# Search results

1. ### Roll a fair die 5 times

Homework Statement Roll a fair die 5 times, find the probability that the first two rolls have the same outcomes. Homework Equations The Attempt at a Solution The total outcomes is 6^5, I think we have 6^2 * 6 choose 2 /6^5 since the first two numbers are fixed and we can choose 2 numbers...
2. ### Expected value and variance of max{Y_1,Y_2}

Homework Statement Let Y_1,Y_2 be independent random variable with uniform distribution on the interval [1,2]. Define X=max{Y_1,Y_2}. Find p.d.f., expected value and variance. Homework Equations The Attempt at a Solution Since $X=\max\{Y_1,Y_2\}$, this tells $Y_1$ and $Y_2$ must at most $x$...
3. ### Conditional Probability and law of total probability

Homework Statement It rains in a city with a chance of 0.4. The weather forecast is not always accurate. When there will be a rain the next day, the forecast predicts the rain with probability 0.8; When there is no rain, the forecast falsely predicts a rain with probability 0.1. You take your...
4. ### Need help for REU personal statement

Hi, all. I am planning to apply REU for math this summer. This is my first time to write a personal statement, I am not sure how it is. Basically, my statement needs to answer why I want to apply this program and what my future goal is. Malcolm X said: “Education is our passport to the future...
5. ### Find all irreducible polynomials over F of degree at most 2

Homework Statement Let F = {0,1,α,α+1}. Find all irreducible polynomials over F of degree at most 2. Homework Equations The Attempt at a Solution To determine an irreducible polynomial over F, I think it is sufficient to check the polynomial whether has a root(s) in F, So far, I got...
6. ### Does f'(0)=0 where x in [-1,1] and f(x)<=f(0)

Homework Statement If the function f:ℝ→ℝ is differentiable and f(x)<=f(0) for all x ∈[-1,1], then f'(0)=0. True or False. Homework Equations The Attempt at a Solution I think the statement is right. Since f(x)<= f(0) for all x in [-1,1], this tells us f is an even function or a symmetric...
7. ### One to one function is monotone?

Homework Statement A one to one function f: ℝ→ℝ is monotone, True or False Homework Equations The Attempt at a Solution I think the statement is false, for example: Let I =[0,1]∪[2,3] f(x)=x if x∈[0,1], f(x)=5-x,x∈[2,3]
8. ### Find the probability of a prize

Homework Statement The state of Oregon wishes to design a new lottery game with the following rules:  1.Each ticket costs $5  2.There will be three prizes:$10, $100 and$1000  3.The probability of the $10 prize will be 20%.  4.The probability of the$100 prize will be 1%  5.Ten thousand...
9. ### Find the percentage rate

Homework Statement If an investment increased by 40% over 3 years and rose by the same percentage rate each of those years, what was the rate? Homework Equations new_value = old_value(1+r)^p The Attempt at a Solution Let value of the investment be 1000, then we have 1000(1+0.4) = 1000(1+r)^3...
10. ### Find the percent of the price

Homework Statement a. Lucy's Lunch and Latte has found that customers are put off by the local tourism tax of 9% that is added to their bill. If Lucy decides to cover the tax herself, rather than adding it to the customer's bill, what percent will the customer see in savings? b. Lucy decides...
11. ### Prove Sup S = c

Homework Statement S≡{x|x∈ℝ,x≥0,x2 < c} Prove Sup S = c Homework Equations The Attempt at a Solution Since x in the set real numbers, there are two cases for x: x < 1 or 0 <= x <=1 if 0 <= x <=1, then x < c + 1 since c is positive. if x < 1, then x^2 < c < x*c + x = x(c+1) thus x < c+1, by...
12. ### Show that Q_F is not a division ring.

Homework Statement Let F be a finite field of characteristic p∈{2,3,5}. Consider the quaternionic ring, Q_F={a_1+a_i i+a_j j+a_k k|a_1,a_i,a_j,a_k ∈ F}. Prove that Q_F is not a division ring. Homework Equations The Attempt at a Solution Let α=1+i,β=1+i+j∈QF. Then...
13. ### Show √ 2 + √ 3 algebraic over Q

Homework Statement Show √ 2 + √ 3 algebraic over Q. Find its degree over Q. Prove the answer. Homework Equations The Attempt at a Solution Let ##\alpha= \sqrt{2}+\sqrt{3}\in \mathbb{R}##, then ##\alpha^4-10\alpha^2+1=0## which is a root of ##f(x)=x^4-10x^2+1## where ##f(x)## in...
14. ### FInd non-zero elements are primitive in a field

Homework Statement Construct $\mathbb{F}_{16}$ as a quotient of $\mathbb{Z}_2[X]$. How many non-zero elements are primitive in this field? Calculate $|GL2_(\mathbb{F}_16)|$. Homework Equations Primitive Theorem The Attempt at a Solution For the first question, I don't know how to construct...
15. ### Show F is isomorphic to F/{0}

Homework Statement Let F be a field. Show that F is isomorphic to F/{0} Homework Equations The Attempt at a Solution By the first ring isomorphic theorem, kernel of the homomorphism is an ideal which is either {0} or I. Hence F isomorphic to F/{0} I think I misunderstood the problem can...
16. ### Show the group of units in Z_10 is a cyclic group of order 4

Homework Statement Show that the group of units in Z_10 is a cyclic group of order 4 Homework Equations The Attempt at a Solution group of units in Z_10 = {1,3,7,9} 1 generates Z_4 3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4 7^0=1 7^1= 7, 7^2= 9 7^3=3 7^4=1...
17. ### Show the range of f is isomorphic to a quotient of z

Homework Statement Let G be any group and a in G, define f: Z → G by f(n) = a^n Apply any isomorphism theorem to show that range of f is isomorphic to a quotient group of Z Homework Equations The Attempt at a Solution The range of f is a^n , then quotient group of Z is Z/nZ Apply the...
18. ### Acceleration of a stirrer

Homework Statement Question: There is a fixed 55 cm long rod (vertical) that is charged uniformly up to +137 nC, Place a 0.5 g plastic 6 cm long stirrer (horizontal) that is charge to +93 nC, 2.3 cm away from the rod Calculate the Electric force and acceleration on the stirrer . Homework...
19. ### Comp Sci Using VPython program to calculate the E-field

Homework Statement Charge a ring of radius R=5.0cm laying in the x-y plane to 50nC. Create a VPython program that will allow you to calculate the E-field due to the ring anywhere in space. Homework Equations E_ring=kQz/(R^2+z^2)^(3/2), point P above the xy-plane The Attempt at a Solution...
20. ### Show (H,+) is isomorphic to (C,+)

Homework Statement Let, M={ (a -b) (b a):a,b∈ℝ}, show (H,+) is isomorphic as a binary structure to (C,+) Homework Equations Isomorphism, Group Theory, Binary Operation The Attempt at a Solution Let a,b,c,d∈ℝ Define f : M→ℂ by f( (a -b) (b a) ) = a+bi 1-1: Suppose f( (a -b) (b a) )= f( (c...
21. ### Find the speed of a galaxy

Homework Statement Taking light collected by the telescope and shine it through a diffraction grating with a 765.5 lines/mm. Using the filter that just the brightest Helium II line. you cast the diffraction grating's diffracted pattern on a screen is 22.000cm away and there are 3 dots, m=-1...
22. ### Gravitational force between 3 masses

Homework Statement What is the magnitude of the net gravitational force on the m2=5kg mass? Assume m1=25kg and m3=15kg. https://session.masteringphysics.com/problemAsset/1383524/6/13_P28.png Homework Equations F=GMm/r^2 The Attempt at a Solution First I find the angle between m1 and m2 which...
23. ### Find the minimum speed

Homework Statement You've taken a summer job at a water park. In one stunt, a water skier is going to glide up the 2.0-m-high frictionless ramp shown, then sail over a 5.0-m-wide tank filled with hungry sharks. You will be driving the boat that pulls her to the ramp. She'll drop the tow rope at...
24. ### Linear Algebra: linear transformation

Homework Statement let A be the matrix corresponding to the linear transformation from R^3 to R^3 that is rotation of 90 degrees about the x-axis Homework Equations find the matrix A The Attempt at a Solution I got stuck on rotating z component. I tried T([e1,e2,e3])=[0 -1 0]...
25. ### Impact speed

Homework Statement A ball is thrown toward a cliff of height h with a speed of 28.0 m/s and an angle of 60.0° above horizontal. What is the ball's impact speed Homework Equations Vf=Vi-at Sf=Si+Vit-1/2gt^2 Vy=sintheta Vx=costheta The Attempt at a Solution first find the vertical velocity =...
26. ### Span of Vectors

As I study today, I read through my textbook that says if Ax=b has non-trival solution, then b span in A. I want to know how can I prove it.
27. ### Find the minimum distance

Homework Statement the minimum distance for a car traveling at a speed of 30m/s is 60 m, including the distance traveled during the driver's reaction time of 0.44s. what is the minimum stopping distance for the same car travelling at the speed of 41m/s? v=30m/s, s=60m drivers' reaction time =...