skip to content

STEP Support Programme

Assignment 1 - STEP Question - c^2 condition

Hi

I'm very stuck on the very last part of the STEP Question in Assignment 1.

I'm not sure what the question even means. There are so many ifs and only ifs and the reasoning seems to be running one way then the other. I can't follow it.

I've done everything up to this point including answering the part about "Show that". But I can't understand the last line. I'm so confused I don't even know what my question is. Anyway here's the relevant part of the question:

Show that this condition can be written
c2=1(a+bab)2


and deduce that it can only hold if 0<c21

Here's my reasoning for "deduce that it [what?] can hold only if 0<c21"

c2 is positive or zero because all the numbers are real
c2 can't be zero because then c = 0 and you have two distinct real roots from part (i)
c2=1A2 with A positive or zero so c21
If a=b then A is zero otherwise it's positive this shows that c2=1 is allowed

Taking these together and tidying up gives
0<c21

But I feel like I should be starting from here not ending here.

Please help!

Actually, what you have written here looks pretty good! Personally I would start with showing that 0c21 (as it is a real number squared, and is also equal to 1 (a real number squared). I would then go on to show that c=0 is not possible, as then we have (ab)2=(a+b)2ab=0, which implies that one of a or b (or both) is zero, which contradicts the information given in the stem,

Then finish by showing that c can be equal to 1.

Many Thanks. I think my confusion was, and to be honest still is, with the "only if" terminology. I shall go away and contemplate. :-)

If we do not have 0<c21 then the condition cannot be true. The condition can only be true if c2 is in the given range, Note that having 0<c21 does not mean that the condition MUST hold, 0<c21 is a necessary condition, but not necessarily a sufficient one!

There is some more about "if" and "only if" in Assignment 10.

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)