m0097 / ГИА-2013. Математика. Тренировочн. раб. 1 (вар. 1-4) 04.12.2012г. (с ответами)

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 1

1

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ

Ɇɨɞɭɥɶ «Ⱥɥɝɟɛɪɚ»

21

ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ:

5N 1 5N 1

.

2 5N

Ɋɟɲɟɧɢɟ

5N 1 5N 1

25 5N 1 5N 1

5N 1(25 1) 24

2,4.

2 5N

2 5N

2 5N

10

Ɉɬɜɟɬ 2,4.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

2

Ɋɟɲɟɧɢɟ

ɞɨɜɟɞɟɧɨ ,ɞɨ

ɤɨɧɰɚ

ɧɨ

ɞɨɩɭɳɟɧɚ

ɨɲɢɛɤɚ ɢɥɢ

ɨɩɢɫɤɚ

1

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɞɚɥɶɧɟɣɲɢɟ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

2

22

ɂɡɜɟɫɬɧɨ,

ɱɬɨ ɩɚɪɚɛɨɥɚ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɭ B 1;

1

ɢ ɟɺ ɜɟɪɲɢɧɚ ɧɚɯɨ-

4

ɞɢɬɫɹ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ. ɇɚɣɞɢɬɟ ɭɪɚɜɧɟɧɢɟ ɷɬɨɣ ɩɚɪɚɛɨɥɵ ɢ ɜɵɱɢɫɥɢɬɟ,

ɜ ɤɚɤɢɯ ɬɨɱɤɚɯ ɨɧɚ ɩɟɪɟɫɟɤɚɟɬ .ɩɪɹɦɭɸ Y

16

Ɋɟɲɟɧɢɟ

ɍɪɚɜɧɟɧɢɟ ɩɚɪɚɛɨɥɵ, ɜɟɪɲɢɧɚ ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɬɫɹ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ:

Y AX2.

ɉɚɪɚɛɨɥɚ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɭ B, ɩɨɷɬɨɦɭ

1

A ( 1)2, ɨɬɤɭɞɚ.

A

1

ɍɪɚɜɧɟɧɢɟ

1

4

4

ɩɚɪɚɛɨɥɵ.

Y

X2

ɛɫɰɢɫɫɵ

ɬɨɱɟɤ

ɩɟɪɟɫɟɱɟɧɢɹ

ɫ ɩɪɹɦɨɣ

Y

16

ɧɚɣɞɺɦ ɢɡ

4

1

ɭɪɚɜɧɟɧɢɹ

X

2

16:

X

8,

X

8.

4

1

2

Ɉɬɜɟɬ

Y

1

X2, (8; 16), ( 8; 16).

4

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 1

2

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ɂɚɞɚɧɢɟ

ɜɵɩɨɥɧɟɧɨ:

ɜɟɪɧɨ

ɜɟɪɧɨ

ɫɨɫɬɚɜɥɟɧɨ

ɭɪɚɜɧɟɧɢɟ ɩɚɪɚɛɨɥɵ

ɢ ɨɛɟ

3

ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɩɚɪɚɛɨɥɵ ɢ ɡɚɞɚɧɧɨɣ ɩɪɹɦɨɣ

Ɋɟɲɟɧɢɟ

ɞɨɜɟɞɟɧɨ

,ɞɨ

ɤɨɧɰɚ

ɧɨ

ɞɨɩɭɳɟɧɚ

ɨɲɢɛɤɚ

ɢɥɢ

ɨɩɢɫɤɚ

2

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɩɪɢ ɧɚɯɨɠɞɟɧɢɢ ɤɨɨɪɞɢɧɚɬ ɬɨɱɟɤ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

3

23

ɇɚɣɞɢɬɟ

ɧɚɢɦɟɧɶɲɟɟ

ɡɧɚɱɟɧɢɟ

ɜɵɪɚɠɟɧɢɹ

(5X 4Y 3)2 (3X Y 1)2 ɢ

ɡɧɚɱɟɧɢɹ X ɢ, Y ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ ɞɨɫɬɢɝɚɟɬɫɹ.

Ɋɟɲɟɧɢɟ

ɉɪɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ

X ɢ

Y

ɢɦɟɟɦ (5X 4Y 3)2 (3X Y 1)2 t 0. Ɂɧɚɱɟɧɢɟ,

ɪɚɜɧɨɟ

0, ɞɨɫɬɢɝɚɟɬɫɹ ɬɨɥɶɤɨ ɜ ɬɨɦ

ɫɥɭɱɚɟ,

ɤɨɝɞɚ

5X 4Y 3

ɢ

3X Y 1 ɪɚɜɧɵ

ɧɭɥɸ

ɨɞɧɨɜɪɟɦɟɧɧɨ

ɋɨɫɬɚɜɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ

­X5

Y4

3

0;

®X3

Y 1

0.

Ɋɟɲɢɜ ɟɺ, ɩɨɥɭɱɢɦ X

¯

1,

Y

2.

Ɍɚɤɢɦ

ɨɛɪɚɡɨɦ,

ɧɚɢɦɟɧɶɲɟɟ

ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ0,

ɪɚɜɧɨ

ɨɧɨ

ɞɨɫɬɢɝɚɟɬɫɹ ɩɪɢ

X 1, Y 2.

Ɉɬɜɟɬ 0, ɩɪɢ X

1,

Y

2.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ɂɚɞɚɧɢɟ ɜɵɩɨɥɧɟɧɨ:

ɜɟɪɧɨ

ɜɟɪɧɨ ɧɚɣɞɟɧɨ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ

4

ɢ ɡɧɚɱɟɧɢɹ X ɢ Y, ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ ɞɨɫɬɢɝɚɟɬɫɹ

Ɋɟɲɟɧɢɟ

ɞɨɜɟɞɟɧɨ

,ɞɨ

ɤɨɧɰɚ

ɧɨ

ɞɨɩɭɳɟɧɚ

ɨɲɢɛɤɚ

ɢɥɢ

ɨɩɢɫɤɚ

3

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɞɚɥɶɧɟɣɲɢɟ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

4

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 1

3

ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ,. ɇɚɣɞɢɬɟ KCB ɟɫɥɢ ABC

20 .

Ɋɟɲɟɧɢɟ

ɬɚɤ ɤɚɤ

ɨɧɢ ɨɩɢɪɚɸɬɫɹ ɧɚ ;ɨɞɧɭ ɞɭɝɭ

ɨɤɪɭɠɧɨɫɬɢ

ɫɥɟɞɨɜɚɬɟɥɶɧɨ,

AKC

AEC,

ɢɦɟɟɦ

BKC

BEA

ɤɚɤ

ɫɦɟɠɧɵɟ ɫ ɧɢɦɢ. ȼ ɱɟɬɵɪɺɯɭɝɨɥɶɧɢɤɟ

BKDE

ɢɦɟɟɦ

BKC

1

(360 90 20 )

125 . ȼ ǻBKC ɢɦɟɟɦ KCB 180 125 20

35 .

2

Ɉɬɜɟɬ 35°.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

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

2

ɨɬɜɟɬ

ɨɞ ɪɟɲɟɧɢɹ,

ɜɟɪɧɵɣ ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ,

ɩɪɚɜɢɥɶɧɨ ɧɨ ɧɟ ɞɚɧɵ

1

ɨɛɴɹɫɧɟɧɢɹ ɢɥɢ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

2

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 1

4

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ.

ȼ ɬɪɟɭɝɨɥɶɧɢɤɚɯ

ABE ɢ

CBF ɢɦɟɟɦ

A C ɤɚɤ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ

ɭɝɥɵ

ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ BEA

CFB ɤɚɤ ɩɪɹɦɵɟ ɭɝɥɵ, ɡɧɚɱɢɬ ɬɪɟɭɝɨɥɶɧɢɤɢ ɩɨɞɨɛɧɵ ɩɨ

ɩɟɪɜɨɦɭ ɩɪɢɡɧɚɤɭ ɩɨɞɨɛɢɹ ɬɪɟɭɝɨɥɶɧɢɤɨɜ.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ

3

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜ ɰɟɥɨɦ ɜɟɪɧɨɟ, ɧɨ ɫɨɞɟɪɠɢɬ ɧɟɬɨɱɧɨɫɬɢ

2

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ

ɛ ɥɥ

3

26

Ⱦɢɚɝɨɧɚɥɢ,

ɱɟɬɵɪɺɯɭɝɨɥɶɧɢɤɚ

ABCD

ɜɟɪɲɢɧɵ

ɤɨɬɨɪɨɝɨ ɪɚɫɩɨɥɨɠɟɧɵ ɧɚ

ɨɤɪɭɠɧɨɫɬɢ

ɩɟɪɟɫɟɤɚɸɬɫɹ

ɜ

ɬɨɱɤɟ M. ɂɡɜɟɫɬɧɨ,

ɱɬɨ

ABC

72 ,

BCD 102 , AMD

110 . ɇɚɣɞɢɬɟ ACD

Ɋɟɲɟɧɢɟ

ɉɭɫɬɶ. ACD X

DMC 180 110

70 ;

DMC DBC BCA;

BCA

102 X;

DBC 102 X

70 ; X

DBC 32 .

DBC ABD

72 ; ABD

X; DBC

72 X; 2X

104 , X

52 .

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 1

5

Ɉɬɜɟɬ 52°.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

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

4

ɨɬɜɟɬ

ɨɞ 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

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 2

1

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ

21

ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ:

10 2N

.

N 1

N 1

2

2

Ɋɟɲɟɧɢɟ

10 2N

10 2N

10 2N

10 2

4.

2N 1 2N 1

4 2N 1 2N 1

(4

1) 2N 1

5

Ɉɬɜɟɬ 4.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

2

Ɋɟɲɟɧɢɟ

ɞɨɜɟɞɟɧɨ ,ɞɨ ɤɨɧɰɚ

ɧɨ

ɞɨɩɭɳɟɧɚ ɨɲɢɛɤɚ

ɢɥɢ

ɨɩɢɫɤɚ

1

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɞɚɥɶɧɟɣɲɢɟ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ

ɛ ɥɥ

2

22

ɂɡɜɟɫɬɧɨ, ɱɬɨ ɩɚɪɚɛɨɥɚ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɭ B 1;

1

ɢ ɟɺ ɜɟɪɲɢɧɚ ɧɚɯɨɞɢɬɫɹ

4

ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ. ɇɚɣɞɢɬɟ ɭɪɚɜɧɟɧɢɟ ɷɬɨɣ ɩɚɪɚɛɨɥɵ ɢ ɜɵɱɢɫɥɢɬɟ, ɜ ɤɚɤɢɯ

ɬɨɱɤɚɯ ɨɧɚ ɩɟɪɟɫɟɤɚɟɬ. ɩɪɹɦɭɸ Y

9

Ɋɟɲɟɧɢɟ

ɍɪɚɜɧɟɧɢɟ

ɩɚɪɚɛɨɥɵ, ɜɟɪɲɢɧɚ ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɬɫɹ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ:

Y AX2.

ɉɚɪɚɛɨɥɚ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɭ B, ɩɨɷɬɨɦɭ

1

A ( 1)2, ɨɬɤɭɞɚ.

A

1

ɍɪɚɜɧɟɧɢɟ

4

4

1

.ɩɚɪɚɛɨɥɵ Y

X2

ɛɫɰɢɫɫɵ ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɫ ɩɪɹɦɨɣ Y

9 ɧɚɣɞɺɦ ɢɡ ɭɪɚɜɧɟɧɢɹ

1

4

X2 9: X

6, X

6.

4

1

2

Ɉɬɜɟɬ

Y

1

X2, (6; 9), ( 6; 9).

4

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 2

2

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ɂɚɞɚɧɢɟ

ɜɵɩɨɥɧɟɧɨ:

ɜɟɪɧɨ

ɜɟɪɧɨ

ɫɨɫɬɚɜɥɟɧɨ

ɭɪɚɜɧɟɧɢɟ ɩɚɪɚɛɨɥɵ

ɢ ɨɛɟ

3

ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɩɚɪɚɛɨɥɵ ɢ ɡɚɞɚɧɧɨɣ ɩɪɹɦɨɣ

Ɋɟɲɟɧɢɟ

ɞɨɜɟɞɟɧɨ

,ɞɨ

ɤɨɧɰɚ

ɧɨ

ɞɨɩɭɳɟɧɚ ɨɲɢɛɤɚ

ɢɥɢ

ɨɩɢɫɤɚ

2

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɩɪɢ ɧɚɯɨɠɞɟɧɢɢ ɤɨɨɪɞɢɧɚɬ ɬɨɱɟɤ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

3

23

ɇɚɣɞɢɬɟ

ɧɚɢɦɟɧɶɲɟɟ

ɡɧɚɱɟɧɢɟ

ɜɵɪɚɠɟɧɢɹ

(5X 4Y 6)2 (3X 4Y 2)2 ɢ

ɡɧɚɱɟɧɢɹ X ɢ, Y ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ ɞɨɫɬɢɝɚɟɬɫɹ.

Ɋɟɲɟɧɢɟ

ɉɪɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ X ɢ

Y ɢɦɟɟɦ (5X Y4 6)2 X (3

4Y 2)2 t 0. Ɂɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ

0, ɞɨɫɬɢɝɚɟɬɫɹ ɬɨɥɶɤɨ ɜ ɬɨɦ

ɫɥɭɱɚɟ,

ɤɨɝɞɚ 5X 4Y 6 ɢ

3X 4Y 2 ɪɚɜɧɵ

ɧɭɥɸ

ɨɞɧɨɜɪɟɦɟɧɧɨ

ɋɨɫɬɚɜɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ

­X5

Y4

6

0;

®X3

Y4

2

0.

Ɋɟɲɢɜ ɟɺ, ɩɨɥɭɱɢɦ X

2,

¯

Y

1.

Ɍɚɤɢɦ

ɨɛɪɚɡɨɦ,

ɧɚɢɦɟɧɶɲɟɟ

ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ0,

ɪɚɜɧɨ

ɨɧɨ

ɞɨɫɬɢɝɚɟɬɫɹ ɩɪɢ

Ɉɬɜɟɬ 0, ɩɪɢ X 2, Y

1.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ɂɚɞɚɧɢɟ ɜɵɩɨɥɧɟɧɨ:

ɜɟɪɧɨ

ɜɟɪɧɨ ɧɚɣɞɟɧɨ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ

4

ɢ ɡɧɚɱɟɧɢɹ X ɢ Y, ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ ɞɨɫɬɢɝɚɟɬɫɹ

Ɋɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ

,ɞɨ

ɤɨɧɰɚ ɧɨ ɞɨɩɭɳɟɧɚ ɨɲɢɛɤɚ ɢɥɢ ɨɩɢɫɤɚ

3

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɞɚɥɶɧɟɣɲɢɟ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

4

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 2

3

ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ,. ɇɚɣɞɢɬɟ ABC ɟɫɥɢ KCB

20 .

Ɋɟɲɟɧɢɟ

ɂɡ ǻCDE ɢɦɟɟɦ DEC

90 20

70 ,

ɬɨɝɞɚ BEA

180 70

110 .

Ⱦɚɥɟɟ

BAE

BCK,

ɬɚɤ

ɤɚɤ

ɨɧɢ

ɨɩɢɪɚɸɬɫɹ

ɧɚ ɨɞɧɭ

ɞɭɝɭ ɨɤɪɭɠɧɨɫɬɢ;

ɫɥɟɞɨɜɚɬɟɥɶɧɨ,

BKC

BEA. ȼ

ɱɟɬɵɪɺɯɭɝɨɥɶɧɢɤɟ

BKDE

ɢɦɟɟɦ

KBE

360 90 2 110

50 .

Ɉɬɜɟɬ 50°.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

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

2

ɨɬɜɟɬ

ɨɞ ɪɟɲɟɧɢɹ,

ɜɟɪɧɵɣ ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ,

ɩɪɚɜɢɥɶɧɨ ɧɨ ɧɟ ɞɚɧɵ

1

ɨɛɴɹɫɧɟɧɢɹ ɢɥɢ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

2

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 2

4

25 ɩɪɨɜɟɞɟɧɵȼɨɫɬɪɨɭɝɨɥɶɧɨɦ ɬɪɟɭɝɨɥɶɧɢɤɟ ABC

ɜɵɫɨɬɵ CE ɢ AD. Ⱦɨɤɚɠɢɬɟ,

ɱɬɨ ǻABD ɩɨɞɨɛɟɧ ǻCBE.

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ.

ȼ ɬɪɟɭɝɨɥɶɧɢɤɚɯ

ABD ɢ CBE

ɭɝɨɥ

B – ɨɛɳɢɣ, BDA

BEC ɤɚɤ

ɩɪɹɦɵɟ,

ɭɝɥɵ

ɡɧɚɱɢɬ, ɬɪɟɭɝɨɥɶɧɢɤɢ ɩɨɞɨɛɧɵ ɩɨ ɩɟɪɜɨɦɭ ɩɪɢɡɧɚɤɭ ɩɨɞɨɛɢɹ ɬɪɟɭɝɨɥɶɧɢɤɨɜ.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ

3

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜ ɰɟɥɨɦ ɜɟɪɧɨɟ, ɧɨ ɫɨɞɟɪɠɢɬ ɧɟɬɨɱɧɨɫɬɢ

2

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ

ɛ ɥɥ

3

26 Ⱦɢɚɝɨɧɚɥɢ,

ɱɟɬɵɪɺɯɭɝɨɥɶɧɢɤɚ

ABCD

ɜɟɪɲɢɧɵ

ɤɨɬɨɪɨɝɨ

ɪɚɫɩɨɥɨɠɟɧɵ ɧɚ

ɨɤɪɭɠɧɨɫɬɢ

ɩɟɪɟɫɟɤɚɸɬɫɹ

ɜ .ɬɨɱɤɟ,

M ɂɡɜɟɫɬɧɨ

ɱɬɨ

ABC

74 ,

BCD 102 , AMD

112 . ɇɚɣɞɢɬɟ ACD

Ɋɟɲɟɧɢɟ

ɉɭɫɬɶ. ACD X

DMC 180 112

68 ;

DBC 102 X

68 ; X DBC 34 .

DBC ABD

74 ; ABD X; DBC 74 X; 2X 108 , X 54 .

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 2

5

Ɉɬɜɟɬ 54°.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

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

4

ɨɬɜɟɬ

ɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɧɨ ɞɨɩɭɳɟɧɚ ɨɞɧɚ

3

ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ

Ⱦɪɭɝɢɟɭɤɚɡɚɧɧɵɦɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ

ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

4

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 3

1

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ

21

ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ:

5N 1 5N 1

.

2 5N

Ɋɟɲɟɧɢɟ

5N 1 5N 1

25 5N 1 5N 1

5N 1(25 1)

24

2,4.

2 5N

2 5N

2 5N

10

Ɉɬɜɟɬ 2,4.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ

2

Ɋɟɲɟɧɢɟ

ɞɨɜɟɞɟɧɨ ,ɞɨ ɤɨɧɰɚ

ɧɨ

ɞɨɩɭɳɟɧɚ ɨɲɢɛɤɚ

ɢɥɢ

ɨɩɢɫɤɚ

1

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɞɚɥɶɧɟɣɲɢɟ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ

ɛ ɥɥ

2

22

ɂɡɜɟɫɬɧɨ, ɱɬɨ ɩɚɪɚɛɨɥɚ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɭ B 1;

1

ɢ ɟɺ ɜɟɪɲɢɧɚ ɧɚɯɨɞɢɬɫɹ

4

ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ. ɇɚɣɞɢɬɟ ɭɪɚɜɧɟɧɢɟ ɷɬɨɣ ɩɚɪɚɛɨɥɵ ɢ ɜɵɱɢɫɥɢɬɟ, ɜ ɤɚɤɢɯ

ɬɨɱɤɚɯ ɨɧɚ ɩɟɪɟɫɟɤɚɟɬ. ɩɪɹɦɭɸ Y

9

Ɋɟɲɟɧɢɟ

ɍɪɚɜɧɟɧɢɟ

ɩɚɪɚɛɨɥɵ, ɜɟɪɲɢɧɚ ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɬɫɹ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ:

Y AX2.

ɉɚɪɚɛɨɥɚ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɭ B,

ɩɨɷɬɨɦɭ

1

A ( 1)2, ɨɬɤɭɞɚ.

A

1

ɍɪɚɜɧɟɧɢɟ

4

4

1

.ɩɚɪɚɛɨɥɵ Y

X2

ɛɫɰɢɫɫɵ ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɫ ɩɪɹɦɨɣ Y

9 ɧɚɣɞɺɦ ɢɡ ɭɪɚɜɧɟɧɢɹ

1

4

X2 9: X

6, X

6.

4

1

2

Ɉɬɜɟɬ

Y

1

X2, (6; 9), ( 6; 9).

4

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 3

2

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ɂɚɞɚɧɢɟ

ɜɵɩɨɥɧɟɧɨ:

ɜɟɪɧɨ

ɜɟɪɧɨ

ɫɨɫɬɚɜɥɟɧɨ

ɭɪɚɜɧɟɧɢɟ ɩɚɪɚɛɨɥɵ

ɢ ɨɛɟ

3

ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɩɚɪɚɛɨɥɵ ɢ ɡɚɞɚɧɧɨɣ ɩɪɹɦɨɣ

Ɋɟɲɟɧɢɟ

ɞɨɜɟɞɟɧɨ

,ɞɨ

ɤɨɧɰɚ

ɧɨ

ɞɨɩɭɳɟɧɚ

ɨɲɢɛɤɚ

ɢɥɢ

ɨɩɢɫɤɚ

2

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɩɪɢ ɧɚɯɨɠɞɟɧɢɢ ɤɨɨɪɞɢɧɚɬ ɬɨɱɟɤ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

3

23

ɇɚɣɞɢɬɟ

ɧɚɢɦɟɧɶɲɟɟ

ɡɧɚɱɟɧɢɟ

ɜɵɪɚɠɟɧɢɹ

(5X 4Y 3)2 (3X Y 1)2 ɢ

ɡɧɚɱɟɧɢɹ X ɢ, Y ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ ɞɨɫɬɢɝɚɟɬɫɹ.

Ɋɟɲɟɧɢɟ

ɉɪɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ

X ɢ

Y

ɢɦɟɟɦ (5X 4Y 3)2 (3X Y 1)2 t 0. Ɂɧɚɱɟɧɢɟ,

ɪɚɜɧɨɟ

0, ɞɨɫɬɢɝɚɟɬɫɹ ɬɨɥɶɤɨ ɜ ɬɨɦ

ɫɥɭɱɚɟ,

ɤɨɝɞɚ

5X 4Y 3

ɢ

3X Y 1 ɪɚɜɧɵ

ɧɭɥɸ

ɨɞɧɨɜɪɟɦɟɧɧɨ

ɋɨɫɬɚɜɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ

­X5

Y4

3

0;

®X3

Y 1

0.

Ɋɟɲɢɜ ɟɺ, ɩɨɥɭɱɢɦ X

¯

1,

Y

2.

Ɍɚɤɢɦ

ɨɛɪɚɡɨɦ,

ɧɚɢɦɟɧɶɲɟɟ

ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ0,

ɪɚɜɧɨ

ɨɧɨ

ɞɨɫɬɢɝɚɟɬɫɹ ɩɪɢ

X 1, Y 2.

Ɉɬɜɟɬ 0, ɩɪɢ X

1,

Y

2.

Ʉɪɢɬɟɪɢɢ ɨ ɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ

Ȼɚɥɥɵ

Ɂɚɞɚɧɢɟ ɜɵɩɨɥɧɟɧɨ:

ɜɟɪɧɨ

ɜɟɪɧɨ ɧɚɣɞɟɧɨ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ

4

ɢ ɡɧɚɱɟɧɢɹ X ɢ Y, ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ ɞɨɫɬɢɝɚɟɬɫɹ

Ɋɟɲɟɧɢɟ

ɞɨɜɟɞɟɧɨ

,ɞɨ

ɤɨɧɰɚ

ɧɨ

ɞɨɩɭɳɟɧɚ

ɨɲɢɛɤɚ

ɢɥɢ

ɨɩɢɫɤɚ

3

ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɞɚɥɶɧɟɣɲɢɟ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ

Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ

0

Ɇ ɤɫ ɦ ɥɶɧɵ ɛ ɥɥ

4

Ɇɚɬɟɦɚɬɢɤɚ 9 ɤɥɚɫɫ ȼɚɪɢɚɧɬ 3

3

ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ,. ɇɚɣɞɢɬɟ ABC ɟɫɥɢ KCB

20 .