Date | November 2019 | Marks available | 1 | Reference code | 19N.1.AHL.TZ0.H_11 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 0 |
Command term | Comment | Question number | H_11 | Adapted from | N/A |
Question
Points A(0 , 0 , 10) , B(0 , 10 , 0) , C(10 , 0 , 0) , V(p , p , p) form the vertices of a tetrahedron.
Consider the case where the faces ABV and ACV are perpendicular.
The following diagram shows the graph of θ against p. The maximum point is shown by X.
Show that →AB×→AV=−10(10−2ppp) and find a similar expression for →AC×→AV.
Hence, show that, if the angle between the faces ABV and ACV is θ, then cosθ=p(3p−20)6p2−40p+100.
Find the two possible coordinates of V.
Comment on the positions of V in relation to the plane ABC.
At X, find the value of p and the value of θ.
Find the equation of the horizontal asymptote of the graph.
Markscheme
→AV=(ppp−10) A1
→AB×→AV=(010−10)×(ppp−10)=(10(p−10)+10p−10p−10p) A1
=(20p−100−10p−10p)=−10(10−2ppp) AG
→AC×→AV=(100−10)×(ppp−10)=(10p100−20p10p)(=10(p10−2pp)) A1
[3 marks]
attempt to find a scalar product M1
−10(10−2ppp)∙10(p10−2pp)=100(3p2−20p)
OR −(10−2ppp)∙(p10−2pp)=3p2−20p A1
attempt to find magnitude of either →AB×→AV or →AC×→AV M1
|−10(10−2ppp)|=|10(p10−2pp)|=10√(10−2p)2+2p2 A1
100(3p2−20p)=100(√(10−2p)2+2p2)2cosθ
cosθ=3p2−20p(10−2p)2+2p2 A1
Note: Award A1 for any intermediate step leading to the correct answer.
=p(3p−20)6p2−40p+100 AG
Note: Do not allow FT marks from part (a)(i).
[8 marks]
p(3p−20)=0⇒p=0 or p=203 M1A1
coordinates are (0, 0, 0) and (203, 203, 203) A1
Note: Do not allow column vectors for the final A mark.
[3 marks]
two points are mirror images in the plane
or opposite sides of the plane
or equidistant from the plane
or the line connecting the two Vs is perpendicular to the plane R1
[1 mark]
geometrical consideration or attempt to solve −1=p(3p−20)6p2−40p+100 (M1)
p=103, θ=π or θ=180∘ A1A1
[3 marks]
p→∞⇒cosθ→12 M1
hence the asymptote has equation θ=π3 A1
[2 marks]