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Date May 2022 Marks available 4 Reference code 22M.1.AHL.TZ1.11
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 1
Command term Show that Question number 11 Adapted from N/A

Question

Consider the three planes

1: 2x-y+z=4

2: x-2y+3z=5

3:-9x+3y-2z=32

Show that the three planes do not intersect.

[4]
a.

Verify that the point P(1, -2, 0) lies on both 1 and 2.

[1]
b.i.

Find a vector equation of L, the line of intersection of 1 and 2.

[4]
b.ii.

Find the distance between L and 3.

[6]
c.

Markscheme

METHOD 1

attempt to eliminate a variable                 M1

obtain a pair of equations in two variables


EITHER

-3x+z=-3 and          A1

-3x+z=44          A1


OR

-5x+y=-7 and          A1

-5x+y=40          A1


OR

3x-z=3 and          A1

3x-z=-795          A1


THEN

the two lines are parallel (-344 or -740 or 3-795)          R1

 

Note: There are other possible pairs of equations in two variables.
To obtain the final R1, at least the initial M1 must have been awarded.

 

hence the three planes do not intersect          AG

 

METHOD 2

vector product of the two normals =-1-5-3  (or equivalent)          A1

r=1-20+λ153  (or equivalent)          A1

 

Note: Award A0 if “r=” is missing. Subsequent marks may still be awarded.

 

Attempt to substitute 1+λ,-2+5λ,3λ in 3                 M1

-91+λ+3-2+5λ-23λ=32

-15=32, a contradiction          R1

hence the three planes do not intersect          AG

 

METHOD 3

attempt to eliminate a variable                M1

-3y+5z=6          A1

-3y+5z=100          A1

0=94, a contradiction           R1

 

Note: Accept other equivalent alternatives. Accept other valid methods.
To obtain the final R1, at least the initial M1 must have been awarded.

 

hence the three planes do not intersect          AG

 

[4 marks]

a.

1:2+2+0=4  and  2:1+4+0=5          A1

 

[1 mark]

b.i.

METHOD 1

attempt to find the vector product of the two normals          M1

2-11×1-23

=-1-5-3          A1

r=1-20+λ153          A1A1

 

Note: Award A1A0 if “r=” is missing.
Accept any multiple of the direction vector.
Working for (b)(ii) may be seen in part (a) Method 2. In this case penalize lack of “r=” only once.

 

METHOD 2

attempt to eliminate a variable from 1 and 2          M1

3x-z=3  OR  3y-5z=-6  OR  5x-y=7

Let x=t

substituting x=t in 3x-z=3 to obtain

z=-3+3t  and  y=5t-7 (for all three variables in parametric form)          A1

r=0-7-3+λ153          A1A1

 

Note: Award A1A0 if “r=” is missing.
Accept any multiple of the direction vector. Accept other position vectors which satisfy both the planes 1 and 2 .

 

[4 marks]

b.ii.

METHOD 1

the line connecting L and 3 is given by L1

attempt to substitute position and direction vector to form L1           (M1)

s=1-20+t-93-2          A1

substitute 1-9t,-2+3t,-2t in 3             M1

-91-9t+3-2+3t-2-2t=32

94t=47t=12          A1

attempt to find distance between 1,-2,0 and their point -72,-12,-1           (M1)

=1-20+12-93-2-1-20=12-92+32+-22

=942          A1

 

METHOD 2

unit normal vector equation of 3 is given by -932·xyz81+9+4           (M1)

=3294          A1

let 4 be the plane parallel to 3 and passing through P
then the normal vector equation of 4 is given by

-932·xyz=-932·1-20=-15             M1

 

unit normal vector equation of 4 is given by

-932·xyz81+9+4=-1594          A1

distance between the planes is 3294--1594           (M1)

=4794=942          A1

 

[6 marks]

c.

Examiners report

Part (a) was well attempted using a variety of approaches. Most candidates were able to gain marks for part (a) through attempts to eliminate a variable with many subsequently making algebraic errors. Part (b)(i) was well done. For part (b)(ii) few successful attempts were noted, many candidates failed to use an appropriate notation "r =" while giving the vector equation of a line. Part (c) proved to be challenging for most candidates with very few correct answers seen. Many candidates did not attempt part (c).

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

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.1—3d space, volume, angles, distance, midpoints
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