РЕШЕНИЯ ЗАДАНИЯ ДЛЯ ВЫПУСКНОГО ЭКЗАМЕНА ПО АЛГЕБРЕ И НАЧАЛАМ АНАЛИЗА 11-класс 7-задание

7-nji iş. Çep tarap
1.Aňlatmany ýönekeýleşdiriň:
m- 32-m- 52m- 52-m-3-m12+m-1m- 12+m-1 = m-2(m 12-m- 12)m-2(m- 12-m-1) -m12+m-1m- 12+m-1 =
= (m 12-m- 12)(m-12+m-1)-(m 12+m-1)(m- 12-m-1)m-1-m-2 =
= 1+m- 12-m-1-m- 32-(1-m- 12+m- 32-m-2)m-1-m-2 = (2m12-1)(m-1-m-2)m-1-m-2 = 2m -1;
Jogaby: 2m -1;
2. Deňlemäni çözüň:
log3x+8-1=log32-log3x-8; log3x+83 = log32x-8 ;
x+8>0; x-8>0, x>0; x+83= 2x-8 ; x2+64 =6; x2=100;
x1=-10, x2=10 x=10; Jogaby: x=10;
3.x2-x-6-x2-3<0 deňsizligiň 1<x<5 şerti kanagatlandyrýan bitin çözüwlerini tapyň.
1<x<5; => x=2,3,4;
5120640792300023647407888500x=2, x2-x-6-x2-3 = 22-2-6-22-3= -4-7 < 0; x=3, x2-x-6-x2-3 = 32-3-6-32-3= 0-12 <0 ;
4. Ýata suwda tizligi 20kmsag deň bolan gaýyk akymyň ugruna 22 km we akymyň garşysyna 36 km aralygy 3 sagatda geçdi. Derýanyň akyş tizligini tapyň.
Gaýygyň ýata suwdaky tizligi 20km/sag. Derýanyň akyş tizligi x km/sag.
2220+x + 3620-x = 3; 2220-x+36(20+x)202-x2 = 3; 3(400-x2)=14x+1160;
3x2+14x-40=0; 3(x-2)(x + 203 )=0; x1=2, x2= - 203 , x2<0 => x=x1=2;
Jogaby: 2km/sag.
5. Toždestwony subut ediň:
1-2 cos2α1+sin2α=1-ctgα1+ctgα ; cos2α+sin2α-2cos2αcos2α+sin2α+2sinαcosα = sin2α-cos2α(cosα+sinα)2 =
= (sinα+cosα)(sinα-cosα)(sinα+cosα)2 = sinα-cosαsinα+cosα = 1-ctgα1+ctgα ; Subut edildi.
6. Integraly hasaplaň:
013x2 dx=01x23dx= x23+123+1 |10 = 35 3x5 |10 = 35 ; Jogaby: 35 ; 7. fx= 12x egriniň üstünde koordinatalar başlangyjyna iň golaý nokady tapyň.
A(x,y), O(0,0) min AO üçin A(x,y) nokady tapmaly.
L(x)=AO= x2+y2 = x2+14x = 4x3+12x; x €(0; +∞);
L(x)= 4x3+12x ; Lˊ(x)= 6x24x3+1 · 2x- 4x3+1 · 1x4x =
= 14x (12x2x4x3+1 -4x3+1x )= 12x3-4x3-14xx(4x3+1) = 8x3-14x(4x4+1) ;
L(x)=0; => x= 12; y= 1212 = 22; minAO= L( 12 )= 32 ; => A(12; 22 );
Jogaby: A(12; 22 ) nokat.7-nji iş. Sag tarap
Aňlatmany ýönekeýleşdiriň:
n- 54+n-2n- 74+n-2-n- 12-n- 32n-1+n- 32 =
= n- 94+n- 114+n-3+n- 72-(n- 94-n- 134+n- 52-n- 72 )n- 114+n- 134+n-3+n- 72= = (n- 114+n-3+2n- 72+n- 134-n- 52 )·n3(n- 114+n- 134+n-3+n- 72 )·n3 =
= (n 14+1+2n- 12+n- 14-n- 12 )(n 14+n- 14+1+n- 12 ) = (n-12+1)(2-n14)(n 14+1)(n- 12+1)(n 14+1 ) =
= 2 - n 14 = 2-4n ; Jogaby: 2-4n 2. Deňlemäni çözüň:
log2x+1-1=log23-log22x+3 ; x0+1>2x+3>0, x € ( - 1; +∞);
log2x+12 = log2 32x+3 ; x+12=32x+3; (x+1)(2x+3) =6;
2x2+5x+3=36; 2x2+5x-33=0; 2(x-3)(x+112 )=0; x1= 3; x2= - 112;
x2 € (-1; +∞), x = 3;Jogaby: x = 3;
3. -x2-2x+3x2+5<0 deňsizligiň 0<x<3 şerti kanagatlandyrýan bitin çözüwlerini tapyň.
0<x<3, x = 1,2; x = 1 , -12-2·1+312+5 = 06 < 0;
x2= 2, -22-2·2+322+5 = -59 <0; => x=2; Jogaby: x = 2;
4. Balykçy gaýykly derýada akymyň garşysyna 6 km we kölde 15 km aralygy geçdi. Özem kölde geçen ýoluna derýada geçen ýoluna garanda 1 sagat köp sarp etdi. Derýanyň akyş tizligi 2kmsag deň. Gaýygyň köldäki tizligini tapyň.
Çözülüşi: Gaýygyň köldäki tizligini x km/sag.
6x-2 = 15x – 1; 6x = (15 – x)( x –2); 6x+x2-17x+30=0;
(x-6)(x-5)=0; x1=6, x2=5; Jogaby: 6 km/sag ýa-da 5 km/sag.
5. Toždestwony subut ediň:
1-2 cos2α1-sin2α=1+ctgα1-ctgα ; 1-2 cos2α1-sin2α = cos2α+sin2α-2 cos2αcos2α+sin2α-2sin αcosα =
= cos2α+sin2α(sinα-cosα)2 = (sinα-cosα)(sinα+cosα)(sinα-cosα)(sinα-cosα) = sinα+cosαsinα-cosα =
= sinα+cosαsinαsinα-cosαsinα = 1+ctgα1-ctgα ; Subut edildi.
6. Integraly hasaplaň:
28 dxx=28x- 13dx= x- 12+112+1 |82 = 2x 12 |82=2( 8 12 - 2 12 )=2·2 12 =22;
Jogaby: 27. fx= x2 - 1 egriniň üstünde koordinatalar başlangyjyna iň golaý nokady tapyň.
A(x,y), O(0,0) minAO xcin A(x,y) nokady tapmaly.
L(x)= AO= x2+(x-1)2 = x4-x2+1 ; x € (- ∞; +∞);
Lˊ(x)= 2x3-xx4-x2+1 ; Lˊ(x)=0; 2x3-x= x(2x2-1)= 0; x1=0, x2= 12 ; x3= - 12L(0)= 1; L(12)= L(- 12)= 12+ 14 = 32 ; min(- ∞; +∞)L(x)= L(x2)= =L(x3)= 32 ; y2=x22 – 1= 12 – 1 = - 12 , y3=x32 – 1= 12 – 1 = - 12;
Gözlenýän nokatlar (12 ; - 12 ) we (- 12 ; 12 ); Jogaby: (12 ; - 12 ) we (- 12 ; 12 );