The Chemistry Maths Book, Second Edition

(Grace) #1

Solutions to exercises 639


Section 7.8














Chapter 8


Section 8.2



  1. 61 − 12 i 2. 4 3. 6 i 4. 181 + 14 i


5.− 81 − 16 i 6. 411 − 138 i 7. 10



  1. (i) 31 + 12 i (ii) 13


(iii)


9.z 1 = 1 ± 11 + 12 i 10.




  1. 12.−i












Section 8.3



  1. (i) 2 (ii)π 22


(iii)2(cos 1 π 221 + 1 i 1 sin 1 π 2 2)



  1. (i) 3 (ii)π


(iii)3(cos 1 π 1 + 1 i 1 sin 1 π)



  1. (i) (ii)−π 24


(iii) (cos 1 π 241 − 1 i 1 sin 1 π 2 4)



  1. (i) 2 (ii)π 26


(iii)2(cos 1 π 261 + 1 i 1 sin 1 π 2 6)



  1. (i) (ii) 3 π 24


(iii) (cos 13 π 241 + 1 i 1 sin 13 π 2 4)



  1. (i) 4 (ii) 4 π 23


(iii)4(cos 14 π 231 + 1 i 1 sin 14 π 2 3)



  1. (i) 1 (ii) 3 π 22


(iii)cos 13 π 221 + 1 i 1 sin 13 π 22



  1. (i)


(ii)


(iii)


3


26 6


cos sin


ππ











i


2


36 6


cos sin


ππ














i


6


5


6


5


6


cos sin


ππ














i


62


62


2


2


12


25


24


25



  • i


5


13


12


13



  • i


5


34


3


34


− i


−±21 3,i


13 ±i


3


1


2


2


1


2


=+−= −(), ()zz


i


zz


A


xxx


+−+−


246

246!!!






1


246


246

−+−+


xxx


!!!






x


xxx


−+ −


4


3


16


5


64


7


357

!!!



  1. (i)


(ii)


(iii)



  1. (i) 81 i (ii)−i 281

  2. 1281 cos


8

1 x 1 − 12561 cos


6

1 x 1 + 11601 cos


4

1 x 1


− 1321 cos


2

1 x 1 + 11


Section 8.4



  1. (i)f(x) 1 = 1 (3x


2

1 + 1 x 1 − 1 2) 1 + 1 i(2x 1 + 1 2)


(ii)g(x) 1 = 10 whenx 1 = 1223 andx 1 = 1 − 1 ,


h(x) 1 = 10 whenx 1 = 1 − 1 ,f(x) 1 = 10 when


x 1 = 1 − 1


(iii)(x 1 + 1 1)


2

(9x


2

1 − 112 x 1 + 1 8)



  1. (i)f(z) 1 = 1 [x


2

1 − 1 y


2

1 − 12 x 1 + 1 3] 1 + 1 i[2y(x 1 − 1 1)]


(ii)h(x, y) 1 = 10 wheny 1 = 1 0 or x 1 = 11 ,


g(x, y) 1 = 10 when (real) x 1 = 11 ,


f(z) 1 = 10 when


Section 8.5



  1. (i) (ii)


(iii)



  1. (i) 2 e


iπ 26

(ii) 2 e


−iπ 26

(iii)(1 2 2)e


−iπ 26


  1. (i) 2 e


iπ 22

(ii) 2 e


−iπ 22

(iii)(1 2 2)e


−iπ 22


  1. (i) 3 e



(ii) 3 e


−iπ

(iii)(1 2 3)e


−iπ








  1. 37.i 38.−i 39.− 1


41.cos 1 a 1 cosh 1 b 1 − 1 i 1 sin 1 a 1 sinh 1 b







  1. (i)


(ii)


Section 8.6


45.± 1 ,±i



  1. 1 ,cos 12 π 251 ± 1 i 1 sin 12 π 25 ,


cos 14 π 251 ± 1 i 1 sin 14 π 25


33


2


1


3


2



  • 3










+−+










i


33


2


1


3


2


− 3










++










i


±−()12i


3 +i


1


2


3


2


− i


3


2


3


2



  • i


12
()

e


iπ 4

2 e


iπ 4

2 e


−iπ 4

zi=± 12


y=± 2


1


512 12


cos sin


ππ











i


5


12 12


cos sin


ππ














i


5


17


12


17


12


cos sin


ππ














i

Free download pdf