082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

328 Chapter 9 Complex numbers
EXERCISES9.2
1 Solve the following equations:
(e) cos ωt +jsin ωt (f) cos ωt −jsin ωt
complex numbers.
1
(c)z=0 (d)z= 1 +j
(a) −11 −8j (b) 5 +3j (c)
2 j (d) −17 2
(f)z=j (e)z=j 2
(a)x 2 +1=0 (b)x 2 +4=0
(g) −0.333j +1
(c)3x 2 +7=0 (d)x 2 +x+1=0 4 Recallfrom Chapter2thatthe polesofarational
(e) x2
2 −x+2=0
functionR(x) =P(x)/Q(x)are those values ofxfor
whichQ(x) = 0.Findany poles of
(f)−x 2 −3x−4=0
x 3x 3
(g)2x 2 (a) (b)
+3x+3=0
x −3 x 2 (c)
+1 x 2 +x+1
(h)x 2 +3x+4=0
2 Solve the cubicequation
5 Solvethe equations 2 +2s +5 = 0.
6 Expressasacomplex number
3x 3 −11x 2 +16x−12=0
(a) j 4 (b) j 5 (c) j 6
given thatoneofthe rootsisx = 2. 7 StateRe(z)andIm(z)where
3 Writedown the complex conjugates ofthe following (a)z=7+11j (b)z=−6+j
Solutions
1 (a) ±j (b) ±2j
(c) ± √ 7/3j (d) −1/2 ± ( √ 3/2)j
(e) 1± √ 3j (f) −3/2 ± ( √ 7/2)j
(g) −3/4 ± ( √ 15/4)j (h) −3/2 ± ( √ 7/2)j
2 2,5/6±( √ 47/6)j
3 (a) −11 +8j (b) 5 −3j (c) − 1 2 j
(d) −17 (e) cos ωt −jsin ωt
(f) cos ωt +jsin ωt (g) 0.333j +1
4 (a)x=3 (b)x=±j
(c)x = −1/2 ± ( √ 3/2)j
5 s=−1±2j
6 (a) 1 (b) j (c) −1
7 (a) 7,11 (b) −6,1 (c) 0, 0
(d)
1
2 , 1 2
(e) 0, 1 (f) −1,0
9.3 OPERATIONSWITHCOMPLEXNUMBERS
Twocomplexnumbersareequalifandonlyiftheirrealpartsareequalandtheirimaginaryparts
areequal.
Example9.5 Findxandyso thatx+6jand 3 −yjrepresentthe samecomplex number.
Solution If both quantitiesrepresentthe same complex number wehave
x+6j=3−yj
Sincethe real parts mustbeequal wecan equatethem,thatis
x = 3