# STEP II curve sketching question 2

Hello,

May someone explain to me how detailed answers are expected to be ?

In fact, I am a French student and I've had some problems with the ways questions are answered.
In the Calculus module (STEP II), for instance in question 3, some implicit assumptions are used : when we say $\frac{dy}{dx} = \frac{-x \pm \sqrt{x^2 +3}}{3}$, and then we solve the differential equation, we assume the expression is the same on all $\mathbb{R}$, but this is not justified (although it could by saying the two expressions are never equal, and thus that if it changes there is at least one point where the derivative does not exist).

More importantly, in STEP II curve sketching question 2, I have shown that $f(\frac{p + q}{2} - a) = f(\frac{p + q}{2} + a) \forall a \in \mathbb{R}$ to justify the symmetry of the curve, and then I have used this result to show that the area $\mathfrak{R}$ was indeed equal to half the rectangle area using integral calculation.
In the solution, these two elements are simply stated without a formal demonstration.

Thus, I have the feeling that I am losing time which I will need on the day of the exam. (The curve sketching question took me 50 minutes ...)

Do you have any tips to know what we have to demonstrate and what we have to admit ? Is a curve sketch considered a demonstration ? Do we assume the functions we search are $\mathcal{C}^1$ ?

I'm a bit at a loss as, in France, we don't use graph sketching, neither do we solve differential equations using separation of variable (at least in Maths, we do so in Physics).

Thank you very much,

### It depends

As the saying goes "How long is a piece of string?" (i.e. it depends!).

As a rule of thumb, if by assuming something you could do the question in 5 minutes, then you should probably not assume it. Conversely, if by proving something the question takes you about an hour (not allowing for false starts etc) then perhaps you could have assumed it.

In Q2 on the curve sketching module, the question says "describe" the symmetry (rather than "justify" or "prove that the curve has rotational symmetry") which usually implies that a description (just saying what it is) is enough.

STEP often uses words like "write down" (i.e. just write it down) etc to show when not much (or even no) explanation is needed.

### Typo

Just in case OP is confused, the second sentence should be "you probably shouldn't assume it" rather than "you should" if the question is doable in 5 minutes with the assumption.

### Thanks!

I'll change my previous comment (which might make this one and Zacken's look a little odd!)

### Thank you

Thank you for your help (and, although funny, the typo wasn't confusing).

I should definitely pay more attention to the verbs used. As a post-baccalaureate foreign student, I sometimes find it hard to understand how detailed a point is expected to be answered ... (although it should not be an issue on the day of the exam as I can just avoid these questions, and that anyway I usually never lose more than 15 minutes on these)

There's just one last question I hope you may answer : although I am striving to answer the problems as they are expected to be answered, I always fear that I may use a theorem or a technique or a tool that is out of the syllabus. As long as it is valid, does not make the problem trivial and is not out of place (eg ignoring a "hence" or any such thing), would it probably be tolerated ?

### Techniques

From the admissions testing service website "The marking scheme for each question will be designed to reward candidates who make good progress towards a complete solution. Correct answers always receive full marks, whatever the method used. "

So basically as long as you don't ignore a "hence" or it makes a problem trivial, and especially if by using it you are actually assuming what you are trying to prove (some STEP questions are leading up to proving results that are expected to be new to the student - if you have actually met it before and use the result to prove the result that will not go down well) then you should be fine!

There are examiner reports and "hints and solutions" documents for at least the last 10 years of papers which should give you a guide as to what sorts of things were expected.

### Thank you for drawing my

Thank you for drawing my attention to these documents. I wasn't quite aware yet that they existed, as I have not yet started doing past papers, but they are indeed quite useful !

### No problem!

:-)

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