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PAT 1 (.. 58) 1
PAT 1 (.. 58) 71 (PAT 1) 31 2558 13.00 - 16.00 .
1 1 - 30 6 1. , ( ) (~ ~) 1. ( ) 2. ( ) ( ) 3. ( ) ( ) 4. ( ~) ( ) 5. ( ) ( ~)
2. () , ( ) = , () = 12 , () = 15 , () = 16 , ( ) = 20 ( ) = ( ) = ( ) 1. ( ) = 10 2. ( ) = 11 3. ( ) = 4 4. (( ) ) = 12 5. (( ) ) = 5
3. || 1| 1| < 1 1+1
22
23+2
1. (5, 1) 2. (3, 1) 3. (1, 3) 4. (0, 4) 5. (1, 5)
29 Feb 2016
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2 PAT 1 (.. 58)
4. (3 4 sin2 9)(3 4 sin2 27)(3 4 sin2 81)(3 4 sin2 243) 1. 0 2. 1 3. 2 4. tan 9 5. cot 9
5. 2 cot 2
= (1 + cot )2 0 < < 2 (1+sin) sec
2
cos2
1. 0.125 2. 0.25 3. 1 4. 2 5. 4
6. sec2(arctan 2) + cosec2(arccot 3) + cosec (2 arccot 2 + arccos 35)
1. 33524
2. 35124
3. 37524
4. 38524
5. 39924
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PAT 1 (.. 58) 3
7. = arcsin (cos 3) 0 < <
2
sin2 + sin2( + ) + sin2(5 + ) 1. 0 2. 1 3. 3
2 sin2
4. 32 cos 2 5. 3
2 2cos 2
8. 2 log( 2) = log 3 + log2( + 2) (log
2 )(log ) = 1 + log + 1. 183 2. 210 3. 216 4. 225 5. 239
9. 1. 1 = { (, ) | + 1 = 0 } 2. 2 = { (, ) | = tan }
3. 3 = { (, ) | 2 = 2 + 1 } 4. 4 = { (, ) | = |2 | }
5. 5 = { (, ) | 2 =
+1 }
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4 PAT 1 (.. 58)
10. 2 + 2 + 6 12 = 0 > 0 4 + 3 = 71 14 = 7
1. 2 4 + 4 16 = 0 2. 2 + 4 + 4 16 = 0 3. 2 + 4 4 + 20 = 0 4. 2 + 4 + 8 + 44 = 0 5. 2 + 4 + 8 36 = 0
11. 2 4 + 40 236 = 0 (4, 6)
1. 42 + 92 + 8 36 + 140 = 0 2. 42 + 92 + 8 + 36 140 = 0 3. 42 + 92 8 36 140 = 0 4. 92 + 42 36 8 180 = 0 5. 92 + 42 + 36 8 + 180 = 0
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PAT 1 (.. 58) 5
12. () 83 4 1 = 0 () 84 82 + + 1 = 0 () 3 + 2 > 0 () [() ()] () [() ()] () [() ()] 1. () 2. () 3. () 4. () () () 5. () () ()
13. 1
limx
1
1(1
23
2+1)
1. 0 2. 0.5 3. 1 4. 2 5. 4
14. = 2 + | 1| (0, 2) (0, 2)
1. (1, 3) 2. (1, 5) 3. (2, 5) 4. (3, 2) 5. (3, 4)
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6 PAT 1 (.. 58)
15. 1() = 2+1
() 2(4) (4) = 3 () ((4)) = ((4)) () (0, 2) 1. () () () 2. () () () 3. () () () 4. () () () 5. () () ()
16. = 16 + = 8 + || = || 60 ( + )2
1. 8 2. 16 3. 64 4. 192 5. 320
17. 4 4
1. 170
2. 135
3. 435
4. 17 5. 2
7
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PAT 1 (.. 58) 7
18. () = {9 , 07 , > 4
() = { + 2 , < 1 4 , 1
() 0 ( )() = 9 4 () 4 < 6 ( )() = 3 () > 6 ( )() = 9 1. () () () 2. () () () 3. () () () 4. () () () 5. () () ()
19. (1 + i) (97i)()3+i
= 6 2i i2 = 1 (conjugate)
() | + 8| = 2 () | + 3i| = 10 () |i + 2| = 8 1. () () () 2. () () () 3. () () () 4. () () () 5. () () ()
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8 PAT 1 (.. 58)
20. = 23
32+1 = 1, 2, 3,
1n
1. 83 2. 4
3. 24 4. 643
5.
21. 1 1 + 4 , 2 + 4 , , 20 + 4 2 21 + 4 , 22 + 4 , , 220 + 4 1, 2, , 20 1 50 1 10 2 1. 96 400 2. 96 576 3. 100 400 4. 104 400 5. 104 576
22. 25% 15.87 85 47.6
0
1. 34.46 2. 18.41 3. 13.57 4. 11.51 5. 9.68
0.4 0.9 1.0 1.1 1.2 1.3
0.1554 0.3159 0.3413 0.3643 0.3849 0.4032
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PAT 1 (.. 58) 9
23. = 1+2+22+23+ +2
32 = 1, 2, 3,
n
lim (1 + 2 + 3 + + )
1. 29 2. 1
8 3. 9
56
4. 27 5. 25
56
24.
= { (, ) | = 2+2
4 2+1 } = { 2 | }
1. 6 2. 10 3. 19 4. 29 5. 30
25. + 2 4 , 1 , + 1 , 0 0 1. = 2 + 2 2. = 3 + 2 3. = 2 + 3 4. = + 4 5. = 4 +
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10 PAT 1 (.. 58)
26. = [1 22 1
] = [
] , , = 9 1 = 1 det() = 24 + + +
1. 5 2. 6 3. 7 4. 8 5. 9
27. () | | = || ||
() | + |2 = ||2 + ||2 () + || = || 1. () () () 2. () () () 3. () () () 4. () () () 5. () () ()
28. , , , + 4 = + + 5 = + + 1 = + 2 = + + 3 () + < + () < < < () + < + 1. () 2. () 3. () 4. () () () 5. () () ()
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PAT 1 (.. 58) 11
29.
5 1. 3.8 2. 4.3 3. 4.8 4. 4.9 5.
30. 60
() 1.5 () () 1. () 2. () 3. () 4. () () () 5. () () ()
0 2 3 3 5 5 6 8
9 11 3
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12 PAT 1 (.. 58)
2 31 - 45 8 31. 150 80 60 30
32. 3(9 + 3||+|+4|) = 3|+4| + 3||+4 = { 59 | }
33. :
2
limx
2+6
1+()3 = 6 1 + () 0
6 = 4 = () = 2 (2)
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PAT 1 (.. 58) 13
34. () = {3 , < 1
+ , 1 < 1
32 + 2 , 1
2
2
()
35. > 1 () = log2( ) = 1, 2, 3, 1
(1)+
1
(2)++
1
(10) = 77
36. |1 0 4 15
| = 17
|5 + 2 2 58 + 2 2 0
|
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14 PAT 1 (.. 58)
37. {} 1 + 3 + 5 + + 49 = 2 + 4 + 6 ++ 50 = 1275 100 = 200 51 + 52 + 53 + + 100
38. 1, 2, 3 1 1 2 1 3 2
39. 8 () ()
8
1
i
= 40 , 8
1
i
= 48 , 8
1
i
2 = 210 ,
8
1
i
2 = 380 ,
8
1
i
= 270
3 = 1 < 2 < < 8 < 10 4
() : 1 2 3 4 5 6 7 8 () 1 2 3 4 5 6 7 8
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PAT 1 (.. 58) 15
40. 2 + 2 + + = 21 2 + 1 = 0 (asymtote) (1 + 25, 3) 2 + 2 + 2 + 2
41. (, ) , 11
=
1
10
42. 21+ + 1
< 1
2 + 2
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16 PAT 1 (.. 58)
43. (, ) 2 + = 2 3 log4 162 = 6 + 6 log2 = { 2 + 2 | (, ) }
44. 5 1, 2, 3, 4, 5 5
1
i
2 = 214
5
1
i
( )2 = 34
> 0 5 1 + 22 , 2 + 23 , 3 + 24 , 4 + 25 , 5 + 21
16 12 , 23 , 34 , 45 , 51
45. + = 143 , {1, 2, 3 , , 9}
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PAT 1 (.. 58) 17
1. 2 11. 3 21. 1 31. 230 41. 4 2. 5 12. 3 22. 4 32. 126 42. 1.5 3. 3 13. 1 23. 5 33. 5 43. 2 4. 2 14. 1 24. 4 34. 9.25 44. 78.7 5. 5 15. 2 25. 5 35. 32 45. 429 6. 4 16. 4 26. 4 36. 68 7. 4 17. 3 27. 3 37. - 8. 2 18. 4 28. 1 38. 160 9. 3 19. 2 29. 2 39. 3 10. 5 20. 3 30. 1 40. 117
1. 2 ( ) (~ ~) ~ ~ T T T ~ ~ ~T ~ F ~ F F F ~ ~ ~F ~ T ~ ~
F 1. 2.
3. 4.
5.
2.
2. 5
( ) 0
~
( ) F F F
F
( ) ( ) (F ) ( F) T ( F) T
( ) ( ) ( ) (F ) ( ) F F
( ~) ( ) ( ~F) ( ) ( T ) ( ) T ( ) T
( ) ( ~) ( ) (F ~) T T T
T F T T
( ) = ( ) =
=
0
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18 PAT 1 (.. 58)
( ) = ( ) = ( ) , , ,
= = = () = 12
Inclusive Exclusive
(1) (2) :
= 10 , = 1
1. 10 2. 11 3.
4. ( ) 12 5. ( )
5.
0
( ) = ( ) = ( )
+ = + = + = =
0
( ) = () + () + () ( ) ( ) ( ) + ( ) 20 = 12 + 15 + 16 ( + ) ( + ) ( + ) + 20 = 43 + 2 + 3 = 23 (2)
0 + + + = 12 + 2 = 12 (1)
+ 2 = 12 (1) 2 + 3 = 23 (2)
2 (1) : 2 + 4 = 24 (3)
(3) (2) : = 1 (1) : + 2(1) = 12 = 10
0 1
1 1 10 () = 15
() = 16
0 1
1 1 10
3
4
0 1
1 1 10
3
4
0 1
1 1 10
3
4
= = 4
0 1
1 1 10
3
4
0 1
1 1 10
3
4
= ( ) = ( ) 4
0 1
1 1 10
3
4
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PAT 1 (.. 58) 19
3. 3 : || 1| 1| < 1
0 < | 1| | 1| < 2
: (1, 3) 3 (1, 3) (1, 3) 3 ( )
:
1
(1, 3) 3
4. 2 sin3 = 3 sin 4 sin3 sin
sinsin
sin3
= (sin9)(34sin2 9)
sin9(sin27)(34sin2 27)
sin27(sin81)(34sin2 81)
sin81(sin243)(34sin2 243)
sin243
= 3 sin94sin3 9
sin 9
3 sin274sin3 27
sin27
3sin814sin3 81
sin81
3sin2434sin3 243
sin243
= sin3(9)
sin9
sin3(27)
sin27
sin3(81)
sin81
sin3(243)
sin243
= sin27
sin9
sin81
sin27
sin243
sin81
sin729
sin243 sin729
sin9
729 360 9 sin729sin9
= sin9
sin9 = 1
1 < | 1| 1 < 1 0 < | 1| < 2
1 = 0 1
2 < 1 < 2 1 < < 3
1
+1
22
23+2
1
+1
2(1)
(2)(1)
1
+1
2(1)
(2)(1)
1
+1
2
2 ; 1
0 2
2
1
+1
0 2(+1)1(2)
(2)(+1)
0 2+2+2
(2)(+1)
0 +4
(2)(+1)
4 1 2
+ +
4 1 2 1
1 1 3
4 1 2 1 3
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20 PAT 1 (.. 58)
5. 5
tan 2
= sin
1+cos cot
2 =
1+cos
sin
(1+sin) sec2
cos2 =
(1+sin30) sec2 30
cos60 =
(1+1
2)(2
3)2
1
2
= (3
2) (
4
3) (
2
1) = 4
6. 4 arc arc Q1
sec2(arctan 2) :
cosec2(arccot 3) :
cosec (2 arccot 2 + arccos 35) =
1
sin(2arccot2+arccos3
5)
= 1
sin(2arccot2) cos(arccos3
5)+cos(2 arccot2) sin(arccos
3
5) ()
sin(2 arccot 2) = 2 sin(arccot 2) cos(arccot 2)
= 2 (1
5) (
2
5) =
4
5
cos(2 arccot 2) = 2 cos2(arccot 2) 1
= 2 (2
5)2 1 =
8
5 1 =
3
5
2 cot
2 = (1 + cot )2
2(1+cos)
sin = (1 +
cos
sin)2
2+2cos
sin = (
sin+cos
sin)2
2+2cos
sin =
sin2 +2sin cos+cos2
sin2
2+2cos
sin =
1 +2sin cos
sin2
2 + 2 cos = 1+2sin cos
sin
2 sin + 2 sin cos = 1 + 2 sin cos 2 sin = 1
sin = 1
2
= 30
sin2 + cos2 = 1
0 < <
2
cosec(arccot 3) = 101
= 10
cosec2(arccot 3) = 10
sec(arctan 2) = 51
= 5
sec2(arctan 2) = 5
2
1
arctan 2
= 22 + 12
= 5
2
1
arctan 2
1
3
arccot 3
= 12 + 32
= 10
1
3
arccot 3
= 12 + 22
= 5
1
2
arccot 2 sin2 = 2 sin cos cos 2 = 2 cos2 1
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PAT 1 (.. 58) 21
cos (arccos3
5) =
3
5
sin (arccos3
5) =
4
5
() cosec (2 arccot 2 + arccos 35) = 1
(4
5)(3
5)+(
3
5)(4
5) =
124
25
= 25
24
= 5 + 10 + 2524
= 360+25
24 =
385
24
7. 4
= arcsin (cos 3) = arcsin (
1
2) = 30
sin2 + sin2( + ) + sin2(5 + )
8. 2
= 52 32 = 4
5
3
arccos3
5
= sin2 + sin2(30 + ) + sin2(5(30) + ) = sin2 + (sin 30 cos + cos 30 sin)2 + (sin150 cos + cos 150 sin)2 = sin2 + (sin 30 cos + cos 30 sin)2 + (sin30 cos cos 30 sin)2
= sin2 + (1
2cos +
3
2sin)
2
+ (1
2cos
3
2sin)
2
= sin2 + cos2
4+3cos sin
2+3 sin2
4 +
cos2
43cos sin
2+3 sin2
4
= sin2 + 2cos2
4 +
6sin2
4
= sin2 + 1sin2
2 +
3 sin2
2
= 2sin2+1sin2+3sin2
2
= 4 sin2 + 1
2
= 4(1cos2
2) + 1
2
= 22cos2+1
2 = 3
2 cos 2
cos 2 = 1 2 sin2 2sin2 = 1 cos 2
sin2 = 1cos 2
2
log( 2) = log 3 + log2( + 2) log( 2) =
1 1
2
log 3 + 1
2log( + 2)
log( 2) = 2 log 3 + 1
2log( + 2)
log( 2) = log 32
+ log( + 2)1
2
log( 2) = log 3 + log + 2 log( 2) = log(3)( + 2) 2 = (3)( + 2)
2 4 + 4 = (9)( + 2) 2 13 14 = 0 ( 14)( + 1) = 0 = 14 , 1
log = 1
log
log = log
log + log = log
( )2 = 2 2+ 2
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22 PAT 1 (.. 58)
+ = 196 + 14 = 210
9. 3 ( ) 3. 3. 2 = 1 1 2 = (1)2 + 1 = 1 + 1 = 2
= 2
(2, 1) (2, 1) 3. 2. tan 1. 4.
5. (1, 1) (2, 2) 2 =
+1
1 = 2 1 = 2
(log2 )(log ) = 1 + log
(log )2(log ) = 1 +
11
2
log
(log )2 (
1
log ) = 1 + 2 ( 1
log )
( )2 ( 1
) = 1 + 2(
1
)
2 = + 2 2 2 = 0 ( 2)( + 1) = 0 = 2 , 1
log = 2 , 1 log14 = 2 , 1 = 142 , 141
= 196 , 1
14
log = 1
log
log = 1
log
log =
= 14
> 2
= |2 |
= | 2| (2, 0)
1 = 2
12 = 22
1
1+1 =
2
2+1
12 + 1 = 12 + 2
1 = 2
+ 1 = 0 = 1
12 =1
1+1 22 =
2
2+1
1
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PAT 1 (.. 58) 23
10. 5
( 2, 3) ()
( 2, 3)
|4(
2)+3(3)71|
42+32 =
|262|
5
= 14
> 0 = 4 (*) ( 42, 3) = (2, 3)
F(2, 3) = 7 V F V(2, 3+7
2) = V(2, 5) = 2 = 5
= 5 3 = 2
11. 3
( )2 = 4( ) V(, ) = (6, 2) = = 10 F(6 10, 2) = F(4, 2)
2 + 2 + 6 12 = 0
(2 + ) + (2 6) = 12
(2 + + (
2)2) + (2 6 + 32) = 12 + (
2)2+ 32
( +
2)2
+ ( 3)2 = 21 + 2
4
2 2 + 2+ 2 = ( + )2
4 + 3 = 71 4 + 3 71 = 0
(, ) + + = 0
|++|2+2
|262|
5 = 14
|2 62| = 70 2 62 = 70 , 70 2 = 132 , 8 = 66 , 4
( )2 = 4( )
( (2))2
= 4(2)( 5)
( + 2)2 = 8( 5) 2 + 4 + 4 = 8 + 40 2 + 4 + 8 36 = 0
F(2, 3)
= 7
V( , )
2 4 = 40 + 236 2 4 + 4 = 40 + 236 + 4 ( 2)2 = 40 + 240 ( 2)2 = 40( 6) ( 2)2 = 4(10)( 6)
V(6, 2) F
10
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24 PAT 1 (.. 58)
F1(6, 2) F2(4, 2)
(, ) = ( 6+(4)2
, 2 ) = (1, 2) = 6 1 = 5
A AF1 + AF2 = A(4, 6) A F1(6, 2) F2(4, 2)
= 65 = 652
= 35
12. 3 1. [() ()] () () () 83 4 1 +1 83
154
= 12 ()
AF1 + AF2 = (4 6)2 + (6 2)2 + (4 (4))2 + (6 2)2
= 4 + 16 + 64 + 16
= 20 + 80
= 25 + 45 = 65
F1
= 5
F2
(1, 2) (6, 2) (4, 2)
2 = 2 + 2
(35)2
= 2 + 52
45 = 2 + 25 20 = 2
()2
2 +
()2
2 = 1
(1)2
(35)2 +
(2)2
20 = 1
22+1
45 +
24+4
20 = 1
4(22+1)+9(24+4)
180 = 1
42 8 + 4 + 92 36 + 36 = 180 42 + 92 8 36 140 = 0
83 4 1 = 0 83 + 1 1 4 1 = 0 (83 + 1) 4 2 = 0 ((2)3 + 13) 2(2 + 1) = 0 (2 + 1)(42 2 + 1) 2(2 + 1) = 0 (2 + 1)(42 2 + 1 2) = 0 (2 + 1)(42 2 1) = 0
+1 1
3 + 3 = ( + )(2 + 2)
= 1
2 =
(2)(2)24(4)(1)
2(4)
= 220
8 =
15
4
24
2
2 + 1
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PAT 1 (.. 58) 25
= 12 () 8 ( 1
2)4 8(
1
2)2+ (
1
2) + 1 =
1
2 2
1
2+ 1
= 1 0 () () 1.
2. [() ()] () () () 1+ 1
= 1 , 12 () ()
= 1 : (1)3 + (1)2 = 1+ 1 = 0 0 () = 1 () () 2.
3. [() ()] () () 1. = 1
2 () ()
= 12 : (
1
2)3+ (
1
2)2
= 1
8+1
4 =
1+2
8 > 0 () 3.
13. 1
11
(1 23
2+1) = 1
1(2+123
2+1) = 2
3+2+1
1 (2+1)
= 1 00 1
1 22 1
23+2+1
1 (2+1) =
(1)(221)
1 (2+1) =
(1)(221)
1 (2+1)
= (1)
2(221)
1 (2+1)
= 1 (221)
(2+1)
= 1 11 (2(12)11)
(12+1) =
(0)(2(12)11)
2 = 0
84 82 + + 1 = 0 82(2 1) + ( + 1) = 0 82( 1)( + 1) + ( + 1) = 0 ( + 1)(82( 1) + 1) = 0 ( + 1)(83 82 + 1) = 0 ( + 1)(83 1 + 1 82 + 1) = 0 ( + 1)(83 1 82 + 2) = 0 ( + 1)((83 1) 2(42 1)) = 0
( + 1)((2 1)(42 + 2 + 1) 2(2 1)(2 + 1)) = 0
( + 1)(2 1)((42 + 2 + 1) 2(2 + 1)) = 0
( + 1)(2 1)(42 + 2 + 1 4 2) = 0 ( + 1)(2 1)(42 2 1) = 0
1 + 1
3 3 = ( )(2 + + 2) 2 2 = ( )( + )
+ 1 2 2 = ( )( + )
= 1
2 1
= 1
2 ()
1 2 1 0 1 2 1 1
2 1 1 0
1
-
26 PAT 1 (.. 58)
14. 1 = 2 + | 1| (0, 2) (0, 2) 0 1 | 1| = ( 1) (0, 2)
(0, 2) = 0 2 + 1 (0, 2) = 0 = 0 2(0) + 1 = 1 = 1 1 = 1 ( 1 1 = 1) (0, 2) 1
1 = 2
0 = 1
+ = 2 1. 1 + 3 = 2 2. 1 + 5 2 3. 2 + 5 2 4. 3 + (2) 2 5. 3 + 4 2 1.
15. 2 () () () = 4
1() = 2+1
1 = ( 2+1)
() = 2
()
() = (2)(1)()(1)(2)2
() (4) = 442
= 2
() (4) = 2(42)2
= 1
2
() () () = 2( 2)2 = 4( 2)3
( 2)
= 4( 2)3 1 =
4
(2)3 ()
((4)) = (2) = 4(22)3
= 1
16
((4)) = ( 4(42)3
) = (1
2) =
1
2
1
2 2
= 1
2
5
2
= 1
5
|| = { 0 < 0
= 2 + (( 1))
= 2 2 +
2 = + = 2
2 = ( )
= 2+1
+ = 2 2 = ( 2) =
=
2
= 2
(2)2
= 2
(2)2 ()
(
) =
( )(
)
()2
2(4) (4) = 2 (12) (2) = 3 ()
()
()
()
-
PAT 1 (.. 58) 27
() () () () = 2
(2)2 ( 2)2
2(2)2
( = 2 ) () (0, 2) ()
16. 4 || = || |16 + | = |8 + |
() = 0 : = 2 :
= 0 = 192
( + )2 = (0 + (192))2 = (192)
2 = 192
17. 3 8 = 8! ( ) = () + () ( ) = + 1 : 4 4! 2 : 5 1 5 2 4 ( ) 3 3 4 2 = 4! 5 4 3 2 = 4! 5! 4 = 4! 5!
= |||| cos (16 + ) (8 + ) = |||| cos 60
(16)(8) + = ||2
1
2
128 + = 162 + 22
1
2
256 + 2 = 256 + 2 0 = 2 2 0 = ( 2)
= 0 = 2
|| = || 60
162 + 2 = 82 + 2 256 + 2 = 64 + 2 ()
256 + 02 = 64 + 2 192 = 2
192 =
256 + (2)2 = 64 + 2 256 + 42 = 64 + 2 32 = 192
(32 )
__ __ __ __ __
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28 PAT 1 (.. 58)
1 : 8 2 : 2 4 3 : 3 2
1 3 8 4 3 3 2 2 1 1 = 8 4! 3!
= 2(4! 5!) 8 4! 3!
= 2(4! 5!) 84!3!8!
= 2(5!) 83!
8765 =
5 1
75 =
4
35
18. 4 () 0 () () = 9 ( )() = (()) = (9 ) 0
() 4 < 6 () () = 7 ( )() = (()) = (7 ) 4 < 6
() > 6 () () = 7 ( )() = (()) = (7 ) > 6
19. 2 = + i = i
0 9 9 9 3 (9 ) () = 4
(9 ) = 9 4 ()
4 > 6 7 4 > 7 7 6 3 > 7 1
(7 ) () = 4 (7 ) = 7 4 = 3 ()
< 6 7 < 7 6 7 < 1 (7 ) () = + 2
(7 ) = 7 + 2 = 9 ()
= ( i) ( + i) = i i = 2i
-
PAT 1 (.. 58) 29
+ 9 = 10 (1)
() | + 8| = |8+ 2i + 8| = |2i| = 2 () () | + 3i| = |8 + 2i + 3i| = |8 + 5i| = (8)2 + 52 = 89 () () |i + 2| = |i(8+ 2i) + 2| = |8i 2 + 2| = |8i| = 8 ()
20. 3
1n
= 123
33+226
35+329
37+4212
39+
( = 23
32)
123
33+226
35+329
37+4212
39+ = (1)
126
35+229
37+3212
39+4215
311+ =
23
32 (2)
(1) (2) : 123
33+126
35+129
37+1212
39+ =
23
32
8
27
1 8
9
= 8
9
8
279
1 =
1
9
24 =
21. 1 2 1 4 2 4
(1 + i)( i) (97i)(2i)
3+i = 6 2i
(3 + i)(1 + i)( i) (9 7i)(2i) = (6 2i)(3 + i) ( 2 + 4i )( i) + 18i + 14 = 20 + 0i 2 2i + 4i + 4 + 18i + 14 = 20 2 + 18 + (4 + 16)i = 20 + 0i + 9 + (2 + 8)i = 10 + 0i
3 + i
2
2 + 8 = 0 + 4 = 0 (2)
(1) (2) : 5 = 10 = 2 (2) : + 4(2) = 0
= 8
= 8 + 2i
23
32
1 = 123
33 =
8
27 = 2
3
32 =
8
9
|| < 1 = 1
1
1 + 4 , 2 + 4 , , 20 + 4 1 , 2 , , 20 21 , 22 , , 220 21 + 4 , 22 + 4 , , 220 + 4
4 2 4
-
30 PAT 1 (.. 58)
: ( 4 2 4)
1 = 50 : 50 4 46
2 92
4 96 2 = 96
: ( 2 2 )
2 = 10 : 10 4 10
2 20
4 20 2 = 20
= 2 2 = 202 = 400
22. 4
=
= 25% = 25100
= 14 4 = ()
15.87 85 15.87% 85
0.5 0.1587 = 0.3413 = 1.0
=
1.0 = 85
= 47.6
=
47.6 68
17 =
20.4
17 = 1.2
1.2 = 0.3849
= 0.5 0.3849 = 0.1151 11.51% 11.51
23. 5 1 + 21 + 22 + 23 + + 2 1 = 1 = 2 = + 1
1 + 21 + 22 + 23 + + 2 = 1(12+1)
12
= 2+1 1 =
2+1 1
32
= 0.5 0.1587 = 0.3413
85
= 85 = 85 4 5 = 85 = 17
()
() = 4(17) = 68
0.3849
1.2
= 0.5 0.3849 = 0.1151
1.2
1 =
1(1)
1
0.1587
85
-
PAT 1 (.. 58) 31
n
lim (1 + 2 + 3 + + )
24. 4 = { 2 | }
= 2+2
4 2+1 0
12 4 1
0, 2, 3, 4 = { 02 , 22 , 32 , 42 } = 02 + 22 + 32 + 42 = 29
= 1 + 2 + 3 +
= 22 1
32 +
23 1
34 +
24 1
36 +
= 22
32
1
32+23
34
1
34+24
36
1
36 +
= (22
32+23
34+24
36+ ) (
1
32+
1
34+
1
36+ )
= 2+1 1
32
|| < 1 =
1
1
1 = 22
32 =
4
9
= 2
32 =
2
9
1 = 1
32 =
1
9
= 1
32 =
1
9
= ( 4
9
12
9
) ( 1
9
11
9
)
= ( 4
99
7 ) (
1
99
8 )
= 4
7
1
8 =
25
56
4 0 4
2 + 1 0
1
2
4 2 + 1 0
4 2 + 1 4 2 + 1 3 3 1
-
32 PAT 1 (.. 58)
25. 5 (0, 0)
4 A(0, 1) , B(0, 2) , D(1, 0) C L1 L2
A(0, 1) , B(0, 2) , C(2, 1) , D(1, 0) 5
= 9 = 4 + 5
26. 4
L1 : + 2 4
(0, 0) (0, 0)
0 4 2 0
L2 : 1
(0, 0) (0, 0)
0 1 1 0
L3 : + 1
(0, 0) (0, 0)
0 1 1 0
0 0 Q1
L1
L2
L3
L1
L2
L3
A
B
C
D
L1 : + 2 = 4 (1) L2 : = 1 (2) (1) (2): 3 = 3 = 1 (2): 1 = 1 = 2 C(2, 1)
A(0, 1) B(0, 2) C(2, 1) D(1, 0) 2 + 2 - - - - 3 + 2 - 4 8 3 2 + 3 - 6 7 2 + 4 - 8 6 1 4 + - 2 9 4
2 + 2 3 + 2
A(0, 1) B(0, 2)
1 = 1 1 = 1 = 1 = 1 =
1
1
-
PAT 1 (.. 58) 33
= [1 22 1
] = [
] =
= =
= 1 2 = 2 + 8 = 12 + 8 = 9 = 3 = = 3 = = 1 + + + = 3 + 1 + 3 + 1 = 8
27. 3 () | | || || | | || || () ()
= 0 () () +
[
] [1 22 1
] = [1 22 1
] [
]
[ + 2 2 + + 2 2 +
] = [ + 2 + 22 + 2 +
]
[ + 2 2 + + 2 2 +
] = [ + 2 + 22 + 2 +
]
+ 2 = + 2 =
2 + = + 2 =
+ 2 = 2 + =
2 + = 2 + =
= 9 22 = 9 2(2 + 8) = 9 4 + 82 9 = 0 (2 + 9)(2 1) = 0
2 = 1 = 1 , 1
2 = 9 ( )
(2 ) , , ,
det() = 24 (det )(det) = 24 (det )(det ) = 24
|1 22 1
| |
| = 24
(1 4)( ) = 24 = 8 2 2 = 8 2 = 2 + 8
det det = det
= , =
= , =
| + |2
= ||2 + ||2
||2 + ||2+ 2 = ||2 + ||
2
2 = 0
= 0
| + |2 = ||2 + ||2 + 2
0
( + ) ( ) = 0 + = 0 ||2 ||
2 = 0
||2 = ||2
|| = || ()
= ||2
-
34 PAT 1 (.. 58)
28. 1
5 4 , , , (1) = 9 (3)
(4) = + 5 (2)
= + 1 , = 9 , = + 5 , = 6 () + < + ( 9) + ( 6) < ( + 1) + ( + 5) 15 < 6 () 9 < 6 < < + 1 < + 5 < < < < () + < + + ( + 5) < ( + 1) + ( 9) 5 < 8
29. 2
= 7+53+11
= 5
+ 4 = + + 5 = + + 1 = + 2 = + + 3
= + 9 (1)
= 4 (2)
= 3 (3)
= + 5 (4)
= + 5 4 = + 1
9 = 3 6 =
() ()
0 2 3 = 0+2
2 = 1 = (3)(1) = 3
3 5 5 = 3+5
2 = 4 = (5)(4) = 20
6 8 7 = (7)() = 7
9 11 3 10 = (3)(10) = 30
+ 11 7 + 53
+ 3
+ 3
+ 3
7+53
+11 = 5
7 + 53 = 5 + 55 2 = 2 = 1
-
PAT 1 (.. 58) 35
= 1
= 122
= 6
6 2 2
= + (
2
)
= 2.5 + (6 3
5) (3) = 2.5 + 1.8 = 4.3
30. 1 60% 40% 60% % 60 % 40% % 40 %
()
() ()
0 2 3 3 3 5 5 8 6 8 1 9
9 11 3 12
60 40
=
60
= 60 + 40
+
(60 )( + ) = (100 )() 60 + 60 2 = 100 2 60 + 60 = 100 60 = 40
= 2
3 ()
= 1.5 ( ) 60 = 1.5 (40 ) 60 = 1.5 (40 2
3)
60 = 60 ()
()
1.5
>
60 60
> 40 40
60 3
> 40
2
3
2
120 2 > 120 2
()
()
> >
2
3 >
()
()
-
36 PAT 1 (.. 58)
31. 230 30 1 = ( ) Inclusive Exclusive
= 230 , , = 0
32. 126 3|+4| 3||
= (59 (1)) + (59 (7)) = 60 + 66 = 126
33. 5 6 = 4 = () = 2 = 2
(2, 8) = () (2) = 8 ()
( ) = () + () + () ( ) ( ) ( ) + ( ) = 150 + 80 + 60 ( ) ( ) ( ) + 30
30
30
, , ( ), ( ), ( )
= 150 + 80 + 60 ( + 30) ( + 30) ( + 30) + 30 = 150 + 80 + 60 30 30 30 + 30 = 230 230
, , , , 0
30
0
0 0
120 50
30
3(9 + 3||+|+4|) = 3|+4| + 3||+4
3(9 + 3|| 3|+4|) = 3|+4| + 3|| 34
3(9 + ) = + 34 27 + 3 = + 81 3 81 + 27 = 0 (3 1) 27(3 1) = 0 ( 27)(3 1) = 0
3|+4| = 3|| =
= 27
3|+4| = 33 | + 4| = 3 + 4 = 3 , 3 = 1 , 7
= 13
3|| = 31 || = 1
()
6(2) = 4 8 = (2, 8)
-
PAT 1 (.. 58) 37
2
limx
2+6
1+()3 = 2 2
2+26
1+(2) 3 =
0
1+8 3 =
0
0
00
2
limx
2+6
1+()3 = 6 30
(2) = 6 (2) = 30
6 = 5
34. 9.25
() = {3 , < 1
+ , 1 < 1
32 + 2 , 1
, () = {3 , < 1
3 + 2 , 1 < 1
32 + 2 , 1
2
2
() 3 ()
= 1
2
3 + 1
1
3 + 2 + 2
1
32 + 2
= 4
4 |1 2
+ 32
2+ 2 |
1 1
+ 3 + 2 |2 1
= ((1)4
4(2)4
4) + ((
3(1)2
2+ 2(1)) (
3(1)2
2+ 2(1))) + ((23 + 2(2)) (13 + 2(1)))
= 1
4 4 +
3
2 + 2
3
2 + 2 + 12 3 = 9.25
= 1 (1)3 = (1) + 1 = + = 1 (1)
(1) + = 3(12) + 2 + = 5 (2)
= 1
(1) + (2) : 2 = 6 = 3
3 + = 5 = 2
(2) :
()
2limx
2+6
1+()3 =
2limx
2+6
1+()3
= 2
limx
2+1
1
2(1+())
12
(1+())
= 2
limx
2+1
1
2(1+())
12 ()
= 2(2)+1
1
2(1+(2))
12 (2)
= 5
1
2(1+ 8 )
12 (2)
= 5
1
6 (2)
= 30
(2)
= 2
()
-
38 PAT 1 (.. 58)
35. 32
1()
= 2
log2 1
(1)+
1
(2)++
1
(10) =
12
log2 +
22
log2 +
32
log2 +
102
log2
= 12 + 22 + 32 + + 102
log2
=
10(10+1)(2(10)+1)
6
log2
= 5(11)(7)
log2
1(1)
+1
(2)++
1
(10) = 77
36. 68
det
2 det 2 1 2
det 3 1
() = log2( ) = log2(1/) = 1/
log2
= log2 2
5(11)(7)
log2 = 77
5 = log2 25 = 32 =
|1 0 4 15
| = 17
|1 5 4 0 1
| = 17
|2 52 4 0 1
| = 34
| 2 54 2 1 0
| = 34
|2 2 58 2 2 0
| = 68
|2 + 5 2 58 + 2 2 0
| = 68
3
12 + 22 + 32 + + 2 = (+1)(2+1)
6
det
1 2 2 2
2 det 2
-
PAT 1 (.. 58) 39
37. -
= 0 100 = 200 200 1 = 2 = 3 = = 200 = 200 1 + 3 + 5 + + 49 = 1275
38. 160 3 1 2 3 2 3 3 3 3 5 1 2 2 = 25 = 32 1 1 2 1 ( 1 2) 32 2 1 2 (11111 22222) 1 2 = 32 2 = 30 3 1: 3 5 5 2: 4 1 2 2 = 24 = 16 2 ( 1 2 ) = 16 2 = 14 = 5 14 = 70 3
1: 3 5 = (52) =
5 4
2 1 = 10
2: 3 1 2 2 = 23 = 8 2 ( 1 2 ) = 8 2 = 6 = 10 6 = 60 = 30 + 70 + 60 = 160
2 = 1 + (1) 4 = 3 + (2) 6 = 5 + (3) 50 = 49 + (25)
2 + 4 + 6 + + 50 = 1 + 3 + 5 + + 49 + 25
25
0 = 25 = 0
-
40 PAT 1 (.. 58)
39. 3 = +
= 9 + 3 4 = 9 + 3(4) = 3 .
40. 117 F1(1 + 25 , 3) (, 3) (, 3)
(1, 3) F1(1 + 25, 3) F1 = 25 = 25 2 = 2 + 2
= + = 2
2
= 2 = 2 ()
n
i 1
= +
n
i 1
(1)
n
i 1
=
n
i 1
+
n
i 1
2 (2)
48 = (8) + (40) (1) 270 = (40) + (210) (2)
(1) 5 : 240 = (40) + (200) (3)
(2) (3): 30 = (10) 3 = (1) : 48 = (8) + 3(40) 6 = + 3( 5 ) 9 =
2 + 1 = 0 2 3 + 1 = 0 2 = 2 = 1
2 + 2 = (25)2
2 + 2 = 20 ()
=
=
2 + 1 = 0 2 + 1 =
2 + (2)2 = 20 2 + 42 = 20 52 = 20 2 = 4 = 2
= 2 = 2(2) = 4
-
PAT 1 (.. 58) 41
, , ,
2 + 2 + + = 21 = 4 , = 1 , = 8 , = 6 2 + 2 + 2 + 2 = 42 + (1)2 + (8)2 + 62 = 16 + 1 + 64 + 36 = 117
41. 4 =
+ 10 10 10 1, 2, 3, , 9 100 1, 2, 3, , 9
4 4
1
1
=
1
10
10 10 = 10 10 = 0 (10 ) 10 = 0 (10 ) + 100 10 = 100 (10 ) + 10(10 ) = 100 ( + 10)(10 ) = 100
10
100
+ 10 10 100 1 50 2 25 4 20 5
= 90 , = 9 = 40 , = 8 = 15 , = 6 = 10 , = 5
()2
2()2
2 = 1
(1)2
22(3)2
42 = 1
22+1
426+9
16 = 1
4(22+1) (26+9)
16 = 1
42 8 + 4 2 + 6 9 = 16 42 2 8 + 6 = 21
-
42 PAT 1 (.. 58)
42. 1.5
0
() 1 < 1+22+2
( 1+2
2+2 < 1 )
= 1+22+2
= 1
1 + 0 1
1 0 1
2 (
2 2
2)
+ <
2 22
21
+ <
2 ()(+)
2 21
+ <
2 <
< 2
< 2 +
< (2 + 1)
1 + = 1 + = 2 (1) 1 = 1 = 2 (2) (1) (2) : 2 = 2 2
= 2 2
2
2
1+ + 1 < 1
2
+ <
1 + = 1 =
1 + < (2 + 1)(1 )
1 + < (2 + 1)2(1 )
1 + < (2 + 22 + 1)(1 )
1 + < (3 + 22 )(1 )
1 + < 3 3 + 22 22 4 + 22 < 2 + 22 2 + 2 < 1 + 2 (2 + 2) < 1 + 2 < 1+2
2+2 ()
,
2
2
1 1
2 + 2 = (1+22+2
)2
+ (1)2
= 1+22+2
4+42+2 + 1
= 3+22
6+42 + 1
= 3+22
2(3+22) + 1
= 1
2 + 1 = 1.5
-
PAT 1 (.. 58) 43
43. 2 , log
=
= = 1
2 + 2 = 12 + 12 = 2 : = || = = 1
4 , = 1
4
44. 78.7
2 2 ()2
1
2 2
1
= (1+22)+(2 + 23)+(3+24)+(4+25)+(5+21)5
= 31+32+33+34+35
5
= 3 (1+2+3+4+5
5) = 3
= 3(6) = 18
2 2 2
1
= (1+22)
2+(2 + 23)2+(3+24)
2+(4+25)2+(5+21)
2 5(182)
51
= (12+412+42
2)+(22+423+43
2)+(32+434+44
2)+(42+445+45
2)+(52+451+41
2) 1620
4
2 + = 2
2 = 2
2 = 4 4 +
4 = 2 + 2
2 = 1 +
0 = 2 + 1
0 = 2 2 + 1
0 = ( 1)2
= 1
= 1
3 log4 162 = 6 + 6 log2
log4 162 = 2 + 2 log2
log4 16 + log4 2 = 2 + 2 log2
2 + log22 2 = 2 + 2 log2
1
2
2
2log2 =
2
2log2
log2 = log2 =
3 log = log + log
log =
log
( )2 = 2 2 + 2
2 2+ 2 = ( )2
2 1 + 1 = 2 1 + 1
()2
1 =
2 2
1
34
51 =
214 52
51
52 = 180 2 = 36 = 6
log log4 2 2 log4|| ( )
-
44 PAT 1 (.. 58)
= 512+52
2+532+54
2+552 + 412+423+434+445+451 1620
4
= 5(1
2+22+3
2+42+5
2) + 4(12+23+34+45+51) 1620
4
= 5( 214 ) + 4(12+23+34+45+51) 1620
4
= 1070 + 4(12+23+34+45+51) 1620
4
= 4(12+23+34+45+51) 550
4
= 12 + 23 + 34 + 45 + 51 137.5
= 16
12 , 23 , 34 , 45 , 51 = 12+23+34+45+51
5 =
393.5
5 = 78.7
45. 429 = 10 + = 10 + + = 143
, {1, 2, 3 , , 9} , + = 13 4 + 9 , 5 + 8 , 6 + 7 , 9 + 4, 8 + 5 , 7 + 6
= { 49 , 94 , 58 , 85 , 67 , 76 } = 49 + 94 + 58 + 85 + 67 + 76 = 143 + 143 + 143 = 429
Gtr Ping GTRmath Athassawat Kammanee Totsaporn Suwannaruang 081-8285490
2 = 214
12 + 23 + 34 + 45 + 51 137.5 = 162 12 + 23 + 34 + 45 + 51 137.5 = 256 12 + 23 + 34 + 45 + 51 = 393.5
(10 + ) + (10 + ) = 143 11 + 11 = 143 + = 13
11