Vectors in Two Dimensions Structured Questions | Cambridge O Level Additional Mathematics

Question 1 of 4 · 10 marks

Vectors in Two Dimensions · 3 - Vector Geometry & Ratios

4 parts

Additional Mathematics 4037 · Question 1Paper 2 · Calculator[10 marks]

In the trapezium $OABC$ , $\vec{OA} = \mathbf{a}$ and $\vec{OC} = \mathbf{c}$ . The side $\ce{CB}$ is parallel to $OA$ such that $\vec{CB} = \frac{3}{2}\mathbf{a}$ . Point $M$ lies on $\ce{OB}$ such that $OM : MB = 3 : 2$ . Point $N$ lies on $AB$ such that $AN : NB = 1 : 4$ .

(a)

Express $\vec{OB}$ in terms of $\mathbf{a}$ and $\mathbf{c}$ .

[2]

(b)

Express $\vec{OM}$ and $\vec{MN}$ in terms of $\mathbf{a}$ and $\mathbf{c}$ .

[4]

(c)

Show that $\vec{MN}$ can be written in the form $k(\mathbf{a} - 2\mathbf{c})$ , stating the value of scalar $k$ .

[2]

(d)

Point $P$ lies on $AC$ such that $\vec{AP} = \frac{2}{5}\vec{AC}$ . Deduce whether $M, N$ and $P$ are collinear.

[2]