Assign-07-H8152.tex 17/7/2006 16: 9 Page 286Complex numbers and Matrices and Determinants
Assignment 7
This assignment covers the material contained
in Chapters 23 to 26.
The marks for each question are shown in
brackets at the end of each question.- Solve the quadratic equationx^2 − 2 x+ 5 =0 and
show the roots on an Argand diagram. (9) - IfZ 1 = 2 +j5,Z 2 = 1 −j3 andZ 3 = 4 −jdeter-
mine, in both Cartesian and polar forms, the value
ofZ 1 Z 2
Z 1 +Z 2+Z 3 , correct to 2 decimal places.(9)- Three vectors are represented byA,4. 2 ∠ 45 ◦,B,
5. 5 ∠− 32 ◦andC,2. 8 ∠ 75 ◦. Determine in polar
form the resultantD, whereD=B+C−A.
(8) - Two impedances, Z 1 =(2+j7) ohms and
Z 2 =(3−j4) ohms, are connected in series to
a supply voltage V of 150∠ 0 ◦V. Determine the
magnitude of the currentI and its phase angle
relative to the voltage. (6) - Determine in both polar and rectangular
forms:
(a) [2. 37 ∠ 35 ◦]^4 (b) [3. 2 −j 4 .8]^5(c)√
[− 1 −j3] (15)In questions 6 to 10, the matrices stated are:A=(
− 52
7 − 8)
B=(
16
− 3 − 4)C=(
j3(1+j2)
(− 1 −j4) −j 2)D=⎛⎝2 − 13
− 510
4 − 62⎞⎠ E=⎛⎝− 130
4 − 92
− 571⎞⎠- DetermineA×B (4)
- Calculate the determinant of matrixC (4)
- Determine the inverse of matrixA (4)
- DetermineE×D (9)
- Calculate the determinant of matrixD (6)
- Solve the following simultaneous equations:
4 x− 3 y= 17
x+y+ 1 = 0
using matrices. (6) - Use determinants to solve the following simul-
taneous equations:
4 x+ 9 y+ 2 z= 21
− 8 x+ 6 y− 3 z= 41
3 x+y− 5 z=− 73 (10) - The simultaneous equations representing the cur-
rents flowing in an unbalanced, three-phase,
star-connected, electrical network are as follows: - 4 I 1 + 3. 6 I 2 + 4. 8 I 3 = 1. 2
− 3. 9 I 1 + 1. 3 I 2 − 6. 5 I 3 = 2. 6 - 7 I 1 + 11. 9 I 2 + 8. 5 I 3 = 0
Using matrices, solve the equations forI 1 ,I 2
andI 3 (10)