Given that complex u and v,  prove that: ||u| - |v|| ≤ |u - v |

Q: Q: For any two complex numbers z, and zz, prove that z-z|+|z+!z|=z-z-2+z-z+

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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: The segment addition postulate states that ifA, B, and C are collinear with B between A and C, then…

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Q: Show that there exists a complex number z such that z" +cos(| z |):(1+31z*)=0

A: The complete solutions are given below

Q: The complex numbers an, n > 0, are defined by the condition that Σan (z-2i)" = 1+ 2z+3z² + Σ(n+1)z"…

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Q: the basis of P2 cons tive to B.

A: Introduction: For some n, consider V to be a subspace of Rn. A basis for V is a collection of…

Q: Explain why the fundamental theorem of algebra does not apply to f(x) = Vx + 3. That is, no complex…

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Q: 21) Find the area of a triangle fosmed from the twovectors a = L'+2j-k-.

A: Given, a ⇀=i+2j-kb⇀ =2i-j+4k

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.

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Q: Say that E = Ãek™, where the components of A and k are real and constant, i.e. they aren't imaginary…

A: A vector is a quantity that has both magnitude and direction. In general, vectors are represented as…

Q: Prove that there is no complex number z such that 1 |z|«anda,z' = 1, where|a,| < 2. 3 r=1

A: In this question we have to prove that there is no complex number z such that z&lt;13 and ∑r=1n…

Q: AABC has inradius r, altitudes of length ha, hb, hc, and exradii ra, b, c. 1 1 1 + r (a) Prove that…

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Q: Let z be a complex number such that |z|=1, that is z∙z ̅=1. Compute 〖|1+z|〗^2+〖|1-z|〗^2.

A: Given that -  |z| = 1 we have to compute the following -  1+z2 + 1-z2

Q: Find the largest subset of the complex plane on which f(z) = cos(z)e-² is holomorphic, and the…

A: Given :  f(z) = cos(z¯) e-z¯ To find : We have to find the largest subset of complex plane on which…

Q: 1-Show that У +їх ave X + iy =/=1 where Complex numbers ? X+iy and

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Q: Prove that R is symmetric

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

A: Given (2-i2)4 &amp;   (1-i)6

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: For complex number z and w, prove that zw-wz=z - w if and only if z =w or, zw = 1.

A: We know that , for a complex number|z|2=zz¯and when z is a real numberz=z¯

Q: Suppose AE R, v#0, and (A- A)v#0. Use the fact that A is symmetric show that it is impossible to…

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Q: graup Consider a dibedral 2. Du= carb: d=b-(ab)"%=D1> Discuss all passible Commulators 'and verify…

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Q: Theorem 11. Let A E Mn be positive semidefinite . Then A has a real root. Moreover ,the square root…

A: Given: To explain the given statement as follows,

Q: 13. Prove that the three points (-3, 2, 4), (6, 1, 2), and (-12, 3, 6) are collinear by using the…

A: To solve the following problem

Q: Hi, there are two questions a) and b) attached  Thanks so Much!!!!

A: Given that z is a complex number satisfying |2z – 1| = |z – 2|. We need to prove that |z| = 1 by two…

Q: 1. If z = x + iy is a complex number. (a) Verify that v2|z| > |Re(z)| + |Im(2)| [Hint: Reduce this…

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Q: Prove that eie ei(0+2K) ,where k = 0,±1, +2,...., = D

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Q: For all complex numbers Zq,Z2 prove that |z, +z2| <]z|+]z2l

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Q: 4) Let z = x+ iy and w = u+iv. Prove the following properties of complex numbers. a) z+w =Z+ w b) zw…

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Q: Let 3 be a complex number algebraic Q. Prove that 3i is algebraic Q. over over

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Q: Let x and y be positive real numbers. Prove that a + y > 2/xy.

A: See solution below

Q: Find the indicated power using De Moivre's Theorem. (Express your fully simplified answer in the…

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Q: What theorem do I use to prove that m is parallel to n?

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Q: Let z and w be complex numbers such that |z|= 1 and |w| # 1. Prove that = 1. - zw

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Q: Let z be a complex mumber and fe be any fusction. If C isa closed contour and 4 r)dz =0 dhen %3D

A: By Cauchy integral formula, let f(z) be any function and C be a closed contour satisfying the given…

Q: T is self - adjoint then (Tx, y) is complex O normal O real O prime o

A: T is self adjoint

Q: Find the source Q cx) Such that the So lutiim Uckit) of + Q(x) 2 ə x² 0 < x <∞ | U(x,o) = 0

A: given equation ∂u∂t=k∂2u∂x2+Qx ,       -∞&lt;x&lt;∞            ...iux,0=0 claim- find the source Qx…

Q: In ABCD, AB = BC = CD = DA. Prove that AC and DB bisect each other at right angles. Use two-column…

A: We have to prove that AC and BD bisect each other at 90°

Q: The Canonical form of a hyperbolic PDE is F (ξ, η , u, ug , un ) %3D False True

A: To check the given statement is true or false. Given that the canonical form of a hyperbolic PDE is…

Q: A (real or complex) scalar u is said to be an nth rootof unity if un = 1. Let ωn = e 2πi n . Use…

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Q: For all complex numbers Zq,Z2 prove that |Z,+z2| <]z¡\+]z2l

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Q: The points A, B, C and O represent the numbers z, =, 1 and 0 respectively in the complex plane.…

A: We will treat A,B,C as position vectors to do the proof

Q: Explain the relationship between ideal and subring

A: A subring S of a ring R is a subset of R which is a ring under the same operations as R.   An ideal…

Q: What are the perpendicular generators on (P) to 4 %3D of the surface

A: Given equation s are, x216-y24=z....1x3=y2=z-4...2 Equation of generators of the hyperboloid (1) are…

Q: Show that, for any complex number z, zz = |2|², z+z = 2Re(z) and Re(z) < |z|. Henc show that i. [21…

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Q: For all complex numbers Z1,Z, prove that |2z|+|*z|s]°z+z[

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Q: The complex numbers an, n > 0, are defined by the condition that Σan (z − 2i)” = 1+2z + 3z² + · · ·…

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Q: Z Is any complex number interior to any simple closed contour dz 2 Ti, n= 1 prove that for all…

A: It can be solved using cauchy integral formula

Q: a) Show that u(x,y) = x3 – 3xy2 + 3x2 – 3y2 is harmonic and find its harmonic conjugate

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