Find the expansion of fix) = In(1+e"? Seiect one: O si In(2) +z/2+/8+z/192 +. O In(2) + z/2+z² /8+z*/192+. O In(2) +a/2+z"/8 – z*/192 +... O s In(2) + z/2 + z² /8 - z*/192 + ....

Q: Without doing any calculations in Aut(Z105), determine how manyelements of Aut(Z105) have order 6.

A: Given: Automorphism, AutZ105To find: Number of elements of AutZ105 having order 6.

Q: Which of [0], [1], [2], [3], [4] is equal to [3]^2023 in Z_5? What is [15]^1989 congruent to in…

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Q: Let A = {4n^2 + 1|n ∈ Z}. Show that A is a subset of Z but not equal.

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Q: Show that M_2(Z) under multiplication is not isomorphic to N under addition

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Q: IfA =[-3,5] U {6, 7} U [8, 00), then Find each of the following (1) inf (A) (2) sup (A) (3) int (A)…

A: Let A=-3,5∪6,7∪[8,∞). (1) inf A is computed as computed as follows, (-∞,-3) is a lower bound of A.…

Q: why are the polynomials x^4 + 2x^2 + x + 1 and x^4 + 1 are not irreducible over Z3 (mod3)

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Q: Find all n ∈ Z for which 3x^2 + 2nx + 12 is irreducible over Q.

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Q: Prove that for a # -1, a>0,n EN IS (1+a)^ n >=1+na.

A: Here let A=(1+a)n, a≠-1, a>0, n∈N and B=1+na

Q: ²*6²²y dy) dr 3

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Q: Find the degree of the splitting field of x^2-3 over Q.

A: Degree of splitting field

Q: For all a, b in R, show that -max{a^2, b^2} <ab max{a^2, b^2}.

A: To prove that for all (a,b)in(R), [max(−a2b2,ab)≤ab≤max(a2b2)] We'll consider three cases: 1.…

Q: If In x +In(x-3)=In(6x), then X=

A: The equation is given as ln x+ln x-3=ln 6x

Q: Let f and g be permutations. Show that the order of h=fgf^−1 is the same as the order of g.

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Q: 2) Prove that: d? L„0) = n;(n – 1) %3D dx²

A: Consider the given information: Let Laguerre polynomial is given by, Lnx=∑k=0n(-1)kk!nkxk…

Q: 2 ľ xe2xdx

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Q: 8. The Laurent's expansion of (2² – 1) (2² + 4 for |z|<1 is 5z2 2125 1). 4 64 16 2) + 21 4 4 4 3) -…

A: None of the option 1,2,3 and 4 is correct. Please check it.

Q: If Pn: 49^n +16^n + k is divisible by 64 for all natural numbers is true, then the least negative…

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Q: Which ring is fx=2x^2-1 irreducible? Z[x] Q[x] Z2[x] Z3[x

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Q: Prove the following using a direct proof.  If a,b, and c are integers such that b is a multiple of…

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Q: Show that the set A = {n^2/ (n+1): ne N} is unbounded in R.

A: To prove that the set A={n2n+1:n∈N} is unbounded in R,

Q: 12 So cx 202x2+3x+1

A: Given  To find antiderivative , we need to apply partial fractions Factor the denominator part

Q: 36. Use the binomial theorem to expand (2x - y)* . A. 16x* – 32x³y + 24x²y² – 8xy³ + y* С. 2x - 4х3у…

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Q: R^n×n is invertible, show that AB is similar to BA for all B ∈ R^n×n.

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Q: 25. Find Uge B and Be B for each collection B. BE 4 (a) + *+{[11] N}*(0) + {[L1 =): N} N}₁ B+ :n (c)…

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Q: state formally the Heine-Borel Theorem in R^k

A: Let S be a subset of ℝk. Then the Heine-Borel theorem states :

Q: Find the order of 175 in Z_245

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Q: 5 М — 11 Find formulas for the entries of M", wheren is a positive integer. M" =

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Q: State De-Moivre's theorem. Compute n" derivative of the function y th x² +a?

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Give the ring of integers and discriminants of the following number fields. Q(e^(2*pi*i)/7)

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Q: Q3)Find the following: 1- 0={2,1}, J={1,2,4,6,5,3}, and , R={5,4,2},T={6,5,1}Prove (OUR)c = OcnRc.…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Chasidethe 好声ERsueh the X茶X学义节4X作3: 的用(X 3)/4 持2次个3:

A: We have to choose the correct answer