| Date | None Specimen | Marks available | 4 | Reference code | SPNone.1.hl.TZ0.7 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Show that | Question number | 7 | Adapted from | N/A |
Question
Consider the following system of equations:
\[x + y + z = 1\]
\[2x + 3y + z = 3\]
\[x + 3y - z = \lambda \]
where \(\lambda \in \mathbb{R}\) .
Show that this system does not have a unique solution for any value of \(\lambda \) .
(i) Determine the value of \(\lambda \) for which the system is consistent.
(ii) For this value of \(\lambda \) , find the general solution of the system.
Markscheme
using row operations, M1
to obtain 2 equations in the same 2 variables A1A1
for example \(y - z = 1\)
\(2y - 2z = \lambda - 1\)
the fact that one of the left hand sides is a multiple of the other left hand side indicates that the equations do not have a unique solution, or equivalent R1AG
[4 marks]
(i) \(\lambda = 3\) A1
(ii) put \(z = \mu \) M1
then \(y = 1 + \mu \) A1
and \(x = - 2\mu \) or equivalent A1
[4 marks]
