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, 7., 14 (SSH).,,,.,,,.,., 1.. 2.,,. 3.,,. 4...,, 14 16, 17 21, 22 26, 27( ), 28 32 SSH,,,, ( 7 9 ), ( 14 16 SSH ), ( 17 21, 22 26, 27 SSH ),,,. 2,,,., 2.,,. 1, 2,. 30 4

1. 2.

i 1 1 1.1..................... 1 1.1.1....................... 1 1.1.2....................... 3 1.2............................ 4 1.3....................... 7 1.4........... 9 1.5....................... 11 2 15 2.1............................... 15 2.2............................... 15 2.3............................... 19 2.4 ( )...................... 22 2.5............................. 23 3 27 3.1............................ 27 3.2...................... 27 3.2.1........................... 27 3.2.2................... 28 3.2.3........................ 30 3.3........................ 31 3.3.1............................ 31 3.3.2............................ 32 3.3.3............... 34 4 39 4.1....................... 39

ii 4.2...................... 39 4.3................. 41 4.4.................. 41 4.5 SI ( )...................... 44 4.5.1...................... 44 4.5.2 SI ( ).................... 45 4.6....................... 47 4.6.1......................... 47 4.6.2........................ 49 4.6.3........................... 51 4.6.4............................. 52 4.6.5........................ 53 5 61 5.1................................ 61 5.2.......................... 63 5.3 sin cos.......................... 65 5.4 0, 90.......................... 66 5.5.................... 68 5.6............................. 69 6 73 6.1....................... 73 6.2........................... 75 6.2.1 a = b.................... 75 6.2.2......................... 75 6.2.3................... 75 6.3........................... 76 6.4........................... 77 6.5............................. 77 6.6........................ 79 6.7....................... 80 6.8........................... 80 6.9.......................... 84 6.10............................ 85

iii 6.11................................. 87 6.11.1...................... 87 6.11.2...................... 88 7 93 7.1........................... 94 7.2............................... 97 7.2.1....................... 97 7.2.2....................... 98 7.2.3...................... 99 7.2.4...................... 100 7.3............................. 102 7.3.1........................... 102 7.3.2................... 106 7.3.3 ( )....................... 109 8 115 8.1................................ 115 8.2................................. 116 8.3.......................... 116 8.4.......................... 118 8.5................................ 120 8.5.1............................. 120 8.5.2.................... 121 9 125 9.1.......................... 125 9.1.1,?........... 125 9.1.2................... 126 9.1.3 1........................... 128 9.1.4................... 128 9.1.5......................... 129 9.2...................... 130 9.2.1........................ 130 9.2.2 ( ).................. 132

iv 9.3 2...................... 136........................ 139 143, 146 149 150 180 186

1 1 1.1,.,,.,,,,. 1.1.1. 1 2 5 3 = 5., 6,..,, 2 a, 1 b, 3 c, 5 d,. b a d c = b d a c = b d a c,., a+3 b 2 x 2,. a b ( ),,.,. 3 2 a, y 5 x 3 x 2, y 5

2 1. 1 5 5 2 = 2.,. 25 1 5 5 2 = 1 5 2 5 = 2 25,.,,. b a d c = b a c d = b c a d,,. b a d c = b a d c = b c a d (b c),, (a d).,,,.. b a d c b a d c = b d a c = b d a c = b a c d = b c a d, b a d c = b a d c = b c a d 1.. (,,,.) (1) (5) 2 5 3 4 1 a b 2 b c a (2) 3 6 1 2 (6) b a x 2 (3) a c y b 1 a 1 b (7) (4) x a 2 b c y x w z y a c

1.1. 3 1.1.2. 1 5 + 3 5 = 4., 5,..,,.,. b a + c a = b + c a b a c a = b c a,. 1 2 + 2 3 = 7 6.,. 1 2 + 2 3 = 1 2 3 3 + 2 3 2 2 = 3 6 + 4 6 = 7 6.,,,., 1 2 3, 2 3 6, 6 2, 3. 2 3 6, 2.,,.,.,,, b a + d c = b a c c + d c a a = b c a c + a d a c b a d c = b a c c d c a a = b c a c a d a c = b c + a d a c = b c a d a c. a c,.

4 1 (1) b a + c a = b + c a b a c a = b c a (2) b a + d c = b a c c + d c a a = b c a c + a d a c = b c + a d a c b a d c = b a c c d c a a = b c a c a d a c = b c a d a c 2.. (1) (4) (7) 1 4 + 2 5 h g j i 2 a b c x 2 1 (2) 10 9 2 (5) 1 x x + 1 3 b c a x 2 (8) (3) (6) 1 3 a b 2 1 4 a 2 b q p + s r 1 a + 1 1 a 1 1.2, 1., x w = z y, x, x =., x w, w, x w w = z y w, w, x, x =. x = = wz y z y w

1.2. 5, a =,.. b a = d c, a.., a, b., 2. 1, a b =. b a a = b = d c a a d c., d c a d c, d c c d. b c d = a d c, a.. 2 b c d = a a = b c d c d,, b. b a b = d c b b a 1 = d b c 1 b 1 a = d b c.,,,. a = b c d

6 1,,., 1 a = 1 b + 1 c,,,., 1 a = 1 b + 1 c = c b c + a = b c b + c b b c = b + c b c ( 1 ),,.,. 1 S 10 = 0 S = 5 2 x a + x b = 1 a c x + b x = c 3 b x + c b x + 3 5 = 2 b x + 2 c 3 b x + 5 = 2 x = a + x b + 1 1 a = 1 b + 1 c + 1 d x (b + 1) = a + x a = b c d b + c + d 3.. (1) I = V (I : [A], V : [V], R : [Ω]) R R =. I = 0.3 [A], V = 3 [V], R. (2) b x 3 = 4 x c x =. ( b c 1 ) (3) b x + 3 = c x b x =. ( b c )

1.3. 7 (4) b x 3 = 4 c x c (5) 3 x a (3 x + c) = (6) b c = (7) a x = 1 x + 1 x 2 a x x =. ( b 1 ) 4 b 4 a b c x =. ( c 0) x =. ( b c ) x =. ( a 0 ) 1.3, x =. x 2 = 1 (x > 0) 3 1, x =., x 2 3,.,, a 2 = c b (a > 0, b > 0, c > 0). c a 2 = b c a = b, x =. x 2 = 3, x = 12., ( : ), 2.,, a = 2, b = 3, c = x a = c b (a > 0, b > 0, c > 0)

8 1. c., 2 a 2 = c b, b a 2 b = c b b a 2 b = c c = a 2 b. a, b, c x = 2 2 3 = 12. 2,. a 2 = c b a = c b ( a > 0, b > 0, c > 0), 2. a = c b a 2 = c b ( a > 0, b > 0, c > 0) ( 2 ),,.,. x 2 + 9 = 4 x + 3 = 2 v 2 = 2 g h + v 0 2 v = 2 g h + v 0 4.. (1) x 2 = 1 4 x. ( x > 0 ) (2) 10 = 5 x x.

1.4. 9 (3) 3 = x 5 x. (4) y 2 = x + 3, x =. (5) y = c x, x =. ( c 0 ) (6) y = (7) z x 1 x 1, x =. ( y 0 ) = 2 x. (8) x 2 = 1 y, x =. ( x > 0, y > 0, z > 0 ) z2 (9) f = 1 2π LC, C =. ( f 0 ) 1.4 2 y. x = 2 1 y = x + 3 2 y = 5., x = 2 2 x 1 y = 2 + 3 y = 5., x = t y = t + a (a ) y = x + a., y t. 2, x, y, t 3.,, x, y, 2 t.

10 1 1. x y, t, t y x. x = t 1 1 y = 2 t + 3 2, 1 t =. t = x + 1 1, 1 2 t. y = 2 (x + 1) + 3 y = 2x + 2 + 3 y = 2x + 5. 5. (1) 2 t, y, x. x = 2 t y = 4 t (2) 2 t, y, x. ( a, b, a 0 ) x = a t y = b t (3) 2 t, y, v. ( g, g 0 ) v = g t y = 1 2 g t2 (4) 2 t, y, x. x = 2 t 1 y = 4 t 2 + 6 t + 3 (5) 2 t, y, x. ( a, b ) x = t a y = 3 t + b

1.5. 11 1.5,.,. 1 1, R 1 [Ω] R 2 [Ω], E [V]., 1.. 1 1 1 1, R 1 [Ω] R 2 [Ω], 1 1 R 1 [Ω] R 2 [Ω] 1 ( ) R [Ω], R [Ω] R 1 [Ω], R 2 [Ω],. 1 R = 1 R 1 + 1 R 2,. 1 1, I 1 = E R 1 1, I 2 = E R 2 2. I [A], ( ), I = I 1 + I 2 3

12 1, 3 1, 2, E I = + E R 1 R 2 ( 1 = E + 1 R 1 R 2 ) 4., 1 1,, I = E R 5. 4 I [A] 5 I [A], 4 5 E R ( 1 = E + 1 R 1 R 2 ) 6 1, 6 E ( E ) E R 1 ( 1 E = E + 1 ) 1 R 1 R 2 E, 1 R = 1 R 1 + 1 R 2 7., 7 R =, R = R 1 R 2 R 1 + R 2 8. 6. (1) 1 1, R 1 = 20 [Ω], R 2 = 30 [Ω], R. (2) 7, 8. (1),. (3) 1 1, R 1 = a [Ω], R 2 = 2 a [Ω], R.

1.5. 13 7., 3 R 1, R 2, R 3, E [V]. 1.5,. (1) R 1 I 1, E, R 1., I 2 E, R 2,, I 3 E, R 3,. (2) I, I 1, I 2, I 3. (3) R, I, E, R. (4) (2), (1). (5) (3) (4), R [Ω] R 1 [Ω], R 2 [Ω], R 3 [Ω],. ( 1 R =. ) (6) R 1 = 10 [Ω], R 2 = 15 [Ω], R 3 = 30 [Ω], R. (7) R 1 = a [Ω], R 2 = b [Ω], R 3 = 3 b [Ω], R.

14 1 1. (1) 1 a = 1 b + 1 c + 1 d a =. (2) P V = w M (3) (4) 1 a + 1 b = 1 f 4 b x + c = 2 b x + c R T M =. b =. ( P 0, V 0 ) x =. ( b 0 ) x (5) a + 2 = x =. ( a 1 ) b + x l 2. T = 2π g =. ( T 0 ) g 3. v > 0, F x = 1 2 m v2 v =. 4. 2 W = 1 2 Q V 1 ( F > 0, x > 0, m > 0 ) Q = C V 2,. (1) W C, V. (2) W C, Q. ( C 0 ) 5. 2 M a = M g T m a = T m g 1 2 ( M > m > 0 ),. (1) a M, m, g. (2) T M, m, g.

15 2,, ( : 345000 = 3.45 10 5, 0.000678 = 6.78 10 4 ).,,. 2.1 a n a n, a n. a a = a 2, 2 2 2 = 2 3, X X X X = X 4., a 1, a 2, a 3, a 4, a ( ), a ( ), a 1, 2, 3, 4,. 2, 3, 10. 2 1 = 2 3 1 = 3 10 1 = 10 2 2 = 4 3 2 = 9 10 2 = 100 2 3 = 8 3 3 = 27 10 3 = 1000 2 4 = 16 3 4 = 81 10 4 = 10000 2 5 = 32 3 5 = 243 10 5 = 100000 2.2.

16 2 m, n (1) a m a n = a m+n (2) a m a n = (3) (a m ) n = a m n (4) (a b) n = a n b n a m n (m > n) 1 (m = n) a 0 1 a n m (m < n) (5) ( b a ) n = bn a n a 0 (1) (5),. (1) a m a n = m n {}}{{}}{} a a {{ a a a } m + n = a m+n 1. 3 2 3 4 = 3 2+4 = 3 6 = 729 4 4 4 = 4 4 4 1 = 4 4+1 = 4 5 = 1024 (2) 1 m > n a m a n = 2 m = n am a n = = m {}}{ a a a a a }{{} n n m n {}}{{}}{ a a a a a } {{ a } n a m a n = m {}}{ a a a a a a }{{} m = m n {}}{ a a = a m n = 1

2.2. 17 m = n, 1,. a m n = a m m = a 0 = 1 a 0 = 1 2. 5 3 5 3 = 5 3 3 = 5 0 = 1, 10 4 10 4 = 10 4 4 = 10 0 = 1 3 m < n a m a n = m {}}{ a a a a } a {{ a } a } {{ a } m n m = 1 a n m 1 a m n 1 = a n m, m = 0, a 0 n 1 = a n 0, a n = 1 a n. a 1 = 1 a, a 3 = 1 a 3., m, n. 3. 2 3 2 5 = 1 2 5 3 = 1 2 2 = 1 4 ( ) 2 3 2 5 = 2 3 5 = 2 2 = 1 2 2 = 1 4 10 4 10 7 = 1 10 7 4 = 1 10 3 = 1 1000 ( ) 10 4 10 7 = 10 4 7 = 10 3 = 1 10 3 = 1 1000 a a 4 = 1 a 4 1 = 1 a 3 ( ) a a 4 = a 1 4 = a 3 = 1 a 3 (3) m m m (a m ) n {}}{{}}{{}}{ = ( a a) ( a a) ( a a) }{{} m n (m n ) = a m n 4. (a 3 ) 2 = a 3 2 = a 6 (10 2 ) 2 = 10 2 2 = 10 4 = 10000 (2 4 ) 2 = 2 4 2 = 2 8 = 256

18 2 (4) (a b) n = n {}}{ (a b) (a b) (a b) = (a } a {{ a } ) (b } b {{ } b) = a n b n n n 5. (2 3) 3 = 2 3 3 3 = 8 27 = 216 (4 x 2 ) 2 = 4 2 (x 2 ) 2 = 4 2 x 2 2 = 16 x 4 (5) ( b a ) n = = n ({ ) ( )}} ( ){ b b b a a a n {}}{ b b b a a a }{{} n = bn a n 6. ( ) x 3 = x3 3 ( ) 3 a 2 = 4 3 3 = x3 27 (3 a)2 4 2 = 9 a2 16 1.. (1) 5 3 5 2 (2) (6 2 ) 3 (3) 7 4 7 2 (4) 9 3 9 5 (5) (6) 10 3 10 4 (7) 10 5 10 2 (8) 10 3 10 2 ( 3 4 ) 4 2. a, b, x, y 0,. (1) a 3 a (2) a 4 a 3 a 2 (3) a 2 a 5 (4) (a b 2 ) 3 (a 2 b) 2 (5) 4 x 2 (2 x y 3 ) 2 (6) 5 a 3 3 a 4 (7) 28 a 2 (7 a 3 ) (8) (a 4 ) 3 (9) (a 2 ) 3 (10) a 5 a 0 3. a, b, c, x, y 0,. (1) (4) a 3 a 3 (2) (a 3 ) 2 (a 4 ) 3 (3) (a 2 b 1 ) 2 a (6 x 2 y) 2 15 a 2 b 3 c 9 x 3 y 3 (5) ( 5 a b c 2 ) 2

2.3. 19 4. a, b, c, x, y 0,. (1) 3 a 3 b 2 (2 a b 2 ) 3 6 a 3 b 2 (2) (2 x y 2 ) 3 6 x 4 y 2 (3) 60 a 2 7 b c 48 a3 c 3 35 b 3 ( ) 2 a c 2 2 (4) 5 b 2 (7 a 2 b) 2 21 x 3 y 3 3 x 2 y 35 (a b 2 ) 2 5. r, h. b2 a, a b, V, V. 2.3,,, 63800 = 6.38 10 4, 0.00000638 = 6.38 10 6., x x = a 10 n ( 1 a < 10, n )., n x., (1) n 0, x n+1. (2) n, x n 0. ( n n ),,. 6. a 10 n ( 1 a < 10, n ). (1) 4470 (2) 0.023 (3) 857600 (4) 0.00038 (5) 76358 7., ( ). (1) 3.59 10 4 (2) 2.93 10 5 (3) 4.2 10 3 (4) 2.738 10 3 (5) 5.14 10 2

20 2 1. a 10 n ( 1 a < 10, n )., (2), (3), (4), a 2. (1) 20 4 = (2 10) 4 = 2 4 10 4 = 16 10 4 = 1.6 10 10 4 = 1.6 10 5 (2) 328100 0.0004892 5062 = (3.281 10 5 ) (4.892 10 4 ) (5.062 10 3 ) = (3.281 4.892 5.062) 10 5 4+3 = 81.2 10 4 = 8.12 10 5,,. (3) 541300 7583 = (5.413 10 5 ) (7.583 10 3 ) = 5.413 7.583 105 3 = 0.714 10 2 = 7.14 10 (4) 0.06425 3 = (6.425 10 2 ) 3 = 6.425 3 10 6 = 265 10 6 = 2.65 10 4 8., 1. (1) 4.8 3500 0.0024 (2) 0.00011 3 (3) 0.108 540000 (4) 0.0035 0.035 3 (5) (8 10 3 ) 2 (2 10 3 ) 2 (6) (8.1 10 2 ) 3 (9.0 10 2 ) 6 9. 200 [g], 6.1 10 23., 1.00 [g],., 197.0 [g] 6.02 10 23.

2.3. 21 10 3 10 4 ( ),. A = 3234589124.,.,, ( ), 3 ( ).., 3. 3, A = 3, 234, 589, 124.,, 3., 10 0 (one), 10 1 (ten), 10 2 (hundred), 10 3 (thousand : ), 10 6 (= 10 3 10 3, million : ), 10 9 (= 10 6 10 3, billion : ), 10 3, 3.,, 10 0 ( ), 10 1 ( ), 10 2 ( ), 10 3 ( ), 10 4 ( ) 10 8 (= 10 4 10 4 : ), 10 12 (= 10 8 10 4 : )., 10 4, 4. A = 32, 3458, 9124, 2., A = 32 3458 9124 (3 ) (4 ),.

22 2 2.4 ( ), 2 a, 3 a, a 2 ( ), a 3 ( ). n, n a a n. 2, 3, 4,,., 2 2 = 4, ( 2) 2 = 4 2 2 4 2., 2 3 = 8, ( 2) 3 = 8 2 8 3 ( 1 ), 2 8 3 ( 1 )., x 2 = 2, 2 2., a n,. a > 0 a < 0 n. n n., n n a., ( n a ) n = a. n = 2 2 a a. 7. 2 9 = 9 = 3 2 = 3 3 125 = 3 5 3 = 5 3 8 = 3 ( 2) 3 = 2 4 16 = 4 2 4 = 2 10.. (1) (4) 2 49 (2) 3 64 (3) 5 32 (5) 3 64 (6) 4 81 5 1

2.5. 23 2.5, n a,. ( n a ) n = a = a 1, (3) (a m ) n = a m n, n a = a 1 n., ( n ( ) a ) n = a 1 n 1 n = a n n = a 1 = a., a a m n a m = (a m ) 1 n = a m 1 n = a m n. a > 0, m, n, a 1 n = n a a m n = n a m,,. 8. 3 10 = 10 1 3 5 10 2 = 10 2 5 11., a m n. (1) 4 10 (2) 7 15 (3) 5 3 4 (4) 3 16 (5) 3 3 4 3 3 5 (6) 3 2 6 2 12. 125 [cm 3 ]. a [cm].

24 2 n a n a. nth root of a., a n., n, a, a n, a n,.,.,, n a.,.,,,,,.,,,,,,,., 3. 6..,.

2.5. 25 1. 2.00 [mm] 1.00 [km],.,, ρ = 1.7 10 8 [Ω m]. (, R [Ω], l [m], S [m 2 ].,,, R = ρ l S [Ω]. S = πr 2 [m 2 ], π = 3.14., 1 2.), 1.0 [mm],. 22 [Ω/km],, 0.4 0.5 [mm],. 100 [Ω/km]. 2.., c = 3.0 10 8 [m/s], l = 1.50 10 11 [m]. 3. 63.5 [g], 6.02 10 23., ρ = 8.93 [g/cm 3 ]. 1 cm 3. 3. ( : ρ ) 4 [ C], 1 [ ], 1.00 [g/cm 3 ].,.. [g/cm 3 ] 19.3 7.87 10.5 2.69 8.93 4., 6378136 [m]. a 10 n ( 1 a < 10, n ) [km].

26 2 5. 1990 5292000 ( ), 123611 ( )., %. P.20 1. ( :, 10 3. ) 6. 9.11 10 31 [kg], 1.67 10 27 [kg].,. P.20 1. ( :,,,. )

27 3,,,,,,,..,.,,,,. 3.1..,, 2, 3, 4.,,,, 179 [mm] 179.0 [mm].,,.,.,,..,.,.,. 3.2 3.2.1,., 1., (L) 10

28 3 1 [mm] ( ), 179.3 [mm],, 179.25 [mm] L < 179.35 [mm]., 1 [mm],., 179.3 [mm].,, 0 ( ), 0,., 4 179.3 [mm], 4., 179.30 [mm], 5., 4 5,., 1 (d) 3 3, 5 5. 3.2.2,. 3 1

3.2. 29 1 (30 [m] ) 23.568 [m] 1 [cm] [cm] 10 1 176.3 [mm] 1 [mm] [mm] 10 1 (1 ) 20.25 [mm] 1 [mm] [mm] 20 ( ) 1 (1 ) 19.992 [mm] 100 [mm] 1 [mm] 1000 (1),. 1 10 [cm] (2),,, 0.1 [mm],, 0.1 [mm].,,. 10, 0.1 [g] 4.5 [g] 0.01 [g] 4.52 [g] 0.0001 [g] 4.5230 [g] 2, 3, 5., 5 4.523 [g].. (1). : 4.5230 [g], 15.290 [m] 5 ( 2, 0,.) (2) 0. : 0.0179 [m] (17.9 [mm] ) 2.15 [mv] (0.00215 [V] ) 3 ([mm] 17.9 [mm] [m], 1 1 0.)

30 3 (3) 0 0. : 1.01 [g], 101 [m] 3 ( 0,. ) 1.01 [g] [mg], 1010 [mg]., 0., 1.01 10 3 [mg], 0 0. 1.. (1) 0.6 (2) 8.7 (3) 3.07 (4) 0.028 (5) 2.050 (6) 0.009070 (7) 4.30 10 6 (8) 7020 3.2.3,., 24.37 3, 2 7 24.4.,, 45 2, 126 3., 24.65, 4., n. 1. 26.86573, 3,, 5., 26.866., 5,, 7. n, (n+1). (1) (n+1), n 1 1 2. : 1.444 3 1.44 (2) (n+1), n 1 1 2 1. : 144.6 3 145 n

3.3. 31 JIS JIS (Japanese Industrial Standard),, (1), (2). (3) (n+1), n 1 1/2, (i), (ii). (i) n. : 14.45 3 14.4 (ii) n. : 14.55 3 14.6,,,.,,. 3.3 3.3.1, 1.,., ( ).,.,. (1) ( ),,. (2). (3),,.

32 3 1. 3, A, B, C 3. A 0.5005 [g], B 3.01 [g], C 2.055 [g]. g.. (1) B 2, 3. A 0.5005 0.501 ( 4 ) B 3.01 3.01 C 2.055 + 2.055 (2) 5.566 (3) 5.57 [g] ( 3 ) 2. A B 10 1. 2,. A 10 2, 4.41 [g]. B 10 3, 4.416 [g]. 2 10., A 2, B 3, 4.41 + 4.416 = 8.826 [g]., (, A 2 ), 3 8.83 [g]. 2.. (1) 10.50 [kg] + 2555 [g] (2) 1.253 [m] + 0.5246 [m] + 3.12 [m] (3) 0.2025 [m] + 1.237 [cm] + 0.3225 [cm] 3.3.2 (1) 2,, 1.,.,.

3.3. 33 3. 125 31 125.1 31.1 1 2 5. 1 1 2 5 3 1. 1 3 1 1 2 5 1 1 2 5 1 2 5 1 + 3 7 5 + 3 7 5 3 3 8 7 5 3 8 9 0 6 1 3.9 10 3 ( 2 ) 3.89 10 3 ( 3 ) (..) 4., 10.17 [m], 5.11 [m]. 10.17 5.11 = 51.96 52.0 [m 2 ] ( 10.2 5.11 = 52.12 52.1 [m 2 ] ), 10.17. 3 (5.11) 4 (10.17), 3. (2),. 5, 5.000000., 5 1. 5. 125 [cm] 3 [cm]. 125 3 = 375, 375 [cm]. (3), 1. 1,. 6. 10.2 [m] S. π = 3.141592., (3 ) 1., π = 3.1415 5, π = 3.142 4 S = π r 2 = 3.142 (10.2) 2 = 326.89368 S = 327 [m 2 ].

34 3 3. (1),,,, 2.00 [m], 1.00 [m], 5.2 [mm]. V [m 3 ]. (2) 10.00 [cm] V [cm 3 ]. π = 3.141592.,. 1111 [m] 11 [m] 1111 [m] + 11 [m] = 1122 [m] = 1.122 10 3 [m] 1.122 10 3 [m] 1110 [m] 11 [m] = 12210 [m 2 ] 1.2 10 4 [m 2 ] 1.2 10 4 [m 2 ] 3.3.3 (1). (2). (3). (4). 7., 0.50 [m], 2.08 [m] 1.05 [m]. π = 3.141592. (1) 0.50 2.08 + (1.05) 2 3.142 = 1.04 + 3.464055 ( 3, π 4 )

3.3. 35 (2) ( ) 1.0 + 3.46 ( 2, 3 ) (3) = 4.46 (4) ( ) 4.5 4.5 [m 2 ] 4. (1), 1.011 [m], 2.02 [m] 0.50 [m]. π = 3.141592. (2), 20.0 [cm], 11.11 [cm] 1 56 [cm 2 ] 10. (3), 10.20 [cm], 11.010 [cm] 10.00 [cm]. π = 3.1415926.,.,. (,. ) 3, 3.,. 8 10..,,. (CD),.. CD, 0.1 [µm],

36 3 1 [mm]. 10000, 0.04 [µm].,,.,.,,,.,,,, 5. 0.2, 0.5, 1.0, 1.5, 2.5. (,,. ),, 0.5, ( ) ±0.5 % (, 10 [ma], ±50 [µa] ), 3.,, ( ), 4., 1, 1.0110 [g].,, 2., 20.15 [ml]., [mm].,, [cm].,.

3.3. 37 1... (1) 127 9.7 (2) 145.6 + 2.0 10 (3) 22.4 765.3 (4) 14 273 (5) 121.5 10.0 + 21.0 100 2. D51 1.400 [m]. m., π = 3.141592. 3. 6.00 [Ω] (R), (I), 10.1 [ma]. (V ) (P )., V = R I, P = R I 2. P [W] ( ). 600 [W], 400 [W]. 4. 1 [A], 1 ( ) (Q), 1 [C] ( )., 1 N., 1 1 1.6 10 19 [C], 2 1.602 10 19 [C],. Q = I t., Q [C] = I [A] t [ s ] ([ s ] : )., Q = N e ( = 1 ). 5. 20 [ C] 13.5462 [g/cm 3 ]. 1.000 [L]., 1 [L] = 1000 [cm 3 ].

38 3 6. 10.0 [m],, 29.00.., tan 29.00 = 0.5543,, 1.500 [m].,,.,, ( [ ] ), ( [ ] ).

39 4 4.1,,,.., l = 2 π r, v = V sin(ω t + θ).. ( ),., ( ).,.,. 4.2,, 4 1.

40 4 4 1 (, ) A * α alpha, B * β beta, Γ * γ gamma, * δ delta, E ε epsilon, Z ζ zeta, H η eta,, Θ * θ theta,, I ι iota, K κ kappa, Λ λ lamda, M * µ mu, N ν nu, Ξ ξ xei, xi,, O o omikron, Π * π pei, pi, P ρ rho, Σ σ sigma, T τ tau, Υ υ y(u)psilon, Φ φ phei, phi, X χ khei, chi, Ψ ψ psei, psi, * Ω * ω omega,. 1. (1) 3. (2)..

4.3. 41,,. TABEPNA. P [ ].. ( ). 4.3,.,,,,.,,,.., (unit)., 1, 4 4, 1 0.5.,,. 4.4,. 1, 1, 1., 1 1000 1000 1 (kilo) (mili).,,. 4 2.

42 4 4 2 1000000000000000000 10 18 E 1000000000000000 10 15 P 1000000000000 10 12 T 1000000000 10 9 G 1000000 10 6 M * 1000 10 3 k * 100 10 2 h 10 10 1 da 1 10 0 ( ) 0.1 10 1 d 0.01 10 2 c 0.001 10 3 m * 0.000001 10 6 µ * 0.000000001 10 9 n 0.000000000001 10 12 p 0.000000000000001 10 15 f 0.000000000000000001 10 18 a *.,. }{{} (k) }{{}}{{} (h) (da) }{{} (m) }{{} (d) }{{}}{{}.. (c) (m),,. Ω,,. µ.. 4 2, 10 6. k, m, c.

4.4. 43 1. 1457 [m],,,.,., 1 1000 k ( ), 1 1 1000 m ( )., [km] [mm]. 1457 [m] = 1.457 [km] 1457 [m] = 1457000 [mm], a 10 n (, 1 a < 10, n ), k ( ) = 1000 = 10 3, m ( ) = 1 1000 = 10 3, 1457 [m] = 1.457 10 3 [m] 1457 [m] = 1457000 [mm] = 1.457 10 6 [mm]. a 10 n (, 1 a < 10, n ),,, 10 n,. 2., (. ), (1) 1230 [m] ( ) (2) 450 [mm] ( ) (3) 1.2 [kv] ( ) (4) 0.00079 [A] ( ) (5) 0.56 [MΩ] ( ) 3., (. (1) 2.94 [km] ( [m] ) (2) 38.4 [µm] ( [m] ) (3) 457 [kg] ( [g] ) (4) 9.86 [µω] ( [Ω] ) (5) 6.31 [MΩ] ( [Ω] ) ) 4.,. (1) 1.58 10 6 [Ω] (2) 8.65 10 6 [g] (3) 2.88 10 3 [m] (4) 5.43 10 3 [A] (5) 3.37 10 4 [V]

44 4 15 I II III IV V VI VII VIII IX X, 1 2 3 4 5 6 7 8 9 0.,,,..,.,. 4.5 SI ( ) 4.5.1,,,,,,.,, 4 1,. 4 1

4.5. SI ( ) 45, (Système International d Unités [ ], SI ),., 1 [m], [kg], [ s ( )] MKS, 2 [cm], [ g ], [ s ] CGS, 3, MKS [A] MKSA, 4 [kgw] ( ). MKS,., [kgw].,, SI ( ). 4.5.2 SI ( ) SI, [J], [Pa]., SI [kgw] [N] [kgw m] [cal], [kcal] [J] [mmhg], [atm], [kgw/cm 2 ], [N/m 2 ] [Pa].,. SI, 7 ( 4 3) 2 ( 4 4), ( 4 5), 10 (P.42 4 2)., SI,.,.

46 4 4 3 SI m kg s A K mol cd 4 4 SI rad sr 4 5 SI m 2 m 3 m/s m/s 2 kg/m 3 rad/s Hz,, J, N m V

4.6. 47 4.6,,,. 3 [km] [m] 3 10 3 [m], 15 [cm 2 ] [mm 2 ] 1.5 10 3 [mm 2 ], 1.,,.,,. 4.6.1. 1 [ g/cm 3 ( )], 1 [cm 3 ] 1 [ g ],. ρ [ g/cm 3 ] = m [ g ] V [cm 3 ] 4 2 1. ( ). 1 [kg/m 3 ] ( g/cm 3 ), 1 [kg] = 1000 [ g ] = 10 3 [ g ] 1 [m] = 100 [cm] = 10 2 [cm] 1 [m 3 ] = 10 6 [cm 3 ], 1 [kg/m 3 ] = 103 [ g ] 10 6 [cm 3 ] = 1 10 3 [ g/cm3 ] = 0.001 [ g/cm 3 ]

48 4 5. ( ). (1) 20.8 [ g/cm 3 ] (kg/m 3 ) (2) 2.43 10 4 [kg/m 3 ] ( g/cm 3 ) 2. d 10 [mm], h 100 [mm]. m. π = 3.14, S S = π ( ) d 2 2 = 3.14 ( ) 1.0 2 2 = 0.785 [cm 2 ], V, V = S h V = 0.785 [cm 2 ] 10 [cm] = 7.85 [cm 3 ] 4 6 ρ 2.69[ g/cm 3 ], m m = ρ V = 2.69 [ g/cm 3 ] 7.85 [cm 3 ] = 21.1 [ g ] 6. 4 3. 4 6 [ g/cm 3 ] 7.86 2.69 19.3 10.5 8.93 13.6 0.49 4 3 7., 1.83 [ g/cm 3 ]. 500 [ g ] [cm 3 ].

4.6. 49 4.6.2, ( v )., [m/s ( ) ]., 1 [m/s ( ) ] 1 [m/s] = 1 [ s ( )] 1 [m].., 1 1, 1 [m/min( )] = 1 [min( )] 1 [m] ( ) 1 [km/h( )] = 1 [ h ( )] 1 [km] ( ). 3. 100 [km/h]. [m/s]. 1 [km] = 1000 [m] 1 [ h ( )] = 3600 [ s ( )], 100 [km/h] = 100 1000 3600 [m/s] = 27.8 [m/s] 8. ( ). (1) 10 [m/s] (m/min) (2) 20 [m/s] (km/h) 9. 100 m 10., km.. ( a ),., t 1 [ s ( )] v 1 [m/s], t 2 [ s ( )] v 2 [m/s].

50 4, 4 4. 4 4,, a = = = v 2 v 1 [m/s] t 2 t 1 [s]., [m/s 2 ( )]., 1 [m/s 2 ] = 1 [ s ( )] 1 [m/s]., a > 0, a < 0. 4. 50 [km/h], 80 [km/h], 5 [ s ].., 50 [km/h] = 50 1000 3600 [m/s] = 13.9 [m/s] 80 [km/h] = 80 1000 3600 [m/s] = 22.2 [m/s], a = 22.2 13.9 [m/s] 5 [ s ] = 1.7 [m/s 2 ]

4.6. 51 10. (1) 36 km, 10 72 km.. (2) 70 km, 7.. 4.6.3 1 [N]. 1 [kg] 1 [m/s 2 ]. =, 1 [N] = 1 [kg] 1 [m/s 2 ] = 1 [kg] 1 [m/s 2 ] = 1 [kg m/s 2 ]., 1 9.8 [m/s]. a = 9.8 [m/s] 0 [m/s] 1 [ s ] = 9.8 [m/s 2 ], 9.8 [m/s 2 ]., 9.8 [m/s 2 ] = g., g.,., 1 9.8 [m/s]. m [kg], m g [kg m/s 2 ]. 1 [N] = 1 [kg m/s 2 ], m g [N].

52 4 5. 800 [ g ],, [N]., 9.8 [m/s 2 ]. m = 800 [ g ] = 0.8 [kg] F = m g = 0.8 [kg] 9.8 [m/s 2 ] = 7.84 [kg m/s 2 ] = 7.84 [N] 11. ρ = 2.5 [g/cm 3 ], 8000 [cm 3 ]., 20 [km/h] 92 [km/h] 5 [ s ].,. (1) m. (2) a. (3) F = m a (, F :, m :, a : ),. 12., 49 [N], [m/s 2 ]. 4.6.4 ( P ), [Pa]. SI 1 [Pa] = 1 [N/m 2 ] ( 1 [m 2 ] 1 [N] ) 6. 490 [N], 0.5 [m 2 ]. [Pa]. 490 [N] 0.5 [m 2 ] = 980 [N/m 2 ] = 980 [Pa] 13. (1) 1 [N/cm 2 ] [N/m 2 ]. (2) 588 [Pa] [kn/m 2 ]. (3) 3500 [N/m 2 ] [kn/m 2 ].

4.6. 53 14. [Pa]., 9200 [kg], 9.8 [m/s 2 ]., 2. 1, 1., 2,.,,.,.,.. 4.6.5 F [N], s [m],,,, W. W = F s. [J], [N m].

54 4 1 [J ( )], 1 [N], 1 [m]. 1 [J] = 1 [N m]. 7. 120 [N] 40 [m] [J]. W [J] W [J] = 120 [N] 40 [m] = 4800 [N m] = 4800 [J] 15., 200 [kg] 10 [m]. [J]., 9.8 [m/s 2 ]. 16. ( ). (1) 300 [J] (N m) (2) 9800 [N m] (kj),.,.. W, t, P, P = W [J] t [ s ], [J/s]., [J/s],, [W( )] ( 1 [W] = 1 [J/s] )., [J ( )] [W h ( )], [W s ( )]., [W h] [W s].

4.6. 55, 1 [J] = 1 [N m] 1 [W] = 1 [J/s] 1 [W s] = 1 [J] 1 [W h] = 1 [W] 3600 [ s ] = 1 [J/s] 3600 [ s ] = 3600 [J]. 8., 2000 [kg] 1 25 [m]. [kw]., 9.8 [m/s 2 ]. 1 [kw] = 1000 [W]. 2000 [kg] 9.8 [m/s 2 ] = 19600 [kg m/s 2 ] = 19600[N] 19600 [N] 25 [m] = 490000 [N m] = 490000 [J], P P = 490000 [J] 60 [ s ] = 8166.7 [J/s] = 8166.7 [W], P [kw] P = 8166.7 1000 [kw] = 8.17 [kw] 17. 120 [N] 15 [ s ] 40 [m]. 18. 2000 [kg], 50 [ s ] 40 [m]. [kw]., 9.8 [m/s 2 ].

56 4 19. 100 [W], 1 [ s ] [J]. 20. ( ). (1) 8.5 [W h] (W s) (2) 20 [W h] (J),.,., SI,, SI [cal], [mmhg], [atm]., 760 [mm], 1,. 1 [atm ( )] = 760 [mmhg ( )] = 1.013 10 5 [Pa],, SI [kgw].,,., g = 9.8 [m/s 2 ]. ( )., ( W ), [kg], SI [N], SI [kgw]., [kgw], ( ) [kg].

4.6. 57.,. 1 [N] 1 [kg] 1 [m/s 2 ]., 1 [N] = 1 [kg] 1 [m/s 2 ] = 1 [kg m/s 2 ], 1 [kgw] 1 [kg] g = 9.8 [m/s 2 ], 1 [kgw] = 1 [kg] 9.8 [m/s 2 ] = 9.8 [kg m/s 2 ]., [N] [kgw] 1 [kgw] = 9.8 [kg m/s 2 ] = 9.8 [N] 1 [N] = 1 [kgw] 9.8 = 0.102 [kgw]

58 4.. (1) 0.5 [kgw] [N]. (2) 0.7 [kn] [kgw]. (3) 800 [g], [gw].,.,, SI., [J] ( [kj], [MJ]). 1 [J( )] = 1 [N] 1 [m],,.

4.6. 59 1843, (J.P.Joule),,,. 1 [cal] = 4.2 [J] 1 [J] = 0.24 [cal].., 1 [cal] = 4.2 [J]., 1 [kcal] = 10 3 [cal] = 4.2 10 3 [J] = 4.2 [kj] 1200 [N], 5 [m], [kcal]. W = 1200 [N] 5 [m] = 6000 [N m] = 6000 [J] = 6 [kj] 1 [kcal] = 4.2 [kj] 6 [kj] = 6 4.2 [kcal] = 1.4 [kcal]. ( ). (1) 30000 [cal] (J) (2) 4.6 10 5 [J] (kcal) (3) 0.4 [kcal] (J)

60 4 1. 45 [m/s]. [km]. 2., 515.4 [km] 2 30. [km/h]. 3., 40 [km/h] 4 [ s ]. [m/s 2 ]. 4. 1.00 [m], 2.00 [m], 2.00 [mm].., [N]., 7.86 [g/cm 3 ]. 5. 50 [cm], 20 [cm] 2 [ t ], [kpa]., 9.8 [m/s 2 ]. 6. 100 [m], 10 [m 3 ]., [kw]., 1000 [kg/m 3 ], 9.8 [m/s 2 ]. 7. 30 [kg] 5 12 [m/s].

61 5,,,.,,,,,,,. ( ),,,.,,. 5.1 C = 90 ABC, BC = a, CA = b, AB = c. a, b, c, b a, a b, c b, b c, a c, c a 6., 3., A = α, B = β. (, α + β = 90. ) a = sin α (, ) c sin, sine. b c a b α sin ( ). a = 3.0 [m], c = 5.0 [m], sin α = 3.0 = 0.60. 5.0 = cos α (, ) cos, cosine. = tan α (, ) tan, tangent., a = 3.0 [m], b = 4.0 [m], tan α = 3.0 = 0.75. 4.0

62 5, sin, cos, tan,. sin, cos, tan.., (,.),,., α,, β ( ). (, β, C,.) sin β = b c cos β = a c tan β = b a, sin, tan.. (1) sin, s. AB BC. sin α = BC AB (2) cos c,., AC. cos α = AC AB (3) tan t., tan α = BC AC

5.2. 63 5.2, 2., 2,,. 2., I, II. I, A = B = 45, C = 90. CA = BC = a,, AB = 2 a,, CA : BC : AB, 1 : 1 : 2, sin 45 = a = 1 2 = 2a 2 2 cos 45 = a = 1 2 = 2a 2 2. tan 45 = a a = 1, II, A = 30, B = 60, C = 90. 2 b, AB = 2 b, BC = b, AC, AC 2 + BC 2 = AB 2,, AC 2 = AB 2 BC 2 AC = AB 2 BC 2 = (2 b) 2 b 2 = 3 b 2 = 3 b, 3 AB : BC : CA, 2 : 1 : 3., sin 30 = b 2 b = 1 2 3 b 3 cos 30 = = 2 b 2 tan 30 = b = 1 3 = 3 b 3 3

64 5, 3 b 3 sin 60 = = 2 b 2 cos 60 = b 2 b = 1 2 3 b tan 60 = = 3 b.,, ( )., sin, cos, tan 3., ( ),.. A = θ ( ). α, β, θ. c a = cosec θ. cosecant ( ) c b = sec θ. secant ( ) b a = cot θ. cotangent ( ) 4.1, cosecθ = 1 sin θ, sec θ = 1 cos θ, cot θ = 1 tan θ. 6. sin, cos, tan.

5.3. sin cos 65 5.3 sin cos, sin, cos, tan., sin α = a c, cos β = a c, sin α = cos β, α + β = 90, sin α = cos(90 α).,. sin β = cos(90 β) cos α = sin(90 α) cos β = sin(90 β), sin 15 = cos 75, cos 50 = sin 40. tan, tan α = a b, tan β = b a 1 tan β = tan α = 1 tan(90 β)., 0 < α < 90, 0 < β < 90., α, β,., tan α = a b,. a c = sin α, b c a b = a c b c = cos α tan α = sin α cos α., α (, tan 90 ).

66 5 5.4 0, 90, α β, 0, 90,., ( ) 0, 90,. (,.),. α 0, β 90 ( α 0, β 90 ). b c, a 0. sin α = a c, α 0, a c. a, c 0., sin 0 = 0., cos α = b c, cos 0 = 1., β 90 b c. b c 1. sin 90 = 1., a 0 a, c 0., cos 90 = 0., tan 0, tan 90. tan α = a b, α 0, a, 0 a., b 0 tan 0 = 0. tan β = b a, β 90, a b, tan β.,, 0, 30, 45, 60, 90.

5.4. 0, 90 67 θ 0 30 45 60 90 1 2 3 sin θ 0 1 2 2 2 3 2 1 cos θ 1 0 2 2 2 3 tan θ 0 1 3 3, θ, (.)., θ,., θ, c, a, b a c = sin θ a = c sin θ b c = cos θ b = c cos θ 2. 1. 3, sin θ, cos θ, sin α, cos α, tan α, sin β, cos β. 2. a, b.

68 5 5.5,., a 2 c 2 a 2 + b 2 = c 2. c 2, + b2 c 2 = 1, sin θ = a c, cos θ = b c a 2 ( a c 2 = c b2 c 2 ) 2 = (sin θ) 2 = (cos θ) 2, (sin θ) 2 + (cos θ) 2 = a2 c 2 + b2 c 2 = a2 + b 2 c 2 = c2 c 2 = 1., (sin θ) 2 = sin 2 θ, (cos θ) 2 = cos 2 θ, sin 2 θ + cos 2 θ = 1.,,., θ. 3. sin 2 θ + cos 2 θ = 1 θ = 0, 30, 45, 90. ( 5.4.) sin α = a c, cos α = b c, tan α = a b, tan α = sin α cos α cos(90 α) = a c = sin α sin(90 α) = b c = cos α tan(90 α) = b a = 1 tan α sin 2 θ + cos 2 θ = 1 a = c sin α, b = c cos α, a = b tan α

5.6. 69 5.6, 5.2 5.4., 0 90., (, ), sin θ, cos θ, tan θ, θ,, θ.,. y = x 2 y x 2.,, sin 15, cos 15., θ = 15,, a, b, c, a c, b c sin 15, cos 15.,.,.. 1 sin, cos, tan 0 90., sin θ, cos θ, tan θ.,. sin 15 = 0.2588, cos 15 = 0.9659, tan 15 = 0.2679. 4.. (1) tan 38 (2) cos 60 (3) sin 75 (4) tan 53 (5) cos 8 5. θ. (1) sin θ = 0.333 (2) cos θ = 0.850 (3) tan θ = 2.36, 0 90., 170 315,,

70 5., 0 90,. 6.. (1) sin 43, cos 25, tan 86. (2) sin 2 37 + cos 2 37 = 1. (3) cos 40 sin 50 sin 70 cos 20. 1. 1. 16.0 [m] A B, 55 (.). 1.5 [m]., C 1.5 [m] D. ABD, BD = tan 55 AD, tan 55,, BD = AD tan 55 = 16.0 1.428 = 22.8 [m] BC = BD + DC = 22.8 + 1.5 = 24.3 [m]

5.6. 71 1. 20 [m] AB A, C, 30. BC x. 2.. (1) B AC D. BD. (2) S [m 2 ]. (3) BC. 3., A = 90, AB > AC ABC, AD BC, BC = 13, AD = 6. (1) BD, CD. (2) cos θ.

72 5 4.,. XX, YY A B. A X, C B BCA = 39. C, X 30.0 [m] D B BDA = 27. AB m (, AB = a, AC = b )., tan 27 = 0.51, tan 39 = 0.81. 5. A C α., A a [m] B β. h. ( : BD = b. α, β, b, h. )

73 6 6.1 6 1,,.,..,,,. 6 2,,.,..,,,. 6 3

74 6, ( ). 6 4 6 5,.,. 6 6, 3 [kg], 1 [kg] 3., 3. 1 [kgw], 1 [kg].,,,. 6 6,,.,,. 6 7, a, a., a, a = a. 6 7 a a,,.,,,, a.,,.

6.2. 75 6.2 6.2.1 a = b 2 a b, 1, 2,. 6 8, 2. 6.2.2 6 8 a 1, 2, a. 6.2.3 n. d = n a, d a n., n = 2 d = 2 a, n = 3 d = 3 a. 6 9, a, 2 a a 2, 3 a a 3. 6 9 a n, (scalar).,,,,,,.,.

76 6 6.3 3 a, b, c, a + b = c, c a b. 6 10 a + b = c a b,. a b,., 6 10 a = OA., a A b. b B, O, B OB c. 2., a b 2. 2, 6 10., O OB a + b = c.

6.4. 77 6.4 a b.,. ( a b = a + ) b, a b. a b, 6 11., ( a b = a + ) b = d. 6 11 a b = d 1. 6 11, d + b = a, a b = d. 6.5. a = OA, a = AB. a a = a + ( a ) = OA + AB (, ), B O. OB,, 0,.,. a a = 0 0, 0.

78 6 1. a, b.,. (1) a + b (2) 2 a (3) 2 b 2. 1 a, b, (,, ). (1) a b (2) a + 2 b (3) 2 a 3 b 2. ABC, AB + BC + CA. AB + BC = AC AB + BC + CA = AC + CA = AC + ( AC) = 0

6.6. 79 3. ABC. (1) BC + CA (2) AB AC (3) AC BC 6.6.,,,. (1) a + b = b + a (2) ( a + b ) + c = a + ( b + c ), m, n (3) (m + n) a = m a + n a (4) m ( a + b ) = m a + m b. 4.. (1) 2 (3 a ) (2) 2 a + 3 a (3) 4 b b (4) 3 ( a + b ) 2 ( a 2 b )

80 6 6.7 2 a, b, 1 c ( = a + b ).., 1 a, 2 a 1, a 2,. a = OP, a 2 OA, OB. P OB, OA, OA, OB P 1, P 2. OP 1 = a 1, OP 2 = a 2, a = a1 + a 2., a OA, OB 2 a 1, a 2. OA, OB, OA, OB, OA x, OB y. 6.8, x, y O a = OP, x, y. x a 1 = OP 1, y a 2 = OP 2. P 1, P 2, P (a 1, 0), (0, a 2 ), (a 1, a 2 ), a a a = a 1 2 + a 2 2., a 1 a x, a 2 a y. 6 12, 1, x, y, x i, y j. i e 1, j e 2.

6.8. 81 i, j,, a1 = a 1 i, a2 = a 2 j a = a1 i + a2 j.,, a = (a1, a 2 ) 6 13., i = (1, 0), j = (0, 1)., a 2. a = a1 i + a2 j = ( a1, a 2 ),,. (1) (a 1, a 2 ) = (b 1, b 2 ) a 1 = b 1, a 2 = b 2, a = b, x, y. 2. (2) (a 1, a 2 ) + (b 1, b 2 ) = (a 1 + b 1, a 2 + b 2 ) a, b a + b, x, y. 6 14., k, (3) k (a 1, a 2 ) = (k a 1, k a 2 ), m, n, 6 14 (a 1, a 2 ) + (b 1, b 2 ) (4) m (a 1, a 2 ) + n (b 1, b 2 ) = (m a 1 + n b 1, m a 2 + n b 2 ) (3), (4).

82 6 3. a, b, c, d, e, a, b, c, d, e. ( 1 1 1 ),. a = ( 3, 2 ), a = 3 2 + 2 2 = 13 b = ( 2, 3 ), b = 2 2 + 3 2 = 13 c = ( 4, 2 ), c = ( 4) 2 + ( 2) 2 = 2 5 d = ( 3, 2 ), d = ( 3) 2 + 2 2 = 13 e = ( 0, 4 ), e = 0 2 + ( 4) 2 = 4 4. a = (3, 2), b = (2, 3), a + b,. a + b = (3 + 2, 2 + 3) = (5, 5) a + b = 5 2 + 5 2 = 50 = 5 2 5. a = (3, 2), b = ( 2, 1),,. (1) a + b (2) 3 b (3) 2 a 3 b (4) a + 2 b

6.8. 83 5. 6 15 x, y 2 A, B, AB AB. AB = (b 1 a 1, b 2 a 2 ) AB = (b 1 a 1 ) 2 + (b 2 a 2 ) 2 6 15 6. 6 16 x, y 2 A, B, AB AB. AB = (4 1, 3 2) = (3, 1) AB = 3 2 + 1 2 = 10 6 16 6. x, y 2 A, B, AB, AB. (1) A (4, 2), B ( 1, 5) (2) A (6, 3), B (1, 4) (3) A (0, 5), B (2, 1) 7. O 2 F 1, F 2 ( 1 1 [N] ). (1) F 1, F 2. (2) F 1, F 2 F. (3) F. (4) F F.

84 6 6.9 a = OP x θ. a = a, 6 17 a 1 a = cos θ, a 2 a = sin θ, a a 1 = a cos θ, a 2 = a sin θ,, 6 17 a a = a1 + a 2 = a 1 i + a2 j = a cos θ i + a sin θ j = (a cos θ, a sin θ). x, y,. 7. a = 10, θ = 60, a = (a 1, a 2 ) a 1, a 2. a = (a 1, a 2 ) = ( 10 cos 60, 10 sin 60 ) ( = 10 1 ) 3 2, 10 2 = (5, 5 3) a 1 = 5, a 2 = 5 3 8. a = 5, θ = 30, a = (a 1, a 2 ) a 1, a 2.

6.10. 85 6.10,,.,, F 0,,., 6 18, O 3 F 1, F 2, F 3,,. F 1 + F 2 + F 3 = 0 6 18 3 8. OA θ, OB., W [N] P, B, P., P S, N., P, W, S, N., x, y (x, y. OA x ).

86 6, x, y, S, N x, y,., W,. W = (W x, W y ) = ( W sin θ, W cos θ ) P, 0, x S W sin θ = 0 y N W cos θ = 0 S = W sin θ [N], N = W cos θ [N] 9., P W 10 [N], θ 30, S N, [N]. S = 10 sin 30 = 10 0.50 = 5.0 [N] N = 10 cos 30 = 10 0.87 = 8.7 [N] 9. W = 50 [N]. θ = 45, 0., S [N], N [N].,. (1) N, S, W, θ,. (2) N, S, W, θ,. (3) S [N] N [N]. ( )

6.11. 87 10. 20 [N] P, O 2 OA, OB,. OA, OB S 1, S 2, P W ( 20 [N]),. (1) x, y, S 1, S 2 x, y. (2) x, y 0 ( ) (, W ). (3) W, S 1, S 2 (, 2 = 1.4, 3 = 1.7 ). 6.11,., ( ). v ( V ). 6.11.1, 6 19 A v A. B v B., C B, B.,. 6 19 B

88 6 C A A B v A v B C B v A + v B C B (C B) v, v = va + v B., 6 20.. 6 20 6.11.2 OA, OB 45,., OA P 30 [km/h] v P, OB Q 60 [km/h] v Q., Q P v. 6 21. Q P v Q v P P Q v Q v P,, P Q (P Q). 6 21

6.11. 89., v P = 30 [km/h], v Q = 60 [km/h], OA x, y OA, v P, v P = 30 i = (30, 0), v Q, 6 22 v Q = 60 cos 45 i + 60 sin 45 j = 30 2 i + 30 2 j = (30 2, 30 2 ), 6 22 v Q v = (vx, v y ) = v Q v P,., v, 6 23 α. tan α = v y v x = 42.3 12.3 = 3.43, α. α 74 = (30 2 30, 30 2 0) = (12.3, 42.3) v = v = 12.3 2 + 42.3 2 = 44.1 [km/h] 6 23 v, P Q 74, 44.1 [km/h].

90 6 11. V 1 = 0.5 [m/s], V 2 = 0.3 [m/s] 50 [m].,.,,. 12. 40 [m], 6.0 [m/s]. A B, 8.0 [m/s]..,.,.,.,,,.,.,,,.,.,.

6.11. 91 1. ABC,. (1) AB AC CB. (2). BC BA 2.. 2 ( a 4 b + c ) 2 ( a 2 b + c ) 3.. F 1 = (2, 2), F 2 = (2, 2), F 3 = ( 3, 4) (1) F. (2) F. ( ) (3) i, j F. (4) F x ( ).. 4. 10 [N] P, A 20 [cm], 30., S 1, S 2, P W,. (1) x, y 0 ( ). ( W ) (2) S 1, S 2. ( ) (3) P. h.

92 6 5., 30, 10 [N] 45.,. (1) S, N, W,.,. (2) x y,. (3) (1), (2), S, N. 6. A 36 [km/h]., 100 [m] B 54 [km/h]. A B,. 1 [m/s] = 3.6 [km/h]

93 7, ph ( ) 4 5. ph 1 0.1 (mol/l), ph 5 0.00001 = 10 5 (mol/l).,.,,.,,.,,.,,.,.,, NTC.,., 7 1,,.,.,. 7 2..,.,,. 7 1 7 2

94 7,,,.,,.. 7.1 1 a, N a m = N ( ) m. m, a ( ) N (Logarithm; ) m = log a N ( )., N, a m (log a N, a, N ). a m = N m = log a N ( ) ( ),,,. 1. 2 4 = 16 4 = log 2 16 10 3 = 1000 3 = log 10 1000, a 0 = 1 0 = log a 1,. a 1 = a 1 = log a a (1) log a 1 = 0, log a a = 1 1.. (1) 2 3 = 8 (2) 5 2 = 25 (3) 3 4 = 81 (4) 2 1 2 = 1.41 (5) 16 1 2 = 4

7.1. 95 2. a N. (1) log a 16 = 4 a 4 = 16 a = 2 (2) log 4 N = 1 2 4 1 2 = N N = 4 = 2 2. a N. (1) log 9 N = 2 (2) log a 125 = 3 (3) log a 27 = 3 (4) log 49 N = 1 2 (2) a 1, A, B log a AB = log a A + log a B log a A B = log a A log a B 1 log a A n = n log a A 2 log a n A = 1 n log a A n A = A 1 n. P.23. 1 1, 100 = 100 2 = 10, A = A 2. [ log a AB = log a A + log a B 1 ] log a A = m, log a B = n A = a m, B = a n AB = a m a n = a m+n m + n = log a AB log a A + log a B = log a AB ( )

96 7 [ log a A n = n log a A 2 ] n log a A n = log a A } A {{ A A} n = log a A + log a A } A {{ A} n 1 ( 1 ) = log a A + log a A + log a A } {{ A} n 2 = = log a A + log a A + log a A + + log a A }{{ A} 2 = log a A + log a A + log a A + + log a A + log a A }{{} n = n log a A ( ) 2, log a A n, n log. 2, n. 3. 1, log a A B = log a A log a B. 4. 2, log a n A = 1 n log a A. 5. log a 2 = x, log a 3 = y, x, y. (1) log a 6 (2) log a 3 2 (3) log a 1 2 (5) log a 3 2 (6) log a 6 (7) loga 8a 2 (4) log a 9 8 (3) a, b 1, A log a A = log b A log b a

7.2. 97 log a A = c., a c = A. a c = A b log b a c = log b A c log b a = log b A log a A log b a = log b A log b a 0. log a A = log b A log b a 6. log b a log a b = 1. 7.2 10.,., N, 10, log 10 N = log N.,. 7.2.1 N log N., log 10 1 = 0, log 10 10 = 1 log 1 log 10 log 1 = 0, log 10 = 1, 1 10 (N) (m), 0 1.

98 7 10 a m = N m = log a N ( ) ( ) a = 10 ( 10 ), 10 m = N m = log N ( m :, N : )., 10 m = 2.00 m = log 2.00,, 2.00 0.3010, 10 0.3010 = 2.00 0.3010 = log 2.00. (N) (m) 10 m = N m = log N ( ) (N) 1.00 N < 10.00, ( ) (m) 0.0000 m < 1.0000. 10. 7.2.2. log 3 = 0.4771, 30, 300, 0.3. 3. log 3 = 0.4771 log 30. log 30 = log( 3 10 ) = log 3 + log 10 = 0.4771 + 1.0 = 1.4771 ( log A B = log A + log B ) 4. log 3 = 0.4771 log 0.3. log 0.3 = log(3 10 1 ) = log 3 + log 10 1 = 0.4771 + ( 1) = 0.5229

7.2. 99 7.., log 2 = 0.30, log 3 = 0.48, log 7 = 0.85. (1) log 9 3 (2) log 5 14 (3) log 2 9 log 3 4 8. log 2 = 0.3010, log 3 = 0.4771. (1) log 200 (2) log 0.0003 (3) log 6 (4) log 5 log ln log, 10, e (2.718 ), log e ln.., e. e.. 7.2.3 7 1, t [ C], R [kω] R = b 10 at ( a, b,, 7 2.., R = b 10 at ( log ). log R = log ( b 10 at ),,, log R = log b + log 10 at log R = log b + a t log 10 log R = log b + a t y = a x + b., log,. 9., (. ) 1 at (1) 5 y = 2 x (2) N = b 2

100 7 7.2.4,,.,, f., 3...,. (1). (2) log,. (3).

7.2. 101 f [Hz] 262 294 330 349 392 440 494 log f 2.42 2.47 2.52 2.54 2.59 2.64 2.69 f [Hz] 523 587 659 698 784 880 988 log f 2.72 2.77 2.82 2.84 2.89 2.94 2.99 f [Hz] 1047 1175 1319 1397 1568 1760 1976 log f 3.02 3.07 3.12 3.15 3.20 3.25 3.30. 3..,,.

102 7 7.3 7.3.1,.,,. ( Hipparchus ; BC 120 ), 1, 6,. ( N. P. Pogson ; 1829 1891 ), 6 1 1 100. 5 1 100. 1 10000. 10.. 1 15. 1000000.. 20..,.,,.... y = a x x = log a y 2 4 = 16 4 = log 2 16,., 1000000 1000000 = 10 6, 10 ( ), log 10 1000000 = 6, 1000000 6 1. 10 20. 20.

7.3. 103,., 1 1000000 = 10 6 ( 10 ), log 1 1000000 = 6, 6. 1 1000000.... 1, 10, 11. 1 10000 11 1 = k log 1 10000, k., 10 1, log 10000 log 1 = 4. 10000. 10 = k ( 4) k = 2.5,. m, F, F 0 1, m 1 = 2.5 log F F 0., ( ), ( ),, UFO. 4, 1., 1.

104 7, 4, 1,., 4 1 = 2.5 log F F 0 5 = 2.5 log F F 0 log F F 0 = 5 2.5 = 2 1,, F F 0., log F F 0 = 2 F F 0 = 10 2 ( log X = a X = 10 a ) 1 100.,,., 2.,., 2.., F 1 2 = D 2 F 2 = 2 D 1 =., ( D2 D 1 ( D1 D 2 ) 2 ) 2 m 1 1 = 2.5 log F 1 F 0 m 2 1 = 2.5 log F 2 F 0 ( m 1 m 2 = 2.5 log F 1 log F 2 F 0 F 0 )

7.3. 105. log x y = log x log y m 1 m 2 = 2.5 { ( log F 1 log F 0 ) ( log F 2 log F 0 ) } = 2.5 ( log F 1 log F 2 ) = 2.5 log F 1 F 2. F ( 1 D1 = F 2 D 2 ( D1 m 1 m 2 = 2.5 log D 2 ) 2 ) 2. log X n = n log X., m 1 m 2 = 2.5 ( 2) log D 1 D 2 = 5 log D 1 D 2.,..,. 1. 4.85 10 6 [pc] ( parsec : ) 26.7., 10 [pc],. (, [m],, [pc].), log 4.85 = 0.6857. m 1 = 26.7 [ ], D 1 = 4.85 10 6 [pc], D 2 = 10 [pc] m 2 m 1 m 2 = 5 log D 1 D 2 m 2 = m 1 5 log D 1 D 2.

106 7, m 2 = 26.7 5 log( 4.85 10 7 ) = 26.7 5 ( log 4.85 7 ) = 26.7 5 ( 0.6857 7 ) = 4.9 [ ].?? 10 [pc]. 1,!, 1 [pc] = 3.0857 10 16 [m]. ( P. 113 4 ) 10., 10, 2. 7.3.2 ph ( ;, (ph) ; ), 5.8. 4.7 ph, power of Hydrogen ( ) 8.3, 2.3. 6.6, ph, 3.3 ph, 7.4., ph. 12.6,, [H + ] ( [ ] ). (mol/l). ph

7.3. 107 [H + ], 10 1 (mol/l) 10 14 (mol/l).,. ph,,. ph, ph = log 1 [H + ] = log 1 log[h + ] = 0 log[h + ] = log[h + ]. ph ph = log[h + ], [ ]. 2.0 [m], 5.0 [kg], ph, [H + ], (mol/l).,. ph = 1.0 ph = 1.0 = log[h + ] log[h + ] = 1.0 = ( 1) 1.0 = ( 1) log 10 = log 10 1 log [H + ] = 10 1 (mol/l)., ph = 2.0 [H + ] = 10 2 (mol/l),.,.

108 7 ph 1 2 3 5 7 9 11 13 14 10 1 10 2 10 3 10 5 10 7 10 9 10 11 10 13 10 14 2. [H + ]( ) = 10 6 (mol/l) ph. ph, ph = log[h + ], ph = log 10 6 = ( 6) log 10 = 6. 3. ph = 2.5., 10 0.5 = 3.2. ph, ph = log[h + ], 2.5 = log[h + ] log[h + ] = 2.5 log X = a X = 10 a [H + ] = 10 2.5 = 10 3+0.5 = 10 3 10 0.5 = 3.2 10 3 (mol/l).

7.3. 109 11. ph., log 2 = 0.30, log 3 = 0.48, log 7 = 0.85. (1) 4.9 10 3 (mol/l) (2) 5.6 10 11 (mol/l) (3) 6.3 10 7 (mol/l) 12. ph., 10 0.5 = 3.2. (1) ph = 5.0 (2) ph = 3.5 7.3.3 ( ),.,,.,,,. ( SPL : Sound Pressure Level ). (decibel) [db], = 20 log [db].,,, 1000 [Hz] ( 1 1000 ) ( [ ], P 0.). ; P 0 = 2 10 5 [N/m 2 ] = 2 10 5 [Pa]. (N. Pa.),,,,,., ( ) P. P 0 P P, ( P 0 ).,, P = 0.1 [Pa], P = 100 [Pa].

110 7 P 0 = 2 10 5 [Pa], P = 100 [Pa],. 20 log P P 0 = 20 log 100 2 10 5 = 20 log 107 2 = 20 ( 7 log 10 log 2 ) = 20 ( 7 0.3 ) = 134 [db],,, P = P 0 = 2 10 5 [Pa], 20 log P P 0 = 20 log 2 10 5 2 10 5 = 20 log 1 = 0 [db]., 0 [db]. 0 [db] 134 [db], 5 10 6, 500. 13., P = 0.10 [Pa]. [db]., log 2 = 0.30. 14. 64 [db].., 10 0.5 = 3.2, log 2 = 0.30., 50 60 [db],, 70 80 [db], 100 [db].,, ( ), 100 %,.,..

7.3. 111..,, 2 10 5 [Pa], 60 [Pa] 10 6,,, 16 [Hz] 16 [khz] 10 3.,. 1 [khz] 2 10 5 [Pa]. 1, 0 [db] (,, ) 10, 60 [Pa] 130 [db]. 1 [khz] [db] [phon ( ) ],.,, 1 [khz], [phon]..,,. 80 [db].,.., 80 [db], 0.2 [Pa].,. x 80 [db] = 20 log 2 10 5 [Pa] = 20 { log x log( 2 10 5 ) } log x = 4 + log( 2 10 5 ) = log( 2 10 1 ) x = 0.2 [Pa]

112 7.,,.,,.., [db], 20 log P, P 0.,,,., ph,., K K = 2.303 (1/t) ( log C 0 log C ). C 0, C t.,,.

7.3. 113 1.., log 2 = 0.30, log 3 = 0.48, log 7 = 0.85. (1) log 35 (2) log 600 (3) log 0.9 (4) log 2 9 (5) log 6 7 (6) log 3 4.9 2.., 20%, 1%,., 5%, log 2 = 0.3. 3., 10 100., 10, 1, 100., log 2 = 0.3. 4.. A 1.5 2.0 0.1 0.0 26.7 A,., 10 0.4 = 2.5. 5., ph 5.6. 1999, ph = 5.1,. (mol/l)., 10 0.9 = 7.9.

114 7 6. ( ), 1 80 %. 100 [db] 40 [db],. 7. ( ) G, 40 [db]. [V] 1 [V]., G [db], V o [V], V i [V]. G = 20 log V o V i

115 8 8.1,,., 1 360,.,.., θ = XOP, OX ( ), O OP ( ). OX OP,,.,,,., OP 50, 310., 90, 180., OP 1, 50 + 360 = 410., OP,.. OP, 50 360 = 310. OP θ, θ = α + 360 n ( n = 0, ±1, ±2, ). (n )

116 8 1., α. ( 0 α < 360 ) (1) 480 (2) 150 (3) 740 (4) 1230 8.2 X OX, Y OY, XOY, YOX, X OY, Y OX 1, 2, 3, 4 ( x, y, )., θ = 45 1, θ = 150 2, θ = 100 3, θ = 300 4. ( x, y ).. 2.. (1) 380 (2) 250 (3) 235 (4) 420 8.3 OP r, x, θ P ( x, y ). x, y,., sin θ = tan θ = y r y x, cos θ = x r, ( x 0 ) sin θ, cos θ,. 1 ( 0 < θ < 90 ), 2, 3.

8.3. 117 2, θ = 150. x < 0, y > 0 1 sin 150 = y r = 2 r = 1 r 2 = 0.5, sin 30 = 0.5., sin 150 = sin 30, 150 =180 30, θ, sin(180 θ ) = sin θ. 3 cos 150 = x r = 2 r 3 = r 2 3 cos 30 =, cos 150 = cos 30., θ, 2 cos(180 θ ) = cos θ. 3, θ = 210. 3 x < 0, y < 0 sin 210 = y 1 r = 2 r = 1 r 2 sin(180 + 30 ) = sin 30., 3 cos 210 = x r = 2 r 3 = r 2 cos 210 = cos(30 + 180 ) = cos 30. 3. tan ( ). 4. 4 sin θ, cos θ, tan θ.

118 8 8.4 θ y = sin θ, y = cos θ, y = tan θ. 1 ( ) sin θ, cos θ, tan θ. 2 4,. 1 y = sin θ ( 0 θ 360 ) θ, sin θ. θ 0 30 45 60 90 120 135 150 sin θ 0.00 0.50 0.71 0.87 1.00 0.87 0.71 0.50 θ 180 210 225 240 270 300 315 330 360 sin θ 0.00 0.50 0.71 0.87 1.00 0.87 0.71 0.50 0.00 360 y = sin θ, θ = α + 360 n,.,. 2 y = cos θ ( 0 θ 360 ),. θ 0 30 45 60 90 120 135 150 cos θ 1.00 θ 180 210 225 240 270 300 315 330 360 cos θ, sin θ 90.,.

8.4. 119 cos θ 360. 3 y = tan θ ( 0 θ 360, 90, 270 ) tan θ, sin θ, cos θ.,. θ 0 30 45 60 90 120 135 150 tan θ 0.00 θ 180 210 225 240 270 300 315 330 360 tan θ 5. (1) 2 sin θ = 0.40, cos θ, tan θ. (2) 4 tan θ = 2.00, sin θ, cos θ. 6.,. (1) sin 115 (2) cos 223 (3) tan( 38 ) (4) cos 403 (5) sin( 412 ) (6) tan 949

120 8 8.5 8.5.1, ( ). 30, 420, 130. 90, 1 360 60 (, )., ( ). r, r AB ( AB ), AB AOB= α., 2 π r, α 360 = AB 2 π r = r 2 π r = 1 2 π, α = 360 2 π = 180 180 57.3 π 3.14 ( ). α (radius) (angle), (radian). radian [rad]. 1 [rad], 57.3., 1 [rad] = 180 π, π [rad] = 180, 2 π [rad] = 360.,, [rad].,. sin 30 30 sin, sin 30, 30 [rad] sin. 7. [rad]. (1) 30 (2) 45 (3) 60 (4) 90 (5) 270 (6) 450 8.. (1) sin π 6 (4) tan π 3 (2) cos π (3) sin π 4 2 (5) cos 5 π (6) tan ( 4 ) 3 π

8.5. 121 60,,, (,, ).,,, 60, (,, [rad].).., 90?, ( ), 1 360. 1 (1 ) 360, 1, 1., 1 360, 1 4 (90 )., 6. 360 6 = 60, 60. 1 60 1 ( ), 1 = 60, 1 60 1 ( ), 1 = 60,. 8.5.2 60,. 1. ; l AOB = 1 [rad], AOC = θ [rad], 1 r = θ l, l = r θ., l = 2 π r θ 2 π, l = r θ

122 8, 60 AOC = θ, l 2 π r = θ 360, l = π r θ 180, l,. 2. ; S AOB = θ [rad], S = π r 2 θ 2 π = 1 2 r2 θ, AOB= θ, S = π r 2 θ 360 = π r2 θ 360, [rad]. 2,,,, [rad].,. [rad] [rad].,, [rad]., [ : ω ( ) ], 1 (second). 60 360 ( = 2π [rad] ), ω = 2 π 60 = π 30 [rad/s]. ( [ s ] second )., [rpm] revolution per minute ( 1 ). [rad/s]., 600 [rpm], [rad/s]

8.5. 123,.,,.,?.,,,.,.,,,,.,.,,....

124 8 1. θ ( sin θ, cos θ, tan θ ),., [ ] [rad], [rad] [ ]. (1) θ = 24 (2) θ = 481 (3) θ = 4 3 π [rad] (4) θ = 13 ( π [rad] (5) θ = 3 7 ) π [rad] 4 6 2., θ, ( 0 θ < 360 ). 3 (1) θ 2 sin θ =, cos θ tan θ 2 (2) θ 4 cos θ = 0.4226, sin θ tan θ 3. y = sin 2θ ( 0 θ 360 ). θ 0 15 30 45 60 75 90 105 sin 2θ 0 θ 120 135 150 165 180 195 210 225 sin 2θ θ 240 255 270 285 300 315 330 345 360 sin 2θ

125 9,,.,, ( )., 1 2,,. 9.1 9.1.1,?,.,,. (1) ( ).,., ( ) ( ),. (2), 1,.,.

126 9 (3), ( ).,,. 9.1.2,. 1. 9 1 9 1,,, 1.00, 2.00, 3.00, 4.00, 5.00 [ma],. 1. [ma] 0 1.00 2.00 3.00 4.00 5.00 [V] 0 1.05 1.90 2.95 3.80 5.50.

9.1. 127 9 2.,.,,.,,.,,,.,.,,,,..,. (,,,.)

128 9 9.1.3 1 9 2, 1. 1,. y = a x + b ( a 0, a, b ), y x 1, y x 1. a, b a = y x b y. (, x, y, x y.) 2 (x 1, y 1 ), (x 2, y 2 ). y 1 = a x 1 + b y 2 = a x 2 + b 1 2 2 1 y 2 y 1 = a x 2 a x 1 = a (x 2 x 1 ) x 2 x 1 a = y 2 y 1 x 2 x 1 = y x 3 1, a, b,. 9.1.4 9 2, 9.1.1. (1), 1., 5.00 [ma] 5.50 [V]. (2), 9.1.3, a y b,. ( x I, y V ), y b 0., 2, (0, 0) A (4.50, 4.40), 3. a = y x = V I = 4.40 [V] 0 [V] 4.50 [ma] 0 [ma] = 0.978 10 3 [Ω] = 0.978 [kω] = 4.40 [V] 4.50 10 3 [A] a, R

9.1. 129, V = R I ( R = 0.978 [kω] ).,., R. V = R I R. [ma] 0 1.00 2.00 3.00 4.00 5.00 [V] 0 1.05 1.90 2.95 3.80 5.50 [kω] 0 1.05 0.950 0.983 0.950 1.10 R 1 ( 1.05 + 0.950 + 0.983 + 0.950 + 1.10 ) = 1.01 [kω] 5,., 5.00 [ma], 4.75 [V], 0.977 [kω].,,.,,. (3),. 10.00 [ma],. 10.00 [ma],.,,., 9.78 [V]. 9.1.5,.., ( ).,,.,.,.,.

130 9 9.2 9.2.1., x F,. F = k x, k,. 1. L = 30 [cm] 1 ( w [N] ), x [cm],., L 2 ( 2 L = 60 [cm] ) 2,. L w [N] x [cm] 0 0 1.0 1.2 2.0 2.3 4.0 4.9 6.0 7.3 7.0 8.4 2 L w [N] x [cm] 0 0 1.0 2.3 2.0 4.9 3.0 7.1 4.0 9.7 5.0 12.0 (1). (2) k, x w. (3) 1, 3.0 [N],. (4) 2,.

9.2. 131 (1) L (2) (i) k = w x = 5.0 0 0.060 0 (ii) k = w x = 5.8 0 0.140 0 2L = 8.3 10 [N/m], F = 83 x = 4.1 10 [N/m], F = 41 x (3) 3.0 [N] ( 8.3 10 [N/m] ) = 0.036 [m] = 3.6 [cm] (4) 2, 2. y = a x + b 1, w = 0 [N], y = 0 [cm], b = 0, y = a x. 1.,. w, l,. w [N] 0 0.10 0.20 0.30 0.35 0.45 l [cm] 0 2.3 4.5 7.2 8.5 11.2 (1). (2) k., l w. (3) 1.00 [N],.

132 9 9.2.2 ( ),., 1 [ C], 2., ( : α ).. 2. 30.0000 [cm],,. t [ C] 20 45 70 95 120 l [cm] 30.0000 30.0104 30.0212 30.0300 30.0380 (1). (2) 25 [ C],.,,. (α) = [m] [K] [m] ( [1/K] ) (3) 20 [ C],. (4) 20 [ C] 25.000 [m]. 10 [ C], 50 [ C] [cm].,, (2). (5). 100 [km], 60 [ C], [m].,,.

9.2. 133 (1) (2) (α) = [m] [K] [m] ( [1/K] ). 20 [ C], 120 [ C] 2,, α = 0.300420 0.300000 120 20 0.300000 = 0.000014 [1/K]., 1.4 10 5 [1/K]. (3) 20 [ C] t [ C] l, 20 [ C] l 0 l = l 0 { 1 + ( t 20) 1.4 10 5 }. (4) 10 [ C] l 0 = 25 [m],, 50 ( 10) = 60 [ C]. l = 25 [ 1 + { 50 ( 10) } 1.4 10 5 ] = 25.021 2.1 [cm]. (5) l = 1.00 10 5 ( 1 + 60 1.4 10 5 ) = 1.00084 10 5 84 [m].

134 9 2. 30.00000 [cm]. t [ C] 20 40 60 80 100 l [cm] 30.00000 30.00222 30.00491 30.00722 30.00944 (1). (2) 20 [ C],. (3) 20 [ C],. (4) 0 [ C] 50.00000 [cm]. 500 [ C]., (2).,.,.,.,. 9 1 Pt 0.090 Au 0.142 Ag 0.193 Cu 0.162 Al 0.237 Fe 0.138 Ni 0.125 W 0.045 ( 10 4 [1/K], 0 [ C] 100 [ C] ).

9.2. 135 3.. t [ C] 20 40 60 80 100 ρ [µω cm] 6.900 7.020 7.144 7.272 7.360. (1). (2). ( 1 [ C] ) (3). (4) 0 [ C] 500 [ C].. ρ = ρ 0 { 1 + ( t 20) k } k :. (,, ),.,,.,,.,., 82 : 18. 20 [ C] 100 [µω cm], 0.0012.,, 14 15.

136 9 9.3 2,. 2. y = a x 2 + b x + c ( a, b, c a 0 ) 2, y = a (x 2 + ba ) x + c { ( = a x + b ) } 2 b2 2 a 4 a 2 + c ( = a x + b ) 2 b2 4 a c 2 a 4 a. p = b 2 a, q = b2 4 a c 4 a, 2, y = a (x p) 2 + q 2. y = a x 2 x p, y q, y = a x 2 + b x + c y = a x 2 x p = b 2 a, y q = b2 4 a c 4 a.,, ( b ) 2 a, b2 4 a c 4 a., a > 0, a < 0. 3. 144 [km] ( 40 [m] = 40 [m/s] ).. t [ s ] 0 2 4 5 6 7 8 l [m] 0 60 80 75 (1) l y, t x, y 2.

9.3. 2 137 (2) y = a (x p) 2 + q, p q. (3) 6, 7, 8. (4) ( ),. (1) y, x, 2 y = a x 2 + b x + c. 0 x = 0, y = 0., c = 0, y = a x 2 + b x. 2, 4 x = 2 y = 60 = 4 a + 2 b 1 x = 4 y = 80 = 16 a + 4 b 2 1 2 2 b 120 = 8 a + 4 b ) 80 = 16 a + 4 b 40 = 8 a a = 5 1 b = 40 y = 5 x 2 + 40 x 3 (2) y = 5 x 2 + 40 x = 5 (x 2 8 x) = 5 (x 2 2 4 x) = 5 (x 2 2 4 x + 16 16) = 5 { (x 4) 2 16 } = 5 (x 4) 2 + 80, 4, 80 [m]. (,.) (3) 3 x = 6, 7, 8, 60, 35, 0 [m].. t [ s ] 0 2 4 5 6 7 8 l [m] 0 60 80 75 60 35 0

138 9 (4). 4. 12 [m] 4, x [m] y [m 2 ]. (1) 2. (2) y = a (x p) 2 + q, p q., x. (3) x 1, 2, 3, 4, 5, 6 [m]. (4),. 5. 40 [cm] y [cm 2 ] x [cm]. (1) y, 2. (2) y = a (x p) 2 + q, p q., x. (3) x 2, 4, 6, 8, 10, 12, 14, 16, 18 [cm]. (4),.

9.3. 2 139,..,.,..,,.,.,. [ 1 ] ( ).,,.,,.,.,,.. t [ C] 19.2 30 40 50 60 70 80 R [Ω] 2000 1500 1000 700 480 300 200,,.,.,, log R, log R = a + b x. log R = y y = a + b x. 1, a y, b., log R = a + b x log R = 10 a+bx,. a, log 4600 = 3.66 2, x 1 = 0 y 1 = log 4600 = 3.66, x 2 = 100 y 2 = log 100 = 2., y x = y 2 y 1 x 2 x 1 = 2 3.66 100 0 = 0.0166

140 9, R = 10 (3.66 0.0166x) = 10 3.66 10 0.0166x = 4600 10 0.0166x. ( ) ( ),. [ 2 ] [ 1 ],,. y = a x b.,. (.),., y, x y = a x b.,, log y = log a + b log x. Y = log y, A = log a, X = log x Y = A + b X, 1. Y A = log a, b.

9.3. 2 141,.., 5.,,.,,.,., (, ), (C) ( y ),. y = k C n ( k ), 5 (C), ( y ). (C).,. log y = log k + n log C, log y = Y, log k = a, log C = x Y = a + n x. 1,, n, Y a, k.,,.

142 9 1. 1 ( L = 30 [cm] ) 2,,. (1) k. (2) 2, W [N]. 2. 1.,. L = 30 [cm] 2 W [N] x [cm] 0 0 2.0 1.2 4.0 2.2 8.0 4.8 12.0 7.2 3. 2 k, x W.,, 14.0 [N],.

,,., force, f F. (Newton) [N],., v [m/s], /, /. [m/s]. ( ) α, β, γ, θ, [ ], [rad] angle l, L [m] length A, S [m 2 ] area, square V [m 3 ] volume t [s] time u, v, w, c [m/s] velocity a [m/s 2 ] acceleration ω [rad/s] angular velocity α [rad/s 2 ] angular acceleration, f, ν [Hz] frequency 143

λ [m] wave length m [kg] mass ρ [kg/m 3 ] density p [kg m/s] momentum f, F [N] force P [Pa] pressure A, W [J] work E, U, W [J] energy P [W] power T, Θ [K] thermodynamic temperature t, θ, [ C] Celsius temperature Q [J] calory,calolie C [J/K] heat capacity c [J/(kg K)] specific heat 144

I [A, C/s] electric current Q [C] electric charge V, φ [V] electric potential, V [V] voltage difference E [V] electromotive force R [Ω] resistance ρ [Ω m] resistivity ε [F/m] dielectric constant E [V/m] strength of electric field µ [H/m] permeability H [A/m] strength of magnetic field L, M [H] inductance C [mol/l] molarity 145

,,.,. A :.. [ ].. ( ) α α, α, α β β, β, β γ γ, γ δ ε ε η η : ; η = 80 % θ, θ : ; θ = 60 λ λ ( ) : ; λ = 4.0 [m] µ µ µ : ; µ = 0.5 ( ( 10 6 [m] ), ( 10 6 ) ) ν ν π π : π = 3.141 ρ ρ σ σ τ τ φ φ : φ = { } ω ω (.). t = t 2 t 1 Π ( ) Σ ( ) Φ Φ Ω [Ω] : ( ) a b c c 146

d d (distance), d (diameter) e e : e = 1.60 10 19 [C] f f (frequency) g g (gravity) : g = 9.8 [m/s 2 ] g [g] : ( (gram) ) h h (height) i i : i = 1 ( ) j i j : j = 1 ( ) j k k : k = 1.381 10 23 [J/K] l l (length) m m (mass) [m] : ( (meter) ) [m 2 ] : ( ) [m 3 ] : ( ) n n (numbers) q q (electric charge) : ; q = 1.0 10 6 [C] r r (radius) s [ s ] : ( (second), ) t t (time) : ; t = 3.5 [ s ] t : ; t = 50 [ C] v v (velocity) x y z A B C D E F G H I J [A] : ( (ampere) ) Φ S B C. [cal] [ C] : (Celsius) : ; 15 [ C] [C] : ( (coulomb) ) Q S D E (Electric field) E (Electromotive force) F (Force) G (Gravitation) H I I = (Impulse) [J] : ( ) 147

K [K] : ( ) : ; 300 [K] L L M M (mass) M (mutual inductance) N N, [N] : ( ) O O P P (pressure) P (power) P = Q Q (electric charge) : ; Q = 10 [C] Q : ; Q = 500 [J] R N R : ; R = 100 [N] R : ; R = 200 [Ω] S S : ; S = 40 [m 2 ], S : ; S = 50 [N] T S T T : ; T = 300 [K] U U V V (voltage), V (voltage difference) [V] : ( (volt) ) W W (weight) X : ; X ray, X 148

149

150 1 1 1 (1) 2 5 3 4 = 1 5 3 2 = 3 10 (2) 3 6 1 2 = 3 6 2 1 = 1 (3) 1 a 1 b = 1 a b (4) y x w z = y x z w = y z x w 1 (5) (6) (7) 2 a b 2 b c a = 1 a b c a = b a x 2 a c y = b b a x 2 = x a 2 b c y a c = b 2 c a 2 b b a c y a 2 c x 2 y x a 2 b c a c y = x a b y (1) 1 4 + 2 5 = 1 5 4 5 + 2 4 5 4 = 5+8 20 = 13 20 (2) 1 10 9 2 = 1 10 9 5 2 5 = 1 45 10 = 44 10 = 22 5 (3) q p + s r = q r p r + s p r p = q r+p s p r (4) h g j i = h i g i j g i g = h i g j g i (5) 1 x x+1 = (x+1) x x+1 (6) (7) (8) 1 a+1 1 a 1 = 1 x+1 (a 1) (a+1) = (a+1) (a 1) 2 = (a+1) (a 1) 2 a b c x 2 3 b c a x 2 = 2 a a 3 b c b c b c x2 a a x 2 b c = 2 a2 3 b 2 c 2 a b c x 2 1 3 a b 2 1 4 a 2 b = 4 a 3 b 12 a 2 b 2 3 (1) I = V R I R = V R = V I I = 0.3 [A], V = 3 [V] (2) b x 3 = 4 x c c (b x 3) = 4 x b c x 3 c = 4 x R = 3 0.3 = 10 [Ω] b c x + x = 3 c + 4 (b c+1) x = 3 c + 4 x = 3 c+4 b c+1 (3) b x + 3 = c x b b x c x = b 3 (b c) x = (b+3) x = b+3 b c (4) b x 3 = 4 c x c c (b x 3) = 4 c x b c x 3 c = 4 c x b c x + c x = 3 c + 4 (b+1) c x = 3 c + 4 x = 3 c+4 (b+1) c 3 x (5) a (3 x+c) = 4 b 4 a b c (4 a b c) 3 x = 4 b a (3 x+c) 12 a b x 3 c x = 12 a b x + 4 a b c 3 c x = 4 a b c x = 4 a b c 3 c (6) b c = 1 x+1 (b c) (x+1) = 1 (b c) x + (b c) = 1 (b c) x = 1 b + c (7) a x = x2 a x (a x) (a x) = x 2 a 2 2 a x + x 2 = x 2 a 2 2 a x = 0 = 4 a b 3 x = 1 b+c b c a 2 = 2 a x x = a2 2 a = a 2

1 151 4 (1) x 2 = 1 4 ( x > 0 ) { x = a t 1 (2) y = b t 2 1 x = 4 = 1 2 (2) 10 = 5 x 100 = 25 x x = 100 25 = 4 (3) 3 = x 5 9 = x 5 x = 14 (4) y 2 = x+3 (y 2) 2 = x+3 y 2 4 y+4 = x+3 x = y 2 4 y+1 (5) y = c x y 2 = c 2 x z (6) y = x y 2 = z x x = y2 c 2 x y 2 = z x = z y 2 1 (7) = 2 x 1 1 x 1 = 4 1 = 4 (x 1) x = 5 4 (8) x 2 = 1 z 2 y ( x > 0, y > 0, z > 0 ) 1 1 x = z 2 y = 1 y y = y = z 2 z z 1 (9) f = f 2 = 2 π L C 1 4 π 2 L C c f 2 = 1 4 π 2 L 5 { x = 2 t 1 (1) y = 4 t 2 c = 1 t = x 2 2 y = 4 x 2 y = 2 x 1 4 π 2 f 2 L 1 t = x a 2 y = b x a v = g t (3) y = 1 2 g t2 (4) (5) y = b a x 1 2 1 t = v g 2 y = 1 ( ) 2 v 2 g y = v2 g 2 g { x = 2 t 1 1 y = 4 t 2 + 6 t + 3 2 1 t = x+1 2 2 ( ) 2 x+1 y = 4 + 6 x+1 + 3 2 2 y = x 2 + 2 x + 1 + 3 x + 3 + 3 y = x 2 + 5 x + 7 { x = t a 1 y = 3 t + b 2 1 x 2 = t a t = x 2 + a 2 6 y = 3 ( x 2 + a ) + b y = 3 x 2 + 3 a + b (1) 1 R = 1 20 + 1 30 = 3+2 60 = 5 60 = 1 12 R = 12 [Ω] (2) 1 R = 1 R 1 + 1 R 2 = R 2+R 1 R 1 R 2 = R 1+R 2 R 1 R 2 R = R 1R 2 R 1 +R 2 (3) 1 R = 1 a + 1 2 a = 2+1 2 a = 3 2 a R = 2 a 3 [Ω]

152 1 7 (1) I 1 = E R 1, I 2 = E R 2, I 3 = E R 3 (2) I = I 1 + I 2 + I 3 (3) I = E R (4) I = E + E + E ( 1 =E + 1 + 1 ) R 1 R 2 R 3 R 1 R 2 R 3 ( E 1 (5) R = E + 1 + 1 ) R 1 R 2 R 3 1 R = 1 + 1 + 1 R 1 R 2 R 3 (6) 1 R = 1 10 + 1 15 + 1 30 = 6 30 = 1 5 R = 5 [Ω] 1 (7) R = 1 a + 1 b + 1 3 b = 4 a+3 b 3 a b R = 3 a b 4 a+3 b [Ω] 1 (1) 1 a = 1 b + 1 c + 1 d a = b c d b c+b d+c d (2) P V = w M R T (3) (4) M P V = w R T M = w R T P V 1 a + 1 b = 1 f 1 b = 1 f 1 a = a f f a b = a f a f 4 b x+c = 2 b x+c 4 ( b x+c) = 2 (b x+c) = c d+b d+b c b c d = a f a f 4 b x + 4 c = 2 b x + 2 c 2 c = 6 b x x = 2 c 6 b = c 3 b (5) a+2 = x b+x (a+2) (b+x) = x a b + 2 b + a x + 2 x = x a x + x = a b 2 b (a+1) x = (a+2) b x = (a+2) b a+1 l 2 T = 2 π g T 2 = 4 π 2 l g g T 2 = 4 π 2 l 3 F x = 1 2 m v2 v 2 = 2 F x m g = 4 π2 l T 2 v = 4 W = 1 2 Q V 1 Q = C V (1) 2 1 2 W = 1 2 C V V = 1 2 C V 2 2 F x (2) 2 V = Q C 1 W = 1 2 Q Q C = Q2 2 C 5 M a = M g T 1 m a = T m g 2 m (1) 2 T = m a + m g 1 M a = M g (m a + m g) M a + m a = M g m g (M m)g (M +m)a=(m m)g a= M +m (2) 1 m M m a = M m g m T 1 2 M M m a = M T M m g 2 1 2 0 = M m g m T M T + M m g MT +mt = 2Mmg T = 2Mmg M +m

2 153 2 1 (1) 5 3 5 2 = 5 3+2 = 5 5 = 3125 (2) ( 6 2 ) 3 = 6 2 3 = 6 6 = 46656 (3) 7 4 7 2 = 7 4 2 = 7 2 = 49 (4) 9 3 9 5 = 9 3 5 = 9 2 = 1 9 2 = 1 ( ) 81 4 3 (5) = 34 4 4 4 = 81 256 (6) 10 3 10 4 =10 3 4 =10 1 = 1 10 1= 10 (7) 10 5 10 2 = 10 5+( 2) = 10 3 = 1000 (8) 10 3 10 2 =10 3 ( 2) =10 5 =100000 2 (1) a 3 a = a 3 a 1 = a 3+1 = a 4 (2) a 4 a 3 a 2 = a 4 3+( 2) = a 1 = 1 a 1 = 1 a (3) a 2 a 5 = a 2 5 = a 3 = 1 a 3 (4) ( a b 2) 3 ( a 2 b ) 2 = ( a b 2 ) 3 (a 2 b) 2 = a3 b 6 a 4 b 2 = b4 a (5) 4x 2 ( 2xy 3) 2 = 4x 2 4x 2 y 6 = 16x 4 y 6 (6) 5a 3 3a 4 =15a 3+( 4) =15a 1 = 15 a (7) 28a 2 ( 7a 3) = 28a2 7a 3=4a2 ( 3) =4a 5 ( (8) a 4 ) 3 = a 4 3 = a 12 ( (9) a 2 ) 3 = a ( 2) 3 = a 6 = 1 a 6 (10) a 5 a 0 = a 5+0 = a 5 3 a 3 a 3 (1) = a 3+( 3) 1 = a 1 = 1 a a ( (2) a 3 ) 2 ( a 4 ) 3 = a 6 a 12 = a 6 ( (3) a 2 b 1) 2 = a 4 b 2 = b2 a ( 4 6 x 2 y ) 2 (4) 9 x 3 y 3 = 36 x4 y 2 9 x 3 y 3 = 4 x y (5) 15 a 2 b 3 c ( 5 a b c 2 ) 2 = 15 a2 b 3 c 25 a 2 b 2 c 4 = 3 b 5 c 3 4 (1) 3 a 3 b 2 ( 2 a b 2) 3 6 a 3 b 2 = 3 a3 b 2 8 a 3 b 6 6 a 3 b 2 = 4 a 3 b 6 = 4 a3 b 6 (2) ( 2 x y 2) 3 6 x 4 y 2 = 8 x3 y 6 6 x 4 y 2 = 4 y4 3 x 60 a 2 (3) 7 b c 48 a3 c 3 ( ) 2 a c 2 2 35 b 3 5 b 2 (4) 60 a2 = 7 b c ( 7 a 2 b ) 2 35 b3 48 a 3 c 3 4 a2 c 4 21 x 3 y 3 3 x2 y 35 (a b 2 ) 2 25 b 4 = a b 2 = 49 a4 b 2 21 x 3 y 3 3 x2 y 35 a 2 b 4 = a 2 5 b 2 x y 2 5 V = π r 2 h ( ) b V 2 2 = π a r a b h 6 = π b4 a 2 r2 a b h = b3 a π r2 h V V = b 3 a π r2 h π r 2 h (1) 4470 = 4.47 10 3 (2) 0.023 = 2.3 10 2 (3) 857600 = 8.576 10 5 (4) 0.00038 = 3.8 10 4 (5) 76358 = 7.6358 10 4 7 (1) 3.59 10 4 = 35900 = b3 a (2) 2.93 10 5 = 0.0000293 (3) 4.2 10 3 = 4200 (4) 2.738 10 3 = 2738 (5) 5.14 10 2 = 0.0514

154 2 8 (1) 4.8 3500 0.0024 = 4.8 ( 3.5 10 3) ( 2.4 10 3) = 4.8 3.5 10 3 ( 3) = 7 10 6 2.4 (2) 0.00011 3 = ( 1.1 10 4) 3 = 1.1 3 10 12 = 1.331 10 12 (3) 0.108 540000 = ( 1.08 10 1) ( 5.4 10 5) = 1.08 5.4 10 6 = 0.2 10 6 = 2 10 7 (4) 0.0035 0.035 3 = ( 3.5 10 3) ( 3.5 10 2) 3 = ( 3.5 10 3) ( 3.5 3 10 6) = 3.5 3.5 3 10 3 ( 6) = 1 3.5 2 103 = 1 12.25 103 0.0816 10 3 = 8.16 10 ( (5) 8 10 3 ) 2 ( 2 10 3 ) 2 = ( 8 2 10 6) ( 2 2 10 6) = 82 2 2 10 6 ( 6) = 4 2 10 0 = 16 1 = 1.6 10 ( (6) 8.1 10 2 ) 3 ( 9.0 10 2 ) 6 = ( 8.1 3 10 6) ( 9.0 6 10 12) = 8.13 9.0 6 10 6 12 ( ) 81 10 1 3 = 9.0 6 10 18 ( 9 2 10 1) 3 = 9.0 6 10 18 = 96 10 3 9.0 6 10 18 = 1 10 21 10 (1) 2 49 = 49 = 7 (2) 3 3 64 = 4 3 = 4 (3) 4 4 81 = 3 4 = 3 (4) 5 5 32 = 2 5 = 2 (5) 3 64 = 3 ( 4) 3 = 4 (6) 5 1 = 5 ( 1) 5 = 1 11 (1) 4 1 10 = 10 4 (2) 7 1 15 = 15 7 (3) 5 34 = 3 4 5 (4) 3 3 16 = 2 4 = 2 4 3 (5) 3 34 3 3 5 = 3 4 5 4 3 3 3 = 3 3 + 5 3 = 3 3 (6) 3 6 2 2 = 2 1 1 1 3 2 6 = 2 3 + 1 1 6 = 2 2 12 a 3 = 125 a = 3 125 = 125 1 3 = ( 5 3) 1 3 = 5 3 1 3 = 5 1 = 5 [cm] 1 l R = ρ S = ( 1.7 10 8) 1.00 10 3 3.14 ( 2.00 10 3 1 2 ( ) 1.7 10 8 ( 1.00 10 3) = 3.14 ( 1.00 2 10 6) = 1.7 10 5 1.7 6 = 3.14 10 3.14 10 5+6 = 0.54 10 1 = 5.4 [Ω] 2 (1.50 10 11 ) ( 3.0 10 8) ) 2 9 (6.1 10 23 ) 200 = ( 6.1 10 23) ( 2 10 2) = 6.1 2 1023 2 = 3.05 10 21 [ ] = 1.50 3.0 1011 8 = 0.5 10 3 = 5 10 2 [s]

3 155 3 63.5 8.93 ( 6.02 10 23) = 63.5 8.93 1 6.02 10 23 = 63.5 8.93 6.02 1 10 23 = 1.18 10 23 [cm 3 ] 4 6378136 [m] = ( 6378136 10 3) [km] 6378136 10 3 = ( 6.378136 10 6) 10 3 = 6.378136 106 10 3 = 6.378136 10 3 [km] 5 ( 123611 10 3 ) ( 5292000 10 3) 100 = ( 1.23611 10 8) ( 5.292 10 9) 10 2 = 1.23611 5.292 10 8 9+2 = 0.234 10 1 = 2.34 [%] 6 ( 1.67 10 27 ) ( 9.11 10 31) = 1.67 9.11 10 27 ( 31) = 0.183 10 4 = 1.83 10 3 [ ] 3 1 (1) 0.6 1 (2) 8.7 2 (3) 3.07 3 (4) 0.028 2 (5) 2.050 4 (6) 0.009070 4 (7) 4.30 10 6 3 (8) 7020 4 2 (1) 2555 [g] = 2.555 [kg], 10.50, 2.555 10.50, 2.555, 10.50 + 2.555 = 13.055 13.06 [kg] (2) 1.253, 0.5246, 3.12 1.253, 0.525, 3.12, 1.253 + 0.525 + 3.12 = 4.898 4.90 [m] (3) 0.2025 [m] = 20.25 [cm], 20.25, 1.237, 0.3225 20.25, 1.237, 0.323, 20.25 + 1.237 + 0.323 = 21.810 21.81 [cm] 3 (1) 5.2 [mm] = 0.0052 [m], V = 2.00 1.00 0.0052 = 0.0104 V = 0.010 [m 3 ] (2) V = 4 3 π r3 = 4 3 3.1416 (10.00)3 = 4188.8 V = 4189 [cm 3 ]

156 3 4 (1) 1.011 2.02 + (0.50) 2 3.14 = 2.04222 + 0.785,, 2.04 + 0.79 = 2.83 2.83 [m 2 ] (2) 20.0 11.11 + 56 10 (, 10 10 10) = 222.2 + 560,, 222 + 560 = 782 7.8 10 2 [cm 2 ] (3) 10.20 11.010 + (10.00) 2 3.1416 = 112.302 + 314.16,, 1 112.3 + 314.2 = 426.5 426.5 [cm 2 ] (1) 127, 9.7 127, 9.7, 127 9.7 = 117.3 117 (2) 145.6 + 2.0 10 = 145.6 + 20,,, 145.6 + 20 = 165.6 166 (3) 22.4 765.3 = 17142.72 1.71 10 4 (4) 14 = 0.051282 273 0.051 (5) 121.5 10.0 + 21.0 100 = 1215 + 2100 = 3315 3.32 10 3 2 1.400 3.1416 = 4.39824 4.398 [m] 3 10.1 [ma] = 0.0101 [A], V = R I = 6.00 0.0101 = 0.0606 0.0606 [V] P = R I 2 = 6.00 (0.0101) 2 = 0.00061206 0.000612 [W] 4 Q = N e N = Q e, 1 e = 1.6 10 19 [C], 1 N = 1.6 10 19 (, 1 1 1) = 1 1.6 1019 = 0.625 10 19 = 6.25 10 18 6.3 10 18 [ ] 2 e = 1.602 10 19 [C], 1 N = 1.602 10 19 (, 1 1 1) 1 = 1.602 1019 5 = (0.62421 ) 10 19 = (6.2421 ) 10 18 6.242 10 18 [ ] 1.000[L] = 1.000 10 3 [cm 3 ], 13.5462, 1.000 10 3 13.546, 1.000 10 3, 13.546 ( 1.000 10 3) = 13.546 10 3 = 1.3546 10 4 1.355 10 4 [g] 6 10.0 0.5543 + 1.500 = 5.543 + 1.500 = 7.043 7.04 [m]

4 157 4 1 (1) [ ] (2) [ ] 2 (1) 1230 [m] = 1.23 10 3 [m] = 1.23 [km] (2) 450 [mm] = 4.5 10 2 [mm] = ( 4.5 10 2) 10 3 [m] = 4.5 10 1 [m] = ( 4.5 10 5) 10 6 [m] = 4.5 10 5 [µm] (3) 1.2 [kv] = 1.2 10 3 [V] (4) 0.00079 [A] = 7.9 10 4 [A] = ( 7.9 10 2) 10 6 [A] = 7.9 10 2 [µa] (5) 0.56 [MΩ] = 5.6 10 1 [MΩ] = ( 5.6 10 1) 10 6 [Ω] = 5.6 10 5 [Ω] = ( 5.6 10 2) 10 3 [Ω] = 5.6 10 2 [kω] 3 (1) 2.94 [km] = 2.94 10 3 [m] (2) 38.4 [µm] = 3.84 10 1 [µm] = ( 3.84 10 1) 10 6 [m] = 3.84 10 5 [m] (3) 457 [kg] = 4.57 10 2 [kg] = ( 4.57 10 2) 10 3 [g] = 4.57 10 5 [g] (4) 9.86 [µω] = 9.86 10 6 [Ω] (5) 6.31 [MΩ] = 6.31 10 6 [Ω] 4 (1) 1.58 10 6 [Ω] = 1.58 [µω] (2) 8.65 10 6 [g] = 8.65 [Mg] (3) 2.88 10 3 [m] = 2.88 [km] (4) 5.43 10 3 [A] = 5.43 [ma] (5) 3.37 10 4 [V] = ( 3.37 10 1) 10 3 [V] = 3.37 10 1 [kv] = 33.7 [kv] 5 (1) 1 [g] = 0.001 [kg] = 10 3 [kg] 1 [cm 3 ] = (0.01) 3 [m 3 ], 20.8 [g/cm 3 ] = = ( 10 2) 3 [m 3 ] = 10 6 [m 3 ] 20.8 10 3 10 6 [kg/m 3 ] = 20.8 10 3 [kg/m 3 ] (2) 1 [kg] = 1000 [g] = 10 3 [g] 1 [m 3 ] = (100) 3 [cm 3 ], = 6 = ( 10 2) 3 [cm 3 ] = 10 6 [cm 3 ] 2.43 10 4 [kg/m 3 ] ( 2.43 10 4 ) 10 3 10 6 [g/cm 3 ] = 2.43 10 1 [g/cm 3 ] = 24.3 [g/cm 3 ] 40 [mm]=4 [cm], 100 [mm]=10 [cm], 10 [mm]=1 [cm], V = 4 10 1 = 40 [cm 3 ] m = ρ V = 7.86 [g/cm 3 ] 40 [cm 3 ] = 314.4 [g]

158 4 7 m = ρ V V = m ρ, 500 [g] V = 1.83 [g/cm 3 ] = 273.2 [cm3 ] 8 (1) 1 [s] = 1 60 10 [m/s] = 273 [cm 3 ] [min], 10 [m] 1 60 [min] = 600 [m/min] (2) 1 [m] = 1 1 [km], 1 [s] = 1000 3600 [h], 20 1 20 [m/s]= 1000 [km] = 72 [km/h] 1 3600 [h] 9 1 [m] = 1 1 [km], 1 [s] = 1000 3600 [h], 100 [m] 100 1 = 1000 [km] 10 [s] 10 1 =36 [km/h] 3600 [h] 10 (1) 36 [km/h]= 72 [km/h]= a = 36 km 36 1000 [m] =10 [m/s] 3600 [s] 72 1000 [m] =20 [m/s] 3600 [s] (20 10) [m/s] 10 [s] = 1 [m/s 2 ] (2) 70 [km/h]= 70 1000[m] =19.4 [m/s] 3600 [s] 0 [km/h] = 0 1000 [m] 3600 [s] = 0 [m/s] a= (0 19.4)[m/s] = 2.8[m/s 2 ] 7 [s] 11 (1) m = 2.5 [g/cm 3 ] 8000 [cm 3 ] = 20000 [g] = 20 10 3 [g] = 20 [kg] (2) 20 [km/h]= 92 [km/h]= a= 20 1000 [m] =5.6 [m/s] 3600 [s] 92 1000 [m] =25.6 [m/s] 3600 [s] (25.6 5.6) [m/s] =4.0 [m/s 2 ] 5 [s] (3) F = m a = 20 [kg] 4.0 [m/s 2 ] 12 = 80 [kg m/s 2 ] = 80 [N] F = m a a = F m, 49 [N] a= 20 [kg] = 49 [kg m/s2 ] =2.45 [m/s 2 ] 20 [kg] 13 (1) 1 [cm 2 ] =, ( ) 2 1 [m 2 ] = 10 4 [m 2 ] 100 1 [N/cm 2 ] = 1 [N] 1 [cm 2 ] = 1 [N] 10 4 [m 2 ] = 10 4 [N/m 2 ] = 10000 [N/m 2 ] (2) 1 [N]= 1 1000 [kn]=10 3 [kn], 588 [Pa] = 588 [N/m 2 ] = = 588 10 3 [kn] 1 [m 2 ] 588 [N] 1 [m 2 ] = 0.588 [kn/m 2 ] (3) 1 [N]= 1 1000 [kn]=10 3 [kn], 3500 [N/m 2 ] = 3500 [N] 1 [m 2 ] = 3500 10 3 [kn] 1 [m 2 ] = 3.5 [kn/m 2 ]

4 159 14 9200 [kg] 9.8 [m/s 2 ] 2 2 [m 2 ] 15 = 22540 [N/m 2 ] W = ( 200 [kg] 9.8 [m/s 2 ] ) 10 [m] 16 = (200 9.8) [N] 10 [m] = 19600 [N m] = 19600 [J] (1) 300 [J] = 300 [N m] (2) 9800 [N m] = 9800 [J] 17 P = = = 120 [N] 40 [m] 15 [s] 4800 [N m] 15 [s] 4800 [J] 15 [s] = 320 [J/s] = 320 [W] = 9.8 10 3 [J] = 9.8 [kj] 18 ( 2000 [kg] 9.8 [m/s 2 ] ) 40 [m] P = 50 [s] = 19600 [kg m/s2 ] 40 [m] 50 [s] = = = 19 19600 [N] 40 [m] 50 [s] 784000 [N m] 50 [s] 784000 [J] 50 [s] = 15680 [J/s] = 15680 [W] = 15.68 10 3 [W] = 15.68 [kw] 100 [W] = 100 [J] 1 [s], 1 [s] 100 [J],. 20 (1) 8.5 [W h] = 8.5 [W] 3600 [s] = 30600 [W s] (2) 20 [W h] = 20 [W] 3600 [s] = 20 [J/s] 3600 [s] = 72000 [J] 1 (1) 0.5 [kgw] = 0.5 [kg] 9.8 [m/s 2 ] = 4.9 [kg m/s 2 ] = 4.9 [N] (2) 0.7 [kn]=0.7 10 3 [N] = 0.7 103 9.8 (3) 800 [g] 9.8 [m/s 2 ] = 0.8 [kg] 9.8 [m/s 2 ] = (0.8 9.8) [kg m/s 2 ] = (0.8 9.8) [N] = 0.8 9.8 9.8 [kgw]=71.4 [kgw] [kgw] = 0.8 [kgw] = 0.8 10 3 [gw] = 800 [gw] 2 (1) 30000 [cal] = 30000 4.2 [J] = 126000 [J] (2) 4.6 10 5 [J] = ( 4.6 10 2) 10 3 [J] = 4.6 10 2 [kj] = 4.6 102 4.2 [kcal] = 1.1 10 2 [kcal] (3) 0.4 [kcal] = 0.4 10 3 [cal] = ( 0.4 10 3) 4.2 [J] = 1.68 10 3 [J]

160 4 1 45 [m] 45 [m/s] = = 45 10 3 [km] 1 [s] 1 [s] ( ) 45 10 3 3600 [km] = 3600 [s] ( ) 45 10 3 3600 [km] = 1 [h] = 162 [km/h] 162 [km] 2 2 30 = 2.5, 3 552.8 [km] 2.5 [h] 40 [km/h] =, a = 4 = 221.12 [km/h] 40 [km] 1 [h] = 40 103 [m] 3600 [s] = 40 103 3600 [m/s] 40 10 3 3600 [m/s] 4 [s] = 10 103 3600 [m/s2 ] = 2.8 [m/s 2 ] 1.00 [m]=100 [cm], 2.00 [m]=200 [cm], 2.00 [mm]=0.200 [cm],, 100[cm] 200[cm] 0.200[cm] 7.86[g/cm 3 ] = 31440 [g] = 31.44 [kg], 31.44 [kg] 9.8 [m/s 2 ] = 308 [kg m/s 2 ] = 308 [N] 5 50 [cm] = 0.5 [m], 20 [cm] = 0.2 [m], 2 [t] = 2000 [kg],, 2000 [kg] 9.8 [m/s 2 ] 0.5 [m] 0.2 [m] = 19600 [kg m/s2 ] 0.1 [m 2 ] = 19600 [N] 0.1 [m 2 ] = 196000 [N/m 2 ] = 196000 [Pa] = 196 [kpa] 6 ( 10[m 3 ] 1000[kg/m 3 ] ) 9.8[m/s 2 ] 100[m] 1 [s] = 10000 [kg] 9.8 [m/s2 ] 100 [m] 1 [s] = 98000 [kg m/s2 ] 100 [m] 1 [s] = = = 98000 [N] 100 [m] 1 [s] 9800000 [N m] 1 [s] 9800000 [J] 1 [s] = 9800000 [J/s] = 9800000 [W] = 9800 [kw] 7 F = m a = 30 [kg] 12 [m/s] 5 [s] = 30 [kg] 2.4 [m/s 2 ] = 72 [kg m/s 2 ] = 72 [N]

5 161 5 1 52 4 2 = 9 = 3, sin θ = 3 5, cos θ = 4 5 172 15 2 = 64 = 8, sin α = 15 17, cos α = 8 17 82 6 2 = 28 = 2 7, sin β = 6 8 = 3 4, cos β = 2 7 8 2 a 4 = sin 30, a = 4 sin 30 = 4 1 2 = 2 2 3 b b = 2 3 3 = tan 30, tan 30 = 2 3 3 3 θ = 0 sin 2 0 +cos 2 0 = 0 2 + 1 2 = 1, tan α = 15 8 = 7 4 = 2 3 3 3 = 6 θ = 30 ( ) 2 ( ) 2 1 3 sin 2 30 +cos 2 30 = + =1 2 2 θ = 45 ( ) 2 ( ) 2 2 2 sin 2 45 +cos 2 45 = + =1 2 2 θ = 90 sin 2 90 +cos 2 90 = 1 2 + 0 2 = 1, sin 2 θ + cos 2 θ = 1. 4 (1) tan 38 = 0.7813 (2) cos 60 = 0.5000 (3) sin 75 = 0.9659 (4) tan 53 = 1.3270 (5) cos 8 = 0.9903 5 (1) sin θ = 0.333 θ 19 (2) cos θ = 0.850 θ 32 (3) tan θ = 2.36 θ 67 6 (1) sin 43 = 0.6820 cos 25 = 0.9063 tan 86 = 14.3007 (2) sin 37 = 0.6018, cos 37 = 0.7986, sin 2 37 + cos 2 37 = (0.6018) 2 + (0.7986) 2 = 0.9999 1 (3) cos 40 = 0.7660, sin 50 = 0.7660 1 cos 40 = sin 50 sin 70 = 0.9397, cos 20 = 0.9397 sin 70 = cos 20 BAC = 90 30 = 60, 2 x = 20 tan 60 = 20 1.7321 = 34.642 35 [m] (1) BD = AB sin BAD = 18.0 sin 34 = 18.0 0.5592 = 10.1 [m] (2) DBA = 90 34 = 56, DBC = 66 56 = 10 AC = AD + DC = 18.0 cos 34 + 10.1 tan 10 = 18.0 0.8290 + 10.1 0.1763 = 16.7 [m] S = AC BD 1 2 = 16.7 10.1 1 2 = 84.3 [m2 ] BD (3) = cos CBD, BC 10.1 = cos 10 BC BC = 10.1 cos 10 = 10.1 = 10.3 [m] 0.9848

162 5 3 (1) CAD = 90 C = B, ACD BAD, DC AD = DA, BD = x, BD 13 x = 6 6 x x (13 x) = 36 x 2 13 x + 36 = 0 (x 4) (x 9) = 0 x = 4, 9 AB>AC BD>DC x = 9 BD = 9, CD = 13 9 = 4 (2) AB = BD 2 + DA 2 = 9 2 + 6 2 = 117 = 3 13 cos θ = BD AB = 9 3 13 = 3 = 3 13 13 13 4 a = b tan 39 1 a = (b+30.0) tan 27 2 1, 2, b tan 39 = (b+30.0) tan 27 b tan 39 = b tan 27 + 30.0 tan 27 b (tan 39 tan 27 ) = 30.0 tan 27 5 BD = b, h = b tan β h = (b+a) tan α 1, 2, b tan β = (b+a) tan α 1 2 b tan β = b tan α + a tan α b (tan β tan α) = a tan α b = 1, h = = a tan α tan β tan α a tan α tan β tan α tan β a tan α tan β tan β tan α b =, 30.0 tan 27 tan 39 tan 27 = 30.0 0.51 0.81 0.51 = 30.0 0.51 = 51.0 [m] 0.30 a = 51.0 tan 39 = 51.0 0.81 = 41.3 [m]

6 163 6 1 6 11, d + b = a, a b = d d d + b = a,. 2 3 ABC, (1) BC + CA = BA (2) AB AC = AB + CA = CA + AB = CB (3) AC BC = AC + CB = AB 4 ( (1) 2 3 ) a = 6 a (2) 2 a + 3 a = 5 a (3) 4 b b = 3 b ( (4) 3 a ) ( + b 2 a ) 2 b = 3 a + 3 b 2 a + 4 b = a + 7 b 5 a = (3, 2), b = ( 2, 1), (2) 3 b = 3 ( 2, 1) = ( 6, 3) 3 b = ( 6) 2 + 3 2 = 3 5 (3) 2 a 3 b = 2 (3, 2) 3 ( 2, 1) 2 a 3 b = (6, 4) ( 6, 3) = (12, 7) = 12 2 + ( 7) 2 = 193 (4) a + 2 b = (3, 2) + 2 ( 2, 1) a + 2 b = ( 3, 2) + ( 4, 2) = ( 7, 4) = ( 7) 2 + 4 2 = 65 6 (1) AB = ( 1, 5) (4, 2) (2) (3) = ( 1 4, 5 2) = ( 5, 3) AB = ( 5) 2 + 3 2 = 34 AB = (1, 4) (6, 3) = (1 6, 4 3) = ( 5, 7) AB = ( 5) 2 + ( 7) 2 = 74 AB = (2, 1) (0, 5) = (2 0, 1 5) = (2, 4) AB = 2 2 + ( 4) 2 = 2 5 7 (1) F 1 = (3, 3), F 2 = ( 4, 2) (2) (1) a + b = (3, 2) + ( 2, 1) a + b = (1, 1) = 1 2 + ( 1) 2 = 2

164 6 (3) (4) F = F1 + F 2 = (3, 3) + ( 4, 2) = ( 1, 5) F = ( 1) 2 + 5 2 = 26 8 a = (a1, a 2 ) = 5 cos 30, 5 sin 30 ) ( ) 3 = 5 2, 5 1 2 ( 5 ) 3 5 =, 2 2 a 1 = 5 3 2, a 2 = 5 2 9 { S W sin θ = 0 (1) N W cos θ = 0 { S cos θ N sin θ = 0 (2) S sin θ+n cos θ W = 0 (3) (1), S = W sin θ = 50 sin 45 2 = 50 2 = 25 2 [N] N = W cos θ = 50 cos 45 2 = 50 2 = 25 2 [N] [ ] (2) S, N 1, S = W sin θ, N = W cos θ. 10 (1) S 1 = (S 1x, S 1y ), S 2 = (S 2x, S 2y ), S 1x = S 1 sin 30 = 1 2 S 1 3 S 1y = S 1 cos 30 = 2 S 1 S 2x = S 2 sin 60 = 3 2 S 2 S 2y = S 2 cos 60 = 1 2 S 2 (2) x 1 3 2 S 1 + 2 S 2 = 0 3 y 2 S 1 + 1 2 S 2 W = 0 (3) (2), 1 2 S 1 + 3 3 2 S 2 = 0 1 2 S 1 + 1 2 S 2 = W 2 3 2 2 1 1 2 S 2, 3 S 1 = 2 W 3 1 2 + 2 1 2 S 1, S 2 = 1 2 W W=20 [N], 2=1.4, 3=1.7, 11 S 1 = 1.7 2 S 2 = 1 2 x, V 1 sin θ + V 2 = 0 20 = 17 [N] 20 = 10 [N] sin θ = V 2 V 1 = 0.3 0.5 = 0.6, θ 37, cos θ = 1 sin 2 θ = 1 (0.6) 2 = 0.64 = 0.8, t, y, V 1 cos θ t = 50 t = 50 V 1 cos θ = 50 = 125 [s] 0.5 0.8

6 165 12, (6.0) 2 +(8.0) 2 =10.0[m/s] tan α = 8.0 = 1.3, 6.0 α 53, A 53,,, 8.0 t = 40 t = 5.0 [s], 6.0 5.0 = 30.0 [m] B 30.0 [m] 1 (1) CB = CA + AB ( = ) AC + AB = AB AC (2) BC BA = ( BC ) AB = AB + BC = AC 2 ( 2 a ) ( 4 b + c 2 a ) 2 b + c 4 (1) W = W, S 1 S 2 S 2, S 1, x S 2 S1 sin 30 = 0 1 S 2 S 1 = 0 2 y S 1 cos 30 W = 0 3 2 S 1 W = 0 (2) (1), = 2 S 1 3 W= 2 3 20 3 10= 3 3 S 2 = 1 2 S 1 = 1 2 20 3 3 = 10 3 3 [N] [N] (3) h = 20 20 cos 30 3 = 20 20 2 = 20 10 3 [cm] 5 (1) = 2 a 8 b +2 c 2 a +4 b 2 c = 4 b 3 (1) (2) F = F1 + F 2 + F 3 = (2, 2) + (2, 2) + ( 3, 4) = (1, 4) F = 1 2 + 4 2 = 17 (3) F = (1, 4) = 1 i + 4 j = i + 4 j (4) α, tan α = 4 1 = 4,, α 76 (2) x S cos 45 W sin 30 =0 2 2 S 1 2 W = 0 y S sin 45 +N W cos 30 =0 2 3 2 S + N 2 W = 0

166 7 (3) (2), S = 1 2 2 W = 2 2 W = 2 10=5 2 [N], 2 2 3 2 2 W + N 2 W = 0, 3 1 N = W 2 3 1 = 10 = 5 ( 3 1 ) [N] 2 6 36 [km/h] = 36 [m/s] = 10 [m/s] 3.6 54 [km/h] = 54 [m/s] = 15 [m/s] 3.6 100 (15 10) = 100 = 20 [s] 5 7 1 (1) 2 3 = 8 3 = log 2 8 (2) 5 2 = 25 2 = log 5 25 (3) 3 4 = 81 4 = log 3 81 (4) 2 1 2 = 1.41 1 2 = log 2 1.41 (5) 16 1 2 = 4 1 2 = log 16 4 2 (1) log 9 N = 2 (2) log a 125 = 3 (3) log a 27 = 3 9 2 = N N = 81 a 3 = 125 a = 5 a 3 = 27 a = 3 (4) log 49 N = 1 2 49 1 2 = N 3 N = 49 1 2 = ( 7 2 )1 2 = 7 2 1 2 = 7 1 = 7 log a A B = log a A + log a B 1, B 1 B, log a A 1 B = log a A + log a 1 B ( ), 2, n, log a 1 B = log a B 1 ( ), ( ), = ( 1) log a B = log a B ( ) log a A B = log a A log a B

7 167 4 n A = A 1 n 5, log a n A = log a A 1 n log a 2 = x, log a 3 = y (1) log a 6 = log a (2 3) = 1 n log a A ( 2 ) = log a 2 + log a 3 = x + y (2) log a 3 2 = 2 log a 3 = 2 y (3) log a 1 2 = log a 1 log a 2 = 0 log a 2 = log a 2 = x (4) log a 9 8 = log a 9 log a 8 = log a 3 2 log a 2 3 = 2 log a 3 3 log a 2 = 2 y 3 x (5) log a 3 2 = log a 2 1 3 = 1 3 log a 2 = 1 3 x (6) log a 6 = loga 6 1 2 = 1 2 log a 6 = 1 2 log a(2 3) = 1 2 ( log a 2 + log a 3 ) = 1 2 ( x + y ) (7) log a 8 a 2 = log a 8 + log a a 2 = log a 2 3 + log a a 2 = 3 log a 2 + 2 log a a = 3 x + 2 1 = 3 x + 2 6 log b a, a, 7 log b a = log a a log a b = 1 log a b log b a log a b = 1 log 2 = 0.30, log 3 = 0.48, log 7 = 0.85, (1) log 9 3 = log 10 3 log 10 9 = log 3 log 9 = log 3 log 3 2 = log 3 2 log 3 = 1 2 = 0.5 (2) log 5 14 = log 10 14 log 10 5 = log 10(2 7) log 10 10 2 = log 10 2 + log 10 7 log 10 10 log 10 2 = = log 2 + log 7 1 log 2 0.30 + 0.85 1 0.30 log 5 14 = 1.6 = 1.64 (3) log 2 9 log 3 4 = log 10 9 log 10 2 log 10 4 log 10 3 = log 9 log 2 log 4 log 3 = log 32 log 2 log 22 log 3 = 2 log 3 log 2 2 log 2 log 3 = 2 2 = 4

168 7 8 log 2 = 0.3010, log 3 = 0.4771, (1) log 200 = log ( 2 10 2) = log 2 + log 10 2 = log 2 + 2 log 10 = 0.3010 + 2 1 = 2.3010 (2) log 0.0003 = log ( 3 10 4) = log 3 + log 10 4 = log 3 4 log 10 (3) log 6 = log (2 3) = 0.4771 4 1 = 3.5229 = log 2 + log 3 = 0.3010 + 0.4771 = 0.7781 (4) log 5 = log 10 2 = log 10 log 2 9 = 1 0.3010 = 0.6990 (1) 5 y = 2 x, log 5 y = log 2 x y log 5 = x log 2 ( ) at 1 (2) N = b 2, ( ) at 1 log N = log b 2 ( ) at 1 log N = log b + log 2 log N = log b + a t log 1 2 [ ] log N = log b a t log 2 10 2, m 1 m 2 = 5 log D 1 D 2, D 1 = 10 D 2, 11 m 1 m 2 = 5 log 10 D 2 D 2 = 5 log 10 = 5 1 = 5 2 5 log 2 = 0.30, log 3 = 0.48, log 7 = 0.85, (1) ph = log ( 4.9 10 3) = log ( 49 10 4) = log ( 7 2 10 4) = ( 2 log 7 4 log 10 ) = ( 2 0.85 4 1 ) = 2.3 (2) ph = log ( 5.6 10 11) = log ( 56 10 12) = log ( 8 7 10 12) = log ( 2 3 7 10 12) = ( 3 log 2+log 7 12 log 10 ) = ( 3 0.30+0.85 12 1 ) = 10.25 (3) ph = log ( 6.3 10 7) = log ( 63 10 8) = log ( 9 7 10 8) = log ( 3 2 7 10 8) = ( 2 log 3+log 7 8 log 10 ) = ( 2 0.48+0.85 8 1 ) = 6.19

7 169 12 10 0.5 = 3.2, (1) 5.0 = log [H + ] log [H + ] = 5.0 [H + ] = 10 5.0 = 10 5 (mol/l) (2) 3.5 = log [H + ] 13 log [H + ] = 3.5 [H + ] = 10 3.5 = 10 4+0.5 = 10 4 10 0.5 = 3.2 10 4 (mol/l) 0.10 = 1 10 1, log 2 = 0.30, 20 log P = 20 log 1 10 1 P 0 2 10 5 ( ) 1 = 20 log 2 104 14 = 20 ( log 1 log 2 + 4 log 10 ) = 20 ( 0 0.30 + 4 1 ) = 74 [db] 10 0.5 = 3.2, log 2 = 0.30, 20 log P P 0 = 20 log, log P = 64 2 10 5 P 64 = 2 10 5 20 = 32 10 = 3.2 log P log ( 2 10 5 ) = 3.2 log P ( log 2 5 log 10 ) = 3.2 log P ( 0.30 5 1 ) = 3.2 log P = 1.5 P = 10 1.5 = 10 0.5+( 2) = 10 0.5 10 2 = 3.2 10 2 [Pa] 1 log 2 = 0.30, log 3 = 0.48, log 7 = 0.85, (1) log 35 = log (7 5) = log 7 + log 5 = log 7 + log 10 2 = log 7 + log 10 log 2 = 0.85 + 1 0.30 = 1.55 (2) log 600 = log ( 6 10 2) = log ( 2 3 10 2) = log 2 + log 3 + 2 log 10 = 0.30 + 0.48 + 2 1 = 2.78 (3) log 0.9 = log ( 9 10 1) (4) log 2 9 = log 9 log 2 = log ( 3 2 10 1) = 2 log 3 log 10 = 2 0.48 1 = 0.04 = 2 log 3 log 2 = 2 0.48 0.30 = log 32 log 2 = 3.2 (5) log 6 7 = log 7 log 6 = log 7 log(2 3) log 7 = log 2 + log 3 0.85 = 0.30 + 0.48 = 1.08 (6) log 3 4.9 = log 4.9 log 3 = log ( 7 2 10 1) log 3 = 2 0.85 1 0.48 1.1 = log ( ) 49 10 1 log 3 2 log 7 log 10 = log 3 = 1.45 1.5

170 7 2 20%, 80%, 5 100, n = 3 = = ( 80 100 ) n = 1 100 ( ) n 4 = 1 5 5 ( 4 log 5 ) n = log 1 5 n log 4 5 = log 1 5 log 1 5 log 1 log 5 log 4 = log 4 log 5 5 log 1 log 10 2 log 2 2 log 10 2 log 1 (log 10 log 2) 2 log 2 (log 10 log 2) = 0 (1 0.3) [ log 2=0.3] 2 0.3 (1 0.3) = 0.7 0.1 = 7 [ ] 1, 1 2, 10 2 n = 1000000, 2 n = 100000 = 10 5 log 2 n = log 10 5 n log 2 = 5 log 10 5 log 10 n = log 2 = 5 1 = 16.6 [ log 2=0.3] 0.3 17 [ ] 4 m 1 = 2.5 log F F 0, 1.5 1 = 2.5 log F F 0 2.0 1 = 2.5 log F F 0 1, 2, log F F 0 = 1 1 log F F 0 = 2 5 2 log F log F 0 = 1 1 log F log F 0 = 2 5 1 2, 5 log F log F = 7 5 log F F = 7 5 2 F F = 10 7 5 = 10 1.4 = 10 0.4+1 = 10 0.4 10 1 = 2.5 10 = 25 [ ] 5.1 = log [H + ] log [H + ] = 5.1 [H + ] = 10 5.1 = 10 6+0.9 = 10 6 10 0.9 [ 10 0.4 = 2.5 ] = 7.9 10 6 (mol/l) [ 10 0.9 = 7.9 ]

7 171 6 20 log P 1 P 0 = 100, log P 1 P 0 = 5 1 20 log P 2 P 0 = 40, log P 2 P 0 = 2 2 1 log P 1 log P 0 = 5 1 2 log P 2 log P 0 = 2 2 2 1 log P 2 log P 1 = 3 7 G = 20 log V 0 V i, 40 = 20 log 1 V i, log 1 V i = 2 1 V i = 10 2 = 100 V i = 1 = 0.01 [V] 100 log P 2 P 1 = 3 P 2 = 10 3 P 1 ( ) n 80, = 10 3, 100 ( ) n 4 = 10 3 5 ( ) n 4 log = log 10 3 5 n log 4 5 = 3 log 10 3 log 10 3 log 10 n = log 4 = log 4 log 5 5 3 log 10 = log 2 2 log 10 2 3 log 10 = 2 log 2 (log 10 log 2) = 3 log 10 3 log 2 log 10 = 3 1 3 0.3 1 = 30 [ ]

172 8 8 1 (1) 480 = 120 + 360 = 120 + 360 1 α = 120 (2) 150 = 210 360 = 210 + 360 ( 1) α = 210 (3) 740 = 20 + 720 = 20 + 360 2 α = 20 (4) 1230 = 210 1440 = 210 + 360 ( 4) α = 210 2 (1) 380 = 20 + 360 = 20 + 360 1 0 < 20 < 90, 1 (2) 250 = 110 360 = 110 + 360 ( 1) 90 < 110 < 180, 2 (3) 180 < 235 < 270, 3 (4) 420 = 300 720 = 300 + 360 ( 2) 270 < 300 < 360, 4 3 tan θ = y x ( x 0 ), 1 x > 0, y > 0 tan θ > 0 2 x < 0, y > 0 tan θ < 0 3 x < 0, y < 0 tan θ > 0 4 x > 0, y < 0 tan θ < 0 4, θ =315, x > 0, y < 0, sin 315 = y r = 1 2 r r 1 2 cos 315 = x r r = r tan 315 = y 1 r 2 x = 1 2 r = 1 2 = 2 2 = 1 2 = 2 2 = 1, r > 0, x 0, y tan θ = y x = r x = sin θ cos θ r., 0 < θ < 90 θ, 180 θ 2, sin(180 θ) = sin θ cos(180 θ) = cos θ tan(180 θ) = sin(180 θ) cos(180 θ) = sin θ cos θ = sin θ cos θ = tan θ, 0 < θ < 90 θ, 180 + θ 3, sin(180 + θ) = sin θ cos(180 + θ) = cos θ tan(180 + θ) = sin(180 + θ) cos(180 + θ) = sin θ cos θ = sin θ cos θ = tan θ

8 173, 0 < θ < 90 θ, 360 θ = 180 +(180 θ) 4, sin(360 θ) = sin{180 +(180 θ)} = sin(180 θ) = sin θ cos(360 θ) = cos{180 +(180 θ)} = cos(180 θ) = ( cos θ) = cos θ tan(360 θ) = sin(360 θ) cos(360 θ) = sin θ cos θ = sin θ cos θ = tan θ, sin 315 = sin(360 45 ) 2 = sin 45 = 2 cos 315 = cos(360 45 ) 2 = cos 45 = 2 tan 315 = tan(360 45 ) = tan 45 = 1. 5 (1) 2 θ (90 < θ <180 ), 180 θ = θ θ = 180 θ θ 1 (0 <θ <90 ), { sin θ = sin(180 θ ) = sin θ sin θ = 0.40 sin θ = 0.40, θ 24 cos θ 0.9135, tan θ 0.4452 cos θ = cos(180 θ ) = cos θ 0.9135 0.91 tan θ = tan(180 θ ) [ ] 4 = tan θ 0.4452 0.45 [ ] 2, tan θ = sin θ cos θ sin2 θ+cos 2 θ =1, sin θ = 0.40, (0.40) 2 + cos 2 θ = 1 cos 2 θ = 0.84 cos θ = ± 0.84 θ 2 cos θ < 0, cos θ = 0.84 cos θ 0.92, tan θ = 0.40 0.44 0.84 [ ],. (2) 4 θ (270 < θ <360 ), 360 θ = θ θ = 360 θ θ 1 (0 <θ <90 ), 4, tan θ = tan(360 θ ) = tan θ, tan θ = 2.00, tan θ = 2.00, tan θ = 2.00 θ 63 sin θ 0.8910, cos θ 0.4540, 4, sin θ = sin(360 θ ) = sin θ 0.8910 0.89 cos θ = cos(360 θ ) = cos θ 0.4540 0.45 [ ] 4, tan θ = sin θ cos θ sin2 θ+cos 2 θ =1

174 8, sin 2 θ cos 2 θ + cos2 θ cos 2 θ = 1 cos 2 θ tan 2 θ + 1 = 1 cos 2 θ. tan θ = 2.00, ( 2.00) 2 + 1 = 1 cos 2 θ cos 2 θ = 1 cos θ = ± 1 5 5 θ 4 cos θ > 0, cos θ = 1 cos θ 0.45 5, 6 2.00 = sin θ 1 5 sin θ = 2.00 5 0.89 (1) sin 115 = sin(180 65 ) = sin 65 0.9063 (2) cos 223 = cos(180 +43 ) (3) 38 = 322 360 = 322 +360 ( 1), 38 322, tan( 38 ) = tan 322 = cos 43 0.7314 = tan(360 38 ) = tan 38 [ 4 ] 0.7813 (4) 403 = 43 +360 = 43 +360 1, 403 43, cos 403 = cos 43 0.7314 (5) 412 = 308 720 = 308 +360 ( 2), 412 308, sin( 412 ) = sin 308 = sin(360 52 ) = sin 52 [ 4 ] 0.7880 (6) 949 = 229 +720 = 229 +360 2, 949 229, tan 949 = tan 229 = tan(180 +49 ) [ ] 4 = tan 49 7 1.1504 (1) 2 π 30 360 = π 6 [rad] (2) 2 π 45 360 = π 4 [rad] (3) 2 π 60 360 = π 3 [rad] (4) 2 π 90 360 = π 2 [rad] (5) 2 π 270 360 = 3 2 π [rad] (6) 2 π 450 360 = 5 2 π [rad] 8 (1) (2) (3) π 6 = 360 π 6 2 π = 30, sin π 6 = sin 30 = 1 2 π π 4 = 360 4 cos π 4 = cos 45 = 2 π = 45, 2 2 π 2 2 π = 90, π 2 = 360 sin π 2 = sin 90 = 1

8 175 π π (4) 3 = 360 3 2 π = 60, tan π 3 = tan 60 = 3 (5) 5 π = 360 5 π 2 π = 900, 900 = 180 +360 2, 5 π [rad] 900,, 180, cos 5 π = cos 900 = cos 180 = 1 (6) 4 3 π = 360 4 3 π = 240, 2 π 240 =120 360 =120 +360 ( 1), 4 3 π [rad] 240,, 120, tan ( 43 ) π = tan( 240 ) = tan 120 = tan(180 60 [ ] ) 4 = tan 60 1 = 3 (1) 24 = 336 360 = 336 +360 ( 1), 24 336, 4, 24 = 2 π 24 360 = 2 15 π [rad]. sin( 24 ) = sin 336 = sin(360 24 ) = sin 24 [ 4 ] 0.4067 cos( 24 ) = cos 336 = cos(360 24 ) = cos 24 [ 4 ] 0.9135 tan( 24 ) = tan 336 = tan(360 24 ) = tan 24 [ 4 ] 0.4452 (2) 481 = 121 +360 = 121 +360 1, 481 121, 2, 481 = 2 π 481 360 = 481 180 π [rad]. sin 481 = sin 121 = sin(180 59 ) [ ] 4 = sin 59 0.8572 cos 481 = cos 121 = cos(180 59 ) [ ] 4 = cos 59 0.5150 tan 481 = tan 121 = tan(180 59 ) [ ] 4 = tan 59 1.6643 4 (3) 3 π [rad] 4 = 360 3 π 2 π = 240, 3. sin 4 3 cos 4 3 tan 4 3 π = sin 240 = sin(180 +60 ) [ ] 4 = sin 60 3 = 2 π = cos 240 = cos(180 +60 ) [ ] 4 = cos 60 = 1 2 π = tan 240 = tan(180 +60 ) [ ] 4 = tan 60 = 3

176 8 (4) 13 4 π [rad] = 360 13 4 π 2 π = 585, 585 = 135 720 = 135 +360 ( 2), 585 135, 2. ( sin 13 ) 4 π = sin( 585 ) = sin 135 = sin(180 45 [ ] ) 4 2 = sin 45 = ( 2 cos 13 ) 4 π = cos( 585 ) = cos 135 = cos(180 45 [ ] ) 4 2 = cos 45 = ( 2 tan 13 ) 4 π = tan( 585 )= tan 135 = tan(180 45 [ ] ) 4 = tan 45 = 1 (5) ( 3 7 6 ) π [rad] = 11 6 π [rad] 11 = 360 6 π 2 π = 330, 4 (. sin 3 7 ) π = sin 330 6 = sin(360 30 ) = sin 30 = 1 ( 2 cos 3 7 ) π = cos 330 6 ( tan 3 7 6 = cos(360 30 ) 3 = cos 30 = 2 ) π = tan 330 = tan(360 30 ) 3 = tan 30 = 3 [ ] 4 [ ] 4 [ ] 4 2 (1) 2 θ (90 < θ <180 ), 180 θ = θ θ = 180 θ θ 1 (0 <θ <90 ), sin θ = sin(180 θ ) = sin θ 3 sin θ = 0.8660 2 sin θ 0.8660, θ 30 cos θ 0.5000, tan θ 1.7321 cos θ = cos(180 θ ) = cos θ 0.5000 tan θ = tan(180 θ ) [ ] 4 = tan θ 1.7321 [ ] 2, tan θ = sin θ cos θ sin2 θ+cos 2 θ =1, ( ) 2 3 3 sin θ = 2, + cos 2 θ =1 2 cos 2 θ = 1 cos θ = ± 1 4 2 θ 2 cos θ < 0, cos θ = 1 2, tan θ = 3 2 1 2 = 3 (2) 4 θ (270 < θ <360 ), 360 θ = θ θ = 360 θ θ 1 (0 <θ <90 ), 4, { cos θ = cos(360 θ ) = cos θ cos θ = 0.4226, cos θ = 0.4226, θ 65

9 177 sin θ 0.9063, tan θ 2.1445, 4, 3 sin θ = sin(360 θ ) = sin θ 0.9063 tan θ = tan(360 θ ) = tan θ 2.1445 θ 0 15 30 45 sin 2θ 0.00 0.50 0.87 1.00 θ 60 75 90 105 sin 2θ 0.87 0.50 0.00 0.50 θ 120 135 150 165 sin 2θ 0.87 1.00 0.87 0.50 θ 180 195 210 225 sin 2θ 0.00 0.50 0.87 1.00 θ 240 255 270 285 sin 2θ 0.87 0.50 0.00 0.50 θ 300 315 330 345 sin 2θ 0.87 1.00 0.87 0.50 θ 360 sin 2θ 0.00 9 1 (1) (2) k = w l = 0.40 0 = 4.3 [N/m] 0.093 0 (3) w = 1.00 [N], w = 4.3 l l = w 4.3 = 1.00 = 0.233 [m] = 23.3 [cm] 4.3 2 (1) (2) 20 [ C] 100 [ C] 2, α = 0.3000850 0.3000000 100 20 0.3000000 = 0.0000850 80 0.3000000 = 0.0000035 [1/K] 3.5 10 6 [1/K] (3) t [ C] l, 20[ C] l 0, { l = l 0 1 + (t 20) 3.5 10 6}

178 9 (4) 0 [ C] 50.00000 [cm], 0.5000000=l 0 { 1+(0 20) 3.5 10 6} l 0 = 0.5000000 0.5000000 = 1 7.0 10 5 0.99993, 500 [ C], l= 0.5000000 0.99993 { 1+(500 20) 3.5 10 6} = 0.5000000 0.99993 1.00168 = 0.500875061 [m] = 50.0875061 [cm] 3 (1) 50.08751 [cm] 500 [ C], ρ = 6.900 {1+(500 20) 8.6 10 4} 4 = 6.900 1.4128 = 9.748 [µω cm] (1) y = x (6 x) = x 2 + 6 x (2) y = x 2 + 6 x = ( x 2 6 x ) = { (x 3) 2 9 } = (x 3) 2 + 9 p = 3, q = 9 x=3[m], 9[m 2 ] (3) x=1 [m] y = 1 2 +6 1=5[m 2 ] (4) x=2 [m] y = 2 2 +6 2=8[m 2 ] x=3 [m] y = 3 2 +6 3=9[m 2 ] x=4 [m] y = 4 2 +6 4=8[m 2 ] x=5 [m] y = 5 2 +6 5=5[m 2 ] x=6 [m] y = 6 2 +6 6=0[m 2 ] (2) 20 [ C] 100 [ C] 2, k = = 7.375 6.900 100 20 6.900 0.475 80 6.900 = 0.00086 [1/K] 8.6 10 4 [1/K] (3) t [ C] ρ, 20 [ C] ρ 0, { ρ = ρ 0 1 + (t 20) 8.6 10 4} (4) 0 [ C], ρ = 6.900 {1+(0 20) 8.6 10 4} = 6.900 0.9828 = 6.781 [µω cm] 5 (1) y = x (40 2 x) = 2 x 2 + 40 x (2) y = 2 x 2 + 40 x = 2 ( x 2 20 x ) = 2 { (x 10) 2 100 } = 2 (x 10) 2 + 200 p = 10, q = 200 x = 10 [cm], 200 [cm 2 ]

9 179 (3) x = 2 [cm] y = 2 2 2 + 40 2 = 72 [cm 2 ] x = 4 [cm] y = 2 4 2 + 40 4 = 128 [cm 2 ] x = 6 [cm] y = 2 6 2 + 40 6 = 168 [cm 2 ] x = 8 [cm] y = 2 8 2 + 40 8 = 192 [cm 2 ] x = 10 [cm] y = 2 10 2 + 40 10 = 200 [cm 2 ] x = 12 [cm] y = 2 12 2 + 40 12 = 192 [cm 2 ] x = 14 [cm] y = 2 14 2 + 40 14 = 168 [cm 2 ] x = 16 [cm] y = 2 16 2 + 40 16 = 128 [cm 2 ] x = 18 [cm] y = 2 18 2 + 40 18 = 72 [cm 2 ] 1 (1) 83 2 = 166 166 [N/m]. (2) 6.0 [N] 166 [N/m] = 0.036 [m] = 3.6 [cm], W [N],. 2 (4) 3 k = w x = 12.0 0 0.070 0 = 1.7 102 [N/m] F = 170 x 14.0 = 170 x, x = 14.0 170 = 0.082 [m] 8.2 [cm]

( )................... 103............................. 52 1.................. 103.......................... 115 n.............................. 15 n........................... 22 MKS.......................... 45 MKSA........................ 45...................... 122 ( ).............. 74...................... 6...................... 109........................... 49.................... 139............. 100..................... 80............................... 2....................... 75............................. 38............................. 41........................... 27........................... 31................................ 30 ( )................ 80......................... 11............................. 73............................. 86............................. 27 cos............................... 61.......................... 120........................ 121 ( ).................. 77 sin............................... 61......................... 62....................... 69........................... 62 3............................ 22..................... 68 CGS.......................... 45............................. 53........................... 54............................. 15......................... 16............................ 115........................ 118............................. 73....................... 51....................... 45............................ 116......................... 97............................. 94........................... 27................. 106..................... 30......................... 75............................. 61............................. 61.......................... 115........................ 115 ( )................ 80........................ 132......................... 88........................ 109........................... 27............................. 49 180

..................... 87............................. 94............................. 41....................... 47 tan.............................. 61........................ 130............................. 85............................... 3 ( ).................. 85 ( )....................... 94 ( )....................... 15 db( )................... 109........................... 54............................. 61 rad( )................... 120.................... 139............................. 15........................... 22....................... 77 60......................... 120 log........................ 99 ( ).................. 76 ( )...................... 103............................ 115 2............................ 22........................... 74 ph.............................. 106 ph....................... 107 (2 )........ 75........................ 109............................. 38.................... 130........................... 41.......................... 115........................ 115 ( )................ 80............................ 1......................... 74............................. 31............................. 47............................. 41 ( )................ 74......................... 28 181

182

( ) 2018

,, 1.. 2.,,. 3.,,. 4...,.,.,,. 30 4

1............................................................. 187 2........................................................... 193 3............................................................. 199 4.................................................. 203 5............................................................... 209 6............................................................. 213 7.......................................................221 8............................................................. 225 9........................................................... 229...........................................................231 186

1 187

1.. (1) 1 3 1 2 = (2) 5 4 1 6 = (3) 1 2 + 1 5 = (4) 1 3 + 1 2 = (5) a 2 a 5 = (6) 1 7 + 4 8 = (7) 5 3 6 7 = (8) 3 6 + 5 2 = (9) 4 5 5 7 = (10) 21 7 9 2 = 2.. (1) (2) (3) (4) (5) (6) 1 a 2 3 = b a + 1 b = 2 a 9 2 = b a + c d = 1 a 2 5 = b a c b = 188

3.. (1) b a c x b c y x = (2) 36 x2 z 2 27 a 2 b 2 1 2 = (3) a b c d e b = (4) a x 2 y z b x z = (5) 1 a 1 + a = (6) 1 x + 1 1 x 1 = (7) 1 3 x y 2 1 4 x 2 y = 4.. (1) b a c b = (2) b a b y a x y x = (3) b a + 2 a 3 b 3 a 5 b = 189

5.. (1) 1 3 + 2 5 2 15 = (2) a b + c e b d = (3) 1 a 1 b + 1 c = (4) a + b c c d = (5) b a a d b c = (6) 1 1 a 1 b = 6. x =. (1) x a + x b = 1 c (2) a x 4 = c x b (3) 3 b x + 5 = 2 b x + c (4) x 2 = 3 4 x 2 (5) 2 x 3 x c = 5 7 (6) 1 1 x + 1 y = c x + y (7) x x + 2 = 1 2 (8) 3 x + y = y 190

7. [ ]. (1) b a + 1 b = 1 2 [ a = ] ( b 2) (2) a x + b = c x + 2 [ x = ] ( a c) (3) x a = x b + 1 [ x = ] ( b 0) (4) 1 a = 1 x + 1 y + 1 z [ a = ] (5) 2 = 1 a 1 [ a = ] (6) x + y = a y + b [ y = ] ( a 1) (7) b x a + c y b = 1 2 [ a = ] ( b 2 c y ) (8) a 2 b y = x 2 b y c 2 [ x = ] ( x 0) (9) x + b c y = c x d y [ c = ] ( b y x) (10) x + 2 = a [ x = ] (11) 1 x + 1 = y 5 [ x = ] (12) 1 2 m v2 = 1 2 m v 0 2 + m g h [ v = ] ( v > 0) (13) a 2 b 2 a 2 c + 2 c b = 1 [ b = ] (14) d e = 1 a + 1 [ a = ] ( d e) (15) a 2 = c + 3 [ c = ] (16) 1 a 1 = 3 [ a = ] 191

8. 2 x, b =. { (1) a = 2 x 1 b = 4 x 2 + 6 x + 3 (2) { a = x a b = 3 x + d 9. 2 x, y., b + c e 0. { a = b x + c y d = e x y 10. 2 M = a x + y 1, Q = b M c x 2,. (1) b M, Q, a, c, y. M 0, a 0. (2) y Q, a, b, c, x. b 0. 11. 2 v y = g t 1, y = 1 2 g t2 2,. (1) 1 t =. (2) (1) t 2, v y =. 192

2 193

1.. (1) 3 3 2 4 = (2) (3 2 ) 3 = (3) 3 3 2 2 = (4) a 2 b 3 c 2 ( a b c) 2 = (5) ( a 2 b 3 ) 2 = (6) (y 2 ) 3 y 2 = (7) 2 2 3 2 = (8) (2 a b) 2 a b 3 = (9) 2 x 2 y 3 x 3 y 2 = (10) 3 a 2 b 3 (2 a 2 b) 2 = 2.. (1) 36 x2 z 2 27 a 2 b 2 57 a b c 56 x y 2 z 3 = (2) ( 2 a 2 b) 3 = (3) ( 3 a b) 2 ( 2 a 3 b) = (4) ( 3 x 3 y 2 ) 2 ( 3 x y 3 ) 3 = (5) (9 a 3 b c 8 a b 2 c 2 + 12 a b c 3 ) ( 6 a b c) = 194

3.. (1) 2 2 3 2 = (2) (2 2 ) 3 = (3) ( ) 1 2 2 2 = 2 (4) a 3 a 2 = (5) ( a b c) 2 (a 2 b) 2 = (6) 2 x 2 y 6 x y 2 (7) (x 3 ) 2 x 3 = (8) (a 2 b 1 ) 2 = (9) (8 a b 2 ) 2 (2 a b c) 3 = (10) 4 a 2 b ( 2 a b) 3 ( 2 b c 2 ) 2 = 4.. (1) x y 2 (x 2 y) 2 = (2) x 0 x 5 x 2 = (3) (x 2 y 1 ) 2 = (4) (2 a b 2 ) 2 6 a 4 b 2 = 195

5. a 10 n ( 1 a < 10, n ). (1) 802000 = (2) 0.0027 = (3) 694000 = (4) 0.000832 = 6. a 10 n ( 1 a < 10, n ). (1) 220 30 = (2) 50 25000 = (3) 2 10 = (4) 30 4 = (5) 0.06 3 = (6) 0.104 52000 = (7) 50 2 2 10 2 = (8) 123 5000 = (9) (4 10 3 ) 2 (2 10 3 ) 2 = (10) 1600 40 4 = 196

7.,. (1) 2 2 2 3 = (2) 4 5 4 6 = (3) 3 6 3 2 = (4) x 2 x 5 = (5) 8 a 2 b 3 ( 2 b 2 c) 2 = (6) 3 4 3 = (7) 4 3 2 3 = (8) 7 16 = (9) 3 3 3 9 = (10) 15 3 5 6 3 5 5 = 197

8.. (1) 3 27 4 = (2) { ( ) 4 3 } 2 3 4 9 = (3) a 3 2 a 2 a 1 4 = (4) 3 5 a 1 3 b 2 3 c 1 2 5 6 a 2 3 b 1 3 c 1 2 = (5) 3 2 2 2 2 3 = (6) 4 ( 4) 4 = (7) 3 125 = (8) 3 a 4 3 a 5 = 198

3 199

1.. (1) 1.0005 (2) 0.060 (3) 51.00 10 2 (4) 3.200 10 12 (5) 128.00 (6) 0.045 (7) 50.09 (8) 25.0 10 6 (9) 0.02298 (10) 4.300 10 12 (11) 0.60 (12) 23.050 10 2 2.,. (1) 21.5 [cm] + 131 [cm] = (2) 9.8 [kg] 4144 [ g ] = (3) 3.24 [m 2 ] + 41.2 [m 2 ] = (4) 18.2 [mm] 16 (5) 2.0 [cm] (, π = 3.1415926) 200

3.,. (1) 9844 [ g ] + 4.6 [kg] = (2) 144 [mm] + 12.5 [m] = (3) 39253.4 [mm] + 128.1 [cm] + 25.66 [m] = (4) 15.6 12 = (5) 40.0 [cm] (, π = 3.1415926) 4.,. (1) 5.42 [m 2 ] + 81.1 [m 2 ] = ( ) [m 2 ] (2) 9.6 [kg] 4055 [ g ] = ( ) [kg] (3) 43.2 15 = ( a 10 n ) (4) 4.0 [cm] (, π = 3.1415926) (5) 66.300[kg] + 603.03[ g ] + 6.501 10 4 [ g ] + 3.2266[kg] = ( ) [kg] (6) 25.002 3.21 73.21 60 = ( a 10 n ) 5.,. (1) 25.33 [m] + 3339.1 [cm] + 22953.5 [mm] (2) 51.00 3.0 + 1.035 1.2 = (3) 25.00 3.10 67.2 5.0 = 201

6. ( ), 1 3.0 km. 420 km,.. 7.,, 10.1 [cm], 120.01 [mm], 1.001 [m], 1.0 [m], 10.02 [cm],., π = 3.1415926. 8. 4. A, B, C, D 4. A 56.300 [kg], B 502.03 [ g ], C 5.502 10 4 [ g ], D 3.2255 [kg]. 4,. 9., 4.06 [cm], 21 [cm] S 1 5.0 [cm] S 2, S. 10., 2.08 [cm], 12 [cm] S 1, 1.25 [cm], 6.9 [cm] S 2, S. 202

4 203

1.,, ( ). (1) ( ) (2) ( ) (3) θ ( ) (4) β ( ) (5) π ( ) 2. [ ]. (1) 48.2 [µm] = ( ) [cm] (2) 0.0482 [A] = ( ) [µa] (3) 148.2 [MΩ] = ( ) [Ω] (4) 48 10 2 [g] = ( ) [kg] (5) 0.48 10 5 [V] = ( ) [kv] 3.,. (1) 20 [kv] = ( ) [MV] (2) 50 [mg] = ( ) [g] (3) 365 [kω] = ( ) [Ω] (4) 3 [m] = ( ) [cm] 204

(5) 25 [km] = ( ) [m] (6) 0.9 [V] = ( ) [mv] (7) 40 [kv] = ( ) [MV] (8) 600 [kω] = ( ) [Ω] (9) 25 [mg] = ( ) [g] (10) 0.096 [A] = ( ) [µa] (11) f = 50 [khz], T = ( ) [ s ] f [Hz] 1 f, 1 T [s]., f T,, T = 1 f., f = 50 [Hz], T = 1 = 0.02 [s], 0.02 50 1 50. (12) T = 25 [µs], f = ( ) [Hz] 205

4.,. (1) 8.24 10 6 [g/cm 3 ] = ( ) [kg/m 3 ] (2) 591 10 2 [kg/m 3 ] = ( ) [g/cm 3 ] 5. 2.7 [g/cm 3 ]. 54 [ g ] [cm 3 ]. 6. 50 [mm], 100 [mm], 30 [mm] [ g ]., 2.7 [g/cm 3 ]. 7. 1 [m], 2 [m], 2 [mm]., 2.7 [g/cm 3 ]. (1) [kg]. (2) [N]. 8. 10 [mm], 100 [cm].., 7.9 [g/cm 3 ], π = 3.14. 9.,. (1) 4.0 [km/h] = ( ) [m/s] (2) 600 [m/min] = ( ) [km/h] 206

10. 552 [km] 3. (1) [km/h]. (2) [m/s]. (3) 90 [km] 5 [m/s 2 ]. 11. 16 [ t ] 5 [m/s], 4, [N]. m v,. m v, f t,. f t, m v = f t. 12. 10 60 [km/h]. [m/s 2 ]. 13. 6 [m/min] 5, 18 [m/min]. [m/s 2 ]. 14. 2 [m], 1.5 [m], 4 [m]. [Pa]., 1.0 10 3 [kg/m 3 ]. 207

15. 930 [kg] [kpa]. 0.3 [m 2 ]. 16., 1 [m 2 ] 2.5 10 8 [N]. 4 [mm 2 ], [N]. 17. 4 4, [Pa].,, 1 140 [cm 2 ]. ( 1 [cm 2 ] [m 2 ].) 18. 980 [N], 12 [m] 1 40. [W]. 19. 400 [kj], 20 [kn] [m]. 20. 9800 [N], 12 [m] 4. [W]. 21. 2 [kw h] [J]. 208

5 209

1.,. (1) sin θ = (2) cos θ = (3) tan θ = 2.. (1) (2) 3.,. (1) x y. (2). sin θ = cos θ = tan θ = (3) θ. 210

4. θ.. 5... (1) tan 42, tan 35. (2) tan 42, tan 35 x, y. (3) x, y. 211

6. A, 10 100 [m]. A.,. ( ) 7. 10 [m], 35. [m]., 1.6 [m]. 212

6 213

1. a = (2, 4), b = (3, 1), 3 a 2 b. 2. a = (1, 2), b = (0, 3), 2 a b. 3. a = ( 1, 2), b = (3, 1), 2 a 3 b. 4.. (1) 2 ( a + 3 b ) 3 ( a 2 b ) = (2) 2 ( a 4 b + c ) 2 ( a 2 b + c ) = 214

5. a, b, c., d, e, f.,. (1) d = a + 1.5 b c a b c (2) e = a b c (3) f = 2 a + b c 215

6., 1 1.0 [N]., 3 F = F1 + F 2 + F 3,. 7. 3 A(2, 2), B( 1, 0), C(2, 3),. BC, CA, AB 8. 3 A( 2, 2), B(1, 0), C(2, 3),. BC, CA, AB 216

9. ABC,. (1) AB AC CB. (2) BC BA. 10. F 1 = F 2 = 10 [N] 2. θ F. (1) θ = 30 (2) θ = 60 (3) θ = 90 217

11. 30, 1 [N] 45.,. (1) S, N, W,,.. (2) x y. (3) S, N. 218

12. 1 [N]. 30, F [N]. S [N],. ( ) (1). (2) F S. 13.,, 30., 30. S [N], T [N],,. W [N]. 219

14. 60 [m], 3.0 [m/s], A, B S 1 S 2, S 3.. ( ) (1) S 3 6.0 [m/s]. S 3 [m/s]. (2) A 4.0 [m/s], B. A S 1 θ, tan θ., S 1 [m/s]. (3) S 1 B [ s ]., A B [m]. 15. a = 2 i 2 [m/s]. b, c, d, e, ( )., i, j.,,. (1) b = 5 j (2) c = 2 i 2 j (3) d = 3 i + 3 j (4) e = i + 0.404 j 220

7 221

1.. (1) 3 2 = 9 (2) 5 2 = 0.04 (3) 2 3 = 8 (4) 2 4 = 0.0625 2. log a 2 = x, log a 3 = y. x, y. (1) log a (a 6) = (2) log a 128 = (3) log a (9 a 2 8) = (4) log a 36 = (5) log a 3 6 = (6) log a 2 3 a 2 = 222

3. x. (1) log x + log(x + 2) 1 2 log 9 = 0 (2) log x 1 = log 2 4. log 2.22 = 0.3464, log 222. 5. 24, ( ). 3.9, 2.1.., L 1, L 2 m 1, m 2 m 2 m 1 = 2.5 log L 2 L 1, log 5.3 = 0.72. 6.. 70 [db]., [Pa]., 20 log P [db], P 0 = 2 10 5 [Pa]., log 2 = 0.3, P 0 10 0.2 = 0.63. 223

7.,., log 2 = 0.3, 10 0.7 = 5.1.,, [H + ] [OH ]. [H + ] > [OH ], [H + ] < [OH ]., [H + ] [OH ] [H + ][OH ] = 1.0 10 14 (mol/l) 2. (1) 2.0 10 5 (mol/l). ph. (2) ph 2.3. (mol/l). (3), 5.0 10 2 (mol/l). ph. 224

8 225

1. θ. ( θ, ) (1) 2 sin θ = 0.5, θ =, cos θ =, tan θ = (2) 3 tan θ = 0.4040, θ =, sin θ =, cos θ = 2.,. (1) sin 70 = (2) tan 230 = (3) cos( 160 ) = (4) sin 570 = 3. [rad],. (1) 45 = (2) π 6 = 4.. (1) 542 (2) 89 5 π 5..,. (1) θ 4 tan θ = 3 3, sin θ, cos θ. (2) θ 2 sin θ = 0.5150, cos θ, tan θ. 226

6..,. (1) θ = 37 [ ], sin θ =, cos θ =, tan θ = (2) θ = 392 [ ], sin θ =, cos θ =, tan θ = (3) θ = 8 π 6 [rad] [ ], sin θ =, cos θ =, tan θ = (4) θ = 13 π 4 [rad] [ ], sin θ =, cos θ =, tan θ = 7..,. (1) sin( 240 ) ( ) 8 (2) tan 3 π 8..,,. (1) θ 3 cos θ = 0.5, sin θ, tan θ. (2) θ 4 sin θ = 0.6018, cos θ, tan θ. 9..,,. (1) sin 1560 = (2) tan( 680 ) = 227

10.. ( ) 23 (1) sin 36 π = ( ) 223 (2) cos 180 π = ( (3) tan 19 ) 90 π = ( 403 (4) cos 180 π ( (5) sin 103 45 π ( 949 (6) tan 180 π ) = ) = ) = 11. r = 6 [cm]., π. (1) 1 [rad] l. (2) π 3 [rad] l. (3) 2 π 3 [rad] S. (4) r 2 θ [rad]. 12. 12 40..,, 1 [cm], 2 [cm]. (1), [rad/min]. (2) 12, [cm]. (3) 12, [cm 2 ]. 228

9 229

1. R [Ω] V, I., R. V [V] 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.0 I [ma] 1.18 2.38 3.53 4.72 5.00 7.10 8.20 9.51 10.6 11.8 2.. 30 [cm].,. t [ C] 20 45 70 95 120 0.00 1.04 2.12 3.00 3.80 l 10 2 [cm] 230

1 1. (1) 1 3 1 2 = 1 6 (2) 5 4 1 6 = 5 24 (3) 1 2 + 1 5 = 5 + 2 10 (4) 1 3 + 1 2 = 3 + 2 6 (5) a 2 a 5 = a2 10 = 7 10 = 5 6 (6) 1 7 + 4 8 = 1 7 + 1 2 = 2 + 7 14 (7) 5 3 6 7 = 35 18 21 = 17 21 (8) 3 6 + 5 2 = 1 2 + 5 2 = 6 2 = 3 (9) 4 5 5 7 = 4 5 7 5 = 28 25 (9) 21 7 9 2 = 3 2 9 = 2 3 2. (1) 1 a 2 3 = 2 3 a (2) b a + 1 b = b2 + a a b (3) 2 a 9 2 = 9 a (4) b a + c d = b d + a c a d (5) 1 a 2 5 = 2 5 a (6) b a c b = b a b c = b2 a c = 9 14 3. (1) b a c x b c y x = b a c x x b c y (2) 36 x2 z 2 27 a 2 b 2 1 2 = 2 x2 z 2 3 a 2 b 2 = 1 a c 1 c y = 1 a c 2 y (3) a b c d e b = a b e b d c c e b (4) a x 2 y z = a b2 e c d b c e b x z = a x 2 y z = a b x y x 2 y z (5) 1 a 1+a = (1+a) a 1+a (6) (7) 4. = 1 x+1 1 x 1 = 2 (x+1)(x 1) 1 3 x y 2 1 4 x 2 y b x y x z x y = 1 1+a (x 1) (x+1) = (x+1)(x 1) [ = 2 ] x 2 1 4 x 3 x y 2 4 x 3 y 4 x 2 y 3 y = 4 x 3 y 12 x 2 y 2 (1) b a c b = b a b c = b2 a c (2) b a b y a x y x = b a a x b y y x = 1 (3) b a + 2 a 3 b 3 a 5 b = b 15 b a 15 b + 2 a 5 a 3 b 5 a 3 a 3 a 5 b 3 a = 15 b2 + a 2 15 a b 231

5. (1) 1 3 + 2 5 2 15 = 5 15 + 6 15 2 15 = 9 15 = 3 5 (2) a b + c e b d = a d b d + c e b d = a d + c e b d (3) 1 a 1 b + 1 c = b c a b c a c a b c + = b c a c + a b a b c (4) a + b c c d = a + b d = a d d 1 1 a 1 b a b a b c + b d = a d + b d (5) b a a d b c = b a a d c b = c d (6) = = a b b a 6. (1) x a + x b = 1 c ( 1 a + 1 ) x = 1 b c b+a a b x = 1 c 1 b a a b (2) a x 4 = c x b b (a x 4) = c x a b x 4 b = c x a b x + x = 4 b + c x = a b (a+b) c (a b+1) x = 4 b + c x = 4 b+c a b+1 3 (3) b x+5 = 2 b x+c 3 ( b x+c) = 2 (b x+5) 3 b x + 3 c = 2 b x + 10 3 c 10 = 5 b x x = 3 c 10 5 b (4) x 2 = 3 4 x 2 2 (x 2) = 3 4 x 2 x 4 = 3 4 x 6 x = 7 x = 7 6 2 x (5) 3 x c = 5 7 14 x = 5 (3 x c) 14 x = 15 x 5 c 5 c = x x = 5 c 1 (6) 1 x + 1 = c x + y y (7) 1 y+x x y = c x + y x y x + y = c x + y x y = c x x + 2 = 1 2 x = c y 2 x = x + 2 x = 2 x = 4 (8) 3 x + y = y 7. 9 (x + y) = y 9 x + 9 y = y 9 x = 8 y x = 8 9 y (1) b a + 1 b = 1 2 (, b 2 ) b a = 1 2 1 b b a = b 2 2 b 2 b 2 = (b 2) a b 2, a = 2 b2 b 2 232

(2) a x + b = c x + 2 (, a c ) a x c x = 2 b (a c) x = 2 b a c, (3) x a = x b + 1 (b+1) (x a) = x b x + x a b a = x b x a b a = 0 b x = a b + a x = 2 b a c b 0, x= a b+a b (4) 1 a = 1 x + 1 y + 1 z, a = = (5) 2 = 1 1 x + 1 y + 1 z 1 yz +xz+xy x y z 1 a 1 2 a 1 = 1 a 1 = 1 2 a 1 = 1 4 (, b 0 ) = a = 5 4 [ = a(b+1) ] b x y z yz+xz+xy (6) x + y = a y + b (, a 1 ) x b = a y y x b = (a 1) y a 1, (7) b x a + c y b = 1 2 b x a = 1 2 c y b y = x b a 1 (, b 2 c y ) b x a = b 2 c y 2 b b x = b 2 c y a 2 b b 2 c y, a = 2 b2 x b 2 c y (8) a 2 b y = x 2 b y c 2 (, x 0) a 2 + c 2 = x 2 x 0, x = a 2 + c 2 (9) x + b c y = c x d y (, b y x) b c y c x = x d y (b y x) c = (x+d y) b y x, c = x+d y b y x (10) x+2 = a (11) [ = x+d y x b y x + 2 = a 2 x = a 2 2 1 x+1 = y 5, x+1 = 5 y (12) x + 1 = 25 y 2 x = 25 y 2 1 = 25 y2 y 2 1 2 m v2 = 1 2 m v 0 2 + m g h (, v > 0) v 2 = v 0 2 + 2 g h v > 0, (13) a 2 b 2 a 2 c + 2 c b = 1 a 2 b b = 2 a 2 c 2 c + 1 ( a 2 1 ) b = 2 a 2 c 2 c + 1 b = 2 a2 c 2 c+1 a 2 1 ] v = v 02 + 2 g h [ = 2 ] a2 c 2 c+1 (a+1)(a 1) 233

(14) d e = 1 a+1 (, d e) d e,, 1 d e = a + 1 a = 1 d e 1 = 1 (d e) d e (15) a 2 = c+3 (a 2) 2 = c + 3 c = 3 (a 2) 2 = 1 d+e d e = 3 ( a 2 4 a + 4 ) = a 2 + 4 a 1 1 (16) = 3 a 1, 1 a 1 = 3 a 1 = 1 a = 10 9 9 8. { a = 2 x 1 1 (1) (2) 1 b = 4 x 2 + 6 x + 3 a + 1 = 2 x x = a+1 2 2 1 1 2, ( ) 2 a+1 b = 4 + 6 a+1 + 3 2 2 = (a+1) 2 + 3 (a+1) + 3 = a 2 + 2 a + 1 + 3 a + 3 + 3 = a 2 + 5 a + 7 { a = x a 1 1 b = 3 x + d a 2 = x a 2 x = a 2 + a 1 1 2, b = 3 ( a 2 + a ) + d = 3 a 2 + 3 a + d 9. { a = b x + c y 1 d = e x y 2 (, b + c d 0 ) 1 + 2 c y, a + c d = b x + c e x a + c d = (b+c e) x b + c d 0, x = a+c d b + c e 1 e 2 b x, a e b d = c e y + b y a e b d = (b+c e) y b+c d 0, 10. { M = a x + y 1 Q = b M c x y = a e b d b + c e 2 (1) 1 c + 2 a x, c M + a Q = c y + a b M c M + a Q c y = a b M M 0, a 0, b = c M +a Q c y a M (2) 2 1 M, Q = b (a x + y) cx Q = a b x + b y cx Q (a b x cx) = b y Q (a b c) x = b y b 0, y = Q (a b c) x b 234

11. v y = g t y = 1 2 g t2 1 2 2 1. (1) 3 3 2 4 = 27 16 = 432 (1) 1, t = v y g (2) (1) 2, y = 1 2 g ( vy g y = v y 2 2 g 2 g y = v y 2 ) 2 v y = 2 g y (2) ( 3 2) 3 = 3 2 3 = 3 6 = 729 (3) 3 3 2 2 = 1 3 3 1 2 2 (4) a 2 b 3 c 2 ( a b c) 2 = a 2 b 3 c 2 ( a b c) 2 = a 2 b 3 c 2 a 2 b 2 c 2 = 1 27 1 4 = 1 27 4 1 = 4 27 = a 2 2 b 3 2 c 2 2 = a 4 b 1 c 0 = b a 4 (5) ( a 2 b 3) 2 = ( 1) 2 a 4 b 6 = b 6 ( 1) 2 a 4 = (6) ( y 2) 3 y 2 = y 6 y 2 b6 a 4 = y 6 ( 2) = y 4 = 1 y 4 (7) 2 2 3 2 = 1 2 2 32 (8) (2 a b) 2 a b 3 = = 1 2 2 1 3 2 = 1 36 (2 a b)2 a b 3 = 4 a2 b 2 a b 3 = 4 a b (9) 2 x 2 y 3 x 3 y 2 = 6 x 1 y 3 = 6 y3 x (10) 3 a 2 b 3 ( 2 a 2 b ) 2 = 3 a 2 b 3 2 2 a 4 b 2 = 3 2 2 a2 b 5 = 12 a 2 b 5 235

2. (1) 36 x2 z 2 27 a 2 b 2 57 a b c 56 x y 2 z 3 = 4 x 3 a b 57 c 56 y 2 z = 19 c x 14 a b y 2 z (2) ( 2 a 2 b ) 3 = ( 2) 3 a 6 b 3 = 8 a 6 b 3 (3) ( 3 a b) 2 ( 2 a 3 b) = ( 3) 2 a 2 b 2 ( 2 a 3 b) = 18 a 5 b 3 (4) ( 3 x 3 y 2) 2 ( 3 x y 3 ) 3 = ( 3) 2 x 6 y 4 ( 3) 3 x 3 y 9 = ( 3) 1 x 3 y 5 = x 3 ( 3) y 5 = x3 3 y 5 (5) ( 9a 3 b c 8a b 2 c 2 +12a b c 3) ( 6abc) 3. = 9a3 b c 8a b 2 c 2 +12a b c 3 6abc = 9a3 b c 6abc 8a b2 c 2 12a b c3 + 6abc 6abc = 3 2 a2 + 4 3 b c 2 c2 (1) 2 2 3 2 = 4 9 = 36 (2) ( 2 2) 3 = 2 6 = 64 ( ) 2 1 (3) 2 2 = ( 2 1) 2 2 2 2 = 2 2 2 2 = 2 4 = 1 2 4 = 1 16 (4) a 3 a 2 = a 1 = a (5) ( a b c) 2 ( a 2 b ) 2 = ( 1) 2 a 2 b 2 c 2 a 4 b 2 = 1 a 2 c 2 = c2 a 2 (6) 2 x 2 y 6 x y 2 = 12 x 3 y 3 (7) ( x 3) 2 x 3 = x 6 x 3 = x 3 (8) ( a 2 b 1) 2 = a 4 b 2 = b2 a 4 (9) ( 8 a b 2) 2 (2 a b c) 3 = 8 2 a 2 b 4 2 3 a 3 b 3 c 3 = 8 a 1 b c 3 = 8 b a c 3 (10) 4 a 2 b ( 2 a b) 3 ( 2 b c 2) 2 4. = 4 a 2 b ( 2) 3 a 3 b 3 ( 2) 2 b 2 c 4 = 2 a 1 c 4 = 2 c4 a (1) x y 2 ( x 2 y ) 2 = x y 2 x 4 y 2 = x 3 = 1 x 3 (2) x 0 x 5 x 2 = x 0 5 ( 2) = x 3 = 1 x 3 (3) ( x 2 y 1) 2 = x 4 y 2 = y2 x 4 (4) ( 2 a b 2) 2 6 a 4 b 2 = 2 2 a 2 b 4 6 a 4 b 2 5. (1) 802000 = 8.02 10 5 (2) 0.0027 = 2.7 10 3 (3) 694000 = 6.94 10 5 = 4 6 a 2 b 2 = 2 b2 3 a 2 (4) 0.000832 = 8.32 10 4 6. (1) 220 30 = 6600 = 6.6 10 3 (2) 50 25000 = 50000 25000 1000 = 2 1000 = 2 10 3 (3) 2 10 = 1024 = 1.024 10 3 (4) 30 4 = (3 10) 4 = 3 4 10 4 = 81 10 4 = 8.1 10 5 236

(5) (0.06) 3 = ( 6 10 2) 3 = 6 3 10 6 = 216 10 6 = 2.16 10 2 10 6 = 2.16 10 4 (6) 0.104 52000 = ( 104 10 3) ( 52 10 3) = 2 10 6 (7) 50 2 2 10 2 = (5 10) 2 2 10 2 = 5 2 10 2 2 10 2 = 5 2 2 = 50 = 5 10 1 (8) 123 5000 = ( 1230 10 1) ( 5 10 3) = 246 10 4 = 2.46 10 2 10 4 = 2.46 10 2 (9) ( 4 10 3) 2 ( 2 10 3 ) 2 = 4 2 10 6 2 2 10 6 = 64 10 12 = 6.4 10 1 10 12 = 6.4 10 11 (10) 1600 40 4 = ( 16 10 2) ( 4 4 10 4) = 1 16 10 2 = 0.0625 10 2 = 6.25 10 2 10 2 = 6.25 10 4 7. (1) 2 2 2 3 = 2 5 (2) 4 5 4 6 = 4 1 (3) 3 6 3 2 = 3 4 (4) x 2 x 5 = x 7 (5) 8 a 2 b 3 ( 2 b 2 c ) 2 = 8 a 2 b 3 ( 2) 2 b 4 c 2 = 2 a 2 b 1 c 2 (6) 3 4 3 = ( 4 3) 1 3 = 4 1 (7) 4 3 2 3 = ( 3 2) 1 4 3 1 2 (8) 7 16 = ( 2 4) 1 7 = 2 4 7 = 3 1 2 3 1 2 = 3 1 (9) 3 3 3 9 = 3 1 2 3 1 ( 3 2) 1 3 = 3 1 2 +1+2 3 = 3 3+6+4 6 = 3 13 6 (10) 15 3 5 6 3 5 5 = 5 3 5 6 5 5 = 5 1 ( 5 6) 1 3 ( 5 5) 1 2 = 5 1+2 5 2 = 5 1 2 8. (1) 3 { (3 27 4 3 = ) 4 }1 3 = 3 3 4 1 3 = 3 4 = 81 { (4 ) 3 }2 { 3 4 (2 ) } 2 3 2 3 4 (2) = 9 3 = ( ) 2 ( 3 2 4) 2 ( ) 3 1 2 = = 1 3 3 2 3 (3) a 3 2 a 2 a 1 4 (4) = a 3 2 +2 1 4 = a 6+8 1 4 = a 13 4 3 5 a 1 3 b 2 3 c 1 2 5 6 a 2 3 b 1 3 c 1 2 = 1 2 a 1 b 1 c 1 = 1 2 a b c (5) 3 2 2 2 2 3 = 2 1 3 2 2 ( 2 3)1 2 = 2 1 3 +2 3 2 = 2 2+12 9 6 = 2 5 6 (6) 4 ( 4) 4 = 4 ( 1) 4 4 4 = 4 1 4 4 = 4 4 4 = 4 (7) 3 125 = 3 ( 5) 3 = 5 (8) 3 a 4 3 a 5 = ( a 4)1 3 ( a 5)1 3 = a 4 3 + 5 3 = a 3 = 3 2 237

3 1. (1) 1.0005 5 (2) 0.060 2 (3) 51.00 10 2 4 (4) 3.200 10 12 4 (5) 128.00 5 (6) 0.045 2 (7) 50.09 4 (8) 25.0 10 6 3 (9) 0.02298 4 (10) 4.300 10 12 4 (11) 0.60 2 (12) 23.050 10 2 5 2. (1) 21.5[cm], 131[cm], 21.5 + 131 = 152.5 153[cm] (2) 4144[g]= 4.144[kg], 9.8[kg], 4.144[kg] 9.8[kg], 4.14[kg], 9.8 4.14 = 5.66 5.7[kg] (3) 3.24[m 2 ], 41.2[m 2 ], 3.24 + 41.2 = 44.44 44.4[m 2 ] (4) 18.2 16 = 291.2 2.9 10 2 (5) 3.14 (2.0) 2 = 12.56 13[cm 2 ] 3. (1) 9844[g]= 9.844[kg], 9.844, 4.6 9.84, 4.6, 9.84 + 4.6 = 14.44 14.4[kg] (2) 144[mm]= 0.144[m], 0.144, 12.5 0.14, 12.5, 0.14 + 12.5 = 12.64 12.6[m] (3) 39253.4[mm]= 39.2534[m], 128.1[cm]= 1.281[m], 39.2534, 1.281, 25.66 39.253, 1.281, 25.66, 39.253 + 1.281 + 25.66 = 66.194 66.19[m] (4) 15.6 12 = 187.2 1.9 10 2 (5) 40.0 2 3.142 = 251.36 4. (, 2 2 2 ) 2.51 10 2 [cm] (1) 5.42, 81.1, 5.42 + 81.1 = 86.52 86.5[m 2 ] (2) 4055[g]= 4.055[kg], 9.6, 4.055 9.6, 4.06, 9.6 4.06 = 5.54 5.5[kg] (3) 43.2 15 = 648 6.5 10 2 (4) (4.0) 2 3.14 = 50.24 5.0 10[cm 2 ] (5) 603.03[g]= 0.60303[kg], 6.501 10 4 [g]= 65.01[kg], 66.300, 0.60303, 65.01, 3.2266 66.300, 0.603, 65.01, 3.227, 66.300 + 0.603 + 65.01 + 3.227 = 135.14 135.14[kg] (6) 25.002 3.21 73.21 60 = 80.25642 1.220166, 80.3 1.2 = 79.1 7.91 10 1 238

5. (1) 3339.1[cm]= 33.391[m], 22953.5[mm]= 22.9535[m], 25.33, 33.391, 22.9535 25.33, 33.391, 22.954, 25.33 + 33.391 + 22.954 = 81.675 (2) 51.00 3.0 + 1.035 1.2 = 17 + 1.242, 17 + 1.2 = 18.2 18 (3) 25.00 3.10 67.2 5.0 6. = 77.5 13.44, 77.5 13 = 64.5 65 420[km] 3.0[km/s]= 140[s] 81.68[m] 8. 9. 502.03[g]= 0.50203[kg], 5.502 10 4 [g]= 55.02[kg], 10. 56.300, 0.50203, 55.02, 3.2255 56.300, 0.502, 55.02, 3.226, 56.300 + 0.502 + 55.02 + 3.226 = 115.048 115.05[kg] 4.06 21 = 85.26 S 1 = 85[cm 2 ] (5.0) 2 3.14 = 78.5 S 2 = 79[cm 2 ] 85 + 79 = 164 S = 164[cm 2 ] 2.08 12 = 24.96 S 1 = 25[cm 2 ] 1.25 6.9 = 8.625 S 2 = 8.6[cm 2 ] 25 + 8.6 = 33.6 S = 34[cm 2 ] 7. 140 60 = 2.33 2.3 10.1[cm]= 0.101[m], 120.01[mm]= 0.12001[m], 10.02[cm]= 0.1002[m],, 0.101 0.1200 1.001 +(1.0) 2 3.14 0.100 = 0.01213212 + 0.314, 0.0121 + 0.31, 0.012 + 0.31 = 0.322 0.32[m 3 ] 239

4 1. (1) ω (2) γ (3) θ (4) β (5) π 2. (1) 48.2 [µm] = 48.2 10 6 [m] = 48.2 10 4 10 2 [m] = 48.2 10 4 [cm] = 4.82 10 3 [cm] (2) 0.0482 [A] = 0.0482 10 6 10 6 [A] = 0.0482 10 6 [µa] = 4.82 10 4 [µa] (3) 148.2 [MΩ] = 148.2 10 6 [Ω] = 1.482 10 8 [Ω] (4) 48 10 2 [g] = 48 10 2 10 3 10 3 [g] = 48 10 2 10 3 [kg] = 48 10 1 [kg] = 4.8 [kg] (5) 0.48 10 5 [V] = 0.48 10 2 10 3 [V] = 0.48 10 2 [kv] = 4.8 10 1 [kv] 3. (1) 20 [kv] = 20 10 3 [V] = 20 10 3 10 6 10 6 [V] = 20 10 3 10 6 [MV] = 20 10 3 [MV] = 2 10 1 10 3 [MV] = 2 10 2 [MV] (2) 50 [mg] = 50 10 3 [g] = 5 10 1 10 3 [g] = 5 10 2 [g] (3) 365 [kω] = 365 10 3 [Ω] = 3.65 10 2 10 3 [Ω] = 3.65 10 5 [Ω] (4) 3 [m] = 3 10 2 10 2 [m] = 3 10 2 [cm] (5) 25 [km] = 25 10 3 [m] = 2.5 10 1 10 3 [m] = 2.5 10 4 [m] (6) 0.9 [V] = 0.9 10 3 10 3 [V] = 0.9 10 3 [mv] = 9 10 1 10 3 [mv] = 9 10 2 [mv] (7) 40 [kv] = 40 10 3 [V] = 40 10 3 10 6 10 6 [V] = 40 10 3 10 6 [MV] = 40 10 3 [MV] = 4 10 1 10 3 [MV] = 4 10 2 [MV] (8) 600 [kω] = 600 10 3 [Ω] = 6 10 2 10 3 [Ω] = 6 10 5 [Ω] 240

(9) 25 [mg] = 25 10 3 [g] = 2.5 10 1 10 3 [g] = 2.5 10 2 [g] (10) 0.096 [A] = 0.096 10 6 10 6 [A] = 0.096 10 6 [µa] = 9.6 10 2 10 6 [µa] = 9.6 10 4 [µa] (11) f = 50 [khz], f = 50 10 3 [Hz] = 5 10 4 [Hz], T = 1 f T = (12) T = 1 f 4., 1 5 10 4 = 1 5 10 4 = 0.2 10 4 = 2 10 5 [s] f T = 1, T = 25 [µs], f = 1 T T = 25 10 6 [s] = 2.5 10 5 [s], f = 1 2.5 10 5 = 1 2.5 105 = 0.4 10 5 = 4 10 4 [Hz] (1) 8.24 10 6 [ g/cm 3 ] = 8.24 106 [g] 1 [cm 3 ] = 8.24 10 6 [g] 1 [cm] 1 [cm] 1 [cm] = 8.24 103 10 3 [g] (1 [cm]) 3 = 8.24 103 [kg] ( 1 10 2 [m] ) 3 = 8.24 103 [kg] 1 3 10 6 [m 3 ] = 8.24 10 9 [kg/m 3 ] (2) 591 10 2 [ kg/m 3 ] 5. 6. = 591 102 [kg] 1 [m 3 ] = 591 10 2 [kg] 1 [m] 1 [m] 1 [m] = 591 102 [kg] (1 [m]) 3 = 591 102 10 3 [g] ( 1 10 2 [cm] ) 3 = 591 105 [g] 1 3 10 6 [cm 3 ] = 591 10 1 [ g/cm 3 ] = 5.91 10 1 [ g/cm 3 ] 54 [ g] 2.7 [ g/cm 3 ] = 54 2.7 [ cm3 ] = 20 [ cm 3 ] = 2.0 10 1 [ cm 3 ] 5.0[cm] 10.0[cm] 3.0[cm] 2.7[g/cm 3 ] = 405 [g] 7. (1) 100[cm] 200[cm] 0.2[cm] 2.7[g/cm 3 ] = 10800 [g] = 10.8 [kg] (2) 10.8 [kg] 9.8 [m/s 2 ] 8. = 105.84 [kg m/s 2 ] = 105.84 [N] 5 [mm], 106 [N] 3.14 (0.5[cm]) 2 100[cm] 7.9[g/cm 3 ] = 620.15 [g] 620 [g] 241

9. (1) 4.0 [km/h] = (2) 600 [m/min] = 10. (1) 552 [km] 3 [h] 4000 [m] 3600 [s] = 1.1 [m/s] 0.6 [km] 1 60 [h] = 36 [km/h] = 184 [km/h] = 3.6 10 [km/h] (2) 184[km/h]= 184 103 [m] =51.1[m/s] 3600[s] (3) 90 [km/h] = 90 103 [m] 3600 [s] 11. 12. (25 0) [m/s] 5 60 [s] m v = f t, = 25 [m/s] = 8.3 10 2 [m/s 2 ] 16 10 3 [kg] 5 [m/s] = f 4 [s] f = 2 10 4 [kg m/s 2 ] = 2 10 4 [N] 60 [km/h] = 60 103 [m] 3600 [s] 13. (16.7 0) [m/s] 10 [s] 6 [m/min] = 6 [m] 60 [s] 18 [m/min] = 18 [m] 60 [s] (0.3 0.1) [m/s] 5 [s] = 16.7 [m/s] = 1.67 [m/s 2 ] = 0.1 [m/s] = 0.3 [m/s] = 0.04 [m/s 2 ] 14. 2 1.5 4[m 3 ] 1.0 10 3 [kg/m 3 ] 9.8[m/s 2 ] 2 1.5 [m 2 ] = 3.92 10 4 [N/m 2 ] = 3.92 10 4 [Pa] 15. 930 [kg] 9.8 [m/s 2 ] 0.3 [m 2 ] 16. 17. = 30380 [N/m 2 ] = 30380 [Pa] = 30.380 [kpa] 30.4 [kpa] 2.5 10 8 [N] 4 10 6 [m 2 ] 1 [m 2 ] = 1 10 3 [N] 4 10 3 [kg] 9.8 [m/s 2 ] 1 4 140 10 4 [m 2 ] = 7 10 5 [N/m 2 ] = 7 10 5 [Pa] 18. 980 [N] 12 [m] 100 [s] = 117.6 [J/s] = 117.6 [W] 19. 400 10 3 [J] 20 10 3 [N] 20. 9800 [N] 12 [m] 4 60 [s] 21. = 490 [J/s] = 490 [W] 2 [kw h] = 2 10 3 [W h] = 20 [m] = 2 10 3 [J/s] 3600 [s] = 7.2 10 6 [J] 242

5 1. 132 5 2 = 144 = 12 (1) sin θ = 12 13 (2) cos θ = 5 13 (3) tan θ = 12 5 2. (1) 8 ( 5 sin 60 ) 1 2 ( ) 3 = 8 5 1 2 2 = 10 3 [cm 2 ] (2) 10 2 ( 8 sin 45 ) 1 2 = 10 ( 2 8 1 ) 1 2 2 = 40 [cm2 ] 3. (1) x = 5 2 3 2 = 16 = 4 y = 4 4 3 = 16 3 6. 7. 0.2002 y = 7.002 y = 7.002 0.2002 = 34.97, x = 0.9004 7.002 0.2002 = 31.49 x 31.5 [m], y 35.0 [m], 100 sin 10 = 100 0.1736 = 17.36 17.4 [m], 100 cos 10 = 100 0.9848 = 98.48 98.5 [m] 1.6 + 10 tan 35 = 1.6 + 10 0.7002 = 1.6 + 7.002 = 8.602 8.6 [m] (2) sin θ = 4 5, cos θ = 3 5, tan θ = 4 3 (3) sin θ = 4 5 = 0.8 θ 53 4. cos θ = 6 10 = 3 5 = 0.6 θ 53 5. (1) tan 42 = 0.9004, tan 35 = 0.7002 (2) tan 42 = x y, tan 35 = x y+10 (3) 0.9004 = x y x = 0.9004 y 0.7002= x x=0.7002 (y+10) y+10 0.9004 y = 0.7002 (y+10) 243

6 1. 3 a 2 b = 3 (2, 4) 2 (3, 1) = (6, 12) (6, 2) = (0, 14) 3 a 2 b = 0 2 + 14 2 = 14 5. (1) d = a + 1.5 b c 2. 2 a b = 2 (1, 2) (0, 3) = (2, 4) (0, 3) = (2, 7) 2 a b = 2 2 + ( 7) 2 = 53 (2) e = a b c 3. 2 a 3 b = 2 ( 1, 2) 3 (3, 1) = ( 2, 4) (9, 3) = ( 11, 1) 2 a 3 b = ( 11) 2 + 1 2 = 122 4. ( (1) 2 a ) + 3 b = 2 a + 6 b ( 3 a ) 2 b ) ( 3 a 6 b (3) f = 2 a + b c = a + 12 b ( (2) 2 a ) ( 4 b + c 2 a ) 2 b + c = 2 a 8 b +2 ( c 2 a 4 b +2 ) c = 4 b 244

6. F = F1 + F 2 + F 3 F = ( 1, 3), F = ( 1) 2 + 3 2 = 10 7. BC = (2, 3) ( 1, 0) = (3, 3) BC = 3 2 + ( 3) 2 = 3 2 CA = (2, 2) (2, 3) = (0, 5) CA = 0 2 + 5 2 = 5 AB = ( 1, 0) (2, 2) = ( 3, 2) AB = ( 3) 2 + ( 2) 2 = 13 8. BC = (2, 3) (1, 0) = (1, 3) BC = 1 2 + 3 2 = 10 CA = ( 2, 2) (2, 3) = ( 4, 5) CA = ( 4) 2 + ( 5) 2 = 41 AB = (1, 0) ( 2, 2) = (3, 2) AB = 3 2 + 2 2 = 13 9. (1) CB = CA + AB = AC + AB = AB AC (2) BC BA = ( BC ) AB 10. = AB + BC = AC (1) θ =30 F 1 = (10, 0) F 2 = (10 cos 30, 10 sin 30 ) ( ) 3 = 10 2, 10 1 2 ( = 5 ) 3, 5 ( F = (10, 0) + 5 ) 3, 5 ( = 10 + 5 ) 3, 5 F = (10+5 3 ) 2 +5 2 = 200+100 3 = 10 2+ 3 4+2 3 = 10 2 3+1+2 3 1 = 10 2 ( 3+ ) 2 1 = 10 = 10 2 3+ 1 2 6+ 2 = 10 2 ( ) = 5 6+ 2 245

(2) θ =60 F 1 = (10, 0) F 2 = (10 cos 60, 10 sin 60 ) ( = 10 1 ) 3 2, 10 2 ( = 5, 5 ) 3 ( F = (10, 0) + 5, 5 ) 3 ( = 15, 5 ) 3 F = 15 2 +(5 3 ) 2 = 300 = 10 30 (3) θ =90 F 1 = (10, 0) F 2 = (10 cos 90, 10 sin 90 ) = (10 0, 10 1) = (0, 10) F = (10, 0) + (0, 10) F = (10, 10) = 10 2 +10 2 = 200 = 10 2 11. (1) (2) x S cos 45 = W sin 30 y S sin 45 + N = W cos 30 2 (3) S 2 = W 1 2 12. 2 S = W 2 3 S 2 + N = W 2 1, 2 S + 2 N = 3 W S = 1 2 W = 2, 2 2 W 2 2 2 W + 2 N = 3 W 3 1 N = W 2 (1) S cos 30 = F S sin 30 = 1 1 2 (2) S 1 2 = 1 S = 2 [N] 3 3 F = S 2 = 2 2 = 3 [N] 246

13. 14. S cos 60 = T cos 30 S sin 60 = W + T sin 30 (1) 6.0 3.0 = 3.0 [m/s] (2) tan θ = 4 3 S 1 (3.0)2 + (4.0) 2 = 5.0 [m/s] (3) 15. 60 [m] 4.0 [m/s] = 15 [s] 5.0 [m/s] 15 [s] = 75 [m] (1) b = 5 j 5 [m/s] (2) c = 2 i 2 j 2 2 [m/s] (3) d = 3 i + 3 j 3 2 [m/s] (4) e = i + 0.404 j tan θ = 0.404 θ 22, 1 12 + 0.404 2 = 1.163216 22 1.163216 [m/s] 7 1. (1) 3 2 = 9 2 = log 3 9 (2) 5 2 = 0.04 2 = log 5 0.04 (3) 2 3 = 8 3 = log 2 8 (4) 2 4 = 0.0625 4 = log 2 0.0625 2. (1) log a ( a 6 ) = log a a log a 6 = log a a (log a 2 + log a 3) = 1 x y (2) log a 128 = log a 2 7 = 7 log a 2 = 7 x (3) log a ( 9 a 2 8 ) = log a 9 a 2 log a 8 = 2 log a 3 + 2 log a a 3 log a 2 = 2 y + 2 3 x (4) log a 36 = log a 2 2 3 2 = 2 log a 2 + 2 log a 3 = 2 x + 2 y (5) log a 3 6 = 1 3 log a 6 = 1 3 ( log a 2 + log a 3 ) = 1 3 ( x + y ) (6) log a 2 3 a 2 = log a 2 log a 3 a 2 = log a 2 ( log a 3 + 2 log a a ) = x y 2 247

3. (1), { x > 0 x+2 > 0 x > 0 1 log x + log (x+2) 1 2 log 9 = 0 log x (x+2) = log 3 x (x+2) = 3 x 2 + 2 x 3 = 0 (x 1) (x+3) = 0 1, 2 x = 1 (2), x > 0 1 4. 5. log x 1 = log 2 log x log 10 = log 2 log x = log 2 + log 10 log x = log 20 x = 20 2 1, 2 x = 20 log 222 = log ( 2.22 10 2) = log 2.22 + 2 log 10 = 0.3464 + 2 1 = 2.3464 L 1, x = 1, 3 2 [ log 10 = 1 ] L 2, ( 3.9) ( 2.1) = 2.5 log L 2 L 1 1.8 = 2.5 log L 2 L 1 log L 2 L 1 = 0.72 log 5.3 = 0.72 L 2 L 1 = 5.3 [ ] 6. P 20 log = 70 2 10 5 P log 2 10 5 = 7 2 7. log P log ( 2 10 5) = 7 2 log P ( log 2 5 log 10 ) = 7 2 log P = log 2 5 log 10 + 7 2 = 0.3 5 1 + 3.5 = 1.2 P = 10 1.2 = 10 1 0.2 = 10 0.2 10 1 = 0.63 10 1 = 6.3 10 2 [Pa] (1) [ H +] = 2.0 10 5 (mol/l), ph = log [ H +] = log ( 2.0 10 5) = ( log 2.0 5 log 10 ) = ( 0.30 5 1 ) = 4.7 (2) ph = log [ H +] = 2.3, log [ H +] = 2.3 [ H + ] = 10 2.3 = 10 0.7 3 = 10 0.7 10 3 = 5.1 10 3 (mol/l) (3) [ OH ] = 5.0 10 2 (mol/l), [ H + ][ OH ] = 1.0 10 14 [ H + ] 5.0 10 2 = 1.0 10 14 [ H + ] = 0.2 10 12 = 2.0 10 13 (mol/l) ph = log [ H +] = log ( 2.0 10 13) = ( log 2.0 13 log 10 ) = ( 0.30 13 1 ) = 12.7 248

8 1. (1) sin θ = 0.5, 1 θ = 30 cos 30 = 0.8660, tan 30 = 0.5774 2 cos θ < 0, tan θ < 0, θ = 180 30 = 150 cos θ = 0.8660, tan θ = 0.5774 (2) tan θ = 0.4040, 2. 1 θ = 22 sin 22 = 0.3746, cos 22 = 0.9272 3 sin θ < 0, cos θ < 0, θ = 180 + 22 = 202 sin θ = 0.3746, cos θ = 0.9272 (1) sin 70 = 0.9397 (2) 230 = 180 + 50 3 tan 230 > 0, tan 230 = tan 50 = 1.1918 (3) 160 = 200 360 = 200 + 360 ( 1) = 180 + 20 + 360 ( 1) 3 cos( 160 ) < 0, cos( 160 ) = cos 20 = 0.9397 (4) 570 = 210 + 360 = 180 + 30 + 360 1 3 sin 570 < 0, sin 570 = sin 30 = 0.5000 3. (1) 45 = 2 π 45 360 = π 4 [rad] (2) 4. π π 6 = 360 6 2 π = 30 (1) 542 = 178 720 = 178 + 360 ( 2) 90 < 178 < 180, 2 (2) 89 5 π = 9 5 π + 16 π 3 2 π < 9 5 = 9 5 π + 2 π 8 π < 2 π, 4 5. 3 (1) tan θ = 3, tan 2 θ + 1 = 1 ( 3 3 4 3 = 1 cos 2 θ cos 2 θ, ) 2 + 1 = 1 cos 2 θ cos 2 θ = 3 4 θ 4 cos θ > 0, 3 cos θ = 2, tan θ = sin θ cos θ, 3 3 = sin θ sin θ = 1 3 2 2 (2) sin θ = 0.5150, 1 θ = 31 cos 31 = 0.8572, tan 31 = 0.6009 2 cos θ < 0, tan θ < 0, ( θ = 180 31 = 149, ) cos θ = 0.8572, tan θ = 0.6009 249

6. (1) θ = 37 = 323 360 = 323 + 360 ( 1) 270 < 323 < 360, 4 1, sin 37 = 0.6018, cos 37 = 0.7986 tan 37 = 0.7536 4 sin θ < 0, cos θ > 0, tan θ < 0, sin θ = 0.6018, cos θ = 0.7986 tan θ = 0.7536 (2) θ = 392 = 32 + 360 = 32 + 360 1 0 < 32 < 90, 1 1, sin θ = 0.5299, cos θ = 0.8480 tan θ = 0.6249 (3) θ = 8 π 8 6 = 360 6 π 2 π = 240, 180 < 240 < 270, 3 3 sin θ = sin 240 = 2 cos θ = cos 240 = 1 2 tan θ = tan 240 = 3 (4) θ = 13 π = 360 13 4 π 4 2 π = 585 = 135 720 = 135 + 360 ( 2) 90 < 135 < 180, 2 2 sin θ = sin 135 = 2 2 cos θ = cos 135 = 2 tan θ = tan 135 = 1 7. (1) 240 = 120 360 (2) 8., = 120 + 360 ( 1) sin( 240 ) = sin 120 = 8 8 3 π = 360 3 π 2 π = 480 = 120 + 360 1 3 2, ( ) 8 tan 3 π = tan 120 = 3 (1) cos θ = 0.5 = 1 2, 3 θ = 240 3 sin θ = sin 240 = 2 tan θ = tan 240 = 3 (2) sin θ = 0.6018, 9. 1 sin θ = 0.6018 θ = 37 cos 37 = 0.7986, tan 37 = 0.7536 4 cos θ > 0, tan θ < 0, cos θ = 0.7986, tan θ = 0.7536 (1) 1560 = 120 + 1440, = 120 + 360 4 sin 1560 = sin 120 = (2) 680 = 40 720 3 2 = 40 + 360 ( 2),, tan( 680 ) = tan 40 = 0.8391 250

10. (1) 23 23 36 π = 360 36 π 2 π = 115 = 180 65 8 4, sin( 180 θ ) = sin θ,, ( ) 23 sin 36 π = sin 115 = sin( 180 65 ) = sin 65 = 0.9063 (2) 223 223 180 π = 360 180 π 2 π = 223 = 180 + 43 8 4, cos( 180 + θ ) = cos θ,, ( ) 223 cos 180 π = cos 223 = cos( 180 + 43 ) = cos 43 = 0.7314 (3) 19 90 π = 360 19 90 π 2 π = 38 = 322 360 = 322 + 360 ( 1) 8 4, tan( 360 θ ) = tan θ,, ( tan 19 ) 90 π = tan( 38 ) = tan {322 + 360 ( 1)} = tan 322 = tan( 360 38 ) = tan 38 = 0.7813 (4) 403 403 180 π = 360 180 π 2 π = 403 = 43 + 360 1,, ( ) 403 cos 180 π = cos 403 = cos (43 + 360 1) = cos 43 = 0.7314 (5) 103 45 π = 360 103 45 π 2 π = 412 = 308 720 = 308 + 360 ( 2) 8 4, sin( 360 θ ) = sin θ,, ( sin 103 ) 45 π = sin( 412 ) = sin {308 + 360 ( 2)} = sin 308 = sin( 360 52 ) = sin 52 = 0.7880 (6) 949 949 180 π = 360 180 π 2 π = 949 = 229 + 360 2 8 4, tan( 180 + θ ) = tan θ,, ( ) 949 tan 180 π = tan 949 = tan (229 + 360 2) = tan 229 = tan( 180 + 49 ) = tan 49 = 1.1504 251

11. (1) l = r θ = 6 1 = 6 [cm] (2) l = r θ = 6 π 3 = 2 π [cm] 9 1. (3) S = 1 2 r2 θ (4) 12. = 1 2 62 2 π 3 = 12 π [cm2 ] 1 2 62 θ = 6 2 θ = 2 [rad] (1), 2 π [rad] 60 [min], = π 30 [rad/min] 2 π 1 12 [rad] = π 60 [min] 360 π [rad/min] ( (2) 2 2 π 40 ) = 8 60 3 π [cm] (2) 1 ( 2 12 2 π 1 12 40 ) = π 60 18 [cm2 ] 2. R, R = V I = 8.3 0 0.010 0 = 8.3 102 [Ω] α, 0.300400 0.300000 120 20 α = 0.300000 = 0.000013 [1/K] = 1.3 10 5 [1/K] 20 [ C], t [ C] l, 20 [ C] l 0, l = l 0 { 1 + (t 20) 1.3 10 5 }. 252

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