Understanding Engineering Mathematics

12.11 Answers to reinforcement exercises

1. (i) j (ii) 3 (iii) 3j

(iv) −1(v)1(vi)−j

(vii) − 1

2. (i) ± 5 j (ii) − 2 +j,− 2 −j (iii) 2,−2,j,−j

(iv) 1,−

1

2

+

√

7

2

j,−

1

2

−

√

7

2

j (v) −1,

1

2

±j

√

3

2

(vi) −( 1 ±

√

2 )j

3.(i) −j (ii) −j (iii) 1+ 5 j (iv) − 64

(v) − 1 + 2 j (vi) − 1 (vii) − 1 − 4 j

4.(i) 2, 0 (ii) 7,− 1 (iii) 7, 24 (iv) −44, 117

5. (i) 5− 3 j (ii)

1

3 + 4 j

(iii)

j

j− 2

(iv)

j+ 2

2 j− 3

(v)

(

j+ 2

2 j− 3

) 4

(vi)

j

( 2 + 5 j)^2

6. (i) 23+ 14 j (ii) 3+ 29 j (iii)

3

24

+

4 j

25

(iv) −

18

17

−

13 j

17

(v) −

11

65

+

23 j

65

7. (i) 8/25, real, sum of complex conjugates.

(ii) 4j/5, imaginary, difference of complex conjugates.

(iii) 1/34, real, product of complex conjugates.

8. 
   

1

2

+

1

2

j Modulus=

1

√

2

Argument=

π

4

9.(i) imaginary (ii) real (iii) complex

10. (i) 2, 0 (ii) 1,π (iii) 3,π/ 2

(iv) 1,−π/ 2 (v) 2,π/3(vi)2,− 5 π/ 6

(vii) 2

√

2, 3π/ 4 (viii) 3

√

2,− 3 π/ 4