Date | November 2019 | Marks available | 6 | Reference code | 19N.1.AHL.TZ0.H_8 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 0 |
Command term | Show that | Question number | H_8 | Adapted from | N/A |
Question
A straight line, Lθ, has vector equation r =(500)+λ(5sinθcosθ), λ, θ∈R.
The plane Πp, has equation x=p, p∈R.
Show that the angle between Lθ and Πp is independent of both θ and p.
Markscheme
a vector normal to Πp is (100) (A1)
Note: Allow any scalar multiple of (100), including (p00)
attempt to find scalar product (or vector product) of direction vector of line with any scalar multiple of (100) M1
(100)∙(5sinθcosθ)=5 (or (100)×(5sinθcosθ)=(0−cosθsinθ)) A1
(if α is the angle between the line and the normal to the plane)
cosα=51×√25+sin2θ+cos2θ (or sinα=11×√25+sin2θ+cos2θ) A1
⇒cosα=5√26 or sinα=1√26 A1
this is independent of p and θ, hence the angle between the line and the plane, (90−α), is also independent of p and θ R1
Note: The final R mark is independent, but is conditional on the candidate obtaining a value independent of p and θ.
[6 marks]