N2017 Specimen Paper P2 Q3

N2017 Specimen Paper P2 Q3

13 marks
Past Year National Exams
Free
  1. The angle between the vectors \(3\mathbf{i}-2\mathbf{j}\) and \(6\mathbf{i}+d\mathbf{j}-\sqrt{7}\mathbf{k}\) is \({{\cos }^{-1}}\left( \frac{6}{13} \right)\). Show that \(2{{d}^{2}}-117d+333=0\).[3]
  2. With reference to the origin \(O\), the points \(A\), \(B\), \(C\) and \(D\) are such that \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\), \(\overrightarrow{AC}=5\mathbf{a}\) and \(\overrightarrow{BD}=3\mathbf{b}\). The lines \(AD\) and \(BC\) cross at \(E\) (see diagram).
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    1. Find \(\overrightarrow{OE}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).[6]
    2. The point \(F\) divides the line \(CD\) in the ratio \(5:3\). Show that \(O\), \(E\) and \(F\) are collinear, and find \(OE:OF\).[4]

Video Solution:

Default solution

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Answer:(a) \(2d^2-117d+333=0\). (b)(i) \(\overrightarrow{OE}=\frac{18}{23}\mathbf a+\frac{20}{23}\mathbf b\). (ii) \(O,E,F\) are collinear and \(OE:OF=8:23\).

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