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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)λθR.

The plane Πp, has equation x=ppR.

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)=(0cosθsinθ))       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=526 or sinα=126       A1

this is independent of p and θ, hence the angle between the line and the plane, (90α), 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]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.14—Vector equation of line
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Topic 3— Geometry and trigonometry » AHL 3.18—Intersections of lines & planes
Topic 3— Geometry and trigonometry

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