[Math] How to work faster on the GRE.

advicearithmeticcalculussoft-question

I'm fixin' to go to grad school next year, so lately I've been studying for the mathematics subject GRE. What I have heard, from MSE and other places, is that the most important thing is to learn to work quickly, since the test is so long. I am not good at this. I've spent the last 2 years studying grad level material in algebra almost exclusively, so calculus/differential equations is not my strong point, and those problems (i.e. half the exam) take me way too long. I realize this is a pretty malleable question, but I was hoping somebody might have general tips on how to get faster at these types of computational problems.

An example of the type of thing that's giving me trouble:

Let $P$ be the tangent plane to the surface $$y^2z-2xz^2+3x^2y=2$$ at the point $Q=(1,1,1)$. Which of the following poitns also lies on $P$?

(a) $(4,5,-3)$

(b) $(6,-4,3)$

(c) $(3,-1,5)$

(d) $(5,3,-2)$

(e) $(-2,4,2)$

I got the expression for the plane, then went through the answers and plugged them in one by one, until finally (e) turned out to be the correct one. This turned out to be exactly how the answer key said to do it, but despite knowing precisely what I was doing going in, the calculation took forever due to messing up arithmetic and general slowness… something like $15$ minutes.

Is there any trick to cutting down on time with computational questions like this, other than not being inept at arithmetic?

Best Answer

It is hard to give anything more than general advice, which is to go through practice papers and calculus type questions. Doing calculus type questions refreshes you on the techniques and doing enough of them gets you faster, while doing practice papers also gets you used to using in-exam methods, taking advantage of the nature of these multiple choice questions.

For example, rather than finding the actual equation of the plane and plugging them in, I would have computed the gradient vector, which is $$ ( -2z^2+6xy, 3x^2+2zy, y^2-4xz)\mid_{(1,1,1)}= (4,5,-3).$$ Any point $P$ that lines on the plane in question must satisfy $(P- (1,1,1))\cdot (4,5,-3)=0$ so subtract 1 from each coordinate and just compute dot products until you find one that is $0.$ You wouldn't even have to compute (a) or (d) properly, you'll notice by signs that the dot product will be positive. It is quite similar to finding the equation of the plane first and plugging them in, but seems to save at least a little bit of time.

General Tips:

  • Don't spend more than 2 minutes on any question, after your two minutes if you haven't got the answer, you should at least be able to rule out 2 answers. Mark off those 2 answers for now.
  • Dimensional analysis is a good trick. There are some calculus type rates questions (A car uses this much fuel per mile when travelling at some speed...). The options might be some integrals with various integrands, and only one of them even has the correct units.
  • If you can rule out 2 options, good, and if you can rule out 3 even better, guessing is benefical. Don't guess if you can't rule anything out, because there is only 1/5 chance of getting the right answer, and an incorrect answer subtracts 0.25.
  • Some things they seem to like testing, so worth revising are: Green's theorem, and using it to find areas, Volumes of surfaces of revolution, finding volumes by setting up double integrals, Decompositions of finite abelian groups, basic statistics and probability was something I had to revise, Residue theorem, Cauchy-Riemann equations. Also, many of the incorrect answer choices for questions on Lebesgue theory seem to come from incorrectly thinking uncountable implies 0 measure. Keep in mind the Cantor set for these questions.

Good luck!

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