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Date November 2021 Marks available 2 Reference code 21N.2.AHL.TZ0.11
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Hence and Find Question number 11 Adapted from N/A

Question

Three points A3, 0, 0, B0, -2, 0 and C1, 1, -7 lie on the plane Π1.

Plane Π2 has equation 3x-y+2z=2.

The plane Π3 is given by 2x-2z=3. The line L and the plane Π3 intersect at the point P.

The point B(0,-2, 0) lies on L.

Find the vector AB and the vector AC.

[2]
a.i.

Hence find the equation of Π1, expressing your answer in the form ax+by+cz=d, where a, b, c, d.

[5]
a.ii.

The line L is the intersection of Π1 and Π2. Verify that the vector equation of L can be written as r=0-20+λ11-1.

[2]
b.

Show that at the point P, λ=34.

[2]
c.i.

Hence find the coordinates of P.

[1]
c.ii.

Find the reflection of the point B in the plane Π3.

[7]
d.i.

Hence find the vector equation of the line formed when L is reflected in the plane Π3.

[2]
d.ii.

Markscheme

attempts to find either AB or AC             (M1)

AB=-3-20 and AC=-21-7             A1

 

[2 marks]

a.i.

METHOD 1

attempts to find AB×AC             (M1)

AB×AC=14-21-7             A1


EITHER

equation of plane is of the form 14x-21y-7z=d   2x-3y-z=d               (A1)

substitutes a valid point e.g 3, 0, 0 to obtain a value of d             M1

d=42  d=6


OR

attempts to use r·n=a·n             (M1)

r·14-21-7=300·14-21-7  r·14-21-7=42               A1

r·2-3-1=300·2-3-1  r·2-3-1=6


THEN

14x-21y-7z=42  2x-3y-z=6               A1

 

METHOD 2

equation of plane is of the form xyz=300+s-3-20+t-21-7               A1

attempts to form equations for x, y, z in terms of their parameters             (M1)

x=3-3s-2t , y=-2s+t , z=-7t               A1

eliminates at least one of their parameters             (M1)

for example, 2x-3y=6-7t2x-3y=6+z

2x-3y-z=6               A1

 

[5 marks]

a.ii.

METHOD 1

substitutes r=0-20+λ11-1 into their Π1 and Π2 (given)             (M1)

Π1: 2λ-3-2+λ--λ=6  and  Π2: 3λ-3-2+λ+2-λ=2            A1


Note: Award (M1)A0 for correct verification using a specific value of λ.

so the vector equation of L can be written as r=0-20+λ11-1            AG


METHOD 2

EITHER

attempts to find  2-3-1×3-12                    M1

=-7-77


OR

2-3-1·11-1=2-3+1=0  and  3-12·11-1=3-1-2=0                    M1


THEN

substitutes 0,-2,0 into Π1 and Π2

Π1: 20-3-2-0=6  and  Π2: 30--2+20=2            A1

so the vector equation of L can be written as r=0-20+λ11-1            AG

 

METHOD 3

attempts to solve 2x-3y-z=6 and 3x-y+2z=2             (M1)

for example, x=-λ, y=-2-λ, z=λ               A1


Note: Award A1 for substituting x=0 (or y=-2 or z=0) into Π1 and Π2 and solving simultaneously. For example, solving -3y-z=6 and -y+2z=2 to obtain y=-2 and z=0.

so the vector equation of L can be written as r=0-20+λ11-1            AG

 

[2 marks]

b.

substitutes the equation of L into the equation of Π3             (M1)

2λ+2λ=34λ=3            A1

λ=34            AG

 

[2 marks]

c.i.

P has coordinates  34,-54,-34       A1

 

[1 mark]

c.ii.

normal to Π3 is n=20-2               (A1)


Note: May be seen or used anywhere.


considers the line normal to Π3 passing through B0,-2,0               (M1)
r=0-20+μ20-2                A1


EITHER

finding the point on the normal line that intersects Π3
attempts to solve simultaneously with plane 2x-2z=3               (M1)

4μ+4μ=3

μ=38                A1

point is 34, -2,-34


OR

2μ-2-2μ-34-54-34·20-2=0               (M1)

4μ-32+4μ-32=0

μ=38               A1


OR

attempts to find the equation of the plane parallel to Π3 containing B' x-z=3 and solve simultaneously with L               (M1)

2μ'+2μ'=3

μ'=34               A1


THEN

so, another point on the reflected line is given by

r=0-20+3420-2               (A1)

B'32,-2,-32               A1

 

[7 marks]

d.i.

EITHER

attempts to find the direction vector of the reflected line using their P and B'               (M1)

PB'=34-34-34


OR

attempts to find their direction vector of the reflected line using a vector approach               (M1)

PB'=PB+BB'=-3411-1+3210-1


THEN

r=32-2-32+λ34-34-34 (or equivalent)                A1


Note:
Award A0 for either 'r=' or 'xyz=' not stated. Award A0 for 'L'='

 

[2 marks]

d.ii.

Examiners report

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

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.12—Vector definitions
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Topic 3— Geometry and trigonometry

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