1.11: Past-paper question 11

4PM1/2/November/2023 — Question 11 · 9 marks

1.11 diagram 1
Figure 3 shows quadrilateral OABC\displaystyle OABC where
OA→=4p+5qOB→=3p+qOC→=2p−4q\overrightarrow{OA} = 4\mathbf{p} + 5\mathbf{q} \quad \overrightarrow{OB} = 3\mathbf{p} + \mathbf{q} \quad \overrightarrow{OC} = 2\mathbf{p} - 4\mathbf{q}
The point M\displaystyle M is the midpoint of OC\displaystyle OC
(a) Find MA→\displaystyle \overrightarrow{MA} as a simplified expression in terms of p\displaystyle \mathbf{p} and q\displaystyle \mathbf{q}
(3)
The point N\displaystyle N lies on OB\displaystyle OB such that M,N\displaystyle M, N and A\displaystyle A are collinear.
(b) Find the ratio MN:NA\displaystyle MN : NA
(6)