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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 Π1 has equation 3xy+z=13 and the line L 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.

31-3λ-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=μ3-11

substituting into equation of the plane       M1

9μ+μ+μ=-13

μ=-1311 =-1.18       A1

distance =1332+-12+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·-31-1-32+12+-12       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)

eg12-2 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 cosθ3-11·6251165        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.
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b.
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c.
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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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