:If z is a complex number satisfying the equation (-1)" = =", where n e N. Show that the locus of z is a straight line parallel to the imaginery axis.

Q: To convert a complex number from rectangular form, z = a + bi, to polar form, z = r(cos e + i sin…

A: We fill the blank spaces as,

Q: In an Argand diagram a set of points representing complex numbers z is defined by the equation |z-…

A:

Q: Find all the solutions to z6 = 1 − i√3, and plot them on the complex plane.

A: We will find out the required values.

Q: Find the fourth roots of the complex number 3i − 7 in exact principal argument form, and plot them…

A: We have the formula for rational root of complex number

Q: As in the previous question, suppose z is a complex number (a point in the plane) with angle 30…

A:

Q: Plot the following complex numbers z = 6 – 3i and w = 2 + 5i and find the midpoint of the line…

A:

Q: In an Argand diagram, O is the origin and the points P and Q represent the complex numbers z1 and z2…

A:

Q: Find all solutions of z E C of the equation i(z + 2i)4 = -81 and hence show via the sketch all these…

A:

Q: If Zo is any complex number interior to any simple closed contour prove that dz 2πί, η- 1 Jc (z-…

A:

Q: When a complex number z is written in the polar formz = r(cos θ + i sin θ), the nonnegative number r…

A: Given- When a complex number z is written in the polar form z=rcosθ+isinθ, the nonnegative number r…

Q: Sketch the in the complex curve plane given by.- i) |2 -31 12+3i/ %3D

A:

Q: Find the midpoint of the line segment joining the points corresponding to the complex numbers in the…

A:

Q: Find the midpoint of the line segment joining the points corresponding to the complex numbers in the…

A:

Q: To convert a complex number from rectangular form,z = a + bi, to polar form, z = r(cos θ + i sin…

A: The rectangular form of a complex number is z=a+ib the polar form is z=rcosθ+isinθ

Q: Q2- The complex number z satisfies the equation 4Z- 32 %3D 2-1 1-181 Then Z = ......... ..*......

A:

Q: Find all the roots of the following complex numbers and sketch them on the Argand plane (-v3 – 1)/

A: Let  z=(-3-j)3/4 ⇒z=(-3-j)314…

Q: Given two complex numbers z 3-3j, and w =1+7j, %3D which of the following options equals ?

A:

Q: According to the Fundamental Theorem of Algebra, the graph of fx) = -x2 - 2x +1, ha v two complex…

A: According to question We have to calculate the number of complex roots using fundamental theorem of…

Q: The set of all complex numbers is denoted by C={x+iy:x,y€R}

A: A complex number is the combination of real and imaginary number in the form of x+iy , x, y are real…

Q: Use the help of the polar representations of complex numbers to express (V2 + i)? in the form of a +…

A: Given, i=-1 2+i2

Q: To convert a complex number z = x + yi from standard form into trigono metricform, use the…

A: The modulus of a complex number z=x+yi is r=x2+y2 & the amplitude or argument is,…

Q: ii. ?² + ?̅²= 2

A: I am providing the solution of (ii) only.

Q: If 'a'is a complex number such that z = 1. Find the value of a, so that the equation az? +z+1=0 has…

A: Explanation of the answer is as follows

Q: Use the parallelogram law to verify the triangle inequality |u + v| < Ju] + ]v[ for the given…

A:

Q: Find the centre and radius of the circle formed by all the points represented by z = x+ iy…

A:

Q: (a) Find all roots of the quintic equation, in the complex plane and plot them. 25 -1 = 0,

A:

Q: If z is a complex number that satisfies the condition: |z − 2 − 3i| = 2, then show that the…

A:

Q: The complex numbers z1 and z2 are connected by the relation 1 21 = 22 + -. 22 If the point…

A: Let z2 = x+iy Given points z2 satisfy: x2 + y2 = a2 Equation in the given question: Hence point…

Q: Find the roots of the given complex number 1. z^5 = 4i

A:

Q: Let z be a complex number and f(z) be any function. If C is a contour and Sc f(z)dz = 0 then

A: By cauchy theorem in complex integration:-

Q: Find the centre and radius of the circle formed by all the points represented by z = x+ iy…

A:

Q: and of b is Writing the complex number z=2 e6 in the form a+ ib, the value of a is

A:

Q: Given that z is a complex number, sketch the common region shared by the following locus: |z – 2| 6

A: let z=x+iy

Q: Show that root of following quadratic equation are complex numbers a. 2x2 + x + 3 = 0 b. 7x2 + 5x +…

A: Compare the equation 2x2+x+3=0 with the general form of the quadratic equation ax2+bx+c=0, to find…

Q: From the following rectangular complex number Z-4+3i, Z-2-3i and Z,--5+2i Find the poler form?

A: Given:     z1=4+3i      z2=2-3i      z3=-5+2i For a complex number z, expressed in Cartesian form as…

Q: z = 45+j3

A: By Bartleby policy I have to solve only first one as these are all unrelated problems.These are all…

Q: Determine all the fourth roots of z = -1+ 7j to 4 decimal places and plot them in the complex plane.

A:

Q: The reciprocal of a complex number x+jy is

A:

Q: Z, = 2 cis 30° Z2 = 3elli %3D Zz = In (1 + i)

A: Given: Z1=2cis30° Z2=3eπi Z3=ln1+i Z=Z1Z2+Z3 Z1=2cis30°     =2cos30°+isin30°      =232+i2…

Q: 5. Determine all roots of the equation z4 – 9 = 0, and sketch them on the complex plane.

A: Given equation:                            z4-9=0

Q: Find the 3 cube roots of 1, that is find   11/3 And Plot them in the complex plane

A: The graph can be drawn with real part as x axis  and imaginay part as y axis

Q: Find all values of the following roots and plot them in the complex plane. V32. V-1

A: Given:  -15=-115Formula: -1=cis(2k+1)π=ei (2k+1)π  for k=0,1,...,n-1

Q: Q4\ If z and z are conjugate complex numbers, find two complex numbers, z = z, and z = z,, that…

A: The given equation is: 3zz¯+2z-z¯=39+i12 Let z=a+ib be a complex number satisfying the equation,…

Q: : If z is a complex number satisfying the equation (=-1) = =", where ne N. Show that the locus of z…

A:

Q: How many complex numbers satisfies both equations, z - i = 2 and |z– 4i| ='1?

A: z=x+iyThen,z-i=2x+iy-i=2x+iy-1=2…

Q: Find all complex number solutions to the equation x^y =y^x , where x and y are both nonzero.

A: We need to find all complex number solutions to the equation xy=yx, where x and y are both nonzero.…

Q: It is given that W = 2+ 2i, z + 2i = iz + X, Imw = 8, where z and w are complex numbers, and is a…

A:

Q: How many complex numbers satisfies both equations, z – i| = 2 and |z – 4i| = 1?

A: We know that z=x+iy, then…