M1 June 2014 Q6

Notes

In this question a misquoted Cosine Rule is M0.

The question asks for both answers to 3 SF but only penalise under or over accuracy once in this question.

First M1 for a complete method to give an equation in \(X\) only i.e. producing two components and usually squaring and adding and equating to \((3X)^2\) (condone sign errors and consistent incorrect trig. in the components for this M mark BUT the \(x\)-component must be a difference)

First A1 for a correct unsimplified equation in \(X\) only

e.g, allow \((\pm(X - 20\cos 60^\circ))^2 + (\pm(20\cos 30^\circ))^2 = (3X)^2\)

Second A1 for any correct fully numerical 3 term quadratic = 0

Second M1(independent) for solving a 3 term quadratic

Third A1 for 5.93

Alternative using cosine rule:

First M1 for use of cosine rule with cos60\(^\circ\) (M0 if they use 120\(^\circ\))

First A1 for a correct equation unsimplified e.g, allow cos60\(^\circ\) and \((3X)^2\)

Second A1 for any correct fully numerical 3 term quadratic = 0

Second M1(independent) for solving a 3 term quadratic

Third A1 for 5.93

Alternative using 2 applications of the sine rule:

First M1 for using \(3X / \sin 60 = X / \sin a\) AND

Either: \(X / \sin a = 20 / \sin(120^\circ - a)\)

Or: \(3X / \sin 60^\circ = 20 / \sin(120^\circ - a)\)

(These could be in terms of \(b\) where \(b = (120^\circ - a)\))

First A1 for two correct equations

Second A1 for \(a = 16.778..^\circ\) (or \(b = 103.221..^\circ\))

Second M1 for solving: \(X / \sin a = 20 / \sin(120^\circ - a)\) or \(3X / \sin 60^\circ = 20 / \sin(120^\circ - a)\) with their \(a\) or \(b\), to find \(X\)

Third A1 for 5.93

6a alt

Notes

First M1 for use of cosine rule unsimplified with cos120\(^\circ\) (M0 if they use 60\(^\circ\))

First A1 for a correct expression for \(|\mathbf{P} - \mathbf{Q}|\) in terms of \(X\) (does not need to be substituted)

Second M1, dependent on first M1, for substituting for their \(X\) and solving for \(|\mathbf{P} - \mathbf{Q}|\)

Second A1 for 23.5

Alternative using components:

First M1 for a complete method i.e. producing two components and squaring and adding (no square root needed) (condone sign errors and consistent incorrect trig. in the components for this M mark BUT the \(x\)-component must be a sum)

First A1 for a correct expression for \(|\mathbf{P} - \mathbf{Q}|\) (e.g, allow \((\pm(X + 20\cos 60^\circ))^2 + (\pm(20\cos 30^\circ))^2\)

Second M1, dependent on first M1, for substituting for their \(X\) and solving for \(|\mathbf{P} - \mathbf{Q}|\)

Second A1 for 23.5

6b alt