b) By using De Moivre theorem, calculate in the form of a + bi: 4. i) (VZ – iv2)* | ii) (1– i)6

Q: Use De Moivre's theorem to find the following: 4 i. [1+ v3i]" -3 ii. [-1+ v3 i]

A: Given that: i. [1+3i]4 ii. [-1+3i]-3

Q: use the inter medicte Value theorem to prove that 2 the equetion V2x+s -4 + x´= 0 has a root in…

A: intermediate value theorem: if the function  is continuous on and if  and  then the function has a…

Q: Given: WZ 1 bis. XY Prove: AWXY is isosceles. (Hint: This proof can be written in three steps by…

A: 2. given  WZ↔ ⊥bis XY¯ as WZ is bisector of XY XZ=ZY......(1)

Q: Evaluate the integral O 03/20 Ⓒ100 C 87² dx using the two-point Gaussian quadrature rule. 1+16z4

A:

Q: 6. Evaluate (7xi - zk) · ndA over the sphere S : x² + y² + z² = 4 (u) by the Divergence Theorem, (b)…

A: The divergence theorem states that  ∬SF·ndA=∭V∇·FdV Where, F is the vector field, V is the given…

Q: Prove that cosh-z = In (z + vz? - 1) %3D

A:

Q: use the definition prove that s d sinZ %3D Tal

A:

Q: By using the residue theorem, prove that 1 dx 4a3 where x E R, and a >0 is a parameter.

A: Given

Q: Q: Show that the functi on f:R →R given by f (x) = ax +b, where ab e R, a #0 is a bijection.

A:

Q: 2.3 Use the Cauchy integral formulae to evaluate 22 – 1 (9 – 22)(z + 2i))3 where C is a circle |z +…

A:

Q: Let C:2-1 in the counterclockwise direction and I= 4. Lnzdz. J= 4. ždz, and K %3D $ tanzdz. For…

A:

Q: 8 Use the Cauchy integral formulae to evaluate 22 – 1 - (9 – 22)(z + 2i))³ | where C is a circle |z…

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…

Q: Use the definition. that sinz •prove Z = UJZ %3D

A: Using definition of derivatives.

Q: Use the reflaction principle lio find the mumber of 2. paths a So =2 for Bimple randon malk from to…

A:

Q: of the suface the in tasechon of what is the fomed by y?+ z?=l and x?+z?=\

A:

Q: Can there exist a AROS in which the trisectors of 20 intersect RS at D and E, with RD = 1, DE = 2.…

A: Given: ∠O intersect RS  at D and E in a △ROSwith the data  RD=1;   DE=2;…

Q: By using the Residue theorem, compute the integral eiz -dz, where I is the circle |2| = 3 traversed…

A:

Q: 3. If C is the unit circle in the complex plane C, and f(2)= =², show that f(z) dz = 0 using two…

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Q: 4. Determine the integral of f(z) = 1/[(z – a)(z – b)], where a and b are in general complex…

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Q: Show that the finite difference formula 1 Dzu(î) = [Зи (т) — 4u(г — h) + u(т — 2h)] 2h is a…

A:

Q: Discuss the cosets of { (x, y) | x, y E R, x = y } in (R × R, +).

A: {(x,y) / x,y∈ℝ , x=y} in (ℝ×ℝ,+)

Q: .Use the De Moivre's Formula to show that if z = (cose + isine), then z" + ; ÷= 2cos (n8) and (금) 3…

A:

Q: 3. Consider the ra greater across z =1-r² - y² plane and the s and a paramet

A:

Q: 2.3 Use the Cauchy integral formulae to evaluate 22 -1 J (9 – 2ª)(2 + 2i))* where C is a circle |2 +…

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…

Q: Complete a two-column proof. Given: ZRSU VST Prove: ZRSV LUST

A:

Q: 1. Find the area of the part of the planer+ 2u + 3z = 1 that lies inside the cylinder x2 + y? = 3.

A:

Q: 2. Prove this using indirect proof in two-column form. Given: MP 1 NO M Prove: ZNPM is not a…

A:

Q: Find the residues of each of the fellowing functièns at its. poles. e •f(z)- 22+3 4. ans : ri=7/4 ?…

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Q: Evaluate the integral S 3012 2/3 de using the two-point Gaussian quadrature rule. 1+16z4

A: Given integral is ∫−121221+16x4dx. We have to evaluate the given integral by using the two-point…

Q: 5. Use De Moivre's Theorem to evaluate (1 – V3i)%.

A: Demoivre's Theorem

Q: Write a paragraph proof showing that AJFG is congruent to AJIH. Given: FG || HI and J is the…

A: This is a congruency of triangle problem

Q: Let C:z-1 in the counterclockwise direction and / = . Lnzdz, J = , zdz, and K = $. cotzdz. For which…

A:

Q: Show that for each 37ER?, ū=, J =, the bilirear form Lūl>= 2e,Vzt U, Vz2t Uz V,t Qus Vz defines an…

A:

Q: Complete the two-column proof. Given: ZB = ZR, ZBPC = ZRCP Prove: ABPC =ARCP R Statements Reasons I

A: Answer and explanation is given below...

Q: Evaluate the integral 108 - 2y/E+ 4ydzdy, 4 where A is the parallelogram with vertices (0,0), (),…

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Q: 3. 2. If A is 2x2 matri x aud Á = A, but %3D A+ A. show that A= 0

A: The solution is :

Q: 4-j2 z = tanh¯(4e-j3)*

A:

Q: 3] By using the properties of dot and cross product, prove the following: a) (w x ū). ū = 0.

A: To prove- w→×v→.v→=0 The cross product of two vectors…

Q: prove that tr(aA+bB)=atr(A)+btr(B)for any A,B belongs to Mmxn(F)

A: Let A=aij and traceA=trA=∑i=1maii.. Note that, aA+bB=aaij+bbij. Then,…

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…

Q: Find the angle between x and x4 in C[0,1] with its standard inner product.

A:

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: Let 1 an(r) : u(reie)e ino de %3D Prove that:

A: .

Q: (c) Evaluate d Gosh: where C: |z| = 2 by using Cauchy's residue theorem.

A: Given integral is ∮C dzcoshz;   C : z=2 As, we know  If z = x+ iy then z =x2+y2 As given that  z=2…

Q: of the following conditions imply that B = se, either prove or give a counterexample. B] AUB = AUC…

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Q: (§5.4, Q12) Evaluate the integral using FTC I: (5u* + u? – u) du.

A:

Q: Prove,J w' Sin wx du %3D COSZ 2

A:

Q: Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises 1-24. 2. а. rV x3…

A: