m0097 / ГИА-2013. Математика. Тренировочн. раб. 1 (вар. 5-8) 04.12.2012г. (с критер. оцен

Ɇɨɞɭɥɶ "Ƚɟɨɦɟɬɪɢɹ"

9 ɋɪɟɞɧɹɹ ɥɢɧɢɹ ɪɚɜɧɨɫɬɨɪɨɧɧɟɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ABC ɪɚɜɧɚ

8

ɫɦ. ɇɚɣɞɢɬɟ ɩɟɪɢɦɟɬɪ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ.

14

Ʉɭɪɢɧɵɟ ɹɣɰɚ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɢɯ ɦɚɫɫɵ ɩɨɞɪɚɡɞɟɥɹɸɬ ɧɚ ɩɹɬɶ ɤɚɬɟɝɨɪɢɣ:

,

,ɜɵɫɲɚɹ ɨɬɛɨɪɧɚɹ ɩɟɪɜɚɹ, ɜɬɨɪɚɹ ɢ ɬɪɟɬɶɹ. ɂɫɩɨɥɶɡɭɹ,

ɞɚɧɧɵɟ ɩɪɟɞɫɬɚɜɥɟɧɧɵɟ

ɜ ɬɚɛɥɢɰɟ, ɨɩɪɟɞɟɥɢɬɟ,

ɤ ɤɚɤɨɣ ɤɚɬɟɝɨɪɢɢ ɨɬɧɨɫɢɬɫɹ ɹɣɰɨ, ɦɚɫɫɨɣ 49,2

ɝ.

Ʉ ɬɟɝɨ

ɹ

Ɇ ɫɫ ɨɞɧɨɝɨ ɹ , , ɧɟ ɦɟɧɟɟ ɝ

ȼɵɫɲɚɹ

75,0 ɢ ɜɵɲɟ

Ɉɬɛɨɪɧɚɹ

65,0 - 74,9

ɉɟɪɜɚɹ

55,0 - 64,9

ȼɬɨɪɚɹ

45,0 - 54,9

Ɍɪɟɬɶɹ

35,0 - 44,9

15

ɇɚ ɬɪɟɧɢɪɨɜɤɟ ɜ 50-ɦɟɬɪɨɜɨɦ ɛɚɫɫɟɣɧɟ ɩɥɨɜɟɰ ɩɪɨɩɥɵɥɦɟɬɪɨɜɭɸ200-

.

ɞɢɫɬɚɧɰɢɸ

ɇɚ ɪɢɫɭɧɤɟ ɢɡɨɛɪɚɠɺɧ ɝɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ

ɩɥɨɜɰɨɦ ɢ ɬɨɱɤɨɣ ɫɬɚɪɬɚ ɨɬ ɜɪɟɦɟɧɢ ɞɜɢɠɟɧɢɹ ɩɥɨɜɰɚ. Ɉɩɪɟɞɟɥɢɬɟ ɪɚɫɫɬɨɹɧɢɟ

(ɜ ɦɟɬɪɚɯ),

ɤɨɬɨɪɨɟ ɩɪɨɩɥɵɥ ɩɥɨɜɟɰ ɡɚ ɩɟɪɜɵɟ 100 ɫɟɤɭɧɞ ɡɚɩɥɵɜɚ.

16 ɋɩɨɪɬɢɜɧɵɣ ɦɚɝɚɡɢɧ ɩɪɨɜɨɞɢɬ: « ɚɤɰɢɸ Ʌɸɛɨɣ ɫɜɢɬɟɪ ɩɨ ɰɟɧɟ 600 ɪ. ɉɪɢ

ɩɨɤɭɩɤɟ ɞɜɭɯ ɫɜɢɬɟɪɨɜ – ɫɤɢɞɤɚ ɧɚ ɜɬɨɪɨɣ

80%».

ɋɤɨɥɶɤɨ ɪɭɛɥɟɣ ɩɪɢɞɺɬɫɹ

ɡɚɩɥɚɬɢɬɶ ɡɚ ɩɨɤɭɩɤɭ ɞɜɭɯ ɫɜɢɬɟɪɨɜ?

Ɇɚɬɟɦɚɬɢɤɚɤɥɚɫɫ9

ȼɚɪɢɚɧɬ 8

1

ɇɚɣɞɢɬɟ

ɡɧɚɱɟɧɢɹ

ɜɵɪɚɠɟɧɢɣ.

ȼ ɨɬɜɟɬɟ ɭɤɚɠɢɬɟ ɧɨɦɟɪ ɧɚɢɦɟɧɶɲɟɝɨ ɢɡ

ɡɧɚɱɟɧɢɣ

ɧɚɣɞɟɧɧɵɯ

.

1)1, 8

3

2)1

1

:

1

3)

0, 8 0, 3

5

3

6

1, 2

2

ȼɵɛɟɪɢɬɟɜɟɪɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɱɢɫɟɥ

a

b, ɪɚɫɩɨɥɨɠɟɧɧɵɯɧɚ

ɱɢɫɥɨɜɨɣɩɪɹɦɨɣ .

1

0

1) b a 0

2)

ab 0

3)

0

1

4)

a

b

3

ɍɤɚɠɢɬɟɞɜɚ

ɫɨɫɟɞɧɢɯɰɟɥɵɯ ɱɢɫɥɚɦɟɠɞɭ,

ɤɨɬɨɪɵɦɢ ɡɚɤɥɸɱɟɧɨ.ɱɢɫɥɨ

3 5

1) ɢ 3 4

2) ɢ 4 5

3)

ɢ 6 7

4) ɢ 45

46

4

ɇɚɣɞɢɬɟɤɨɪɧɢ ɭɪɚɜɧɟɧɢɹ 2x2 11x 6 0.

Ɉɬɜɟɬ:

Ɇɚɬɟɦɚɬɢɤɚɤɥɚɫɫ9

ȼɚɪɢɚɧɬ 8

5

ɇɚ

ɪɢɫɭɧɤɟ ɢɡɨɛɪɚɠɟɧɵ ɝɪɚɮɢɤɢ ɬɪɺɯ

ɮɭɧɤɰɢɣ, ɡɚɞɚɜɚɟɦɵɯ ɮɨɪɦɭɥɚɦɢ ɜɢɞɚ

y

kx

b. ɍɤɚɠɢɬɟ ɞɥɹ ɤɚɠɞɨɝɨ ɝɪɚɮɢɤɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɟɦɭ ɮɨɪɦɭɥɭ, ɜɵɛɪɚɜ

ɟɺɢɡ ɱɢɫɥɚɧɢɠɟɩ ɪɢɜɟɞɺɧɧɵɯ

.

ȽɊȺɎɂɄɂ

Ⱥ)

Ȼ

)

) ȼ

ɎɍɇɄɐɂɂ

1) y x 1

2) y x 1

3) y x 1

4) y x 1

:

Ɉɬɜɟɬ

Ⱥ

Ȼ

ȼ

6

Ɂɚɩɢɲɢɬɟɜ ɨɬɜɟɬɟ ɧɨɦɟɪɚ ɜɟɪɧɵɯ

ɪɚɜɟɧɫɬɜ.

1)

(4

b)(b

4)

2

16

b

2)

(b

1)(3

b4 )

(1 bb)(4

3)

3)

(b

1)(3 b2 )

3

2

b 2b

7

ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ

a

:

a

ɢ

ɧɚɣɞɢɬɟ

ɟɝɨ ɡɧɚɱɟɧɢɟ ɩɪɢ a 0, 7,

b2

a

2

ab

a2

Ɇɚɬɟɦɚɬɢɤɚɤɥɚɫɫ9

ȼɚɪɢɚɧɬ 8

8

Ɋɟɲɢɬɟɭɪɚɜɧɟɧɢɟ

x 4

x 1.

3

Ɉɬɜɟɬ:

"

ɆɨɞɭɥɶȽɟɨɦɟɬɪɢɹ"

9

ɋɪɟɞɧɹɹ ɥɢɧɢɹ

ɪɚɜɧɨɫɬɨɪɨɧɧɟɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ABC .ɪɚɜɧɚɫɦ 7

ɇɚɣɞɢɬɟ

ɩɟɪɢɦɟɬɪɷ ɬɨɝɨɬ

ɪɟɭɝɨɥɶɧɢɤɚ.

Ɇɚɬɟɦɚɬɢɤɚɤɥɚɫɫ9

ȼɚɪɢɚɧɬ 8

Ʉ ɬɟɝɨ ɹ

Ɇ ɫɫ ɨɞɧɨɝɨɧɟ, ɹ

ɦɟɧɟɟɝ,

ȼɵɫɲɚɹ

ɢ 75,0

ɜɵɲɟ

Ɉɬɛɨɪɧɚɹ

65,0 - 74,9

ɉɟɪɜɚɹ

55,0 - 64,9

ȼɬɨɪɚɹ

45,0 - 54,9

Ɍɪɟɬɶɹ

35,0 - 44,9

Ɇɚɬɟɦɚɬɢɤɚɤɥɚɫɫ9

ȼɚɪɢɚɧɬ 8

15 ɇɚ

ɬɪɟɧɢɪɨɜɤɟ ɜ 50-ɦɟɬɪɨɜɨɦ ɛɚɫɫɟɣɧɟ ɩɥɨɜɟɰ200- ɩɪɨɩɥɵɥ

ɦɟɬɪɨɜɭɸ

ɞɢɫɬɚɧɰɢɸ. ɇɚ ɪɢɫɭɧɤɟ ɢɡɨɛɪɚɠɺɧ ɝɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ

ɩɥɨɜɰɨɦ ɢ ɬɨɱɤɨɣ ɫɬɚɪɬɚ ɨɬ ɜɪɟɦɟɧɢ ɞɜɢɠɟɧɢɹ ɩɥɨɜɰɚ. Ɉɩɪɟɞɟɥɢɬɟ ɪɚɫɫɬɨɹɧɢɟ

ɞɨ ɫɬɚɪɬɚ (ɜ ɦɟɬɪɚɯ) ɱɟɪɟɡ40

ɫɟɤɭɧɞ ɨɬ ɧɚɱɚɥɚ ɡɚɩɥɵɜɚ.

Ɇɚɬɟɦɚɬɢɤɚɤɥɚɫɫ9

ȼɚɪɢɚɧɬ 8

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 5

1

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ

21

ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ

10

2

10

2

.

24

Ɋɟɲɟɧɢɟ.

10

2

10

2

(

10

2)(

10

2)

10

4

6

1

1

.

24

24

24

24

4

2

Ɉɬɜɟɬ:

1

.

2

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

2

ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ

1

ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫɢɦ ɥɶɧɵɣ ɛ ɥɥ

2

Ʉɨɦɦɟɧɬ

ɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ; ɩɪɢ ɢɯ

ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.

22

Ɉɞɢɧ ɢɡ ɤɨɪɧɟɣ ɭɪɚɜɧɟɧɢɹ 5X2

2X

3P

0 ɪɚɜɟɧ 1. ɇɚɣɞɢɬɟ ɜɬɨɪɨɣ ɤɨɪɟɧɶ.

Ɋɟɲɟɧɢɟ.

ɉɨɞɫɬɚɜɢɦ

ɢɡɜɟɫɬɧɵɣ

ɤɨɪɟɧɶ

ɜ

ɭɪɚɜɧɟɧɢɟ:

5

2 3P 0.

ɉɨɥɭɱɢɦ

ɭɪɚɜɧɟɧɢɟ

ɨɬɧɨɫɢɬɟɥɶɧɨ P.

Ɋɟɲɢɦ

ɟɝɨ:

3P

3; P

1.

ɉɨɞɫɬɚɜɢɦ

P

ɜ

ɭɪɚɜɧɟɧɢɟ:

5X2

2X

3

0, ɨɬɤɭɞɚ

X

2 r

4 4 5 3

2 r 8

,

X

1, X

0,6.

10

10

1

2

Ɉɬɜɟɬ:

0, 6.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

3

ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ

2

ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫɢɦ ɥɶɧɵɣ ɛ ɥɥ

3

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 5

2

Ɋɟɲɟɧɢɟ.

ɋɭɦɦɚ

6X Y

5

3X 2Y 1

ɩɪɢɧɢɦɚɟɬ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ,

ɪɚɜɧɨɟ 0,

ɬɨɥɶɤɨ ɜ

ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɛɚ ɫɥɚɝɚɟɦɵɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɜɧɵ 0. ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ

®­ 6X

Y

5

0,

Ɋɟɲɢɦ ɟɺ

¯ 3X

2Y

1

0.

­

6X Y 5 0,

­

3Y 3 0,

­

Y 1,

­ Y 1,

®

6X 4Y 2 0,

®

6X Y

5 0,

®

6X 6 0,

®

¯

¯

¯

¯ X

1.

Ɉɬɜɟɬ: 0; (–1;1).

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

4

ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ

3

ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫɢɦ ɥɶɧɵɣ ɛ ɥɥ

4

Ɉɬɜɟɬ: 26°.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ɋɟɲɟɧɢɟ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

2

Ⱦɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɟɬɨɦ

1

ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫɢɦ ɥɶɧɵɣ ɛ ɥɥ

2

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 5

3

25 ȼ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɟ ABCD ɬɨɱɤɚ M — ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ

AB. ɂɡɜɟɫɬɧɨ, ɱɬɨ

MC

MD. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɞɚɧɧɵɣ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦ — ɩɪɹɦɨɭɝɨɥɶɧɢɤ.

ɉɭɫɬɶ

ɬɨɱɤɚ M – ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ AB ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ABCD – ɪɚɜɧɨɭɞɚɥɟɧɚ ɨɬ ɟɝɨ

ɜɟɪɲɢɧ C ɢ D. Ɍɨɝɞɚ, ɬɪɟɭɝɨɥɶɧɢɤ CMD – ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, ɩɨɷɬɨɦɭ MCD

MDC.

ɉɨɫɤɨɥɶɤɭ

ɩɪɹɦɚɹ

CD ɩɚɪɚɥɥɟɥɶɧɚ

ɫɬɨɪɨɧɟ AB,

ɬɨ

BMC

MCD ɢ

AMD MDC ɤɚɤ

ɧɚɤɪɟɫɬ ɥɟɠɚɳɢɟ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,

+BMC

+AMD ɩɨ

ɩɟɪɜɨɦɭ

ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ( BMC

AMD, AM

BM, MC MD).

Ɂɧɚɱɢɬ, CBM

DAM. ɂɯ ɫɭɦɦɚ ɪɚɜɧɚ 180q, ɬ ɤ. ɷɬɨ ɞɜɚ ɭɝɥɚ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ,

ɩɪɢɥɟɠɚɳɢɟ ɤ ɨɞɧɨɣ ɫɬɨɪɨɧɟ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, CBM

DAM

90 . ɉɨ ɫɜɨɣɫɬɜɭ

ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ɭɝɥɵ BCD ɢ CDA ɬɚɤɠɟ ɩɪɹɦɵɟ. Ɂɧɚɱɢɬ, ABCD – ɩɪɹɦɨɭɝɨɥɶɧɢɤ.

Ʉɨɦɦɟɧɬ

ɢɣ: Ɋɚɜɟɧɫɬɜɨ

ɬɪɟɭɝɨɥɶɧɢɤɨɜ BMC ɢ AMD ɦɨɠɟɬ ɛɵɬɶ

ɞɨɤɚɡɚɧɨ ɢɧɚɱɟ,

ɧɚɩɪɢɦɟɪ, ɩɨ ɬɪɟɬɶɟɦɭ ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ.

Ⱦ ɭɝɨɟ ɜɨɡɦɨɠɧɨɟ ɞɨɤ ɡ ɬɟɥɶɫɬɜɨ:

ɉɭɫɬɶ ɬɨɱɤɚ O –

ɫɟɪɟɞɢɧɚ CD. ɑɟɬɵɪɟɯɭɝɨɥɶɧɢɤ OMBC ɹɜɥɹɟɬɫɹ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɨɦ,

ɩɨɫɤɨɥɶɤɭ

ɟɝɨ

ɫɬɨɪɨɧɵ

OC ɢ

MB ɩɚɪɚɥɥɟɥɶɧɵ ɢ ɪɚɜɧɵ. Ɍɪɟɭɝɨɥɶɧɢɤ MCD –

ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ,

ɩɨɷɬɨɦɭ

OM –

ɟɝɨ ɜɵɫɨɬɚ. Ɂɧɚɱɢɬ,

OMBC–

ɩɪɹɦɨɭɝɨɥɶɧɢɤ,

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ, ɜɫɟ ɲɚɝɢ ɨɛɨɫɧɨɜɚɧɵ

3

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫɨɞɟɪɠɢɬ ɧɟɬɨɱɧɨɫɬɢ ɢɥɢ ɩɪɨɛɟɥɵ, ɧɚɩɪɢɦɟɪ, ɨɬɫɭɬɫɬɜɭɸɬ

2

ɫɫɵɥɤɢ ɧɚ ɫɜɨɣɫɬɜɚ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ ɢɥɢ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫɢɦ ɥɶɧɵɣ ɛ ɥɥ

3

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 5

4

ɱɬɨ AM2 MQ MO. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, QM

AM2

9

4,5.

OM

2

Ɉɬɜɟɬ: 4, 5.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ

4

ɨɬɜɟɬ

ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ

ɜɵɩɨɥɧɟɧɵ

ɩɪɚɜɢɥɶɧɨ, ɧɨ ɞɚɧɵ

3

ɧɟɩɨɥɧɵɟ ɨɛɴɹɫɧɟɧɢɹ ɢɥɢ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫɢɦ ɥɶɧɵɣ ɛ ɥɥ

4

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 6

1

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ

21

ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ

54

.

15

3

15

3

Ɋɟɲɟɧɢɟ.

54

54

54

54

9

3.

15

3

15

3

(

15

3)(

15

3)

15 9

6

Ɉɬɜɟɬ: 3.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

2

ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ

1

ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫɢɦ ɥɶɧɵɣ ɛ ɥɥ

2

Ʉɨɦɦɟɧɬ

ɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ; ɩɪɢ ɢɯ

ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.

22

Ɉɞɢɧ ɢɡ ɤɨɪɧɟɣ ɭɪɚɜɧɟɧɢɹ 4X2

X

3M

0 ɪɚɜɟɧ 1. ɇɚɣɞɢɬɟ ɜɬɨɪɨɣ ɤɨɪɟɧɶ.

Ɋɟɲɟɧɢɟ.

ɉɨɞɫɬɚɜɢɦ

ɢɡɜɟɫɬɧɵɣ

ɤɨɪɟɧɶ

ɜ ɭɪɚɜɧɟɧɢɟ:

4

1 3M 0.

ɉɨɥɭɱɢɦ

ɭɪɚɜɧɟɧɢɟ

ɨɬɧɨɫɢɬɟɥɶɧɨ M.

Ɋɟɲɢɦ

ɟɝɨ:

3M

3; M

1.

ɉɨɞɫɬɚɜɢɦ M ɜ

ɭɪɚɜɧɟɧɢɟ:

4X2

X

3

0, ɨɬɤɭɞɚ

X

1 r 1 4 3 4

1 r 7

, X

1, X

3

.

8

8

1

2

4

3

Ɉɬɜɟɬ:

.

4

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

3

ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ

2

ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫɢɦ ɥɶɧɵɣ ɛ ɥɥ

3

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 6

2

Ɋɟɲɟɧɢɟ.

ɋɭɦɦɚ

3X 4Y 1

X

5Y

6

ɩɪɢɧɢɦɚɟɬ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ 0,

ɬɨɥɶɤɨ ɜ

ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɛɚ ɫɥɚɝɚɟɦɵɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɜɧɵ 0. ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ

®­ 3X

4Y

1

0,

Ɋɟɲɢɦ ɟɺ

¯ X

5Y

6

0.

­

3X

4Y

1 0,

­

19Y

19 0,

­ Y

1,

®

3X

15Y 18 0;

®

5Y 6 0;

®

¯

¯

X

¯ X

1.

Ɉɬɜɟɬ: 0; (–1;1).

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

4

ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ

3

ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫɢɦ ɥɶɧɵɣ ɛ ɥɥ

4