If, doing three FP1 past papers over Xmas, you come across questions pretty much like the ones attached, that’s fine. If you don’t come across questions like these, try them. They’re a little more tricky than most of the examples you’ve done.
Question 1: it’s just a matter of being confident about what it means for a matrix to be “singular”. It means that its determinant is zero; or (in other words) that you can’t invert it; or (in other words again) that it collapses every triangle (every shape, in fact) into something of zero area, a line segment or just a point. The actual working-out is quite easy.
Question 2: Here, the method is simple and easy, as it is with all these proofs by induction by matrices. It’s just that the working is a bit more complicated.
Question 3 and 4: It’s a matter of being alert to when you have to prove by induction and when using standard formulas.
The standard-formulas question here (Q.3) has quite easy working. The method for the proof-by-induction one is entirely standard, something you all know by now, but the working is a bit more complicated than average. Notice that my model solution does a bit of simplifying the thing to be proved in Step 2 before starting on the main working in Step 2.
Question 5: The issue here is whether you remember that you must divide through to make the coefficient of x3 equal to 1 before you use the rules about sum of roots and product of roots.
If you’ve practised enough that you can do all these five questions ok, then you should be confident.