c) Find the range of values for \( \theta \) such that there are two possible positions for M.

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