РЕШЕНИЯ ЗАДАНИЯ ДЛЯ ВЫПУСКНОГО ЭКЗАМЕНА ПО АЛГЕБРЕ И НАЧАЛАМ АНАЛИЗА 11-класс 2-задание

2-nji iş. Çep tarap
1.Aňlatmany ýönekeýleşdiriň:
1-a- 121+a-1+1+a-11-a- 12∙a-1a+1 = 1-2a- 12+a-1+1+2a- 12+a-11-a-1 ∙a-1a+1 =
= 2(1+a-1)1-a-1 ∙a-1a+1 = 2(a+1)a-1 ∙a-1a+1 = 2; Jogaby: 2;
2. Deňlemeler sistemasyny çözüň:
x2+xy=2,y-3x=7. x2-x7+3x=2; x2+7x+3x2-2=0;4x2+7x-2=0; x1= -7+49+328 = -7+818 = -7+98 = 28 = 14;
x2 = -7-98 = -168 = -2; y1= 7+3x1= 7+34 = 314; y2= 7+3x2 = 7(-2) = 1;

Jogaby: (1 4; 314 ); ( -2; 1 );
3.Deňsizligi çözüň:
log2x-1>log1212x-3. log2x-1> log22x-3; =>
=> x-1> 2x-3; x-2 < 0;
x-2 < 0; x-1>0;2x-3>0; x<2 ; x>1;3x>32; x€( 32;2 );Jogaby: x€( 32;2 );4. Abssissalar okunyň üstünde M1;-4 we N3;2 nokatlardan deňdaşlaşan nokady tapyň.
Berlen: M(1;-4) we N3;2 ; L( 0;y ); ML=LN; L(x;0) nokady tapmaly.
ML=(x-1)2+42 ; LN=(3-y)2+22 ;
(x-1)2+42 =(3-y)2+22; x2-2x+17 =x2-6x+13;
x2-2x+17=x2-6x+13; -2x+6x= 13-17; 4x = -4; x=- 1; L(-1;0);
Jogaby: L(-1;0); nokat.
5. Toždestwony subut ediň:
tg2α – sin2α = tg2 α ∙ sin2 α.
Subudy: tg2α – sin2α = sin2αcos2α - sin2α = sin2α- cos2αsin2αcos2α = sin2α(1-cos2α) cos2α=
= sin2αcos2α ·(1-sin2α) = tg2α · sin2α; Subut edildi.
6. fx=ex -1x funksiýa üçin grafigi M(1; 3e) nokatdan geçýän asyl funksiýany tapyň.
F(x)=fxdx=( ex -1x )dx= ex - ln+C; F(1)= 3e ;
F(1)= e-ln1+ C=3e; C=2e; F(x)= ex -lnx+e2;
Jogaby: F(x)= ex -lnx+e2;
7. Birinji goşulyjynyň kwadratynyň bäş essesi bilen ikinji goşulyjynyň kubunyň jemi iň kiçi bolar ýaly edip, 20-ni iki položitel goşulyjynyň jemi görnüşinde ýazyň.
x+y=20; y=20-x; 5x2+y3 ; san iň kiçi. x€(0; 20);
S(x)= 5x2+y3 = 5x2+ (20-x) 3 ; Sˊ(x)= 10x- 3(20-x) 2=
=10x- 3(400-40x+x2) ; Sˊ(x)=0; 3x2-130x+1200=0;
x1= 130+16900-144006 = 130+25006 = 130+506 = 1806 = 30; x1€(0; 20);
130-25006 = 130-506 = 806 = 403 ; x2€(0; 20);
x = 403; y = 20 – x = 20 - 403 = 203; Jogaby: 403 we 2032-nji iş. Sag tarap
1.Aňlatmany ýönekeýleşdiriň:
1+b-1b- 12-1+b- 12-11+b-1∙b-1b+1 = 1+2b- 12+b-1+b-1-2b- 12+1b-1-1 ∙b-1b+1 =
= 2(b-1+1)b-1-1 ∙b-1b+1 = 2(1+b)1-b ∙b-1b+1 = - 2; Jogaby: -2;
2.Deňlemeler sistemasyny çözüň:

x2-xy-y2=19,x-y=7. x2-xx-7-x-72=19; -x2+21x-49-19=0;x2-21x+68=0; x1= 21+441-2722 = 21+1692 = 21+132 = 342 = 17;
x2 = 21-132 = 82 = 4; y1= x1- 7 = 17- 7 = 10; y2= x2- 7 = 4- 7 = -3;

Jogaby: (17; 10); (4; -3);
3.Deňsizligi çözüň:
log32x-1>log1313x-4 ; log32x-1> log33x-4; =>
=> 2x-1> 3x-4; x-3 < 0;
x-3 < 0; 2x-1>0;3x-4>0; x<3 ; x>12;3x>43; x€( 43;3 );Jogaby: x€( 43;3 );4. Ordinatalar okunyň üstünde P-3;2 we K(4;3) nokatlardan deňdaşlaşan nokady tapyň.
Berlen: P(-3;2) we K(4;3); L( 0;y ); PL=LK; L(0;y) nokady tapmaly.
PL=32+(y-2)2 ; LK=42+(3-y)2 ; 9+ (y-2)2 =16+ (3-y)2 ;
y2-4y+13=y2-6y+25; y2-4y-y2+6y= 25-13; 2y = 12; y=6; M(0;6);
Jogaby: M(0;6);
5. Toždestwony subut ediň:
ctg2α – cos2α = ctg2 α ∙ cos2 α.
Subudy: ctg2α – cos2α = cos2αsin2α - cos2α = cos2α- sin2αcos2αsin2α = cos2α(1-sin2α) sin2α=
= cos2αsin2α ·(1-sin2α) = ctg2α · cos2α; Subut edildi.
6. fx=1x+ex funksiýa üçin grafigi M (1; 2e) nokatdan geçýän asyl funksiýany tapyň.
F(x)=fxdx=( 1x+ex )dx= ln+ex +C; F(1)= 2ex ;
F(1)= ln1+e+C=e+C; C=e; F(x)=lnx+ex +e; Jogaby: F(x)=lnx+ex+e;
7. Birinji goşulyjynyň kuby bilen ilkinji goşulyjynyň ikeldilen kwadratynyň jemi iň kiçi bolar ýaly edip, 5-i iki položitel goşulyjynyň jemi görnüşinde ýazyň.
x+y=5; S=x3+2y2 ; y=5-x; x€(0; 5);
S(x)= x3+2(5-x) 2 ; Sˊ(x)= 3x2+4(5-x)·(-1)= 3x2+4x-20=0; Sˊ(x)=0;
3x2+4x-20>0; 3( x-2)(x+103 )=0; x1=2; x2= - 103 ; x1€(0; 5); x2€(0; 5);
x=2; y=5-x=5- 2=3; Jogaby: 2 we 3.