in the form a + ib where a. b ER. hence or 1+21 -5+10i 20. Express the complex number Z. where Z%= otherwise, find a) The modulus and argument of Z b) The square roots of Z.
in the form a + ib where a. b ER. hence or 1+21 -5+10i 20. Express the complex number Z. where Z%= otherwise, find a) The modulus and argument of Z b) The square roots of Z.
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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Question
Solve all Q20, 21 explaining detailly each step
Transcribed Image Text:-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
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