b) If d = 1, prove that (1 if a is odd gcd(a, 2b) %3D if a is even

Q: 18) (SAnn B anlogbro dtiv C AB = 12, BC = 3x + 6, and AC = 5x – 12 a) Find BC b) Find AC

A: Since you have asked  multiple questions in single request  so we  will be answering only first…

Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

Q: 4.3.a Use Theorem 9.4 to show that Y is sufficient for p 4.3.b Find the MVUE for p

A:

Q: 4. Show that if ki = (X; – X)/[Σ(X; – Χ)2], then (a) Σ ki = 0; = (b) Σk? = 1/[Σ(X - X)]; (c) Σk;(X;…

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Q: Show that (n+1)Pn+1(x) = (2n + 1)xp,(x) – npr-1(x) ;n> 1? %3D

A:

Q: Find an example for which (zc)d ≠ zcd with z ≠ 0

A: To find an example for which zcd≠zcd where z≠0 Now, for integers c and d zcd=zc·zc·zc····d…

Q: The Maclaurin expansion of f (x) = e¯7× is %3D

A:

Q: Let p». Prove that, for every e > 0, a.a.s. the largest component of G(n, p) has size at least…

A:

Q: 1 Show that 2-* d.x 2 In 2

A: We can show the answer in the following steps.

Q: Show that the series 3. + + 4 To t (-1) Lonverges absolutely.

A:

Q: Which of the following is homogenous of smallest degree? (A) In x - In y B x+y x-y C √x-y x² + y4 X…

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Q: If (ar + br) = 2n %3! n+1 and k-1 8. E(ak - br) = n+2 - , then ar %3D k-1 k-1

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Q: The characteristic of Z5[x]/ is O 3 5

A:

Q: Find the pmf p(x) of X, the number of casts needed to obtain that first six.

A: Let x be the number of times die thrown until 6 appears. Probability that 6 appears on top is 1/6…

Q: The number of the element(s) of order 2 in Z2 O D6

A:

Q: 2. Solve the given SLE using LU-Factorization X, + x, +x3 = 3 X, - x, +4x; =4 2х, + 3x, — 5х, %3D 0

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Q: Prove that E, k2* = (n – 1)2"+1 +2 for all n > 1. =D1

A:

Q: locally Lipschitz in (t,x)E[a, b]x D.

A:

Q: a) Is a G = Z24 cyclic? Why %3D

A: Since you have asked multiple questions, we will solve first one for you. If you want other question…

Q: Show that for all z1, Z2 E C, Then we have: |Iz1| – |z2|| < |z1 +z2l

A: 2) The inequality can be proved as follows:

Q: 2. Prove that, for all k E N, 2 can be written as the sum of two squares.

A:

Q: A cubic spline S is defined by: So(x) = 3(x – 2) + (x – 2)² – 4(x – 2)ª 32 if 25x53 S(x) s,(+) =…

A: true

Q: Show that x2 + x + 4 is irreducible over Z11.

A:

Q: Show that x2+x+4 is irreducible over Z11.

A:

Q: Suppose a e Z. If a² is not divisible by 4, then a is odd.

A:

Q: 7. (a) Find all the values of z such that e = 1+ i.

A:

Q: Assume that |Xn] 0. (Hint: first, prove that P(|X|>c+1) = 0.)

A: Answer:

Q: 1. Condense and expand. a) Condense as a single quantity: 1 In(3)+- In(x? +1)-3 In(1+x) 3 b) Expand:…

A: Given, (a)  To condense ln3 + 13ln(x2+1) -3ln(1+x) (b) To expand lnx2+1x+2(1+x)5

Q: 3. Let n E Z. Give a direct proof that there exist x, y E Z such that n = 55x + 78y.

A: Objective : To prove for any n∈ℤ ∃x,y∈ℤ such that n=55x+78y Prove for n=1, Greatest common divisor…

Q: Find an example for which (zc)d ≠ zcd with z ≠ 0.

A:

Q: (AB) = 30 (АВ) 3D 10 Calculate the Coefficient of Collignation. Given (aß) = 5 (а.В) %3D 5

A:

Q: Prove that D24 and S4 are not isomorphic

A: Since D24 ={a,b: a 12 =b2 =e (identity) and ab=ba-1 } This implies that D24  has 12 reflection and…

Q: If (ar + br) : 8n and n+1 %3D k-1 (ak - br) +, then E ak = n+2 k-1 k-1 None 2.

A:

Q: 2. What is the largest possible c in (1) if X=R² and x₁=(1,0), x₂ = (0, 1)? If X = R³ and x₁ = (1,…

A: Given: c To find: a) X=R2b) X=R3

Q: If A is a 2 x 2 with det(A) = 3, then det(4A1) = %3D

A: det(A-1)=1det(A) det(4A-1)=4ndet(A-1) where n is the order of the matrix.

Q: 1: Find Z by convolution theorem. (z- 2)

A: convolution theorem

Q: 1. Prove that fraction where m is z-4

A: Given: zn-1z-1 ⇒zn-1nz-1 ----(1) Let apply the limit in equation (1) limz→1zn-1nz-1 ----(2) Let n=pq…

Q: Show that if X gamma(r, A) and Y ~ gamma(s, ) independent, are then X + Y is gamma(r + s, ).

A: X~Gammar,λ fx=λrΓre-λxxr-1,     x&gt;0, r&gt;0, λ&gt;0 Y~Gammas,λ fy=λsΓse-λyys-1,     y&gt;0,…

Q: Is ZXZ cyclic? [Here Z means (Z, +).]

A:

Q: Given the function z = In (2xy + e3x) - 5y, %3D az a. ax

A: Since, we have to find the partial derivatives ∂z∂x and ∂z∂y of functiongiven by z=ln…

Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

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Q: .In Z[V-5], the element 29 is: O Unit O reducible Irreducible None of the choices

A:

Q: 4. Show that if ki = (X; - X)/[Σ(X; – X)2], then (a) Σki = 0; = (b) Σk? = 1/[Σ(X; − x)2]; (c) Σκ;(X;…

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Q: (12) If E(X*) is finite with E(X) = µ and Var(X) =o² then prove that E((X – µ)*) > of. %3D

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

Q: For any n e N, let Bn = (n, ∞0). {B„}1 is descending. Show that Bn = 4. %3D n=1

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Q: Use Euler-Maclaurin's formula to prove that E n² = "(n+ 1)(n +2) 6. %3D 1

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Q: If det A= 0 why is det A+ = O? If A has rank r, why does A+ have rank r?

A: Given Data det A=0   Suppose the column and row of the matrix are linearly independent. The matrix…

Q: . In Z[V-5], the element 29 is: None of the choices Irreducible O Unit O reducible

A:

Q: In Z[V-5], the element 29 is: None of the choices Irreducible Unit reducible

A:

Q: of 3 ty ?

A: