РЕШЕНИЯ ЗАДАНИЯ ДЛЯ ВЫПУСКНОГО ЭКЗАМЕНА ПО АЛГЕБРЕ И НАЧАЛАМ АНАЛИЗА 11-класс 4-задание

4-nji iş. Çep tarap
1.Aňlatmany ýönekeýleşdiriň:
ab∙4aa+24a-1b2-a2+4a2-4 = a12b-12a-14(a+2)a- 14b12 -a2+4a2-4 = aa+2 - a2+4a2-4 =
= a2-2a-a2-4a2-4 = -2(a+2)a2-4 = - 2a2-4 ; Jogaby: - 2a2-4 ;
2.Deňlemeler sistemasyny çözüň:
xy-y2=-2, 2x2-xy-y2=-4. X= 1y (y2-2); 1y2(y2-2)2-2(y2-2) - y2=-4;
-8 + 8y2+2=- 4; =8y2 =2; y2=4; y1= 2 ; y2= -2 ;
x1= 1y1(y12-2 )= 12 (4-2)=1 ; x2 = - 1 ;
Jogaby: (1,2); ( -1, -2);
3.Deňsizligi çözüň:
log2x2-7x+6≤1+log27; log2x2-7x+6≤log214;
x2-7x+6>0;x2-7x+6≤14; (x-6)(x-1)>0;(x+8)(x+1)≤0;
x€(-∞;1)∪(6;+∞);x€-∞;1; x€[-1;1∪6;8]; Jogaby: x€[-1;1∪6;8];4. Gönüburçly üçburçlugyň katetleriniň jemi 23 sm, meýdany 60 sm2 deň. Gönüburçly üçburçlugyň katetlerini tapyň.
a+b=23sm, 12ab= 60sm2; ab -?
a+b=23ab= 120 a(23-a)=120; a2-23a+120=0;
a1= 23+529-4802 = 23+72 = 15; a2= 23-529-4802 = 23-72= 8;
a1=15; a2= 8; Jogaby8 sm we 15sm;
5. Sanlaryň köpeltmek hasylyny tapyň:
z1 = 1 + i we (и) z2 = 8 cos3π8+isin3π8.
Z1 · Z2 = (1 + i ) 8cosπ3+isinπ3 = (1 + i ) 8cos3π412+isin3π412=
= (1 + i ) 81-222+i 1+222 = (1 + i ) 82-22 +i 2-22 =
= 2(1 + i ) (2-2 + i2+2 ) = ( 1 + i )( 2+i6 ) =
= 2-6 + i( 2+i6);
Jogaby: 2-6 + i( 2+i6);
6. Integraly hasaplaň:
π23π2sin2x2dx=π23π212 1-cosxdx= 12(x-sinx)||2π/23π/2 =
= 12( 3π2+1- π2+1 ) = π2+1 ; Jogaby: π2+1 7. Goşulyjylaryň biri beýlekisinden 6 esse kiçi, olaryň üçüsiniň köpeltmek hasyly bolsa iň uly bolar ýaly edip, 21-i üç položitel goşulyjynyň jemi görnüşinde ýazyň.
x+y+z = 21; x = 6y, p = xyz; p(x)=xyz = x· x6 · ( 21 - 7x6) ;
x€[0; 18]; pˊ(x)= x3 ( 27 - 7x6 ) - 76 ·x26 = 7x - 21x236 = 7x( 1 - x12 );
pˊ(x)=0; x1=0; x2= 12;
P(0)=p(18)=0; max[0;18]p12=12·2·7 =168;
Jogaby: 12+2+7 = 21..
4-nji iş. Sag tarap
Aňlatmany ýönekeýleşdiriň:
3a2b∙6bb+36a4b-3-b2+9b2-9. 3a2b∙6bb+36a4b-3-b2+9b2-9 = a23b-13b-16(b+3)a23b-12- b2+9(b-3)(b+3) =
= bb+3- b2+9(b-3)(b+3) = b2-3b-b2-9(b-3)(b+3) = -3(b+3)(b-3)(b+3) =
= - -3b-3) ; Jogaby: - -3b-3) ;
Deňlemeler sistemasyny çözüň:
xy+y2=4,x2-2xy-y2=2. x= 1y(4-y2); 1y2(4-y2)2-2(4-y2)- y2=2;
=16y2 -16+2y2=2; =8y2 -9+y2=0; y ≠ 0; ( y - 8y )( y- 1y ) = 0;
y = 8y = 0; y1= 8 ; y2= -8 ; y = 1y = 0; y3= 1; y4= - 1;
x1= 1y1(4-y12 )= - 48 = - 2 ; x2 = 2 ; x3 = 3; x4 = - 3;
Jogaby: (- 2,8 ); (2, -8 ); (3,1); ( -3, -1);
3.Deňsizligi çözüň:
log0,5-x2+9x-14≥log0,53-1.log0,5-x2+9x-14≥log0,56 ; -x2+9x-14 > 0;
-x2+9x-14>0;-x2+9x-14 ≤6; x2-9x+14>0;x2-9x+20 ≤0; (x-2)(x-7<0;x-4x-5≥0;x€ 2;7; x€( -∞;4)∪( 5; +∞); x€( 2;4)∪( 5;7);
Jogaby: x€( 2;4)∪( 5;7);
4. Gönüburçly üçburçlugyň katetleriniň biri gipotenuzasyndan 3 sm, beýlekisi 6 sm kiçi. Üçburçlugyň gipotenuzasyny tapyň.
a2+b2= c2 ; a= c-3; b = c-6; (c-3)2+(c-6)2 =
= 2c2 – 18c + 45 = c; c2 – 18c + 45 = 0; c1= 15; c2= 3; c2 – däl kök.
Jogaby: 15sm.
5. Sanlaryň köpeltmek hasylyny tapyň:

z1 = -1 + i we (и) z2 = cosπ3+isinπ3.
Z1 · Z2 = (-1 + i ) cosπ3+isinπ3 = 2cos13π12+isin13π12=
= - 121+3 +i 121-3;
Jogaby: - 121+3 +i 121-3;
6. Integraly hasaplaň:
0π/2сos2x2dx =0π2 12 1-cosxdx= 12(x+sinx)|π 22 = 12(π 2 + 1) = π 4 + 12 ;
Jogaby: π 4 + 12 ;
7. 54 san üç položitel goşulyjynyň jemi görnüşinde ýazylypdyr. Birinji goşulyjy ikinjiden iki esse uly. Goşulyjylaryň haýsy bahalarynda olaryň köpeltmek hasyly iň uly baha eýe bolar?
x+y+z = 54; x = 2y, p = xyz; p(x)=xyz = x· x2 · ( 54 - x - x2) =
= x22 ( 54 - 3x2 ); x€[0; 36]; pˊ(x)=x( 54 - 3x2 ) - 34 x2= - 94 x2+54x =
= 9x( -- x4 + 6); pˊ(x)=0; x1=0; x2= 24; x1€[0; 36]; x2 €[0; 36];
P(0)=p(36)=0; max[0;36]p24=24·12·18 =5184;
Jogaby: 24; 12; 18.