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A: We use Cauchy-Gaoursat Theorem.

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Q: matoix Such that show that Let is Then detx = tt mxn

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Q: Consider the function & : [0, π] → [−1, 1] given by o(t) = cos t. Let w be a one-form on R. TRUE or…

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Q: 3) suppose fle) is a defferentiale funchou on the interval E-7,1J Such that H-7) =4 and fli)=1.…

A: Given, A function f(x) is a differential function on the interval -7,1 such that  f(-7)=4 andf(1)=-1…

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Q: It Question : Ins -(第) IRS in Let Xに Xz= Find leng ths f xi, Xz and inner product of Xı, Xz の Frnd…

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Q: 3. a) Prove that the function sin is not continnous on b) Show that the functwoin g(x) x e RiQ is…

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Q: (b) Show that x. y < ||x|| ||y||- [Hint: If x, y # 0, let a = , 6= and use the fact that ||ax + by |…

A: Given that, Euclidean metric d on Rn is defined as d(x,y) = || x - y || for x, y in Rn Part(a): Let…

Q: 3. In R with the usual metrie , find: (i) The interior of 2.3)U(5.6 (ii) The interior of (5, 6) (i)…

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Q: 7. Given RS and TV bisect each other at point x. TR and SV are drawn Prove: TR = SV

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Q: This exercise comes from Stein & Shakarchi, ch 2 # 6. Let be an open subset of C, and let TCN be a…

A: Given that, Ω be an open subset of ℂ and let T⊂Ω be a triangle whose interior is also contained in…

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Q: 3) Consider the topology 7 = {G CR: 0 ¢ G or G = R} on R. Is open T 3= interval (-1,1) compact in…

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Q: a. In (R, d) where d(x,y) = |x – y| (Euclidean Metric space) show that: Q (The set of all rational…

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

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Q: O Consider the set R. Prove: (i) VxER, x 0 = 0; (ii) If x y = 0 then either x = 0 or y = 0, or x = y…

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Q: 3. If ACX such that A# 0 and r (GCX: GnA = 0}U{X} then prove that r is a topology on X. If A {p},…

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Q: Let B=[9,b]xc and 9:0.b]->IR and h:[ed]>R ora continvous funetions f:1B>IR f (x.y)=gG)h(w) be…

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Q: DATE MONTH YEAR - Find two Points - Consis ting 67 one all metrices on n a Set X Consisting f foint-

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Q: zez Use Cauchy residue theorem to evaluate . (z'el + ) dz, C: 4x? + y2 = 16. %3D

A: We know that  z2e1πz is analytic throughout the complex plane. Hence by Cauchy's theorem ∮z2e1πzdz=0…

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A: Intermediate value theorem and Lagrange mean value theorem

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A: It is given that if f is continuous and ∫09f(x)dx=10 We have to find ∫03xf(x2)dx=?

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A: Given that 2&lt;x&lt;6.......(1) We have to show which of the following statements about x is…

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A: The length 'L' of the curve for y=f(x) between the limit x=a and x=b is given by ,…

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Q: Let f : [a,b] discontinuous at infinitely many points. (Hint: try proving it by contradiction) + [a,…

A: Given that f:a,b→a,b such that f(f(x))=x and f(x)≠x for all x. The objective is to show that f is…

Q: 6. 1 Let W = Span -1 4 and y = 1 What is the distance -4 1 10 from y to W ?

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Q: 12 In the diagram below, GI is parallel to NT. and IN intersects GT at A. A G Prove: AGIA~ ATNA

A: Consider the given triangles, In this diagram GI∥NT and IN intersects GT in A.

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Q: Given: G is midpoint of IE 21 = 2 Prove: ATIG = AREG 2 1. 3,

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Q: Consider AGRT and AARE in the figure to the right. Given: R is the midpoint of AG and ET. Prove:…

A: Given: A figure with two triangles EAR and GTR. As R is the midpoint of AG, therefore, GR=AR As,…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 0 a limit…

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