r* – 2 Show, using the definition, that is O(z²). z +1

Q: Y Cn) be defined recursively by Let YCO) = 3, Y (1)=4 and Ycn) =3Y(a-1) – 2 Y(n-2) Ynz2. Y (n)= 2^+2…

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Q: 1. Let ƒ : N × N → N be defined as f(m, n) = 2m−¹(2n — 1). (a) Show that f is a bijection. (b) Using…

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Q: z'dz Evaluate where C is z| by residue theorem (z-1)*(z-3)

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Q: (6) Let f(a), g(n) € Pla] where f(a) is irreducible.

A: Let m = deg(f(x)) and n = deg(h(x)).

Q: Let C1 : y = x² sin 3x, C2 : y = x2 and C3 : y = -a², then (1) C1 touches C2 at infinite points (2)…

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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: a b (6) Identify all of the nonempty submatrices of A = с d

A: Given that:Matrix A = abcd

Q: 3. Find two elements a and b in Z25 such that a and b are units, but a +b is not a unit. Justify…

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Q: ¤1 + ag – r3 = 0, -2r1 – 12 + 5r3 = 0, -#1 + 4.x3 = 0 find the Rank(A), where 1 1. -1 A = -2 -1 -1…

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Q: Let G = {3"6" | m, nE Z} under multiplication. Prove that G is isomor- phic to ZOZ. Does your proof…

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Q: Q3- Give an example of two denes not equal X subsets of (R,T) and (z, of ). ( Explain your answer ).…

A: Answere : The two example are given below :

Q: Solve the folowing equation on [0,2pi) 2sqrt2 +4cosx=0

A: In the given equation, the cosine function is the trigonometric function:   Rewrite the given…

Q: [0 1 1 Given: A = and B 1 1 1 1 l1 0 0- LO 0. a. Find C, the boolean product of A and B. Show your…

A: Given : Matrices A = 001110111100 , B = 011100 and D = 00011011   To find : a. Matrix C the…

Q: {x0.5,x-1/3}

A: I am attaching image so that you understand each and every step.

Q: Suppose f:N→ N has the rule f(n) = 3n² – 1. (a) Determine whether f is surjection onto N and show…

A: a)   The given function is not surjective. Because there is no pre image n of 3 in the domain N such…

Q: ;0<x<0, - 0 <0 <0 Let X,,X,X,r(x;e)=- e.w Discuss the sufficiency by using Factorization Theorem.

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Q: Find the Wronskian for {x2, ex^2, x2ex

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Q: Find a closed-form formula for f(n)=...) k+1 II₁¹. (Write your answer in the form of: k

A: We will use definition of product notation to find closed form of given product form.

Q: Z[i] is an integral domain with : 49 elements 6 elements O 9 elements

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Q: nex}. 2n (b) Let B = :nE 3n+1 i. Prove that inf B = ii. Show sup B =

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Q: AE Mmxn amed suppose be non Let that A=xY is the fall ragak factorization that At =x*y then Show

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Q: -1 2 Define T: R? - R³ by T(x) = Ax, where A =-3 3 1 1 - 6 Find an x whose image under T is – 15 - 5…

A: here we are given that the transformation of R2 to R3 given as T(x)=Ax

Q: Define injections f : N → N and g : N → N by f(n) = 2n and g(n) = 2n. From f and g, the…

A: Given  f:N→N and g:N→N are two injective function defined by f(n)=2n and g(n)=2n Let 2n = k then…

Q: the integret fector of the bifferentiel e qration →き x28x he is 2. O Not found

A: Consider the given differential equation xdy-ydx=x2y3dx and rewrite it again by doing…

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

A:

Q: 5. The Wronskian of {X,X} Select one: a. b. x C. X d. m/N

A: The Wronskian of two functions f(x) and g(x) is given by: W=f(x)g(x)f'(x)g'(x)

Q: 4. The field of quotients of (Z2, +2)-2) is (a) Q2 (b) Z2 (c) Z4 (d) Zg

A: Field of quotient

Q: Q3. Prove or disprove: If A is invertible, then A'A is positive definite.

A: A positive definite matrix is related to the theory of matrix. For a given square matrix A, it is…

Q: elements. Why is Dg not isomorphic to Z/8Z?

A: D8 is not isomorphic to Z/8Z because D8 is not a cyclic group but Z/8Z is cyclic. For Cayley table,…

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

A: convolution theorem

Q: The element (3x-6) is Reducible over QO lucible over Q and R c. Both d. Other O O O O

A: The elements (3x-6) is_____

Q: Find the value of k such that A is singular. 1 05 6 -1 0 4 5 k k. %3D

A: If the determinant of a matrix is zero then the matrix is singular.

Q: 5. Find the residues of the following: a. 2-2 z(1-2) at z = 0; at z = 1

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Q: S = {x2 − y2 ∈ N : x, y ∈ Z, x2 − y2 > 127}. Prove that S has a smallest element.

A: Given S={x2-y2∈N: x,y∈Z, x2-y2&gt;127} Theorem (Well ordering Principle) Every Nonempty subset of…

Q: 9-x2 B: Find the integral o L 18-x2-y2 /x²+y2 xydzdydx.

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Q: { (3i – 2)² = -n(6n² – 3n (b) Prove by induction that | i=1

A: We have to prove that ∑i=1n3i−22=12n6n2−3n−1 for n≥1 by using principal of mathematic induction.…

Q: -3 47 Define T: R? - R³ by T(x) = Ax, where A =|-5 3 - . - 28 Find an x whose image under T is -32

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Q: 7. Let r, y E R such that r 5xy, then 2x < y. [PROOF BY CONTRAPOSITION] (Hint. Consider the…

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Q: V21. Show that there exist constants M and N such that M||x||| < ||||2< N ||x|| -

A: Given ||x||1=∑i=1n|xi|=|x1|+|x2|+...+|xn|       ||x||2=∑i=1n|xi|2=|x1|2+|x2|2+...+|xn|2 Now we have…

Q: Q2\ Prove that Ax (B+C) = A×B + AxC When A= 2i + j-k B=i-j+ 2k Sumer C=i+2j+k

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Q: Q1(i) Prove that Lim + a+ 1 2n²+1 V2n²+2 V2n²+n]

A: (1) Given, g(x)=limn→∞12n2+1+12n2+2+.....+12n2+n suppose any f(x) and g(x)…

Q: Evaluate det(A) by a cofactor expansion along a row or column of your choice. 3 0 0 1 0] 4 4 4 1 0

A: Given that

Q: on the closed interval Co,14], (0,14)such that f'ccF FC14D-FCO) 14-0

A: we have to find the value of c

Q: 1. For positive integers i, j,n such that i+ j = n (a) show that corresponding multinomial…

A: sol(a) we know that multinomial co-efficient is definedby ni,j=n!i!,j!  —(1) where i+j=nnow…

Q: Let z = 3 + 9i and w = 2 – 7i . Find out the following in the x + iy form. 4. z-1 – w-1

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Q: B. Use indirect proof 1. Show that: Vx€z, if x2- 6x+ 5 is even, then x is odd.

A: Given : x2-6x+5 is even for all x∈ℤ   To prove : x is odd As x2-6x+5 is even . So,  x2-6x+5=2m for…

Q: The Wronskian of a fundamental set of solutions of the DE 2y (10) +-y (9) -y=0is given by O None of…

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Q: 3 Z Find the residue of z² - 1 2 Z at z→ ∞00.

A: Use the formula of residue at infinity to solve this problem as in next.

Q: (3) If R is an integral domain has non zero characteristic, then Char(R)=..... (i) 5 (ii) 4 (iii)9

A: Option (i)