m0101 / ГИА-2013. Математика. Тренировочная работа 2. (вар. 1-4) 18.01.2013г. (с отв)

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 3

1

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 3

2

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ

22

Ɏɢɪɦɚ ȼ ɦɨɠɟɬ ɜɵɩɨɥɧɢɬɶ ɡɚɤɚɡ ɧɚ ɩɪɨɢɡɜɨɞɫɬɜɨ ɢɝɪɭɲɟɤ ɧɚ 4 ɞɧɹ ɛɵɫɬɪɟɟ,

ɱɟɦ ɮɢɪɦɚ Ⱥ. ɉɪɢ ɫɨɜɦɟɫɬɧɨɣ ɪɚɛɨɬɟ ɷɬɢ ɮɢɪɦɵ ɡɚ 24

ɞɧɹ ɜɵɩɨɥɧɹɸɬ ɡɚɤɚɡ,

Ɇɨɞɭɥɶ «Ⱥɥɝɟɛɪɚ»

ɜ 5 ɪɚɡ ɛɨɥɶɲɢɣ, ɱɟɦ ɞɚɧɧɵɣ. Ɂɚ ɤɚɤɨɟ ɜɪɟɦɹ ɦɨɠɟɬ ɜɵɩɨɥɧɢɬɶ ɞɚɧɧɵɣ ɡɚɤɚɡ

ɤɚɠɞɚɹ ɮɢɪɦɚ?

21

6

10

10

2a

2

Ɋɟɲɟɧɢɟ.

ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ a

1

1)2

: a2

a

1 .

(a

1

ɉɭɫɬɶ ɮɢɪɦɚ Ⱥ ɦɨɠɟɬ ɜɵɩɨɥɧɢɬɶ ɡɚɤɚɡ ɡɚ x ɞɧɟɣ, ɬɨɝɞɚ ɮɢɪɦɚ ȼ –

ɡɚ (x

4) ɞɧɟɣ. Ɂɚ

Ɋɟɲɟɧɢɟ.

ɨɞɢɧ ɞɟɧɶ ɮɢɪɦɚ Ⱥ ɜɵɩɨɥɧɹɟɬ

1

ɱɚɫɬɶ ɜɫɟɣ ɪɚɛɨɬɵ, ɚ ɮɢɪɦɚ ȼ

1

ɱɚɫɬɶ.

ɋɨɫɬɚɜɢɦ

2

1)

10

:

10

10(a

1)

(a

1)(a

1)

a

1.

ɭɪɚɜɧɟɧɢɟ:

x

x

4

(a 1)2

a2 1

(a 1)2 10

(a 1)2

a 1

1

1

5 ;

5x2 68x 96 0.

2)

6

a 1 2a 2 6 a 1 2a 2 3 3a

3.

x

x

4

24

Ʉɨɪɧɢ ɭɪɚɜɧɟɧɢɹ: 12 ɢ 1,6, ɜɬɨɪɨɣ ɤɨɪɟɧɶ ɧɟ ɩɨɞɯɨɞɢɬ, ɬɚɤ ɤɚɤ x

4.

a

1

a

1

a

1

a

1

a

1

Ɉɬɜɟɬ: –3.

Ɉɬɜɟɬ: ɮɢɪɦɚ Ⱥ ɡɚ 12 ɞɧɟɣ, ɮɢɪɦɚ ȼ ɡɚ 8 ɞɧɟɣ.

Ʉɨɦɦɟɧɬ

ɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ, ɚ ɬɚɤɠɟ ɜ ɩɪɢɦɟɧɟɧɢɢ ɩɪɚɜɢɥɚ ɞɟɥɟɧɢɹ

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ɞɪɨɛɟɣ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ;

ɩɪɢ ɢɯ ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.

Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

3

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ɉɨ

ɯɨɞɭ

ɪɟɲɟɧɢɹ

ɞɨɩɭɳɟɧɚ

ɨɞɧɚ

ɨɲɢɛɤɚ

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ

2

ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

2

ɯɚɪɚɤɬɟɪɚ/ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ

ɉɨ

ɯɨɞɭ

ɪɟɲɟɧɢɹ

ɞɨɩɭɳɟɧɚ

ɨɞɧɚ

ɨɲɢɛɤɚ

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ

1

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

ɯɚɪɚɤɬɟɪɚ/ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

3

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ

ɛ ɥɥ

2

© ɋɬɚɬȽɪɚɞ 2013 ɝ

© ɋɬɚɬȽɪɚɞ 2013 ɝ

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 3

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 3

4

23

ɉɨɫɬɪɨɣɬɟ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ y

x2

3 x

x ɢ ɨɩɪɟɞɟɥɢɬɟ, ɩɪɢ ɤɚɤɢɯ ɡɧɚɱɟɧɢɹɯ

Ɇɨɞɭɥɶ «Ƚɟɨɦɟɬɪɢɹ»

c ɩɪɹɦɚɹ y

c ɢɦɟɟɬ ɫ ɝɪɚɮɢɤɨɦ ɬɪɢ ɨɛɳɢɟ ɬɨɱɤɢ.

24

ɇɚ ɫɬɨɪɨɧɚɯ ɭɝɥɚ BAC, ɪɚɜɧɨɝɨ 40q, ɢ ɧɚ ɟɝɨ ɛɢɫɫɟɤɬɪɢɫɟ ɨɬɥɨɠɟɧɵ ɪɚɜɧɵɟ

Ɋɟɲɟɧɢɟ.

­

ɨɬɪɟɡɤɢ AB, AC ɢ AD. Ɉɩɪɟɞɟɥɢɬɟ ɜɟɥɢɱɢɧɭ ɭɝɥɚ BDC.

x2

y

x2

3 x

x;

y

°

Ɋɟɲɟɧɢɟ

40q : 2

20q; ǻCAD – ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, CDA

(180q

20q) : 2

80q;

®

x2

2x, x

0.

ɂɦɟɟɦ CAD

°

ǻCAD

ǻBAD ɩɨ

ɞɜɭɦ

ɫɬɨɪɨɧɚɦ

ɢ

ɭɝɥɭ ɦɟɠɞɭ

ɧɢɦɢ, ɩɨɷɬɨɦɭ

BDA

CDA

¯

; BDC

2 80q

160q.

Ɉɬɜɟɬ:160q.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ɉɪɚɜɢɥɶɧɨ ɜɵɩɨɥɧɟɧɵ ɜɫɟ ɲɚɝɢ ɪɟɲɟɧɢɹ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

2

Ⱦɥɹ

ɩɨɫɬɪɨɟɧɢɹ ɢɫɤɨɦɨɝɨ

ɝɪɚɮɢɤɚ

ɩɨɫɬɪɨɢɦ

ɝɪɚɮɢɤ

ɮɭɧɤɰɢɢ

y

x2

4x ɧɚ

Ɋɟɲɟɧɢɟ

ɞɨɜɟɞɟɧɨ

ɞɨ

ɤɨɧɰɚ,

ɧɨ

ɞɨɩɭɳɟɧɚ

ɨɲɢɛɤɚ

ɢɥɢ

ɨɩɢɫɤɚ

1

ɩɪɨɦɟɠɭɬɤɟ [0;

f) ɢ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ y

x2

2x ɧɚ ɩɪɨɦɟɠɭɬɤɟ (

f; 0). Ƚɪɚɮɢɤɨɦ

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɞɚɥɶɧɟɣɲɢɟ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ

ɮɭɧɤɰɢɢ y

x2

4x ɹɜɥɹɟɬɫɹ

ɩɚɪɚɛɨɥɚ,

ɜɟɬɜɢ

ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɥɟɧɵ

ɜɜɟɪɯ,

ɜɟɪɲɢɧɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

ɛ ɥɥ

0

ɢɦɟɟɬ

ɤɨɨɪɞɢɧɚɬɵ

(2;

4),

ɬɨɱɤɢ

ɩɟɪɟɫɟɱɟɧɢɹ ɫ

ɨɫɹɦɢ

ɤɨɨɪɞɢɧɚɬ:

(0; 0),

(4; 0).

Ɇ ɤɫ ɦ ɥɶɧɵ

2

Ƚɪɚɮɢɤɨɦ ɮɭɧɤɰɢɢ

y

x2

2x ɹɜɥɹɟɬɫɹ ɩɚɪɚɛɨɥɚ,

ɜɟɬɜɢ ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɥɟɧɵ ɜɜɟɪɯ,

ɜɟɪɲɢɧɚ ɢɦɟɟɬ ɤɨɨɪɞɢɧɚɬɵ (

1;

1),

ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɫ ɨɫɹɦɢ ɤɨɨɪɞɢɧɚɬ:

(0; 0),

( 2; 0). Ƚɪɚɮɢɤ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɢɡɨɛɪɚɠɺɧ ɧɚ ɪɢɫɭɧɤɟ.

ɉɪɹɦɚɹ y

c ɢɦɟɟɬ ɫ ɩɨɫɬɪɨɟɧɧɵɦ ɝɪɚɮɢɤɨɦ ɪɨɜɧɨ ɬɪɢ ɨɛɳɢɟ ɬɨɱɤɢ ɩɪɢ c

0 ɢ

ɩɪɢ c

1.

Ɉɬɜɟɬ: ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ ɢɡɨɛɪɚɠɺɧ ɧɚ ɪɢɫɭɧɤɟ; ɩɪɹɦɚɹ y

c ɢɦɟɟɬ ɫ ɝɪɚɮɢɤɨɦ ɪɨɜɧɨ

ɬɪɢ ɨɛɳɢɟ ɬɨɱɤɢ ɩɪɢ c

0 ɢ ɩɪɢ c

1.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ƚɪɚɮɢɤ ɩɨɫɬɪɨɟɧ ɩɪɚɜɢɥɶɧɨ,

ɜɟɪɧɨ ɭɤɚɡɚɧɵ ɜɫɟ ɡɧɚɱɟɧɢɹ c, ɩɪɢ ɤɨɬɨɪɵɯ

4

ɩɪɹɦɚɹ y

c ɢɦɟɟɬ ɫ ɝɪɚɮɢɤɨɦ ɪɨɜɧɨ ɬɪɢ ɨɛɳɢɟ ɬɨɱɤɢ

Ƚɪɚɮɢɤ ɩɨɫɬɪɨɟɧ ɩɪɚɜɢɥɶɧɨ, ɧɨ ɨɞɧɨ ɢɡ ɬɪɟɛɭɟɦɵɯ ɡɧɚɱɟɧɢɣ c ɧɟ ɭɤɚɡɚɧɨ

3

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ

ɛ ɥɥ

4

© ɋɬɚɬȽɪɚɞ 2013 ɝ

© ɋɬɚɬȽɪɚɞ 2013 ɝ

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 3

5

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 3

6

25

Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɭ ɪɚɜɧɵɯ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ABC ɢ A1B1C1 ɦɟɞɢɚɧɵ, ɩɪɨɜɟɞɺɧɧɵɟ ɢɡ

26

Ⱦɢɚɝɨɧɚɥɢ AC ɢ

BD ɬɪɚɩɟɰɢɢ

ABCD ɩɟɪɟɫɟɤɚɸɬɫɹ

ɜ ɬɨɱɤɟ

O.

ɉɥɨɳɚɞɢ

ɜɟɪɲɢɧ A ɢ A1, ɪɚɜɧɵ.

ɬɪɟɭɝɨɥɶɧɢɤɨɜ AOD ɢ BOC ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ 25 ɫɦ2 ɢ 16 ɫɦ2. ɇɚɣɞɢɬɟ

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ.

ɩɥɨɳɚɞɶ ɬɪɚɩɟɰɢɢ.

Ɋɟɲɟɧɢɟ.

ɉɭɫɬɶ AM ɢ A1M1 — ɦɟɞɢɚɧɵ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ABC ɢ A1B1C1. ȼ ɬɪɟɭɝɨɥɶɧɢɤɚɯ ABM ɢ

A1B1M1

ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɚɜɧɵ ɫɬɨɪɨɧɵ ABɢ A1B1, BM ɢ B1M1, ɚ ɬɚɤɠɟ ɭɝɥɵ B ɢ B1.

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɬɪɟɭɝɨɥɶɧɢɤɢ ɪɚɜɧɵ ɩɨ ɩɟɪɜɨɦɭ ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ.

Ɂɧɚɱɢɬ, AM

A1M1, ɱɬɨ ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɞɨɤɚɡɚɬɶ.

ɉɨ ɭɫɥɨɜɢɸ

S AOD

S BOC,

ɩɨɷɬɨɦɭ AD ɢ BC ɹɜɥɹɸɬɫɹ ɧɟ ɛɨɤɨɜɵɦɢ ɫɬɨɪɨɧɚɦɢ,

ɚ

ɨɫɧɨɜɚɧɢɹɦɢ ɬɪɚɩɟɰɢɢ. Ɍɨɝɞɚ ɬɪɟɭɝɨɥɶɧɢɤɢ AOD ɢ

BOC ɩɨɞɨɛɧɵ ɩɨ ɞɜɭɦ ɭɝɥɚɦ,

ɚ

ɨɬɧɨɲɟɧɢɟ

ɢɯ

ɩɥɨɳɚɞɟɣ

ɪɚɜɧɨ

ɤɜɚɞɪɚɬɭ

ɤɨɷɮɮɢɰɢɟɧɬɚ

ɩɨɞɨɛɢɹ

k.

ɉɨɷɬɨɦɭ

k

5

AO. ɉɨɫɤɨɥɶɤɭ ɬɪɟɭɝɨɥɶɧɢɤɢ ABO ɢ CBO ɢɦɟɸɬ ɨɛɳɭɸ ɜɵɫɨɬɭ, ɩɪɨɜɟɞɺɧɧɭɸ

ɢɡ

4

OC

ɨɬɧɨɲɟɧɢɟ

ɢɯ

ɩɥɨɳɚɞɟɣ ɪɚɜɧɨ

ɨɬɧɨɲɟɧɢɸ ɢɯ ɨɫɧɨɜɚɧɢɣ, ɬ. ɟ.

ɜɟɪɲɢɧɵ

B,

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

S ABO

AO

5

. Ɂɧɚɱɢɬ, S ABO

5

5

20. ɉɥɨɳɚɞɢ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ABD

S CBO

OC

4

4S CBO

4 16

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ, ɜɫɟ ɲɚɝɢ ɨɛɨɫɧɨɜɚɧɵ

3

ɢ ACD ɪɚɜɧɵ, ɬɚɤ ɤɚɤ ɷɬɢ ɬɪɟɭɝɨɥɶɧɢɤɢ ɢɦɟɸɬ ɨɛɳɟɟ ɨɫɧɨɜɚɧɢɟ AD,

ɢ ɢɯ ɜɵɫɨɬɵ,

ɏɨɞ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɜɟɪɧɵɣ, ɧɨ ɨɬɫɭɬɫɬɜɭɸɬ ɧɟɤɨɬɨɪɵɟ ɫɫɵɥɤɢ, ɧɚɩɪɢɦɟɪ,

2

ɩɪɨɜɟɞɺɧɧɵɟ ɤ ɷɬɨɦɭ ɨɫɧɨɜɚɧɢɸ, ɪɚɜɧɵ ɤɚɤ ɜɵɫɨɬɵ ɬɪɚɩɟɰɢɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ

ɧɟ ɭɤɚɡɚɧ ɩɪɢɡɧɚɤ ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ

S AOB

S ABD

S AOD

S ACD

S AOD

S COD.

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

ɉɨɷɬɨɦɭ ɢ S

20; S

25

16

20

20

81 ɫɦ2.

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

COD

0

ABCD

Ɉɬɜɟɬ: 81 ɫɦ2.

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 ɞɨɩɭɳɟɧɚ ɨɞɧɚ

3

ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

4

© ɋɬɚɬȽɪɚɞ 2013 ɝ

© ɋɬɚɬȽɪɚɞ 2013 ɝ

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 4

1

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 4

2

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ

22

Ⱦɜɚ ɨɩɟɪɚɬɨɪɚ, ɪɚɛɨɬɚɹ ɜɦɟɫɬɟ, ɦɨɝɭɬ ɧɚɛɪɚɬɶ ɬɟɤɫɬ ɝɚɡɟɬɵ ɨɛɴɹɜɥɟɧɢɣ ɡɚ 8 ɱ.

ȿɫɥɢ ɩɟɪɜɵɣ ɨɩɟɪɚɬɨɪ ɛɭɞɟɬ ɪɚɛɨɬɚɬɶ

3 ɱ,

ɚ ɜɬɨɪɨɣ 12 ɱ, ɬɨ ɨɧɢ ɜɵɩɨɥɧɹɬ

Ɇɨɞɭɥɶ «Ⱥɥɝɟɛɪɚ»

ɬɨɥɶɤɨ

75% ɜɫɟɣ ɪɚɛɨɬɵ.

Ɂɚ ɤɚɤɨɟ ɜɪɟɦɹ ɦɨɠɟɬ ɧɚɛɪɚɬɶ ɜɟɫɶ ɬɟɤɫɬ ɤɚɠɞɵɣ

ɨɩɟɪɚɬɨɪ, ɪɚɛɨɬɚɹ ɨɬɞɟɥɶɧɨ?

21

a

2

5

5

3a

2

Ɋɟɲɟɧɢɟ.

ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ a

1

1)2

: a2

a

1 .

(a

1

ɉɭɫɬɶ

ɩɟɪɜɵɣ

ɨɩɟɪɚɬɨɪ

ɦɨɠɟɬ

ɜɵɩɨɥɧɢɬɶ ɞɚɧɧɭɸ ɪɚɛɨɬɭ ɡɚ x ɱɚɫɨɜ, ɚ ɜɬɨɪɨɣ

Ɋɟɲɟɧɢɟ.

ɡɚ y ɱɚɫɨɜ. Ɂɚ ɨɞɢɧ ɱɚɫ ɩɟɪɜɵɣ ɨɩɟɪɚɬɨɪ ɜɵɩɨɥɧɹɟɬ 1

ɱɚɫɬɶ ɜɫɟɣ ɪɚɛɨɬɵ, ɚ ɜɬɨɪɨɣ 1 .

5

5

5(a

2

1)

(a

1)(a

1)

a

1

x

y

1) (a

1)2 :

a2

1

(a

1)2 5

(a

1)2

a

1.

ɋɨɫɬɚɜɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ:

­ 1

­

1

1

1

1

1

a 2 a 1 3a 2 a 2 a 1 3a 2

3a 3

°

,

° x

,

3.

x

y

8

2)

y

8

a 1 a 1

a 1

a 1

a 1

®

3

12 3

®

1;

:

°

x

y

;

° 1

4

Ɉɬɜɟɬ

–3.

°

4

° x

y

4

¯

¯

Ʉɨɦɦɟɧɬ

ɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ, ɚ ɬɚɤɠɟ ɜ ɩɪɢɦɟɧɟɧɢɢ ɩɪɚɜɢɥɚ ɞɟɥɟɧɢɹ

­

1

1

1,

ɞɪɨɛɟɣ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ; ɩɪɢ ɢɯ ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.

°

x

y

8

24,

x

12.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

®

1;

y

° 3

ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

2

°

y

8

ɉɨ

ɯɨɞɭ

ɪɟɲɟɧɢɹ

ɞɨɩɭɳɟɧɚ

ɨɞɧɚ

ɨɲɢɛɤɚ

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ

¯

1

Ɉɬɜɟɬ: ɩɟɪɜɵɣ ɨɩɟɪɚɬɨɪ ɡɚ 12 ɱ, ɜɬɨɪɨɣ ɨɩɟɪɚɬɨɪ ɡɚ 24 ɱ.

ɯɚɪɚɤɬɟɪɚ/ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ɇ ɤɫ ɦ ɥɶɧɵ

ɛ ɥɥ

2

Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

3

ɉɨ

ɯɨɞɭ

ɪɟɲɟɧɢɹ

ɞɨɩɭɳɟɧɚ

ɨɞɧɚ

ɨɲɢɛɤɚ

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ

2

ɯɚɪɚɤɬɟɪɚ/ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

3

© ɋɬɚɬȽɪɚɞ 2013 ɝ

© ɋɬɚɬȽɪɚɞ 2013 ɝ

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 4

3

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 4

4

23

ɉɨɫɬɪɨɣɬɟ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ y

x2

3 x

x ɢ ɨɩɪɟɞɟɥɢɬɟ, ɩɪɢ ɤɚɤɢɯ ɡɧɚɱɟɧɢɹɯ

Ɇɨɞɭɥɶ «Ƚɟɨɦɟɬɪɢɹ»

c ɩɪɹɦɚɹ y

c ɢɦɟɟɬ ɫ ɝɪɚɮɢɤɨɦ ɬɪɢ ɨɛɳɢɟ ɬɨɱɤɢ.

24

ɇɚ ɫɬɨɪɨɧɚɯ ɭɝɥɚ BAC, ɪɚɜɧɨɝɨ 20q, ɢ ɧɚ ɟɝɨ ɛɢɫɫɟɤɬɪɢɫɟ ɨɬɥɨɠɟɧɵ ɪɚɜɧɵɟ

Ɋɟɲɟɧɢɟ.

­

ɨɬɪɟɡɤɢ AB, AC ɢ AD. Ɉɩɪɟɞɟɥɢɬɟ ɜɟɥɢɱɢɧɭ ɭɝɥɚ BDC.

x2

y

x2

3 x

x;

y

°

Ɋɟɲɟɧɢɟ

20q : 2

10q; ǻCAD ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, CDA

(180q

10q) : 2

85q;

®

x2

4x, x

0.

ɂɦɟɟɦ

CAD

°

ǻCAD

ǻBAD

ɩɨ ɞɜɭɦ

ɫɬɨɪɨɧɚɦ ɢ ɭɝɥɭ ɦɟɠɞɭ ɧɢɦɢ, ɩɨɷɬɨɦɭ BDA

CDA;

¯

BDC

2

85q

170q.

Ⱦɥɹ

ɩɨɫɬɪɨɟɧɢɹ

ɢɫɤɨɦɨɝɨ

ɝɪɚɮɢɤɚ

ɩɨɫɬɪɨɢɦ

ɝɪɚɮɢɤ

ɮɭɧɤɰɢɢ

y

x2

2x ɧɚ

Ɉɬɜɟɬ

:170 .

ɩɪɨɦɟɠɭɬɤɟ [0;

f) ɢ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ y

x2

4x ɧɚ ɩɪɨɦɟɠɭɬɤɟ (

f; 0).

Ƚɪɚɮɢɤɨɦ

q

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ɮɭɧɤɰɢɢ

y

x2

2x ɹɜɥɹɟɬɫɹ

ɩɚɪɚɛɨɥɚ,

ɜɟɬɜɢ

ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɥɟɧɵ

ɜɜɟɪɯ, ɜɟɪɲɢɧɚ

ɢɦɟɟɬ ɤɨɨɪɞɢɧɚɬɵ

(1;

1),

ɬɨɱɤɢ

ɩɟɪɟɫɟɱɟɧɢɹ ɫ

ɨɫɹɦɢ

ɤɨɨɪɞɢɧɚɬ:

(0;

0),

(2; 0).

ɉɪɚɜɢɥɶɧɨ ɜɵɩɨɥɧɟɧɵ ɜɫɟ ɲɚɝɢ ɪɟɲɟɧɢɹ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

2

Ɋɟɲɟɧɢɟ

ɞɨɜɟɞɟɧɨ

ɞɨ

ɤɨɧɰɚ, ɧɨ ɞɨɩɭɳɟɧɚ ɨɲɢɛɤɚ ɢɥɢ ɨɩɢɫɤɚ

Ƚɪɚɮɢɤɨɦ ɮɭɧɤɰɢɢ

y

x2

4x ɹɜɥɹɟɬɫɹ ɩɚɪɚɛɨɥɚ,

ɜɟɬɜɢ ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɥɟɧɵ ɜɜɟɪɯ,

1

ɜɟɪɲɢɧɚ ɢɦɟɟɬ ɤɨɨɪɞɢɧɚɬɵ (

2;

4),

ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɫ ɨɫɹɦɢ ɤɨɨɪɞɢɧɚɬ:

(0; 0),

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɞɚɥɶɧɟɣɲɢɟ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

( 4; 0). Ƚɪɚɮɢɤ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɢɡɨɛɪɚɠɺɧ ɧɚ ɪɢɫɭɧɤɟ.

Ɇ ɤɫ ɦ ɥɶɧɵ

ɛ ɥɥ

2

ɉɪɹɦɚɹ y

c ɢɦɟɟɬ ɫ ɩɨɫɬɪɨɟɧɧɵɦ ɝɪɚɮɢɤɨɦ ɪɨɜɧɨ ɬɪɢ ɨɛɳɢɟ ɬɨɱɤɢ ɩɪɢ c

0 ɢ ɩɪɢ

c

1.

Ɉɬɜɟɬ: ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ ɢɡɨɛɪɚɠɺɧ ɧɚ ɪɢɫɭɧɤɟ; ɩɪɹɦɚɹ y

c ɢɦɟɟɬ ɫ ɝɪɚɮɢɤɨɦ ɪɨɜɧɨ

ɬɪɢ ɨɛɳɢɟ ɬɨɱɤɢ ɩɪɢ c

0 ɢ ɩɪɢ c

1.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ƚɪɚɮɢɤ ɩɨɫɬɪɨɟɧ ɩɪɚɜɢɥɶɧɨ,

ɜɟɪɧɨ ɭɤɚɡɚɧɵ ɜɫɟ ɡɧɚɱɟɧɢɹ c, ɩɪɢ ɤɨɬɨɪɵɯ

4

ɩɪɹɦɚɹ y

c ɢɦɟɟɬ ɫ ɝɪɚɮɢɤɨɦ ɪɨɜɧɨ ɬɪɢ ɨɛɳɢɟ ɬɨɱɤɢ

Ƚɪɚɮɢɤ ɩɨɫɬɪɨɟɧ ɩɪɚɜɢɥɶɧɨ, ɧɨ ɨɞɧɨ ɢɡ ɬɪɟɛɭɟɦɵɯ ɡɧɚɱɟɧɢɣ c ɧɟ ɭɤɚɡɚɧɨ

3

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ

ɛ ɥɥ

4

© ɋɬɚɬȽɪɚɞ 2013 ɝ

© ɋɬɚɬȽɪɚɞ 2013 ɝ

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 4

5

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 4

6

25

Ⱦɨɤɚɠɢɬɟ,

ɱɬɨ

ɭ

ɪɚɜɧɵɯ

ɬɪɟɭɝɨɥɶɧɢɤɨɜ

ABC

ɢ

A1B1C1

ɛɢɫɫɟɤɬɪɢɫɵ,

26

Ⱦɢɚɝɨɧɚɥɢ

AC ɢ

BD ɬɪɚɩɟɰɢɢ

ABCD ɩɟɪɟɫɟɤɚɸɬɫɹ ɜ ɬɨɱɤɟ

O.

ɉɥɨɳɚɞɢ

ɬɪɟɭɝɨɥɶɧɢɤɨɜ AOD ɢ BOC ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ 16 ɫɦ2 ɢ 9 ɫɦ2.

ɇɚɣɞɢɬɟ

ɩɪɨɜɟɞɺɧɧɵɟ ɢɡ ɜɟɪɲɢɧ A ɢ A1, ɪɚɜɧɵ.

ɩɥɨɳɚɞɶ ɬɪɚɩɟɰɢɢ.

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ.

Ɋɟɲɟɧɢɟ.

ɉɭɫɬɶ AK ɢ A1K1

— ɛɢɫɫɟɤɬɪɢɫɵ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ABC ɢ A1B1C1. ȼ ɬɪɟɭɝɨɥɶɧɢɤɚɯ ABK

ɢ A1B1K1 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɚɜɧɵ ɫɬɨɪɨɧɵ AB ɢ A1B1, ɚ ɬɚɤɠɟ ɭɝɥɵ B ɢ B1, BAK ɢ B1A1K1.

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɬɪɟɭɝɨɥɶɧɢɤɢ ɪɚɜɧɵ ɩɨ ɜɬɨɪɨɦɭ ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ.

Ɂɧɚɱɢɬ, AK

A1K1, ɱɬɨ ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɞɨɤɚɡɚɬɶ.

ɉɨ ɭɫɥɨɜɢɸ S AOD

S BOC, ɩɨɷɬɨɦɭ AD ɢ BC ɹɜɥɹɸɬɫɹ ɧɟ ɛɨɤɨɜɵɦɢ ɫɬɨɪɨɧɚɦɢ, ɚ

ɨɫɧɨɜɚɧɢɹɦɢ ɬɪɚɩɟɰɢɢ. Ɍɨɝɞɚ ɬɪɟɭɝɨɥɶɧɢɤɢ

AOD ɢ BOC ɩɨɞɨɛɧɵ ɩɨ

ɞɜɭɦ ɭɝɥɚɦ,

ɚ ɨɬɧɨɲɟɧɢɟ

ɢɯ

ɩɥɨɳɚɞɟɣ

ɪɚɜɧɨ ɤɜɚɞɪɚɬɭ

ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɨɞɨɛɢɹ

k.

ɉɨɷɬɨɦɭ

k

4

AO. ɉɨɫɤɨɥɶɤɭ ɬɪɟɭɝɨɥɶɧɢɤɢ ABO ɢ CBO ɢɦɟɸɬ ɨɛɳɭɸ ɜɵɫɨɬɭ, ɩɪɨɜɟɞɺɧɧɭɸ

ɢɡ

3

OC