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Date November 2020 Marks available 4 Reference code 20N.2.AHL.TZ0.H_10
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Find Question number H_10 Adapted from N/A

Question

The plane Π1Π1 has equation 3xy+z=133xy+z=13 and the line LL has vector equation

r=(12-2)+λ(-3-14) , λ.

The plane Π2 contains the point O and the line L.

Given that L meets Π1 at the point P, find the coordinates of P.

[4]
a.

Find the shortest distance from the point O(0, 0, 0) to Π1.

[4]
b.

Find the equation of Π2, giving your answer in the form r.n=d.

[3]
c.

Determine the acute angle between Π1 and Π2.

[5]
d.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

3(1-3λ)-(2-λ)+(-2+4λ)=-13        (M1)

λ=3         (A1)

r=(12-2)+3(-3-14)=(8-110)        (M1)

so  P(-8, -1, 10)         A1


Note:
Do not award the final A1 if a vector given instead of coordinates


[4 marks]

a.

METHOD 1

r=μ(3-11)

substituting into equation of the plane       M1

9μ+μ+μ=-13

μ=-1311 (=-1.18)       A1

distance =1332+(-1)2+1211        (M1)

=1311(=131111=3.92)       A1



METHOD 2

choice of any point on the plane, eg (-8, -1, 10) to use in distance formula        (M1)

so distance =(-8-110)·(-31-1)(-3)2+12+(-1)2       A1A1


Note: Award A1 for numerator, A1 for denominator.


=24-1-1011

=1311(=131111=3.92)       A1


[4 marks]

b.

EITHER

identify two vectors        (A1)

eg(12-2) and (-3-14)

n=(12-2)×(-3-14)=(625)        (M1)


OR


identify three points in the plane        (A1)

eg  λ=0,1 gives (12-2) and (-212)

solving system of equations        (M1)


THEN


Π2:r.(625)=0        A1


Note: Accept 6x+2y+5z=0.


[3 marks]

c.

vector normal to Π1 is eg n1=(3-11)

vector normal to Π2 is eg n2=(625)        (A1)

required angle is θ, where cosθ(3-11)·(625)1165        M1A1

cosθ=211165=0.785        (A1)

θ=0.667526

θ=0.668  (=38.2°)      A1

Note: Award the penultimate (A1) but not the final A1 for the obtuse angle 2.47406 or 142°.


[5 marks]

d.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.
[N/A]
d.

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.16—Vector product
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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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