19N.2.SL.TZ0.S_2

pestleMathematicsAASLPaper 219N· sl-2-10-solving-equations-graphically-and-analyticallysource ↗

Consider the lines ${L_1}$ and ${L_2}$ with respective equations

${L_1}\,{\text{:}}\,y = - \frac{2}{3}x + 9$ and ${L_2}{\text{:}}\,y = \frac{2}{5}x - \frac{{19}}{5}$ .

A third line, ${L_3}$ , has gradient $- \frac{3}{4}$ .

Find the point of intersection of ${L_1}$ and ${L_2}$ .

[2]

Write down a direction vector for ${L_3}$ .

[1]

${L_3}$ passes through the intersection of ${L_1}$ and ${L_2}$ .

Write down a vector equation for ${L_3}$ .

[2]

Markscheme / solution

valid approach (M1)

eg ${L_1} = {L_2}$ , $x = 12$ , $y = 1$

$\left( {12{\text{, }}1} \right)$ (exact) A1 N2

[2 marks]

$\left( {\begin{array}{*{20}{c}} { - 4} \\ 3 \end{array}} \right)$ (or any multiple of $\left( {\begin{array}{*{20}{c}} { - 4} \\ 3 \end{array}} \right)$ ) A1 N1

[1 mark]

any correct equation in the form r = a + $t$ b (accept any parameter for $t$ ) where
a is a position vector for a point on ${L_1}$ , and b is a scalar multiple of $\left( {\begin{array}{*{20}{c}} { - 4} \\ 3 \end{array}} \right)$ A2 N2

eg r $= \left( {\begin{array}{*{20}{c}} {12} \\ 1 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} { - 4} \\ 3 \end{array}} \right)$

Note: Award A1 for the form a + $t$ b, A1 for the form $L$ = a + $t$ b, A0 for the form r = b + $t$ a.

[2 marks]

Examiners’ report

[N/A]