Daily GRE

Zai geen, sayonara, goodbye (for now at least).

2011-09-05T21:08:00.000-07:00

With the start of school looming so close, I fear that I won't have enough time to update this blog daily. Thus, I have decided to take a break on posting questions 'til winter vacation comes. I'll still be in the background gathering questions, but they won't be released to the public until December.

Thanks for all the comments, you guys really taught me a lot. I hope I was able to help all of you too (even if it was just the slightest bit).

- Paul

PS: The title of this post is from a song my friend wrote. His alias is LittleColumbus. This text is in white so that I can properly attribute him while not turning this post into a bad advertisement.
PPS: It would be awesome if you guys spread the word about this blog. I'd love to get more people commenting so we can have more discussions.

Math GRE - #34

2011-09-05T20:53:00.001-07:00

$f$ is the function defined by: \[ f(x)=\begin{cases} xe^{-x^{2}-x^{-2}} & \text{if } x\neq0\\ 0 & \text{otherwise.}\end{cases}\] At how many values of $x$ does $f$ have a horizontal tangent line?

None
One
Two
Three
Four

Solution :

Physics GRE - # 34

2011-09-05T20:53:00.000-07:00

Eigenfunctions for a rigid dumbbell rotating about its centre have a $\phi$ dependence of the form $\psi(\phi)=Ae^{im\phi}$, where $m$ is a quantum number and $A$ is a constant. Which of the following values of $A$ will properly normalize the eigenfunction?

$\sqrt{2\pi}$

$2\pi$

$(2\pi)^2$

$\dfrac{1}{\sqrt{2\pi}}$

$\dfrac{1}{2\pi}$

Solution :

Math GRE - #33

2011-09-04T21:43:00.001-07:00

How many integers from 1 to 1000 are divisible by 30 by not by 16?

Solution :

Physics GRE - #33

2011-09-04T21:43:00.000-07:00

A radioactive nucleus decays, with the activity shown in the plot below. The half-life of the nucleus is:

2 min
7 min
11 min
18 min
23 min

Solution :

Math GRE - #32

2011-09-03T19:57:00.001-07:00

Let $\mathbb{R}$ be the set of real numbers and let $f$ and $g$ be functions from $\mathbb{R}$ into $\mathbb{R}$. Which of the following is the negation of the statement:

"For each $s$ in $\mathbb{R}$, there exists an $r$ in $\mathbb{R}$ such that if $f(r)>0$, then $g(s)>0$."

For each $s$ in $\mathbb{R}$, there does not exist an $r$ in $\mathbb{R}$ such that if $f(r)>0$, then $g(s)>0$.
For each $s$ in $\mathbb{R}$, there exists an $r$ in $\mathbb{R}$ such that $f(r)>0$ and $g(s)\leq0$.
There exists an $s$ in $\mathbb{R}$ such that for each $r$ in $\mathbb{R}$ such that $f(r)>0$ and $g(s)\leq0$.
There exists an $s$ in $\mathbb{R}$ and there exists an $r$ in $\mathbb{R}$ such that $f(r)\leq0$ and $g(s)\leq0$.
For each $r$ in $\mathbb{R}$, there exists an $s$ in $\mathbb{R}$ such that $f(r)\leq0$ and $g(s)\leq0$.

Solution :

Physics GRE - #32

2011-09-03T19:57:00.000-07:00

As shown below, a coaxial cable having radii $a$, $b$, and $c$ carries equal and opposite currents of magnitude $i$ on the inner and outer conductors. What is the magnitude of the magnetic induction at a point $P$ outside the cable at a distance $r$ from the axis?

$0$
$\dfrac{\mu_0 ir}{2\pi a^2}$
$\dfrac{\mu_0 i}{2\pi r}$
$\dfrac{\mu_0 i}{2\pi r}\dfrac{c^2-r^2}{c^2-b^2}$
$\dfrac{\mu_0 i}{2\pi r}\dfrac{r^2-b^2}{c^2-b^2}$

Solution :

Math GRE - #31

2011-09-01T22:34:00.000-07:00

Let $\text{F}$ be a constant force that is given by the vector $\left(\begin{array}{c}-1\\0\\1\end{array}\right)$. What is the work done by $\text{F}$ on a particle that moves along the path given by $\left(\begin{array}{c}t\\t^2\\t^3\end{array}\right)$ between time $t=0$ and $t=1$?

$-\dfrac{1}{4}$

$-\dfrac{1}{4\sqrt{2}}$

$\sqrt{2}$

$3\sqrt{2}$

Solution :

Physics GRE - #31

2011-09-01T22:33:00.000-07:00

Which of the following is nearly the mass of the Earth? The radius of the Earth is about $6.4\cdot10^6\,\text{m}$.

$6\cdot10^{24}\,\text{kg}$
$6\cdot10^{27}\,\text{kg}$
$6\cdot10^{30}\,\text{kg}$
$6\cdot10^{33}\,\text{kg}$
$6\cdot10^{36}\,\text{kg}$

Solution :

Math GRE - #30

2011-08-31T23:03:00.000-07:00

What is \[\int_{-3}^3{|x+1|\,dx}?\]

Solution :

Physics GRE - #30

2011-08-31T23:02:00.000-07:00

The energy levels of the hydrogen atom are given in terms of the principal quantum number $n$ and a positive constant $A$ by the expression:

$A\left(n+\frac{1}{2}\right)$
$A(1-n^2)$
$A\left(\frac{1}{n^2}-\frac{1}{4}\right)$
$An^2$
$-\frac{A}{n^2}$

Solution :

Math GRE - #29

2011-08-30T21:52:00.001-07:00

If $z=e^{2\pi i/5},$ then \[1+z+z^2+z^3+5z^4+4z^5+4z^6+4z^7+4z^8+5z^9=\]

$0$
$4e^{3\pi i/5}$
$5e^{4\pi i/5}$
$-4e^{2\pi i/5}$
$-5e^{3\pi i/5}$

Solution :

Physics GRE - #29

2011-08-30T21:52:00.000-07:00

Solid argon is held together by which of the following bonding mechanisms?

Ionic bond only.
Covalent bond only.
Partly covalent and partly ionic.
Metallic bond.
Van der Waals bond

Solution :

Math GRE - #28

2011-08-29T22:07:00.001-07:00

What is the volume of the solid formed by revolving about the x-axis the region in the first quadrant of the xy-plane bounded by the coordinate axes and the graph of the equation: \[y=\frac{1}{\sqrt{1+x^2}}?\]

$\dfrac{\pi}{2}$
$\pi$
$\dfrac{\pi^2}{4}$
$\dfrac{\pi^2}{2}$
$\infty$

Solution :

Physics GRE - #28

2011-08-29T22:07:00.000-07:00

lalalala test element $^A_ZX$ decays by natural radioactivity in two stages to $^{A-4}_{Z-1}Y$. The two stages would most likely be which of the following?

	First stage	Second stage
1.	$\beta^-$ emission with an antineutrino	$\alpha$ emission
2.	$\beta^-$ emission	$\alpha$ emission with a neutrino
3.	$\beta^-$ emission	$\gamma$ emission
4.	Emission of a deuteron	Emission of two neutrons
5.	$\alpha$ emission	$\gamma$ emission

Solution :

Physics GRE - #27

2011-08-28T23:33:00.000-07:00

Suppose that there is a very small shaft in the Earth such that a point mass can be placed at a radius of $R/2$ where $R$ is the radius of the Earth. If $F(r)$ is the gravitational force of the Earth on a point mass at a distance $r$, what is: \[\frac{F(R)}{F(2R)}?\]

Solution :

Math GRE - #27

2011-08-28T23:29:00.000-07:00

Which of the following is the best approximation of $\sqrt{1.5}(266)^{3/2}$?

1000
2700
3200
4100
5300

Solution :

Physics GRE - #26

2011-08-27T22:24:00.001-07:00

An engine absorbs heat at a temperature of 727 degrees Celsius and exhausts heat at a temperature of 527 degrees Celcius. If the engine operates at maximum possible efficiency, for 2000 J of heat the amount of work the engine performs is most nearly:

400 J
1450 J
1600 J
2000 J
2760 J

Solution :

Math GRE - #26

2011-08-27T22:24:00.000-07:00

A total of $x$ feet of fencing is to form three sides of a level rectangular yard. What is the maximum possible area of the yard, in terms of $x$?

$\dfrac{x^2}{9}$
$\dfrac{x^2}{8}$
$\dfrac{x^2}{4}$
$x^2$
$2x^2$

Solution :

Physics GRE - #25

2011-08-26T18:20:00.001-07:00

A 2-kilogram box hangs by a massless rope from a ceiling. A force slowly pulls the box horizontally until the horizontal force is 10 N. The box is then in equilibrium as shown below.

The angle that the rope makes with the vertical is closest to:

$\arctan0.5$
$\arcsin0.5$
$\arctan2.0$
$\arcsin2.0$
$45^\circ$

Solution :

Math GRE - #25

2011-08-26T18:20:00.000-07:00

Let $k$ be the number of roots to the equation $f(x)=e^x+x-2$ in the interval $[0, 1]$, and let $n$ be the number of roots that are not in $[0,1]$. Which of the following is true?

$k=0$ and $n=1$
$k=1$ and $n=0$
$k=n=1$
$k>1$
$n>1$

Solution :

Math GRE - #24

2011-08-25T22:53:00.000-07:00

What is the greatest possible area of a triangular region with one vertex at the centre of a unit circle and the other two vertices on the circle?

$\frac{1}{2}$
$1$
$\sqrt{2}$
$\pi$
$\frac{1+\sqrt{2}}{4}$

Solution :

Physics GRE - #24

2011-08-25T22:52:00.000-07:00

Assume $A$, $T$, $\lambda$ are positive constants. The equation \[y=A\sin\left[2\pi\left(\frac{t}{T}-\frac{x}{\lambda}\right)\right]\] represents a wave whose:

amplitude is $2A$
velocity is in the negative x-direction
period is $\frac{T}{\lambda}$
speed is $\frac{x}{t}$
speed is $\frac{\lambda}{T}$

Solution :

Physics GRE - #23

2011-08-24T03:45:00.001-07:00

A particle is initially at rest at the top of a curved frictionless track.

The x and y coordinates of the track are related in dimensionless units by $y=\frac{x^2}{4}$, where the positive y-axis is in the vertical downward direction. As the particle slides down the track, what is its tangential acceleration?

$0$
$g$
$\frac{gx}{2}$
$\frac{gx}{\sqrt{x^2+4}}$
$\frac{gx^2}{\sqrt{x^2+16}}$

Solution :

Math GRE - #23

2011-08-24T03:45:00.000-07:00

Suppose B is a basis for a real vector space V of dimension greater than 1. Which of the following statements could be true?

The zero vector of V is an element of B.
B has a proper subset that spans V.
B is a proper subset of a linearly independent subset of V.
There is a basis for V that is disjoint from B.
One of the vectors in B is a linear combination of the other vectors in B.

Solution :