GRE problems

galactus · #1 $Old$ October 27th, 2007, 12:02 PM

I am sure some of you are familiar with the GRE exam. I have been looking at old exams in preparation. I plan on taking it sometime in the future before I start graduate school. Here are a few from the exam. It's not a pushover. The questions have a diverse range of topics. Just thought you'd like a peek at some of the questions.

When 20 children in a classroom line up for lunch, Pat insists on being somewhere ahead of Lynn. If Pat's demand is to be satisified, in how many ways can the children line up?.

Suppose f is a differentiable function for which $\lim_{x\rightarrow{\infty}}f(x) \;\ and \;\ \lim_{x\rightarrow{\infty}}f'(x)$ both exist and are finite. Which of the following must be true?.

A) $\lim_{x\rightarrow{\infty}}f'(x)=0$
B) $\lim_{x\rightarrow{\infty}}f''(x)=0$
C) $\lim_{x\rightarrow{\infty}}f(x)=\lim_{x\rightarrow{\infty}}f'(x)$
D)f is a constant function
E)f' is a constant function

In the xyz space, an equation of the plane thangent to the surface $z=e^{-x}sin(y)$ at the point x=0 and $y=\frac{\pi}{2}$ is:

A) x+y=1
B)x+z=1
C)x-z=1
D)y+z=1
E)y-z=1

CaptainBlack · #2 $Old$ October 27th, 2007, 12:34 PM

Quote:

Originally Posted by galactus $View Post$

When 20 children in a classroom line up for lunch, Pat insists on being somewhere ahead of Lynn. If Pat's demand is to be satisified, in how many ways can the children line up?.

This is pretty easy since in half of all orders P is ahead of L

RonL

galactus · #3 $Old$ October 27th, 2007, 12:53 PM

Yes, I got that one pretty quick, myself. $\frac{20!}{2}$

For the second one, I was thinking A was the best answer. What do you think, CB?.

The last one, I got x+z=1. I think that's right.

I used $f_{x}(0,\frac{\pi}{2})=-e^{-x}sin(y)=-1$

and $f_{y}(0,\frac{\pi}{2})=e^{-x}cos(y)=0$

$-(x-0)-(z-1)=0$

$-x-z+1=0$

$x+z=1$

I am kind of spooked about taking it, but maybe it won't be that bad.

0123 · #4 $Old$ October 27th, 2007, 01:44 PM

Quote:

Originally Posted by CaptainBlack $View Post$

This is pretty easy since in half of all orders P is ahead of L

RonL

and why? and how? what's the rule?

Plato · #5 $Old$ October 27th, 2007, 02:28 PM

For the first one: in any queue either Lynn is before Pat or Pat is before Lynn. So half the time Pat is before Lynn.

For number two: consider the function $f(x) = \frac{{x^2 }}{{x^2 + 1}}$ .

For number three:
Given $F = e^{ - x} \sin (y) - z\quad \& \quad P:\left( {0,\frac{\pi }{2},1} \right)$
The tangent plane normal is $F_x (P)i + F_y (P)j + F_z (P)k$ .

Jhevon · #6 $Old$ December 26th, 2007, 08:07 PM

no one gave their thoughts on galactus' answer to the second question. i'm curious as to whether he has the right answer.

Plato gave an example, and indeed, $\lim_{x \to \infty}f'(x) = 0$ , which lines up with galactus' answer (and in so doing wipes out choices C,D, and E). but is A the correct? Plato's example makes B look possible.

my thoughts on it: A is correct.

my reasoning: it is not really the reasoning of a pure mathematician, so bare with me. if the infinite limit of f is finite, it means the function approaches some line y = something asymptotically. in which case, the function would kind of level of giving rise to a slope that tends to zero as we go out to infinity

colby2152 · #7 $Old$ December 27th, 2007, 06:43 AM

If A is true, then B is certainly true. If you try to find a function that tapers off and gives an actual limit, then it's slope is going towards zero. If that doesn't happen it will not have a limit. Also, doesn't the infinite limit of the derivative Plato's example goto zero? IMO, the answer should be A.