Solve all Q29 explaining detailly each step -5+10i20. Express the complex number Z. where Z=in the form a + ib where a. b ER. hence or1+2iotherwise, finda) The modulus and argument of Zb) The square roots of Z.21. i) Given that Z, = i(5+4i), express Z, in the form a + ib and hence find /Z¡/ii) Find the locus of points Z such that arg(Z- 2+ 3i)IT4111) Verify that 2+ 3i is a root of the equation Z – 5z + 17z - 13 = 0. Find the other roots ofthis equationTπ、22. Given that Z = 2 - 3i, express Z/Z* and Z(cos--isin ) in the form a + bi, where a and b are44real constants and Z* is the complex conjugate of Z.23. i) Given the complex numbers Z = 2 + i and Z,= -1+ 2i. Evaluate,|Z,+Z2!a)Z1-Z2b) arg(-)tto 1 decimal place, where Z* is the conjugate of Z.Z2Z-1ii) Given that =ri, where A is a reai parameters, show that the locus of the point P whichIZ+11represents Z on the complex plane, is a circle, stating the coordinates of the centre and theradius.1V3.24.1) Given that Z+V3i and Z = -- Find z'. Hence deduce the value of Z,512 22 2ii) Find the square roots of zi giving the modulus and arguments of each of them.iii) Prove that for two complex numbers Z, and Z, argZ 1-Z22.25.1) Find the square root of the complex number z=5+ 12iii) Find the modulus and argument of the complex number z(1+i)?(-1+i)*iii) Given that z = 1+iv3 represent the complex numbers zz* and -as vectors on anArgand diagram where z* is the complex conjugate of z.26. (i) Find the two complex numbers z,and z2 which simultaneously satisfy the equationsZ+Z2 = --i and z - Z2 = -2+ 5iZ+1(ii) Given that:i find z in the form a + ib, where a and b are realZ--1(iii) Find the complex number z such that: /z! +z = 1+ 2i27. (i) Express in the form a + ib, the complex number z, where a and b are real constants given4-3ithat ( z - (1+ 3i) = 1- 2i .2-i(ii) Verify that the complex numbers Z, = 1 - iV 3 and Z, = 1 + iv3 are roots of p(z) = 0.where p(z) = z* – 3z' + 8z - 24. Hence, find the other roots of p(z) = 0%3D%3D2(1+i)express Z, and Z,Z2 in the form a28. (i) Given the complex numbers Z; = 10 + 5i and Z,3-i+ bi, where a, b EO and Z is the complex conjugate of Z,.(ii) Find the locus of points represented by complex numbers, z such that2/z-3/ /z- Gi/.29. D Find in the form: a+ bi, a, b E R, the complex number z such that:78 (2 – i)z(3 – 4i) = 01+ 2iii) Given that z = x = iy, x,y€ R, find the locus of the points z, on an Argand diagram, forwhich the imaginary part of z +- is zero.(4+3i)(3+4i)30. i) The complex number z is given by: z =(3+i)Express z in the form: a + bi,where a, b E R.1+iii) Another complex number z1, is such that: Z1Find |z,| and arg (z)(.V3+i1 + 2i)( 2 + i)31. (i) Given the complex number z such that z =express z in the form a + bi.1 + 3iwhere a, b e R.Hence find(a) z² – (1 + ±i)(b) arg [z² - (1 +i)]-.22(ii) Find the locus of w, where w = [x + (y – 6)i][(x + 8) – yi], if(a) w is purely real,(b) w is purely imaginary32. (i) Given that z = e, show that z" + z- =Use this resuit to express cos 0 in terms of cosines of multiples of 0.(ii) Given that z, = 1+ iv3 andz, = -1 + i, evaluate(a) \z, z2 ²(b) arg(z,)^("n = 2cos n.2

Answered: Image /qna-images/answer/dcf7d671-f688-4f6d-84a8-840217bfeed9.jpg

Answered: (2 -i)z (3 - 4i) = 0 1+ 2i ii) Given…

(2 -i)z (3 - 4i) = 0 1+ 2i ii) Given that z =x = iy, x,ye R,find the locus of the points z, on an Argand diagram, for which the imaginary part of z +- is zero.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

Solve all Q29 explaining detailly each step

-5+10i
20. Express the complex number Z. where Z=
in the form a + ib where a. b ER. hence or
1+2i
otherwise, find
a) The modulus and argument of Z
b) The square roots of Z.
21. i) Given that Z, = i(5+4i), express Z, in the form a + ib and hence find /Z¡/
ii) Find the locus of points Z such that arg(Z- 2+ 3i)
IT
4
111) Verify that 2+ 3i is a root of the equation Z – 5z + 17z - 13 = 0. Find the other roots of
this equation
Tπ、
22. Given that Z = 2 - 3i, express Z/Z* and Z(cos--isin ) in the form a + bi, where a and b are
4
4
real constants and Z* is the complex conjugate of Z.
23. i) Given the complex numbers Z = 2 + i and Z,= -1+ 2i. Evaluate,
|Z,+Z2!
a)
Z1-Z2
b) arg(-)t
to 1 decimal place, where Z* is the conjugate of Z.
Z2
Z-1
ii) Given that =ri, where A is a reai parameters, show that the locus of the point P which
IZ+11
represents Z on the complex plane, is a circle, stating the coordinates of the centre and the
radius.
1
V3.
24.1) Given that Z
+V3i and Z = -- Find z'. Hence deduce the value of Z,
51
2 2
2 2
ii) Find the square roots of zi giving the modulus and arguments of each of them.
iii) Prove that for two complex numbers Z, and Z, arg
Z 1-Z2
2.
25.1) Find the square root of the complex number z=
5+ 12i
ii) Find the modulus and argument of the complex number z
(1+i)?
(-1+i)*
iii) Given that z = 1+iv3 represent the complex numbers zz* and -as vectors on an
Argand diagram where z* is the complex conjugate of z.
26. (i) Find the two complex numbers z,and z2 which simultaneously satisfy the equations
Z+Z2 = --i and z - Z2 = -2+ 5i
Z+1
(ii) Given that:
i find z in the form a + ib, where a and b are real
Z--1
(iii) Find the complex number z such that: /z! +z = 1+ 2i
27. (i) Express in the form a + ib, the complex number z, where a and b are real constants given
4-3i
that ( z - (1+ 3i) = 1- 2i .
2-i
(ii) Verify that the complex numbers Z, = 1 - iV 3 and Z, = 1 + iv3 are roots of p(z) = 0.
where p(z) = z* – 3z' + 8z - 24. Hence, find the other roots of p(z) = 0
%3D
%3D
2(1+i)
express Z, and Z,Z2 in the form a
28. (i) Given the complex numbers Z; = 10 + 5i and Z,
3-i
+ bi, where a, b EO and Z is the complex conjugate of Z,.
(ii) Find the locus of points represented by complex numbers, z such that
2/z-3/ /z- Gi/.
29. D Find in the form: a+ bi, a, b E R, the complex number z such that:
78

(2 – i)z
(3 – 4i) = 0
1+ 2i
ii) Given that z = x = iy, x,y€ R, find the locus of the points z, on an Argand diagram, for
which the imaginary part of z +- is zero.
(4+3i)(3+4i)
30. i) The complex number z is given by: z =
(3+i)
Express z in the form: a + bi,where a, b E R.
1+i
ii) Another complex number z1, is such that: Z1
Find |z,| and arg (z)(.
V3+i
1 + 2i)( 2 + i)
31. (i) Given the complex number z such that z =
express z in the form a + bi.
1 + 3i
where a, b e R.
Hence find
(a) z² – (1 + ±i)
(b) arg [z² - (1 +i)]
-
.2
2
(ii) Find the locus of w, where w = [x + (y – 6)i][(x + 8) – yi], if
(a) w is purely real,
(b) w is purely imaginary
32. (i) Given that z = e, show that z" + z- =
Use this resuit to express cos 0 in terms of cosines of multiples of 0.
(ii) Given that z, = 1+ iv3 andz, = -1 + i, evaluate
(a) \z, z2 ²
(b) arg(z,)^("
n = 2cos n.
2

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