A couple questions from today's test

Forum for the GRE subject test in mathematics.
dumplinghao123
Posts: 24
Joined: Sat Sep 22, 2012 4:34 pm

A couple questions from today's test

Postby dumplinghao123 » Sat Oct 13, 2012 5:26 pm

Questions removed...
Thought I was doing a nice thing, but it's not allowed...
Last edited by dumplinghao123 on Sun Oct 14, 2012 12:25 am, edited 1 time in total.

mindreader
Posts: 90
Joined: Sat Oct 13, 2012 11:12 am

Re: A couple questions from today's test

Postby mindreader » Sat Oct 13, 2012 8:37 pm

dumplinghao123 wrote:1. (This is actually the last question of the test.)
Q: Z17 is the modular 17 ring. Consider the group of all the invertible elements under multiplication. Which of the following is the generator of such group?
I.5
II.8
III.16
A: I thought they all are (at the test). But the correct answer is 5 only. Could refer to this wiki link for generators:
http://en.wikipedia.org/wiki/Multiplica ... s_modulo_n

2. There is a tank with salty water. The total volume is 100 liter, and 3 grams of salt was initially there. Now pour in the water with 0.02 per liter salty water with speed of 4 liters per minute. Assuming the salt and water mix instantly, and also pouring out the water at 4 liters per minute. In 100 minutes, how many grams of salt are there in this tank?
A: Set the function for salt is f(t). df/dt = 0.08-4*f(t)/100.
Solving this ODE by multiplying e^0.04t.
d(e^0.04t*f)=d(2*e^0.04t)
e^0.04t*f=2*e^0.04t+c
f=2+c/e^0.04t
at t=0, f=3, solve for c, get c=1
thus at t=100, f=2+e^0.04t

3.
Q: There is a line in 3-D space with the following expression:
x=5*cos(t)
y=5*sin(t)
z=t
If there is a point from which to (5,0,0) the length is 26. What’s the distance from this point to the origin?
A: dr = sqrt(26)*dt, so solve it, and get t = -sqrt(26), so the point is sqrt(25+26)=sqrt(51).

4.
Asking for a plane that’s orthogonal to both of the planes.

5.
Giving a matrix, asking the orthonormal basis of the column space.

6.
Q: How many spanning trees are there for 5 vertices under to isormorphism?
A: My answer was 3.

I will post more if I could remember...


Wow, it seems that we did not take the same exam! Are you based in China?

etcetera
Posts: 1
Joined: Sat Oct 13, 2012 9:25 pm

Re: A couple questions from today's test

Postby etcetera » Sat Oct 13, 2012 9:28 pm

Yeah, I instinctively assumed Z17 was of prime order as well.

And I had the same test and am the US.

dumplinghao123
Posts: 24
Joined: Sat Sep 22, 2012 4:34 pm

Re: A couple questions from today's test

Postby dumplinghao123 » Sat Oct 13, 2012 9:28 pm

mindreader wrote:
dumplinghao123 wrote:1. (This is actually the last question of the test.)
Q: Z17 is the modular 17 ring. Consider the group of all the invertible elements under multiplication. Which of the following is the generator of such group?
I.5
II.8
III.16
A: I thought they all are (at the test). But the correct answer is 5 only. Could refer to this wiki link for generators:
http://en.wikipedia.org/wiki/Multiplica ... s_modulo_n

2. There is a tank with salty water. The total volume is 100 liter, and 3 grams of salt was initially there. Now pour in the water with 0.02 per liter salty water with speed of 4 liters per minute. Assuming the salt and water mix instantly, and also pouring out the water at 4 liters per minute. In 100 minutes, how many grams of salt are there in this tank?
A: Set the function for salt is f(t). df/dt = 0.08-4*f(t)/100.
Solving this ODE by multiplying e^0.04t.
d(e^0.04t*f)=d(2*e^0.04t)
e^0.04t*f=2*e^0.04t+c
f=2+c/e^0.04t
at t=0, f=3, solve for c, get c=1
thus at t=100, f=2+e^0.04t

3.
Q: There is a line in 3-D space with the following expression:
x=5*cos(t)
y=5*sin(t)
z=t
If there is a point from which to (5,0,0) the length is 26. What’s the distance from this point to the origin?
A: dr = sqrt(26)*dt, so solve it, and get t = -sqrt(26), so the point is sqrt(25+26)=sqrt(51).

4.
Asking for a plane that’s orthogonal to both of the planes.

5.
Giving a matrix, asking the orthonormal basis of the column space.

6.
Q: How many spanning trees are there for 5 vertices under to isormorphism?
A: My answer was 3.

I will post more if I could remember...


Wow, it seems that we did not take the same exam! Are you based in China?



No, I am in US. What you got? How was your questions, are they difficult?

dumplinghao123
Posts: 24
Joined: Sat Sep 22, 2012 4:34 pm

Re: A couple questions from today's test

Postby dumplinghao123 » Sat Oct 13, 2012 9:30 pm

etcetera wrote:Yeah, I instinctively assumed Z17 was of prime order as well.

And I had the same test and am the US.


How did you do?
Ah, I keep thinking about the test, lol.
I can't wait to know how I did... Hope that wasn't as bad as I thought.

mindreader
Posts: 90
Joined: Sat Oct 13, 2012 11:12 am

Re: A couple questions from today's test

Postby mindreader » Sat Oct 13, 2012 9:59 pm

dumplinghao123 wrote:
mindreader wrote:
dumplinghao123 wrote:1. (This is actually the last question of the test.)
Q: Z17 is the modular 17 ring. Consider the group of all the invertible elements under multiplication. Which of the following is the generator of such group?
I.5
II.8
III.16
A: I thought they all are (at the test). But the correct answer is 5 only. Could refer to this wiki link for generators:
http://en.wikipedia.org/wiki/Multiplica ... s_modulo_n

2. There is a tank with salty water. The total volume is 100 liter, and 3 grams of salt was initially there. Now pour in the water with 0.02 per liter salty water with speed of 4 liters per minute. Assuming the salt and water mix instantly, and also pouring out the water at 4 liters per minute. In 100 minutes, how many grams of salt are there in this tank?
A: Set the function for salt is f(t). df/dt = 0.08-4*f(t)/100.
Solving this ODE by multiplying e^0.04t.
d(e^0.04t*f)=d(2*e^0.04t)
e^0.04t*f=2*e^0.04t+c
f=2+c/e^0.04t
at t=0, f=3, solve for c, get c=1
thus at t=100, f=2+e^0.04t

3.
Q: There is a line in 3-D space with the following expression:
x=5*cos(t)
y=5*sin(t)
z=t
If there is a point from which to (5,0,0) the length is 26. What’s the distance from this point to the origin?
A: dr = sqrt(26)*dt, so solve it, and get t = -sqrt(26), so the point is sqrt(25+26)=sqrt(51).

4.
Asking for a plane that’s orthogonal to both of the planes.

5.
Giving a matrix, asking the orthonormal basis of the column space.

6.
Q: How many spanning trees are there for 5 vertices under to isormorphism?
A: My answer was 3.

I will post more if I could remember...


Wow, it seems that we did not take the same exam! Are you based in China?



No, I am in US. What you got? How was your questions, are they difficult?


I believe that I can't post the questions online but yeah, it's different. I'm based in Asia so I guess I took the exam a little bit earlier than you. But it is indeed different from what I had. Well, the differential equation problem I got is much more difficult than the one you posted...

Legendre
Posts: 217
Joined: Wed Jun 03, 2009 1:05 am

Re: A couple questions from today's test

Postby Legendre » Sat Oct 13, 2012 11:16 pm

We aren't supposed to reveal test questions. :shock:

I am in south east asia and none of your questions appeared in my test. Mine had questions asking about things which I didn't know was part of the syllabus!

Gelor
Posts: 5
Joined: Tue Sep 18, 2012 7:35 am

Re: A couple questions from today's test

Postby Gelor » Sun Oct 14, 2012 7:34 am

Yes, it seems it's not the twst i've taken yesterday.
I wonder if they diversify tasks because of time zones. I mean when the test begins in one part of the world it has already ended in another so there would be a slight possibility of cheating.

c3adv
Posts: 12
Joined: Fri Apr 06, 2012 8:34 pm

Re: A couple questions from today's test

Postby c3adv » Sun Oct 21, 2012 8:03 pm

If some of the questions are already posted, can we discuss them?

If so, is the following analysis correct? For the modular ring, the group of elements invertible under multiplication has order 16. You do not include zero, so 17-1=16 invertible elements, or units, form part of the multiplicative group, which we can call G. These correspond to the equivalence classes created by division modulo 17 when the remainder is not zero. So, the generator must be coprime to 16 in order to generate all of G. So, 5 is the only viable option.




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