Brands math

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แบรนด์ซัมเมอร์แคมป์ 2010คณิตศาสตร์ 2010

Text of Brands math

  • (2)_______________________________ 2010

    { } 1 A B 1) A 2) B *3) A 4) B n(A) A 1. n(U) = n(A) + n(A) 2. n(AU B) = n(A) + n(B) - n(AI B) 3. n(AU BU C) = n(A) + n(B) + n(C) - n(AI B) - n(AI C) - n(BI C) + n(AI BI C) 4. n(A - B) = n(A) - n(AI B)

  • 2010 _______________________________ (3)

    2

    AU B AU C BU C AU BU C AI BI C 25 27 26 30 7

    (AI B)U C *1) 23 2) 24 3) 25 4) 26 3 46

    39 19

    1) 9 2) 10 3) 11 *4) 12 4 50 32 6

    1

    *1) 11 2) 12 3) 17 4) 18 5 A B n(AU B) = 88 n[(A - B)U (B - A)] = 76

    n(A) = 45 n(B) 1) 45 2) 48 3) 53 *4) 55

  • (4)_______________________________ 2010

    6 300 100 100 150 ( 50 )

    A B A B

    A B 7 A = {1, 2} B = {1, 2, 3, 4, 5} A

    B A B A A P(A) 8 A = {1, 2, 3} A , {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} P(A) = {, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} 1. 2. 3. A A 4. A P(A) 5. A B P(A) P(B) 6. A 2n(A) 7. P(A) 2n(A)

  • 2010 _______________________________ (5)

    1. A B A B AU B 2. A B A B AI B 3. A B A B A - B 4. A U A U A A 9 U = {1, 2, 3, ..., 10} A = {1, 2, 4, 8} B = {2, 4, 6, 10} AU B = {1, 2, 4, 6, 8, 10} AI B = {2, 4} A - B = {1, 8} B - A = {6, 10} A = {3, 5, 6, 7, 9, 10} B = {1, 3, 5, 7, 8, 9} 10 A - B = {2, 4, 6}, B - A = {0, 1, 3} AU B = {0, 1, 2, 3, 4, 5, 6, 7, 8} AI B 1) {0, 1, 4, 5, 6, 7} 2) {1, 2, 4, 5, 6, 8} *3) {0, 1, 3, 5, 7, 8} 4) {0, 2, 4, 5, 6, 8}

  • (6)_______________________________ 2010

    2 1. (Inductive Reasoning) 2. (Deductive Reasoning) 1 1) 1. 2. 2) 1. 2. 2 a 1, 4, 9, 16, 25, a 2, 4, 8, 16, 32, a - 1. a A 2. a A

  • 2010 _______________________________ (7)

    3. A B 4. A B 5. A B 6. A B 3 . . *1) 2) 3) 4) 4 1. 2. 300 3. 300 1) 2) *3) 4)

  • (8)_______________________________ 2010

    5

    1 2 47

    2 4 814

    3 6 1221

    . . . a b c77

    2a - b + c 1) 11 2) 22 3) 33 *4) 44 6 . . . 1) 2)

    3) *4)

    7 1. 2. 3. 1) *2) 3) 4)

  • 2010 _______________________________ (9)

    (R)

    (Q) (Q)

    (I) (I)

    (I-) (I0)

    (I+) () (N)

    2 , 5 , - 3 , , 2.17254... 1 . 0 . 0 1) . . 2) . . 3) . . *4) . . 2 3 3 5 1.732

    2.236 . 2.235 + 1.731 5 + 3 2.237 + 1.733 . 2.235 - 1.731 5 - 3 2.237 - 1.733 *1) . . 2) . . 3) . . 4) . .

  • (10)______________________________ 2010

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

    a + b R a b R a + b = b + a a b = b a a + (b + c) = (a + b) + c a (b c) = (a b) c 0

    0 + a = a = a + 0 1 1 a = a = a 1

    a -a (-a) + a = 0 = a + (-a)

    a a 0 a-1 a a-1 = a-1 a = 1

    a(b + c) = ab + ac 3 a b c d . a - b . c - d 1) . . *2) . . 3) . . 4) . .

  • 2010 ______________________________ (11)

    4 . b b + a = 0 = a + b . a b

    ba = 1 = ab 1) . . *2) . . 3) . . 4) . . 1. (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 - 2ab + b2 2. (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a - b)3 = a3 - 3a2b + 3a2b - b3 3. a2 - b2 = (a - b)(a + b) 4. a3 - b3 = (a - b)(a2 + ab + b2) a3 + b3 = (a + b)(a2 - ab + b2) ax2 + bx + c = 0 a, b c , a 0 x = 2a

    4ac b b 2 -- b2 - 4ac > 0 x 2 b2 - 4ac = 0 x 1 b2 - 4ac < 0 x

  • (12)______________________________ 2010

    a, b c 1. a > b b > c a > c 2. a > b a + c > b + c 3. a > b c > 0 ac > bc a > b c < 0 ac < bc 4. a b a < x < b a < x x < b a b a < b 1. (a, b) = {x|a < x < b} a b 2. [a, b] = {x|a x b} a b 3. (a, b] = {x|a < x b} a b 4. [a, b) = {x|a x < b} a b 5. (-, a) = {x|x < a} a 6. [a, ) = {x|x a} a 5 65

    3 1) 30 *2) 36 3) 42 4) 48

  • 2010 ______________________________ (13)

    6 y = ax2 + bx + c a 0 ax2 + bx + c = 0

    1)

    y

    x50

    5

    -5

    -5

    2)

    y

    x50

    5

    -5

    -5

    3)

    y

    x50

    5

    -5

    -5

    *4)

    y

    x50

    5

    -5

    -5

    7 1 3 4

    100 250 50 100 100 20%

    1) 10 *2) 12 3) 14 4) 16

  • (14)______________________________ 2010

    8 -1 2 + 21

    x-

    1 1) [ 2 - 1, 1] 2) [ 2 - 1, 2] *3) [3 - 2 2 , 1] 4) [3 - 2 2 , 2] a |a| =

    a x < -a x > a |x| a x -a x a 9 |x - 7| = 6 1) 10 15 2) 14 *3) 2 4) 3

  • 2010 ______________________________ (15)

    10

    += 0 a a1 |a| |a|1 a x x22

    --

    1) 1 *2) 2 3) 3 4) 4

    11 x3 - 2x = |x| 1) 0 2) 3 *3) 3 - 1 4) 3 + 1

  • (16)______________________________ 2010

    A B A B A B = {(a, b)|a A b B} A = {1, 2} B = {a, b, c} A B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)} B A = {(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2)} A, B C 1. A = A = 2. A B B A 3. n(A B) = n(A) n(B) 4. A (BU C) = (A B)U (A C) (BU C) A = (B A)U (C A) 5. A (BI C) = (A B)I (A C) (BI C) A = (B A)I (C A) 1 A = {1, 2} B = {a, b} A B * 1) (2, b) 2) (b, a) 3) (a, 1) 4) (1, 2) A B r A B r A B r A r A A * A B 2n(A)n(B)

  • 2010 ______________________________ (17)

    2 A = {1, 2, 3, 4, 5, 6} B = {1, 2, 3, ... , 11, 12} S =

    += 2a 2a b B A b)(a,

    S 1) 1 * 2) 2 3) 3 4) 4 3 A = {1, 2, 3, 4} r = {(m, n) A A | m n} r

    1) 8 * 2) 10 3) 12 4) 16 r Dr r

    Dr = {x|(x, y) r} r Rr r

    Rr = {y|(x, y) r} r = {(-2, 4), (-1, 1), (1, 1)} Dr = {-2, -1, 1} Rr = {1, 4} r R R 1. y x x

    y 2. x y y r (x, y) r (x, z) r y = z r (x, y) r (x, z) r y z

  • (18)______________________________ 2010

    1. r r 1 r r1 = {(a, 1), (b, 2), (b, 3), (c, 4)} r1 b 2 3 r2 = {(p, 2), (q, 4), (r, 6)} r2 2. r x x r 1 r x r

    1 r1

    x

    r 1 r2

    4 f = (x, y)|y =

    2 3x xx

    2 ++ +

    1x12x

    2 --

    1) -2 2) -1 * 3) 0 4) 1

    y r1

    x

    y

    r2

    x

  • 2010 ______________________________ (19)

    5 A = {1, 99} A 1) 2) * 3) 4) 6 r

    1 2

    y

    x30

    123

    -3 -2 -1

    -3-2-1

    1) r (1, 1), (2, 2) (3, 3) 2) r * 3) r (3, 3) (3, -1) 4) r (1, 1) (-1, 1) (Linear Function) f(x) = ax + b a, b R (Constant Function) a = 0 X (Quadratic Function) f(x) = ax2 + bx + c a, b, c R a 0 a > 0 a < 0 f(x) = ax2 + bx + c a, b, c R

    2ab f,2a

    b --

    4a b 4ac ,2a

    b 2--

    f(x) = a(x - h)2 + k a, k R a 0 (h, k)

  • (20)______________________________ 2010

    1. x 2. y = a(x + c)2 c > 0 x (-c, 0) x = -c y = a(x - c)2 c > 0 x (c, 0) x = c 3. x x = 2a 4ac bb

    2 -- (Exponential Function) y = ax a > 0 a 1 (Absolute Value Function) y = |x - a| + c a, c R (Step Function) R 7 a y = a(2x) (3, 16) * 1) 2 2) 3 3) 4 4) 5 8 x y = -4x2 - 5x + 6 x * 1)

    31 ,3

    2 -- 2)

    23 ,2

    5 -- 3)

    76 ,4

    1 4)

    23 ,2

    1

    9 a b y1 = 1 + ax y2 = 1 + bx

    0x

    1

    2

    yx

    2 b 1 y +=x

    1 a 1 y +=

    1) 1 < a < b 2) a < 1 < b * 3) b < 1 < a 4) b < a < 1

  • 2010 ______________________________ (21)

    10 x = 3 f(x) = -x2 + (k + 5)x + (k2 - 10) k f

    1) -4 * 2) 0 3) 6 4) 14 11 f(x) = x2 - 2x - 15 1) f(x) -17 x 2) f(-3 - 2 - 3 ) > 0 3) f(1 + 3 + 5 ) = f(1 - 3 - 5 ) * 4) f(-1 + 3 + 5 ) > f(-1 - 3 - 5 )

  • (22)______________________________ 2010

    a b m n k 1. am an = am+n 2. n

    m

    aa = am-n

    3. (am)n = amn 4. (am bn)k = amk bnk 5.

    k

    nm

    ba

    = nkmk

    ba , b 0

    6. a-n = na1 , a 0

    7. a0 = 1, a 0 a n 1 a1/n = n a a m n 1 .. m n 1 n m a = am/n a b 1 m, n 1. am = an m = n 2. am = bm m = 0 a, b 0

    1 22)(- +

    +

    3222 81/2

    1) -1 2) 1 * 3) 3 4) 5

  • 2010 ______________________________ (23)

    2 4

    1258

    =

    x/162516

    x

    1) 43 * 2) 3

    2 3) 23 4) 3

    4 3 1) (24)30 < 220 330 440 2) (24)30 < 230 320 440 * 3) 220 340 430 < (24)30 4) 230 340 420 < (24)30 4 ( 18 + 2 3 125- - 3 4 4 ) * 1) -10 2) 10 3) 2 5 - 5 2 4) 5 2 - 2 5

    5 2

    152

    6