Higher Engineering Mathematics

260 COMPLEX NUMBERS

135 ◦to forceA, ForceC, 12 N acting at an

angle of 240◦to forceA.

[8.394 N, 208◦ 40 ′from forceA]

XC = 20 Ω R 1 = 30 Ω

R 2 = 40 Ω XL = 50 Ω

R 3 = 25 Ω

V = 200 V

l

Figure 23.11

7. A delta-connected impedance ZA is
   given by:

ZA=

Z 1 Z 2 +Z 2 Z 3 +Z 3 Z 1

Z 2

DetermineZAin both Cartesian and polar

form givenZ 1 =(10+j0),

Z 2 =(0−j10)andZ 3 =(10+j10).

[(10+j20),22. 36 ∠ 63. 43 ◦]

8. In the hydrogen atom, the angular momen-
   tum,p, of the de Broglie wave is given

by:pψ=−

(

jh

2 π

)

(±jmψ). Determine an

expression forp.

[

±

mh

2 π

]

9. An aircraftPflying at a constant height has
   a velocity of (400+j300) km/h. Another
   aircraftQat the same height has a veloc-
   ity of (200−j600) km/h. Determine (a) the
   velocity ofPrelative toQ, and (b) the veloc-
   ity ofQrelative toP. Express the answers
   in polar form, correct to the nearest km/h.[
   (a) 922 km/h at 77. 47 ◦

(b) 922 km/h at− 102. 53 ◦

]

10. Three vectors are represented byP,2∠ 30 ◦,
    Q,3∠ 90 ◦andR,4∠− 60 ◦. Determine in
    polar form the vectors represented by (a)
    P+Q+R, (b)P−Q−R.
    [
    (a) 3. 770 ∠ 8. 17 ◦

(b) 1. 488 ∠ 100. 37 ◦

]

11. In a Schering bridge circuit,
    ZX=(RX−jXCX),Z 2 =−jXC 2 ,

Z 3 =

(R 3 )(−jXC 3 )

(R 3 −jXC 3 )

andZ 4 =R 4

whereXC=

1

2 πfC

At balance: (ZX)(Z 3 )=(Z 2 )(Z 4 ).

Show that at balance RX=

C 3 R 4

C 2

and

CX=

C 2 R 3

R 4