Date | May Example question | Marks available | 1 | Reference code | EXM.3.AHL.TZ0.2 |
Level | Additional Higher Level | Paper | Paper 3 | Time zone | Time zone 0 |
Command term | State | Question number | 2 | Adapted from | N/A |
Question
This question will investigate the solution to a coupled system of differential equations when there is only one eigenvalue.
It is desired to solve the coupled system of differential equations
The general solution to the coupled system of differential equations is hence given by
As the trajectory approaches an asymptote.
Show that the matrix has (sadly) only one eigenvalue. Find this eigenvalue and an associated eigenvector.
Hence, verify that is a solution to the above system.
Verify that is also a solution.
If initially at find the particular solution.
Find the values of and when .
Find the equation of this asymptote.
State the direction of the trajectory, including the quadrant it is in as it approaches this asymptote.
Markscheme
M1A1
A1A1
So only one solution AGA1
M1
So an eigenvector is A1
[7 marks]
So M1A1A1
and M1A1
showing that is a solution AG
[5 marks]
M1A1
M1A1A1
Verifying that is also a solution AG
[5 marks]
Require M1A1
A1
[3 marks]
A1A1
[2 marks]
As M1A1
so asymptote is A1
[3 marks]
Will approach the asymptote in the 4th quadrant, moving away from the origin. R1
[1 mark]