M1 June 2016 Q1

EdexcelOld spec10 marksVectors

1. [In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]

Two cars \(P\) and \(Q\) are moving on straight horizontal roads with constant velocities. The velocity of \(P\) is \((15\mathbf{i} + 20\mathbf{j})\) m s\(^{-1}\) and the velocity of \(Q\) is \((20\mathbf{i} - 5\mathbf{j})\) m s\(^{-1}\)

(a) Find the direction of motion of \(Q\), giving your answer as a bearing to the nearest degree. (3)

At time \(t = 0\), the position vector of \(P\) is \(400\mathbf{i}\) metres and the position vector of \(Q\) is \(800\mathbf{j}\) metres. At time \(t\) seconds, the position vectors of \(P\) and \(Q\) are \(\mathbf{p}\) metres and \(\mathbf{q}\) metres respectively.

(b) Find an expression for

(i) \(\mathbf{p}\) in terms of \(t\),

(ii) \(\mathbf{q}\) in terms of \(t\). (3)

Mark scheme (a)

Scheme	Marks
\(\tan\theta = \tfrac{5}{20}\)	M1
\(\theta = 14.036..^\circ\)	A1
\(\theta = 104^\circ\) nearest degree	A1
	(3)

Notes

Allow column vectors throughout

M1 for \(\tan\theta = \pm\tfrac{5}{20}\) or \(\pm\tfrac{20}{5}\) (or any other complete method)

First A1 for \(\pm 14.04^\circ\) or \(\pm 75.96^\circ\)

Second A1 for \(104^\circ\)

Mark scheme (b)

Scheme	Marks
\(\mathbf{p} = 400\mathbf{i} + t(15\mathbf{i} + 20\mathbf{j})\)	M1 A1
\(\mathbf{q} = 800\mathbf{j} + t(20\mathbf{i} - 5\mathbf{j})\)	A1
	(3)

Notes

M1 for clear attempt at either \(\mathbf{p}\) or \(\mathbf{q}\) (allow slip but \(t\) must be attached to the velocity vector and position vector and velocity vector must be paired up correctly)

First A1 \(400\mathbf{i} + t(15\mathbf{i} + 20\mathbf{j})\) “\(\mathbf{p} =\)” not needed but must be clear it’s \(P\)

Second A1 \(800\mathbf{j} + t(20\mathbf{i} - 5\mathbf{j})\) “\(\mathbf{q} =\)” not needed but must be clear it’s \(Q\)

Mark scheme (c)

Scheme	Marks
Equate their \(\mathbf{j}\) components: \(\ 20t(\mathbf{j}) = (800 - 5t)(\mathbf{j})\)	M1
\(t = 32\)	A1
\(\mathbf{s} = 800\mathbf{j} + 32(20\mathbf{i} - 5\mathbf{j})\)	M1
\(= 640\mathbf{i} + 640\mathbf{j}\)	A1
	(4)
	(10 marks)

Notes

First M1 for equating their \(\mathbf{j}\) components; allow \(\mathbf{j}\)’s on both sides

First A1 for \(t = 32\)

Second M1 independent for substituting their \(t\) value into their \(\mathbf{q}\) from (b)

Second A1 for \(640\mathbf{i} + 640\mathbf{j}\)