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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
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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Solve all Q20, 21 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
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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