Date | May 2021 | Marks available | 7 | Reference code | 21M.1.AHL.TZ1.11 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 1 |
Command term | Find | Question number | 11 | Adapted from | N/A |
Question
Consider the line L1 defined by the Cartesian equation x+12=y=3-z.
Consider a second line L2 defined by the vector equation r=(012)+t(a1-1), where t∈ℝ and a∈ℝ.
Show that the point (-1, 0, 3) lies on L1.
Find a vector equation of L1.
Find the possible values of a when the acute angle between L1 and L2 is 45°.
It is given that the lines L1 and L2 have a unique point of intersection, A, when a≠k.
Find the value of k, and find the coordinates of the point A in terms of a.
Markscheme
-1+12=0=3-3 A1
the point (-1, 0, 3) lies on L1. AG
[1 mark]
attempt to set equal to a parameter or rearrange cartesian form (M1)
x+12=y=3-z=λ⇒x=2λ-1, y=λ, z=3-λ OR x+12=y-01=z-3-1
correct direction vector (21-1) or equivalent seen in vector form (A1)
r=(-103)+λ(21-1) (or equivalent) A1
Note: Award A0 if =r is omitted.
[3 marks]
attempt to use the scalar product formula (M1)
(21-1)∙(a1-1)=(±)√6√a2+2 (A1)(A1)
Note: Award A1 for LHS and A1 for RHS
A1A1
Note: Award A1 for LHS and A1 for RHS
A1
M1
attempt to solve their quadratic
A1
[8 marks]
METHOD 1
attempt to equate the parametric forms of and (M1)
A1
attempt to solve equations by eliminating or (M1)
or
Solutions exist unless
A1
Note: This A1 is independent of the following marks.
or A1
has coordinates A2
Note: Award A1 for any two correct coordinates seen or final answer in vector form.
METHOD 2
no unique point of intersection implies direction vectors of and parallel
A1
Note: This A1 is independent of the following marks.
attempt to equate the parametric forms of and (M1)
A1
attempt to solve equations by eliminating or (M1)
or
or A1
has coordinates A2
Note: Award A1 for any two correct coordinates seen or final answer in vector form.
[7 marks]