ติวสบายคณิต (เพิ่มเติม) บทที่ 02...

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  • 1 http://www.pec9.com 2

    1

    2

    2.1

    ( R ) 3 , 35 , 0

    , 0.5 , 3 , o33.

    ( Q )

    8 , 6 , 0 , 43 , 4.3 ,

    o33.

    8 = 18 , 6 = 1

    6 , 0 = 10 , 4.3 = 10

    43 , o33. = 9

    33

    ( Q/ ) 3 , 5 +2 , p 3 = 1.7320508

    p =

    ( C )

    ( R )

    ( Q )

    ( Q)

    ( I ) ( I)

    ( I+ ) ( N )

    ( I ) ( 0 )

    3 2 1 0 1 2 3 3 35 0 0.5 3

    o33.

  • 1 http://www.pec9.com 2

    2

    ( Q ) ( I ) ( I/ ) ( I ) (I) , (I/) (0) (0) ( N )

    1. 8 , 0 , 2 , 3

    4 , 1.2693 , 95. , 3 , 1+ 3 1. 4 2. 5 3. 6 4. 7 2. 95. 6 1. 2. 3. 4. 3( O-net)

    () 6 () 6

    1. () () 2. () () 3. () () 4. () ()

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    3

    2.2

    1. a , b R a + b R 1 , 2 R 1 + 2 R 2. a + b = b + a 1 + 2 = 2 + 1 3. a + ( b + c ) = ( a + b ) + c 1 + ( 2 + 3 ) = ( 1 + 2 ) + 3 4. 0 0 + a = a = a + 0 0 + 5 = 5 = 5 + 0 5. a a

    (a) + a = 0 = a + (a) (3) + 3 = 0 = 3 + (3)

    1. a , b R a .b R 1 , 2 R 1 x 2 R 2. a . b = b . a 1 x 2 = 2 x 1 3. a ( b . c ) = ( a . b ) . c 1 x ( 2 x 3 ) = ( 1 x 2 ) x 3 4. 1 1 . a = a = a . 1 1 x 5 = 5 = 5 x 1 5. a a 0

    a1 (a1) a = 1 = a ( a1 ) ( 3

    1 ) x 3 = 1 = 3 x ( 31 )

    a (b + c ) = ab + ac 3 ( 2 + 5 ) = 3 (2) + 3 (5)

    4. 1. (6) + 0 = 6 2. [2 + (c + d)] + 3(c + d)] = 2 + [(c + d) + 3 (c + d)] 3. 7

    1 + 71 = 0

    4. 2 + (m + 1) = 2 + (1 + m)

  • 1 http://www.pec9.com 2

    4

    5. 1. 3

    1 (2 + 7) = (2 + 7) 31 2. (5) (

    51 ) = 1

    3. 1 x = x x 4. 7 ( 3)

    6. 1. q x 37 = 37 x q 2. (2 + 3) p = 2 p + 3 p 3. 6 + (5 + y) = (6 + 5) + y 4. (2 + 3) (5 + 6) = (5 + 6) (2 + 3)

    7. ( a + b ) 1. a + b 2. a b 3. a b 4. a + b

  • 1 http://www.pec9.com 2

    5

    8. a , b a * b = a b + 1 . a * b = b * a . ( a * b ) * c = a * ( b * c ) . * . * . a a

    1. 1 2. 2 3. 3 4. 4

    2.3

    2.3.1 p(x) p(x) x c p(c) c

    p(x) = x3 + x2 + 4x + 1 x 1 1 x 1 = 0 x = 1

    2 p(x) = x3 + x2 + 4 x + 1 ( x = 1 ) p(1) = (1)3 + (1)2 + 4( 1 ) + 1 p(1) = 7 7

  • 1 http://www.pec9.com 2

    6

    9. x3 7x2 + 11x 4 x 3 1. 5 2. 6 3. 7 4. 8 10. x3 + x2 + 4x + 1 x 1. 1 2. 3 3. 5 4. 5 11. 4x4 3x2 + 8 x2 3 1. 40 2. 35 3. 30 4. 25

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    7

    12. a x 2 x2 7x + a 1. 5 2. 10 3. 15 4. 20 13( ) p(x) = 6x3 + ax2 + bx 1 (x 1) ( x + 1) 24 a b 1. 44 2. 44 3. 66 4. 66

    2.3.2 an x

    n + an1 xn + an+2 x

    n+2 + + a1 x + a0 = 0 n ( 3 )

  • 1 http://www.pec9.com 2

    8

    14. x3 + x2 8x 12 = 0 1. 3 < x < 4 2. 2 < x < 3 3. 1 < x < 5 4. 0 < x < 6

    15. x3 10x2 + 27x = 18 1. { 1 , 2 , 3 , 4 , 5 } 2. { 5 , 6 , 9 , 13 , 15 } 3. { 1 , 3 , 6 , 9 , 12 } 4. { 1 , 2 , 5 , 6 , 9 } 16( ) x3 x2 5x 3 = 0 x2 > 2 1. 3 < x < 0 2. 2 < x < 3 3. 0 < x < 5 4. 1 < x < 1 17. 1 3x2 + 2x3 = 0 1. 1 < x < 2 2. 0 < x < 1 3. 3 < x < 0 4. 0 < x < 6

  • 1 http://www.pec9.com 2

    9

    18. 2x4 13x3 + 28x2 = 23x 6 1. 5 < x < 0 2. 0 < x < 5 3. 4 < x < 2 4. 3 < x < 3 19. x3 + 5x2 + 6x = 0 1. 3 < x < 2 2. 2 < x < 1 3. 4 < x < 1 4. 0 < x < 3

    20. x3 5x2 16x + 80 = 0 1. 5 < x < 0 2. 6 < x < 6 3. 4 < x < 8 4. 0 < x < 10

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    10

    21( PAT1) 2x3 7x2 + 7x 2 = 0 1. 3.0 2. 3.5 3. 4.0 4. 4.5 22( En) 9x3 + 12x2 + x 2 = 0 1. { 2 , 3

    1 , 23 } 2. { 1 , 3

    1 , 21 }

    3. { 1 , 31 , 3

    2 } 4. { 1 , 32 , 3

    1 }

    23. a x a x3 + 2 x2 5x 2 4 a 1. 6 2. 2 3. 3 4. 6

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    11

    2.4

    a , b , c 1) a > b b > c a > c

    10 > 8 8 > 7 10 > 7 2) a > b a + c > b + c 2 > 1 2 + 4 > 1 + 4

    6 > 5 3)

    3.1) a > b a c > b c c > 0 2 > 1 2 x 4 > 1 x 4

    8 > 4 3.2) a > b a c < b c c < 0 2 > 1 2 x (5) < 1 x (5)

    10 < 5 3.3) a > b a c = b c c = 0 2 > 1 2 x 0 = 1 x 0

    0 = 0 4) a < b c < d a + c < b + d

    5) a < b c < d a d < b c 6) 0 < a < b 0 < c > d ac < bd

    7) 0 < a < b 0 < c > d da < cb

    8) 0 < a < b a1 < b1

  • 1 http://www.pec9.com 2

    12

    2.5 2.5.1

    a < b

    (a , b) [a , b]

    (a , b] [a , b)

    (a , ) [a , )

    ( , a) ( , a] ( , )

    { x a < x < b } { x a x b } { x a < x b } { x a x < b } { x x > a} { x x a} { x x < a } { x x a } R

    1. ( 2 , 1)

    2. [ 4 , 7 ] 3. [ 3 , 1 ) 4. ( 3 , 0 ]

    1) 2 < x < 1

    2) 4 x 7

    3) 3 x < 1

    4) 3 < x 0

    -3 -2 -1 0 1 2

    -3 -2 1 0 1 2

    3 4 5 6 7 8

    -3 -2 -1 0 1 2

    -3 -2 -1 0 1 2

    4 5 6 7 8 9

    a b

    a b

    a b

    a b

    a a

    a

    a

    -3 -2 1 0 1 2

    -3 -2 -1 0 1 2

  • 1 http://www.pec9.com 2

    13

    24. A = (1 , 2) , B = [0 , 4] A B 1. ( 1 , 4 ] 2. ( 1 , 5 ] 3. ( 1 , 6 ] 4. ( 1 , 7 ]

    25. A = (1 , 2) , B = [0 , 4] A B 1. [0 , 1) 2. [0 , 2) 3. [0 , 3) 4. [0, 4) 26. A = (1 , 2) , B = [0 , 4] A B 1. (1 , 2) 2. (1 , 1) 3. (1 , 0) 4. (1 , 1) 27. A = (1 , 2) , B = [0 , 4] B A 1. [1 , 2] 2. [1 , 4] 3. [2 , 2] 4. [2 , 4]

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