Let z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2) be complex numbers, then the productz1z2 = ________ and the quotientz1z2= ________ (z1 ≠ 0).

Q: Find z, z, and for the pair of complex numbers using trigonometric form. Z2 Zq = - 20 + 20 i, z2 = -…

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Q: (a) For complex numbers z₁ = 6 + 6j and z₂ = 4 − 4j, π (i) find z = 21. Present your result in polar…

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Q: Find the square roots of the complex number z= 16 (cos 60° + i sin 60°). There may be more than one…

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Q: 4V6(cos- 11 m +i sin 12 Võ(cos +i sin ) 11 m O Option 2 O Option 4 10 m 2V3(cos+ i sin ) V6(cos+i…

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Q: Let z2 = V3 – i, w = -2 + i and p = 5+5i. 1. Find all the complex numbers given by ln z2. 2. Express…

A: Standard formulae for logarithm of complex number is given by: ln(z)=ln|z|+i argz 1)…

Q: b) Find all complex numbers z such that z= V-7+ 24i

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Q: Let z be a complex number. Then the solution(s) of sinhz+coshz=D1/e is (are) * O z=2nni-1 where n is…

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Q: Plot the complex number. 3 + V3i Imaginary Imaginary axis axis 4| 4| 2 2 Real Real -4 -2 4 axis -4…

A: To plot the given complex number

Q: 3 lim Z→2 (Z-2)4 ∞

A: We have to find the limit.

Q: 1. Consider the nonzero complex numbers z1 = 3i, z2 = 6i and z3 = 4i. (a) Evaluate (z,z,z3) (1 + z2…

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Q: (a) Use the formula zk = Vr(co e+ 27tk cos +i sin 0+277K to find the indicated roots of the complex…

A: Given Z=-322(-1+i) Zk=rncosθ+2πkn+sinθ+2πkn, 0≤θ<2π The trigonometric form of the complex number…

Q: Given the two complex numbers: z1 = 6(cos π/2 + i sin π/2) and z2= 3/2(cos π/4 + i sin π/4) Find the…

A: we have to find

Q: ri(cos 01 + i sin 01)· r2(cos 02 + i sin 02) = [cos( ) + i sin ( )] The product of two complex…

A: We know that, the product of two complex number in the polar form is found by multiplying their…

Q: Two complex numbers z1 and z2 are given by z1 = 3 + 2i and z2 = − 1 − 4i. Evaluate, (a) |z1z2| (b)…

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Q: For a positive integer n, the complex number z = r(cos 0 + i sin 0) has exactly n distinct nth roots…

A: for the complex number,  the nth root is given by

Q: Find in the form of red, the six complex numbers which satisfy the equation 2° +32+32( V3 )i = Hence…

A: Given:                       z6+32+323i=0 ....... (*) To find: Solution of (*) of the form reiα.…

Q: 3. How may the following complex numbers be obtained from z = x + iy geometrically? Sketch. а. 2 b.…

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Q: 1. Consider the nonzero complex numbers z1 = 3i, zz = 6i and z3 = 4i. (ə) Evaluate (2,2,z,) Ei++*)…

A: given Data: z1=3i, z2=7i, z3=7i

Q: 1. (a) For z #1, verify the identity 1+2+2 + .. + 1-2+1 1-2 (b) Use part (a) and appropriate…

A: (a) Consider the given identity as 1+z+z2+…+zn=1-zn+11-z for z≠1. We need to prove the above…

Q: The complex number Z1and Zzare given by, Z1 = 3 – 5i and Z, = -1+4i Determine: 221 + 5Z2 O1+ 10i 01-…

A: Given: Z1=3-5i & Z2=-1+4i We know that; a+bi±c+di=a±c+b±di And ca+bi=ca+cbi, where c is scalar.

Q: Given z1 = 2 – 2i, z2 = -V3 + i a. Find the modulus and principal argument of the complex numbers…

A: a) The modulus of z=x+yi is given by r=x2+y2. And the principal argument is given by θ=arctanyx;…

Q: . Two complex numbers a + ib and e + id are equal if and only if a - c and b - d. Use this fact to…

A: Given :             ∫ ea+ibx dx = a-iba2+b2 e(a+ib)x + Cwhere C = C1 + C2  is a complex constant of…

Q: 6. Find the following complex numbers in the form r+ iy, whe and y are real. (i) (3+7i) + (6 – 10i)…

A: we have to find complex numbers in the form of x + i y  Adding real part with real part and…

Q: IV-1 Find the roots of the following complex numbers in polar form, correct to 3 decimal places. [4…

A: Thanks for the question :)And your upvote will be really appreciable ;) Correct Option :- C

Q: Two complex numbers a + ib and c + id are equal if and only if a = c and b = d. Use this fact to…

A: Given: ∫e(a+ib)dx=a-iba2+b2e(a+ib)x+C

Q: 8e2i 5e-0.7i Express the complex number p= to a rectangular form. Determine the quotient of in…

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Q: snip

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Q: e the complex number 8e" in a + bi form X error - invalid form

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Q: 1. Consider the complex number z = -2(1 + i). (a) Compute z' by multiplying z by itself three times.…

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Q: Find all complex numbers z so that z6 = 218 Express your solutions in the form a + bi, evaluating…

A: Given: z6=218 We have z6=236z=23 Let z=rcosθ+isinθ. Then, z to the power 6 gives…

Q: 1. For the complex numbers: 21 = v2 – i, z2 = 1 – iv2. Find the polar and exponential form of the…

A: given: z1=2-i z2=1-i2 we have to find the polar form and exponential form of the complex number…

Q: Find the quotient z1/z2 of the complex numbers.Leave answers in polar form. z1 = 10(cos 10° + i sin…

A: We have given complex numbers z1 = 10(cos 10∘ + i sin 10∘)z2= 5(cos 5∘ + i sin 5∘) We have to find…

Q: Let z2 = V3 – iv2, w = 1+2i and p = 1+ i. 1. Find all the complex numbers given by ln z2.

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Q: find all complex values z satisfying the given equation cosh z = 2+3i

A: We know that - cosh x= (ex-e-x)/2 using it, the explanation is given below-

Q: 3n i Let z = V2 e 4. Write the complex number 1/z + 2° in normal form (in the form a + ib).

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Q: Express the complex number z = - 1 - (square root 3)i in polar form: Z = r ( cos ( θ ) + i sin ( θ )…

A: Explained below

Q: Let the left complex number be z1 and the right complex number be z2. +i sin- 43 - 4i 8 cos a) Write…

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Q: Let z be a complex number. Then the solution(s) of sinhz+coshz=1/e is (are) * O z=2nti-1 where n is…

A: Use formula of hyperbolic function.

Q: Consider the following complex number. Z = 4(cos (280°) + i sin (280°)) (a) Find two square roots…

A: Root of complex number

Q: and z2 = 1– 2i. Write z122 and ž1/22 in the form a al numbers. and w are complex numbers with |2| =…

A: We have to find the following conditions according to the given information

Q: Find the distance between the pair of complex numbers. (7 - 2i) and (5 + 8i) O4/26 V194 O2/2 2/26

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Q: 3. Let z be a complex number and f(z) = sinh( .Then 27 z=0 is an isolated singular point of f z=0 is…

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Q: 2n Express the complex number z = 8 cis 4 in 3 rectangular form a + bi. Select the correct answer…

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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: =(-1= v3i)³

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Q: 74. Two complex numbers a + ib and c + id are equal if and only if a = c and b = d. Use this fact to…

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Q: 3ri Let z V2e. Express the complex number 1/z + 2° in the form a + ib.

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Q: 4. a) Given that z = 2 – 3i, express i'z, and Z/Z* and Z (cos - isin -) in the form a + bi, where a…

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Q: Exer. 21–46: Express the complex number in trigonometric form with 0 so<27. 25 2V3 + 2i 4 cis 37 29…

A: Given: The complex number 25) 23+2i. 29) -20i 34) -5.