SPM.1.AHL.TZ0.8

pestleMathematicsAAHLPaper 1SPM· ahl-3-14-vector-equation-of-linesource ↗

The plane П has the Cartesian equation $2x + y + 2z = 3$

The line L has the vector equation r $= \left( {\begin{array}{*{20}{c}} 3 \\ { - 5} \\ 1 \end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}} 1 \\ { - 2} \\ p \end{array}} \right){\text{,}}\,\,\mu {\text{,}}\,p \in \mathbb{R}$ . The acute angle between the line L and the plane П is 30°.

Find the possible values of $p$ .

Markscheme / solution

recognition that the angle between the normal and the line is 60° (seen anywhere) R1

attempt to use the formula for the scalar product M1

cos 60° = $\frac{{\left| {\left( {\begin{array}{*{20}{c}} 2 \\ 1 \\ 2 \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} 1 \\ { - 2} \\ p \end{array}} \right)} \right|}}{{\sqrt 9 \times \sqrt {1 + 4 + {p^2}} }}$ A1

$\frac{1}{2} = \frac{{\left| {2p} \right|}}{{3\sqrt {5 + {p^2}} }}$ A1

$3\sqrt {5 + {p^2}} = 4\left| p \right|$

attempt to square both sides M1

$9\left( {5 + {p^2}} \right) = 16{p^2} \Rightarrow 7{p^2} = 45$

$p = \pm 3\sqrt {\frac{5}{7}}$ (or equivalent) A1A1

[7 marks]

Examiners’ report

[N/A]