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PAT 1 (.. 57) 1
PAT 1 (.. 57) 71 (PAT 1) 22 2557 13.00 - 16.00 .
1 1 - 30 6 1. , (, , ) , (, , )
(, , ) 1. ( ) ( ) 2. ( ) ( ~) 3. ( ~) ( ) 4. ( ~) ( ~)
2. { | 0 < < 1 }
() [ 2 2 < ] () [ | | < 1 ] 1. () () 2. () () 3. () () 4. () ()
1 Mar 2015
(, , ) T T T T T T F T T F T F T F F F F T T T F T F T F F T T F F F T
2 PAT 1 (.. 57)
3. = 2
() () = 33 92 + 3 = 0 1. () () 2. () () 3. () () 4. () ()
4. R () R R R( + ) , () R R ()R() , , 1. () () 2. () () 3. () () 4. () ()
5. 1 log 4 + log 4 = 9 log 2
log(5) + log (2
)
1. 13.5 2. 11.5 3. 9 4. 7
PAT 1 (.. 57) 3
6. sin25 sin85 sin35sin75
1. tan 15 2. sin15 sin 75 3. cos 20 cos 40 cos 80 4. sec 420
7. () = +
0 = () = 1 (1, 1)
() = 1 ()
1limx
( )() = (22 + 22)
1. () () 2. () () 3. () () 4. () ()
4 PAT 1 (.. 57)
8. = {1, 2, 3, , 15} 4
1. 3455
2. 4455
3. 191
4. 291
9. || + 2 3 = 3 45i || (absolute value) (conjugate) ||2
1. 95 2. 225 3. 245 4. 375
PAT 1 (.. 57) 5
10. 2 22 + 8 6 = 0 = 2 = 2
1. 22 + 2 8 + 42 4 = 0 2. 22 + 2 8 22 + 8 = 0 3. 2 + 22 4 42 + 6 = 0 4. 2 + 22 + 4 + 42 + 6 = 0
11. 2 + 2 + + + = 0 (quadrant) 1 = 2 + + (4, 1) 1
() 2 + 2 + 2 = 133 () 4 + 3 7 = 0 1. () () 2. () () 3. () () 4. () ()
6 PAT 1 (.. 57)
12. 5 12 60 = = = (2 )
1. 64 2. 109 3. 114 4. 124
13. [ 22 + 3 0 | 2| 3 ] 62 51 1 > 0
1. 2. 2 3. ( ) ( ) = (6, 1) 4. (6, 0) ( )
14. 2 log2( 2) + log12
+ log12
= 0
()2
+ 1 1. 2 2. 5 3. 10 4. 17
PAT 1 (.. 57) 7
15. , , , ()
<
+
<
+
()
< ++
1. () () 2. () () 3. () () 4. () ()
16. ( )() = 2 + 5 () = 2 4 + = () = 0
1. + 3 = 0 2. 2 + 7 = 0 3. 3 + 5 = 0 4. 5 + 2 = 0
17. 1 (2, 4) 3 2 1 (0, 13) 2 =
1. 8.5 2. 7.5 3. 6.5 4. 5.5
8 PAT 1 (.. 57)
18. 1 = 5 + 2 2 = 4 + 3 2 + 3 6 , 3 15 , + 4 , 2 + 5 27 , 0 0 1 2 1 2 1 2 1 2 () 1 2 () 1 2 1. () () 2. () () 3. () () 4. () ()
19. () = 43 + 2 + + , 2
2
() = 64
3
() () = () (1) = (0) = (0) = 0 () = () + () 1. 4 43 + 122 6 = 0 2. 4 83 122 6 = 0 3. 34 163 + 482 24 = 0 4. 34 + 83 482 + 24 = 0
PAT 1 (.. 57) 9
20. {} 1 = 1
6 = 1
1
3 = 2, 3, 4,
()
nlim = 0
() 1 + 2 + 3 + 0.75 1. () () 2. () () 3. () () 4. () ()
21. , , = 24 = 8
() > (+1)
<
(+1)
() < (0.01) < (0.05) 1. () () 2. () () 3. () () 4. () ()
22. , , {0, 1, , 9} > > 500
1. 119 2. 117 3. 114 4. 110
10 PAT 1 (.. 57)
23. 22 24 15 16
() 96 () = 9 1. () () 2. () () 3. () () 4. () ()
24.
(3, ) () = 13 () 0.5 1.3 1. () () 2. () () 3. () () 4. () ()
1 2 3 4 5 9 11 17 19
PAT 1 (.. 57) 11
25. 1, 2, , 1 1, 2, , 2 21 + 1 , 22 + 1 , , 2 + 1 () 1 2 () 1 2 1. () () 2. () () 3. () () 4. () ()
26. 2 3 3 2 2 2 = [1 61 14
]
() det() det() = 0
() = [1 21 2
] = [5 76 10
]
1. () () 2. () () 3. () () 4. () ()
12 PAT 1 (.. 57)
27. 160 60 4 84.5 55 0
1. 19.15 2. 15.54 3. 34.46 4. 30.85
28. 5 2.8 15
1. 3.5 2. 5.25 3. 7.5 4. 11.25
29. sin4
5+
cos4
7 =
1
12 > 0 sin
2(2)
5+
cos2(2)
7
1. 1144
2. 25126
3. 29 4. 1
6
0.3 0.4 0.5 1.0 1.1 1.96 2.0
0.1179 0.1554 0.1915 0.3413 0.3643 0.4750 0.4773
PAT 1 (.. 57) 13
30. , , = + , = + = 2 1. < < < 2. < < < 3. < < < 4. < < <
2 31 - 45 8 31. () ,
= , , , () = 20 , () = 12 , () = 9 , () = 15 , (( ) ( )) = 11 (( ) ( )) = 12 (( ) ( ))
32. = cos 15 + cos87 + cos159 + cos 231 + cos 303
= sin (arctan(158) + arccos(
4
5))
+ = ... 1 +
14 PAT 1 (.. 57)
33. 1 2 |1| = 2 , |2| = 3 |1 2| = 1 |1 + 2| ||
34. > 0 > 1 = = 3 20 + 14
35. {} = ( + 1)( + )
= 1, 2, 3, n
lim (+1
12+
+2
23+ +
+
+1) =
1
312 2 + 57
PAT 1 (.. 57) 15
36. [|| 12 ||
] + 2 [ 3
1 ||] = [
10 + 07 7
]
+
37. = {1, 2, 3, 4, 5} (, ) 2
38. {} 1 = 2 1 < 2 < 3 < 2, 4, 8
(11)3+(21)
3+ +(1)3
13+2
3+ +3 =
391
450
16 PAT 1 (.. 57)
39. 32 + 62 + 44 2 = 10 3
... 1 +
40. 8 cos(2) + 8 sec(2) = 65 0 < < 90 160 sin(2) sin(
5
2)
41. (2 1) = 42 10 +
(0) = 12 4
1
()
PAT 1 (.. 57) 17
42. : : () = 2() + 5 ( 1)(1 + ) = ( 1)(1 + ) 2
43. (4 + 2 6)3 = (2 4)3 + (4 2)3
18 PAT 1 (.. 57)
44. : , : : () = + 1 (()) = 2 + 2 1
() = 0
limh
((+))2(())
2
()(1)
45. = {0, 1, 2, } (,) , () (, 0) = + 1 () (0,) = (1, 1) {0} () ( + 1, + 1) = ((, + 1),) , (, 2) = 2557
PAT 1 (.. 57) 19
1. 3 11. 1 21. 3 31. 7 41. 34.5 2. 3 12. 4 22. 4 32. 169 42. 36 3. 2 13. 2 23. 4 33. 9 43. 3.5 4. 3 14. 4 24. 3 34. 66 44. 4 5. 2 15. 4 25. 4 35. 201 45. 1277 6. 2 16. 3 26. 4 36. 3 7. 1 17. 1 27. 3 37. 270 8. 4 18. 1 28. 1 38. 14 9. 2 19. 4 29. 2 39. 11 10. 3 20. 2 30. 1 40. 55
1. 3 4 F T
(, , ) ~ ____ ( ~F ___ T ___ T )
4 (, , ) ~ 3 1.
2.
3. 4.
2. 3
. = 0 < 0 ( = 0.2 = 0.2 , = 0.7 = 0.7 ) .
. ( (0, 1) 1 )
~
~ ( ) [(~ ) (~ )] [ T (~ )] (~ )
~(~ ) (~ ~) ( ~) ~ ~ ( ~) (T ~) ~ [ ( T) ~] ~ [ T ~] ~ ~ ~
~ T ~ ( )
( )
2
~( ~) ( ) ~ ( ) ~
~( ~) ( ~) ~ (~ ~) ~ ~
() ( T 2 )
| |2 < (1 )2 ( )2 < (1 )2 ( )2 (1 )2 < 0 ( + 1 )( 1 + ) < 0
2 2 = ( + )( )
20 PAT 1 (.. 57)
3. 2
. sin sin
=
sin =
sin = 3 = 2 2
sin3 =
sin
3 23 sin4 sin3
=
sin sin 2
34sin2 = 1
= 150 = 3(150) 180 = 30 , = 3(30) = 90 .
. = = sec 30 =
2
3 .
3 ( 833
) 9 (4
3)
2
3+ 3 =
8
3 12
2
3+ 3 0 .
4. 3 ( ) R | . . . | | ( + ) | = = = 1 1 | 1 1 | 1 2 1 . . | | | . ( = , = = ()() | )
5. 2 4 22 log 2 log 2 + 2 log 2 = 9 log 2 log2 :
log log log = log = 1
log(5) + log (2
) = log + log
5 + log 2 log
= 1 + 5 log + 2 log 1
2
, (0, 1)
(2 log 2)(log2 ) + (2 log 2)(log2 ) = (9 log 2)(log2 ) 2 log + 2log = 9 2(log + log ) + 2(log + log ) = 9 2( 1 + log ) + 2(log + 1 ) = 9
2 log + 2 log 5 = 0
2 +2
5 = 0
22 + 2 5 = 0 22 5 + 2 = 0 (2 1)( 2) = 0
= 1
2 , 2 log =