1d) Prove: If z=Zo is a zero of order n of flz), then it is zero of order 2n of

Q: In Z[r], let I = {f(x) E Z[r] | f(0) is even}. (a) Prove that I = (x, 2). (b) Show that I is maximal…

A: Given, In Zx, let I=fx∈Zx | f(0) is even (a) Prove that I=x, 2 Let…

Q: 1- iz If z = x +iy and w =- z-i show that w = 1= zis purely real

A: Use modulus property

Q: If TeZx Such that x is a momber of 27+5x-3=03 and S={x Such that x i. member of 2i+71+233/3 that T…

A:

Q: Prove Theorem 8.2.3 for the case where O lies in the interior of ZAPQ.

A: According to the given information, it is required to prove the given theorem for the case 0 lies in…

Q: Let x, y e Z. Prove that if 3x+7y is odd or 5x + 6y is even, then x or y is even.

A: We have given that for x,y belongs to integers. If 3x+7y is odd or 5x+6y is even then x or y is even…

Q: In the domain of Gaussian integers Z[i], the element 23 is * Irreducible Reducible Unit None of the…

A: (i) Every prime is irreducible in Z[i].

Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

A:

Q: A cubic spline S is defined by: So(x) = 3(x- 1) + 2(x – 1)² – (x – 1) S(x) = s,(x) = 4 + 4(x – 2) –…

A: Natural Cubic Spline is a piece-wise cubic polynomial that is twice continuously differentiable.…

Q: Given: - L KL ON OP NP Prove: ZM L = 2O * View

A:

Q: (b) Prove that xe" 2 for some x in (0, 1). %3D

A: By the intermediate value property of continuous function if a function f:(a, b) --> R be a…

Q: a+ib a) If x+ iy = prove that x? + y² = 1. a-ib

A:

Q: こ be terms Prove Find o of ů, ů. your resubts . in

A: Consider the given vectors: Here the objective is to find the vector w such that

Q: ZW and ZX are supplementary and ZW and ZY are supplementary. Construct a proof that ZX = ZY.

A: <W  and < X  are supplementary angle  the sum of supplementary angles = 180 degree <W  and…

Q: (c) Show that the ideal generated by x² + y² + z² € C[x, y, z] is a prime ideal.

A: If <x2+y2+z2>is not a prime ideal, there should exist g,h∈ℝx, y, z s.t. x2+y2+z2 | gh,…

Q: 2) Prove that the foll owing func tion is not differen tiable at (0,0). 2. 2. f(x.y) = (X.y) = (0,0)…

A:

Q: Show that \{e ^ (- x), x * e ^ (- x)\} is a fundamental set of solutions of y^ prime prime +2y^…

A:

Q: when C: z= Using Cauchy's residue theorem, is c (z-1)(z+2)2 2 O a. 0 O b. 2ni Oc. 2xi 6. O d. ri

A:

Q: Compute 4n — бп? lim Vlan| for a,n %3D 3n2 +7 n-00

A:

Q: Find all positive integers x, y, z satisfying x · y · z² = 5xyz. '*

A: Introduction :Given , positive integers x,y,zwe have asked to find the values of x,y,z which…

Q: (2) Find all the numbers z such that elz = 10i.

A: Consider the given. eiz=10i

Q: 3.2 Consider R2 with (, ·) defined for all x and y in R? as х and 2 (x, y) := x |1 =:A Is (:, :) an…

A: The objective of the question check for inner product.

Q: 7. Find a pair of integers (x,y) such that 4021x + 2014y= 1.

A:

Q: A cubic spline S is defined by: So(x) = 3(x – 1) + 2(x – 1)² – (x – 1)³ if 1< x< 2 S(x) = 1 s, (x) =…

A: Given,Sx=S0x=3x-1+2x-12-x-13                             if 1≤x≤2S1x=4+4x-2-x-22+13x-23…

Q: Prove that Z(Sn) = {e} for n ≥ 3.

A:

Q: %3D n-0o 2n

A: Given. We are given a limit and we need to prove that the limit of the given function at infinity…

Q: 20.* a. Let P(z) = 1 + 2z + 3z² +.. of P(2) are inside the unit disc. + nz"-1, By considering (1 –…

A: Proved by using Rouches theorem from complex analysis.

Q: c If in is an involution of X, show that (r, y)in = (yi", rin) is an involution of X12). %3D

A: Given X is a lattice. Given in  is an involution of X To show: x,yin=yin,xin is an involution of X2…

Q: 13 3. Use the IVT to prove that y = r Inr has at least one zero on 2 2

A: Above function is continuous function in given interval. So we can apply IVT

Q: 10. Prove that Z has no smallest element.

A: we will prove it by contradiction, Suppose x belongs to Z  is the smallest element.

Q: A natural cubic spline S on [0,2] is defined by (So(x) = 1+ 2x – x³ S(x) = Is, (x) = a + b(x – 1) +…

A: Given: Natural cubic spline S on [0,2] is defined as: Sx=S0x=1+2x−x3 if 0≤x≤1S1x=a+bx−1+cx−12+dx−13…

Q: Prove that there are infinitely many triples (x, y, z) of positive integers satisfying x² + y? = z*…

A:

Q: Find the rank and nullity of the 2. 4 9. A z -7 2. -5 4 L7 -9 2 -4 verify the that dimension…

A: The given matrix is  A=20464-7201-5247-92-4 According to dimension theorem for any matrix A of size…

Q: #5 Let x, y, z be nonnegative real numbers such that y + z> 2. Prove that (x + y + z) >4x + 4yz.

A:

Q: Let m and n be positive integers with gcd(m,n) = 1. Prove that Z/mZ x _Z/ nZ = Z/mnZ

A: Suppose m and n are relatively prime.

Q: Find the Zero and its order for the funcation fcz) = 1 -Pit z te.

A: We know that, f(z) = (z-z0)n has zero of order n at z=z0, if f(z0)= 0

Q: A cubic spline S is defined by: (So(x)%D 1+x- x (5,(x) = 2 - 5(x – 1) + (x – 2) if 0 SxS1 if 15xS2…

A: As we all know that Natural Cubic Spline is a piece-wise cubic polynomial that is twice continuously…

Q: In ALMN, LN is extended through point N to point 0, mZMNO = (5x – 13)°- mZNLM = (x – 4)°, and m/LMN…

A: Solution-

Q: (a)Show that,if x is small enough for its cube and higher powers to be 1 neglected, = 1-x +. by…

A: It is given that x is small enough for its cube and higher powers to be neglected. It is required to…

Q: dv Prove that 4u/2n

A: We use Gamma function to prove the given result.

Q: 9) suppose that horx)=1, f,(x)=x ond dz(X) Prove that Ba(f, - where Ba is a Bernyterh Polymomial.

A: Given functions are f0(x)=1, f1(x)=x and f2(x)=x2 Since Bernstein polynomial is…

Q: 1. Let 1, y € R" be given. (a) If ||z|| = ||y|| = 1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1…

A: \documentclass{article}\usepackage{amsmath,amssymb}\begin{document} (a) Norm x=Norm y=1\\ Let angle…

Q: Prove or disprove: If xn ≥ 0 for all n ∈ N and lim sup xn = 0, then xn → 0

A:

Q: (c) Show that 1 and –1 are the only units that have finite order in R*.

A: Units in R*

Q: Use the Cauchy Residue Theorem to find ∫C [3(z+1)] / [z(z-1)(z-3)] for C being z(t) = 2eit

A:

Q: Find-the-leng of AB. 14. Given: JZz = KZ Prove: 2J = LK 15. Prove the converse of Exercise 14. K

A: 14 ) Given: JZ⏜≅KZ⏜ To prove:∠J≅∠KClearly, the sidesof JZ⏜ and KZ⏜…

Q: Q1. Prove that KT = = -.

A:

Q: z, Z2 Prove

A: If z =x+ iy then  z = x- iy. i2 =-1.

Q: Theorem. A rectangle R with side lengths 1 and x, where x is irra- tional, cannot be “tiled" by…

A: Without using the concept of linear algebra you can proof this. Let length of rectangle= 1 Unit…

Q: Let x, y e Z. Prove that if 3x + 7y is odd or 5x + 6y is even, then x or y is even.

A: