Решение задания для выпускного экзамена по алгебре и началам анализа (11-класс 17-задание)

17-nji iş. Çep tarap
Hasaplaň:
12-1-20,2∙1-20,52-0,3 = 2+1(2-1)(2+1)-20,5(1-20,5)= 2+1- 2+2=3;Deňlemeler sistemasyny çözüň:
y2-3xy+x2-x+y+9=0,y-x=2. => (x-2)2-(x-y)-xy+9=0,y-x=2. =>
=> 4+2-xy+9=0,y-x=2. => xy=15 ,y-x=2. => x(x+2) = 15; x2 + 2x = 15;
x2 + 2x + 1 = 16; (x + 1)2 = 16; x1,2 +1 = ± 4; x1,2 = - 1± 4; =>
=> x1= - 5; x2 = 3; => y1= - 3; y2 = 5; Jogaby: (-5; -3); (3; 5);
Deňsizligi çözüň:
2log2x-3logx4-4≤0; log2x = t; 2t – 6 ·1t – 4 ≤ 0;
2t2 – 6 - 4t ≤ 0; 2t2 +2t –6t - 6 ≤ 0; (2t – 6)(t+1)≤ 0;
( t – 3)( t + 1) ≤ 0 ; t€[- 1; 3]; => x€[ 12 ; 8 ]; Jogaby: x€[ 12 ; 8 ];
4. Arifmetik progressiýanyň birinji agzasy 213-e, tapawudy -29-ä deň. -1 şol progressiýanyň agzasy bolarmy?
Berlen: a1= 213-e; d = -29 ; -1€{ak}-? Goý, ak = -2 bolsun.
-1 = a1+(k-1)d = 213-e + (k-1) · (-29 ); -1 = 73-e -29 (k-1) ;
-9 = 21 – 9e – 2k + 2; 2k = 32 – 9e; k = 16 - 92e => bolup bilmez
sebäbi e – irrasional san, k- bitin. Jogaby: bolup bilmez
5. Eger cos2α=-35, π<α<3π2 bolsa, tg α- ny tapyň.
Çözülüşi: 2sin2a=1-cos2a=1--35 = 85 ;2cos2a=1+cos2a=1- 35= 25 ;tg2a = ( sinacosa )2 = 8525 = 4; tg a = 2 ; Jogaby: tg a = 2 ;
6. Berlen çyzyklar bilen çäklenen figuranyň meýdanyny hasaplaň:
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y=x2-4x, y=0, x=-32 ;
S=- 32 0( y1(x) - y2(x))dx =
- 32 0( x2-4x )dx = ( x33 - 2x2 ) │0-32 =
= 0 + 13 · ( 32 )3 + 2 · (- 32 )2 =
= 98 + 92 = 9 · ( 18 + 48 ) = 458 ; Jogaby: 458 ;
7. М (-1 ; 0 ) nokat arkaly y=2x-1 funksiýanyň grafigine geçirilen galtaşýan çyzygyň deňlemesini ýazyň.
F(x)= y0 + yˊ(x0)(x-x0) => yˊ(x0) = 12x0-1 ; 0 =2x0-1 + 12x0-1 (-1 - x0) =>
X0 = 2 ; y0 = 3 ; yˊ(x0) = 13 ; => F(x) = 3 + 13 (x-2) ;
Jogaby: F(x) = 3 + 13 (x-2) ;

17-nji iş. Sag tarap
Hasaplaň:
30,9:30,41-30,5+21-3 = 30,9 · 1-30,530,4+2(1-3(1-3)(1-3) = 30,5 · 3 – 1 - 3 = - 4;
Deňlemeler sistemasyny çözüň:
x+y=3,x2+3xy+y2-x-y=2. => x+y=3,(x-y)2+xy-(x+y)=2. =>
=> x+y=3,xy=-4. => x2+3x-4=0; (x – 4)(x + 1) = 0;
x1 = - 1; x2 = 4; => y1 = 4; y2 = - 4; Jogaby: (-1; 4); (4; -1);
Deňsizligi çözüň:
2logx9-log3x-3≥0; log3x = k; 2 · 2 · 1k – 3 ≥ 0;
4 – k2 – 3k ≥ 0; k2 + 3k – 4 ≥ 0; (k+4)(k-1) ≤ 0; =>
=> k€[ -4; 1]; => x€[ 181; 3]; Jogaby: x€[ 181; 3];
4. Arifmetik progressiýanyň birinji agzasy -212-e, tapawudy 34-e deň. 3 şol progressiýanyň agzasy bolarmy?
Berlen: a1= -213-e; d = 34 =e ; 3€{ak}-? Goý, ak = 3 bolsun.
ak = a1+(k-1)d = 52-e + (k-1) · ( 34 -e); 3 = - 52-e + ( 34 -e)x+e- 34 ;
( 34 -e)x = 134 + 3; x = 134 + 334 -e ∄Z; => 3∄{ak} agzasy bolup bilmez.
Jogaby: bolup bilmez
5. Eger sin 2α=35, π2 <α< π bolsa, tg α- ny tapyň.
cos2a = 1-sin22a = 1-925 = 45; sin2a = 1-cos2a2= 1-452= 110;
cos2a = 1+cos2a2 = 1+452= 910; tg2a = sin2a cos2a = = 110 910 = 19; =>
=> π2<α< π => tga = - 13; Jogaby: tga = - 13;
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6. Berlen çyzyklar bilen çäklenen figuranyň meýdanyny hasaplaň:
y=3x-x2, y=0;
S=0 3( y1(x) - y2(x))dx =
0 3( 3x-x2 )dx = ( 3 x22 - x33 ) │30 =
= 3 · 92 - 273 = 272 - 9 = 92 ; Jogaby: S = 92 ;
7. М (2 ; 0 ) nokat arkaly y=1-x funksiýanyň grafigine geçirilen galtaşýan çyzygyň deňlemesini ýazyň.
F(x)= y0 + yˊ(x0)(x-x0) => yˊ(x0) = -121-x0 ;
0 =1-x0 +(-1)21-x0 (2 - x0) => 0 = 21-x0-2+ x021-x0 ;
2 - 2x0-2+ x0=0 ; x0 = 0 ; y0 = 1-x0 = 1;
yˊ(x0) = - 12 ; => F(x) = 1 - 12 x ;
Jogaby: F(x) = 1 - 12 x ;