Solve all Q8 explaining detailly each step otherwise, find the roots of the equatiön3. i) Given that the complex numbers z,and z, are the roots of the equation: x² - 4i – 3 = 0.Express z,and z, in the form a + bi where a and b are real.ii) Indicate by shaded areas on separate Argand diagrams the regions defined by:a) /Z + 3/<1_b) /Z+4/>/Z+1/ c) {"< ArgZ < "}n {Im Z < 4}%3D32T- isin -)in the form a + bi, where .:34. a) Given that z = 2- 3i, express i'z, and Z/Z* and Z (cos-44a and b are real.Z+1b) Given thatFi, find z in the forms a + bi, where a and b are real.Z-1c) Given that the complex number 2i represents the point A on the Argand diagram and thatthe point B represents the image of A when reflected in the line y = 2x, find the complexnumber which represents the point B.5. i) Find two complex numbers Z, and Z, which satisfy the simultaneous equationsZ, + Z2 = -i, Z, – iZ2= -4 +iii) Given that Z= 1 + iv3 express Z* in the form|a) a+ bi where a and b are real.b) r(cosq + isinq), where r> 0 and - 1

Answered: 8. a) Find in the form a + bi, where a…

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8. a) Find in the form a + bi, where a and b are real, the values of Z for which 3 ii) Z 1 i) /zl=3 and arg = 1. 1-2i

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

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Question

Solve all Q8 explaining detailly each step

otherwise, find the roots of the equatiön
3. i) Given that the complex numbers z,and z, are the roots of the equation: x² - 4i – 3 = 0.
Express z,and z, in the form a + bi where a and b are real.
ii) Indicate by shaded areas on separate Argand diagrams the regions defined by:
a) /Z + 3/<1_b) /Z+4/>/Z+1/ c) {"< ArgZ < "}n {Im Z < 4}
%3D
3
2
T
- isin -)
in the form a + bi, where .
:3
4. a) Given that z = 2- 3i, express i'z, and Z/Z* and Z (cos-
4
4
a and b are real.
Z+1
b) Given that
Fi, find z in the forms a + bi, where a and b are real.
Z-1
c) Given that the complex number 2i represents the point A on the Argand diagram and that
the point B represents the image of A when reflected in the line y = 2x, find the complex
number which represents the point B.
5. i) Find two complex numbers Z, and Z, which satisfy the simultaneous equations
Z, + Z2 = -i, Z, – iZ2= -4 +i
ii) Given that Z= 1 + iv3 express Z* in the form
|
a) a+ bi where a and b are real.
b) r(cosq + isinq), where r> 0 and - 1<q<I
iii) By shading in three separate Argand diagrams show the regions in which the points
representing z can lie when (a) Imz< 2 b) /Z-2i/ < 2.
c) Z--2i/</Z - 2/. Shade in another Argand diagram the region in whichz can lie when
ail the three inequalities apply
6. i) Given thati the complex numbers Z, and Z2 where Z, = (1+i), Z2
1
1
express Z
2+i
3-i
and Za in the form a + ib, where a, b are real.
ii) Solve the simultaneous equations Z; + Z4 = 6, 2Z; - 2iZ4 = 8+ 3i, expressing your
%3D
answer in the form a + ib, where a and b are real numbers.
iii) A regular hexagon ABC DEF is drawn in the Argand diagram so that it centre is at the
origin and the two adjacent vertices A and B are at the points represented by the complex
numbers ZA and ZB = 1 and ZB = ½ + i½v3. Find, in the form a + bi where a and b are real
numbers, the complex numbers which represent the other four vertices.
7. Given that /2Z - t/= /Z-3i/, show that the locus of the complex number Z is a circle, giving
%3D
the radius and the coordinates of the centre.
8. a) Find in the form a + bi, where a and b are real, the values of Z for which
3
i) /z/=3 and arg Z
ii) + =1.
= 1.
1-2i
3
b) Find the roots of the equation: Z* +6Z“ + 25 = 0. Represent these roots as points on the
Arganddiagram.,
9. i) Express each of the complex numbers Z= (2+3i)(1- 2i), Z2 = (3 + 5i)/2 - i in the form a +
%3D
%3D
Z2
ib where a and b are real numbers. Find arg Z2 and arg
gi.ng your answer in degrees
Z1
correct to one decimal place.
76

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