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STEP Support Programme

Foundation Module Assignment 1 Part 3ii

The question below will involve spoilers.

I would really appreciate it if someone could help me understand the solution for the second part of the STEP question. I completed these assignments about a year ago but thought that I should revisit this as revision before starting university and now that I have read more about necessary and sufficient conditions I think I may be overthinking it.

Would this be an appopriate method?

  • Calculate an expression for the discriminant of the quadratic.
  • Substitute the given c^2 into the discriminant.
  • Show that the discriminant evaluates to 0.
  • Show that the solution is not equal to a or b (though I failed to do this).

I suggest the above method as from what I understand the question is to show that c^2 =... is a sufficient condition for the equation in question to have exactly one solution.

The worked solution starts with the fact that the discriminant equals 0 and shows that c^2 must be what is given in the question. I am confused as to whether this answers the question - the problem is probably not the answer (I am pretty sure I am wrong and not the question) but I would love some help understand which way the implication in the question goes and why each solution answers it being either necessary or sufficient.

Many thanks for helping me understand!

Because each line of working here can be reversed, this is actually an if and only if (and the condition is both necessary and sufficient).

In an ideal world a line of working to recognise that the implication works both ways might be a nice idea, however this was a STEP 1 question so perhaps a little less formality is ok. This was also the first assignment so we didn't want to put a lot of formality in there.

Useful Links

Underground Mathematics: Selected worked STEP questions

STEP Question database

University of Cambridge Mathematics Faculty: What do we look for?

University of Cambridge Mathematics Faculty: Information about STEP

University of Cambridge Admissions Office: Undergraduate course information for Mathematics

Stephen Siklos' "Advanced Problems in Mathematics" book (external link)

MEI: Worked solutions to STEP questions (external link)

OCR: Exam board information about STEP (external link)

AMSP (Advanced Maths Support programme): Support for University Admission Tests (external link)