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