8. Let A= (a) Show that A is irreducible. 0 1 1 1 0 1 1 0,

Q: 4 Let A b [² }] Find Conditions on 1 c d a,b,c and d that AX =0 that AX=0 has infinitely many…

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

A: Use modulus property

Q: 5. In which annular domain is f(z) = (a) A(0,0,4) (b) A(0,0,2√2) (a) (b) 1 24 - 16 (d) holomorphic?…

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Q: 22. Show that if F₁ and F₂ are primitives for f: G → C and G is connected then there is a constant c…

A: Given : F1 and F2 be the primitives of f : G→ℂ, where G is connected. To show : There is a constant…

Q: Show that A = (aj) = Cmxn. Then || A||2 ≤ || A||F ≤ √ Rank(A)|| A||2 V

A: Matrix A=(aij)∈Cm×n: This notation indicates that A is a matrix with complex entries, where m and n…

Q: 13. If R = {a + b/2|a, bE Z}, then the system (R, +, ') is an integral domain, but not a field.…

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Q: Let g(x)= x³ - 4x² + 2x - 4 and f(x) = x + 2 be polynomials in Q[r]. so that g(x) = f(x)q(x) +r(x).

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Q: Kailash ben

A: The objective of this question is to prove two properties of a given matrix A with dimensions m x n…

Q: u? + 5u – 7 If g(u) In(u) find g'(u)

A: The given function,    g(u)=u2+5u-7ln(u)

Q: (2) Prove that for all x, y € R, we have 8xy³ ≤ (2x + y³)².

A: A.M>=G.M

Q: In Z[i], show that 3 is irreducible but 2 and 5 are not

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Q: Let D be an integral domain. (i) If aD is maximal,then show that 'a' is irreducible (ii) If D is a…

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Q: 6. Determine the number of zero's for h(z) = 2z* – 3z² + 2z – 5 inside the disk |z| = 2. (Use…

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Q: Theorem 1.24. The set P of all polynomials p(x) = ao + a1r + azx² + .. . + amr" %3D .......... ith…

A: For each pair of natural no. (m, n)  Let Pmn be the set of polynomials of the form…

Q: 1. Find an r> 0 such that all zeroes of 24 + 6z²+z+3 are contained in D,(0).

A:

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: Q1. Expand the following function along 1 < |Z| < 3 1 f(z)= (Z+1)(Z+3) Hint: Use partial fraction

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Q: For complex variables y, z = C, show that Jn(y + 2) = Σ Jm(Y)Jn-m(2). m=-

A: As asked, we shall solve only part (2).

Q: 8. The polynomial x² a. Q and R e. C and z b. R

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Q: Example 1.5. Let X = R? x R² and d1 : X R, such that, (1) di(x, y) = d;((r1, 12), (Y1, Y2)) =| ¤1 –…

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

A: Field of quotient

Q: 12. Evaluate the expression using de Moivre's Theorem. Write your answer in standard form z = x +…

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Q: 3. Express the following in terms of z] and Zz: (a) z* (b) 1/z (c) Re{z} (d) Im{z}

A: In this question, we have to express the following in terms of z and ∠z :  (a) z* (b) 1z (c) Rez (d)…

Q: 3.2 Find and classify zeros of z*(e² – 1).

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Q: Let n e Z. n#-1,3,

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Q: 4) Write the taylor { Cn(x-2) n não series for f(x) = ex about x = 2 as

A: Given that fx=ex centered at x=2. We have to find the Taylor series for fx.

Q: Suppose that A and B are two sets such that n[A] = 5, n[B] = 9, and n[AU B] = 13. If n[U] = 19, then…

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Q: d) Prove: If Z = Zo is a zero of order n of flz), then it is zero of order 2n of (f(z))²

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Q: 21. Show that the set of polynomials {C₁, C₂ (2x - 1), C3 (6x² − 6x + 1)} in P(R) is orthogonal with…

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Q: Let z = x + yi, z, = 1- 0.5i, and z, + z, = 3-1.5i. Find In(z,) - In(z2) C) In(50) + 2nin A) In(25)…

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Q: (5) Prove that the ideal (3,x) = {a+axtor + ₁x² [3 (40] is not a Principal ided of 21 Cx2

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Q: 5z+1 2. Prove that the function f : R – {2} → R – {5} defined by f (x) = is bijective. z-2

A: Consider the given information: The given function is, f:ℝ-{2}→ℝ-{5} f(x)=5x+1x-2 To show that f is…

Q: Let f(x) be an irreducible ow that all roots of f(x) =

A: