If we have three independent equations, we will have a unique solution. In other words, we will not have unique solutions if:
The equations are inconsistent or
Two equations can be combined to give the third
Now, let us move to the statements.
i.
This tells us that the first two equations cannot hold good at the same time.
Think about this:
x + y + z = 3;
2x + 2y + 2z = 5.
Either the first or the second can hold good. Both cannot hold good at the same time. So, this will definitely not have any solution.
ii. and
a_{1}, a_{2}, a_{3} are in GP, b_{1}, b_{2}, b_{3}. This does not prevent the system from having a unique solution.
For instance, if we have
x + 9y + 5z = 11
2x + 3y – 6z = 17
4x + y – 3z = 15
This could very well have a unique solution.
iii. a_{1}, a_{2}, a_{3} are integers; b_{1}, b_{2}, b_{3} are rational numbers, c_{1}, c_{2}, c_{3} are irrational numbers
This gives us practically nothing. This system of equations can definitely have a unique solution.
So, only Statement i tells us that a unique solution is impossible.
Correct Answer: Only Statement 'i'.