ABSTRACT
Z2 − R2 2π f
14. v = √
2 L ρ a c
15. V = k 2 H 2L2
1. a = √(
xy m − n
) 2. R = 4
√( M π
+ r4 )
3. r = 3(x + y) (1 − x − y) 4. L =
mrC R μ− m
5. b = c√ 1 − a2 6. r =
√( x − y x + y
)
7. b = a(p 2 − q2)
2(p2 + q2) 8. v = u f
u − f , 30
9. t2 = t
, 55
10. v = √(
2dgh 0.03L
) , 0.965
11. l = 8S 2
3d + d , 2.725
12. C = 1 ω { ωL −√Z2 − R2
} , 63.1 × 10−6
13. 64 mm 14. λ = 5 √(
aμ
ρC Z4n
15. w = 2R − F L
, 3 kN/m 16. t2 = t1 − Qdk A 17. r = v
ω
( 1 − s
)
18. F = E I (nπ
L
)2 , 13.61 MN
1. 4 or −8 2. 4 or −4
3. 2 or −6 4. 1.5 or −1
5. 1 2
or 1 3
6. 1 2
or −4 5
7. 2 8. 1 1 3
or −1 7
9. 1 1 3
or −1 2
10. 5 4
or −3 2
11. x2 − 4x + 3 = 0 12. x2 + 3x − 10 = 0
13. x2 + 5x + 4 = 0 14. 4x2 − 8x − 5 = 0
15. x2 − 36 = 0 16. x2 − 1.7x − 1.68 = 0
1. −0.268 or −3.732 2. 0.637 or −3.137 3. 1.468 or −1.135 4. 1.290 or 0.310 5. 2.443 or 0.307
1. 0.637 or −3.137 2. 0.296 or −0.792 3. 2.781 or 0.719 4. 0.443 or −1.693 5. 3.608 or −1.108
1. 1.191 s 2. 0.905 A or 0.345 A 3. 19.38 m or 0.619 m 4. 0.0133 5. 1.066 m 6. 86.78 cm 7. 18.165 m or 1.835 m 8. 7 m 9. 12 ohms, 28 ohms 10. 5.73 s or 0.52 s
11. 400 rad/s
1. x = 1, y = 3 and x = −3, y = 7
2. x = 2 5
, y = −1 5
and −1 2 3
, y = −4 1 3
3. x = 0, y = 4 and x = 3, y = 1
1. (a) t > 2 (b) x < 5 2. (a) x > 3 (b) x ≥ 3 3. (a) t ≤ 1 (b) x ≤ 6
4. (a) k ≥ 3 2
(b) z > 1 2
5. (a) y ≥ −4 3
(b) x ≥ −1 2
1. −5 < t < 3 2. −5 ≤ y ≤ −1 3. −3
2 < x <
5 2
4. t > 3 and t < 1 3
5. k ≤ −2 and k ≥ 4
1. −4 ≤ x ≤ 3 2. t > 5 or t < −9 3. −5 < z ≤ 14 4. −3 < x ≤ −2
1. z > 4 or z < −4 2. −4 < z < 4 3. x ≥ √3 or x ≤ −√3 4. −2 ≤ k ≤ 2 5. −5 ≤ t ≤ 7 6. t ≥ 7 or t ≤ −5 7. y ≥ 2 or y ≤ −2 8. k > −1
2 or k < −2
1. x > 3 or x < −2 2. −4 ≤ t ≤ 2 3. −2 < x < 1
2 4. y ≥ 5 or y < −4
5. −4 ≤ z ≤ 0 6. ( −√3 − 3
) < x <
(√ 3 − 3
) 7. t ≥
( 2 +√11
) or t ≤
( 2 −√11
)
8. k ≥ (√
13 4
+ 1 2
) or k ≤
( − √
13 4
+ 1 2
)
1. 4 2. 4 3. 3 4. −3 5. 1
3 6. 3
7. 2 8. −2 9. 1 1 2
10. 1 3
11. 2 12. 10,000
13. 100,000 14. 9 15. 1 32
16. 0.01 17. 1
16 18. e3
1. log 6 2. log 15 3. log 2 4. log 3 5. log 12 6. log 500 7. log 100 8. log 6 9. log 10
10. log1 = 0 11. log 2 12. log 243 or log 35 or 5 log 3 13. log 16 or log 24 or 4 log 2 14. log 64 or log 26 or 6 log 2 15. 0.5 16. 1.5 17. x = 2.5 18. t = 8 19. b = 2 20. x = 2 21. a = 6 22. x = 5
1. 1.690 2. 3.170 3. 0.2696 4. 6.058 5. 2.251 6. 3.959 7. 2.542 8. −0.3272 9. 316.2
1. (a) 0.1653 (b) 0.4584 (c) 22030 2. (a) 5.0988 (b) 0.064037 (c) 40.446
3. (a) 4. (a) 5. 2.739 6. 120.7 m
1. 2.0601 2. (a) 7.389 (b) 0.7408 3. 1 − 2x2 − 8
3 x3 − 2x4
4. 2x1/2+2x5/2+x9/2+ 1 3
x13/2 + 1 12
x17/2+ 1 60
1. 3.95, 2.05 2. 1.65, −1.30 3. (a) 28 cm3 (b) 116 min 4. (a) 70◦C (b) 5 minutes
1. (a) 0.55547 (b) 0.91374 (c) 8.8941 2. (a) 2.2293 (b) −0.33154 (c) 0.13087 3. 8.166 4. 1.522 5. 1.485 6. −0.4904 7. −0.5822 8. 2.197 9. 816.2 10. 0.8274
11. 1.962 12. 3 13. 4 14. 147.9 15. 4.901 16. 3.095 17. t = eb+a ln D = ebea ln D = ebeln Da i.e. t = eb Da
18. 500 19. W = PV ln (
)
20. p2 = 348.5 Pa 21. 992 m/s
1. (a) 150◦C (b) 100.5◦C 2. 99.21 kPa 3. (a) 29.32 volts (b) 71.31 × 10−6 s 4. (a) 2.038 × 10−4 (b) 2.293 m 5. (a) 50◦C (b) 55.45 s 6. 30.4 N, 0.807 rad 7. (a) 3.04 A (b) 1.46 s 8. 2.45 mol/cm3 9. (a) 7.07 A (b) 0.966 s 10. £2424
11. (a) 100% (b) 67.03% (c) 1.83% 12. 2.45 mA 13. 142 ms 14. 99.752% 15. 20 min 38 s
1. 68 2. 6.2 3. 85.25 4. 23.5 5. 11th 6. 209 7. 346.5
1. −0.5 2. 1.5, 3, 4.5 3. 7808 4. 25 5. 8.5, 12, 15.5, 19 6. (a) 120 (b) 26070 (c) 250.5 7. £19840, £223,680 8. £8720
1. 2560 2. 273.25 3. 512, 4096 4. 812.5 5. 8 6. 1 2
1. (a) 3 (b) 2 (c) 59022 2. 10th 3. £1566, 11 years 4. 48.71 M 5. 71.53 g 6. (a) £599.14 (b) 19 years 7. 100, 139, 193, 268, 373, 518, 720, 1000 rev/min
1. (a) 84 (b) 3 2. (a) 15 (b) 56 3. (a) 12 (b) 840 4. (a) 720 (b) 6720
1. x7 − 7x6y + 21x5y2 − 35x4y3 + 35x3y4 − 21x2y5 +7xy6 − y7
2. 32a5 + 240a4b + 720a3b2 + 1080a2b3 + 810ab4 +243b5
1. a4 + 8a3x + 24a2x2 + 32 a x3 + 16x4 2. 64 − 192x + 240x2 − 160x3 + 60x4 − 12 x5 + x6 3. 16x4 − 96x3y + 216x2y2 − 216xy3 + 81y4 4. 32x5 + 160x3 + 320x + 320
x + 160
x3 + 32
x5 5. p11 + 22 p10q + 210 p9q2 + 1320 p8q3 + 5280 p7q4 6. 34749 p8 q5 7. 700000 a4b4 8. (a) 1.0243 (b) 0.8681 9. 4373.88
1. 1 + x + x2 + x3 + . . . , | x | < 1 2. 1 − 2x + 3x2 − 4x3 + . . . , | x | < 1
3. 1 8
[ 1 −
+
4. √
2 (
1+ x 4
− x 2
32 + x
128 −..
