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Date May 2021 Marks available 3 Reference code 21M.1.AHL.TZ2.8
Level Additional Higher Level Paper Paper 1 Time zone Time zone 2
Command term Find Question number 8 Adapted from N/A

Question

Two lines L1 and L2 are given by the following equations, where p.

L1:r=(2p+9-3)+λ(p2p4)

L2:r=(147p+12)+μ(p+44-7)

It is known that L1 and L2 are perpendicular.

Find the possible value(s) for p.

[3]
a.

In the case that p<0, determine whether the lines intersect.

[4]
b.

Markscheme

setting a dot product of the direction vectors equal to zero           (M1)

(p2p4)·(p+44-7)=0

p(p+4)+8p-28=0           (A1)

p2+12p-28=0

(p+14)(p-2)=0

p=-14, p=2          A1


[3 marks]

a.

p=-14

L1:r=(2-5-3)+λ(-14-284)

L2:r=(147-2)+μ(-104-7)

a common point would satisfy the equations

2-14λ=14-10μ

-5-28λ=7+4μ                   (M1)

-3+4λ=-2-7μ 

 

METHOD 1

solving the first two equations simultaneously

λ=-12, μ=12         A1

substitute into the third equation:                   M1

-3+4(-12)-2+12(-7)

so lines do not intersect.                   R1


Note: Accept equivalent methods based on the order in which the equations are considered.


METHOD 2

attempting to solve the equations using a GDC               M1

GDC indicates no solution         A1

so lines do not intersect                  R1


[4 marks]

b.

Examiners report

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

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.11—Vector equation of a line in 2d and 3d
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