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1. Establish or each of the following statements as being true false. Justify each answer fully.
(a) Z5 ⊕ Z12 ≅ Z60
(b) Z5 ⊕ Z10 ≅ Z50
(c) Z12 ⊕ Z4 ≅ Z24 ⊕ Z2
(d) U(180) ≅ U(112)
2. (a) Express U(77) as an external direct product of groups of the form Zn in three different ways.
(b) Express Aut(Z55) as Zm ⊕ Zn for some m and n.
3. Suppose φ is an isomorphism from Z5 ⊕ Z11 to Z55, and φ(2, 3) = 4. Find the element that φ maps to 1.
4. Give an example of an infinite non-Abelian group with precisely six elements of finite order.
5. Find two (distinct) subgroups of order 30 in Z50 ⊕ Z60.
6. Determine the number of elements of order 10 and the number of cyclic subgroups of order 10 in Z20 ⊕ Z15.
The motion of a spring that is subject to a frictional force or a damping force is often modeled by the product of an exponential function and a sine or cosine function. Find the velocity of the point after t seconds.
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Factorize and solve the given quadratic equation and Solving the given quadratic equation by using formula method
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A company wishes to manufacture a box with a volume of 56 cubic feet that is open on top and is twice as long as it is wide. Find the width of the box that can be produced using the minimum amount of material.
Use a direct method to prove that for any string x, xxR is palindromic.
Should the null hypothesis be rejected for the difference between the two groups in change in mobility scores over 12 weeks? Provide a rationale for your answer.
Objective questions based on regression.
a fast food restaurant wants a special container to hold coffee. the restautrant wishes the container to quickly cool the coffee from 200 degrees to 130 degrees fahrenheit and keep the liquid between 110 degrees and 130 degrees as long as possible..
The number of sequence is 17. If the second number is 3, what is the first number?
7- Jensen Enterprises paid $1,300 in dividends and $920 in interest this past year. Common stock increased by $1,200 and retained earnings decreased by $310. What is the net income for the year?
a company just completed a physical inventory count at year-end december 31. items were counted and costed on a fifo
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