Решение задания для выпускного экзамена по алгебре и началам анализа (11-класс 19-задание)

19-njy iş. Çep tarap
Hasaplaň:
6-4∙5160-2+23-1-322∙cos3π4 = (6-4)-2 +( 32 ) - 322 · ( - 22 )=
= (6-4)-2 +( 32 ) - 322 · ( - 22 )= 14 + 32 + 34 = 104 = 52 ; Jogaby: 52 ;
Deňlemäni çözüň:
2x-3x-1=3x-2x2x ; x ≠ 0; x ≠ 1;
2x-3x-1= 3-2x ; 2x-3 = 3x-2x2 – 3 + 2x; x(3-2x)= 0;
x1 ≠ 0; x2= 32 ; Jogaby: x = 32 ;
Deňsizligi çözüň:
3∙93x2+2x-272x>0; 31+ 6x2+4x> 36x ; 1+ 6x2+4x> 6x ;
6x2+4 - 6x2 + x > 0; 4 + x > 0; x > - 4; => x€(-4; +∞); => x€(0; +∞);
Jogaby: x€(0; +∞);
4. Sany ilki 15% artdyrdylar, soňra alnan netijäni ýene 10% artdyrdylar. Netijede ilkibaşdaky sandan 530 san uly bolan san alyndy.Başdaky san näçä deň? Goý, x berlen san.1). x + 15x100 = x + 3x20 = 20x+3x20= 23x20 ; 2). 23x20 + 23x20 · 10100 = 230x200 + 23x200 = 253x200;
3). x + 530 = 253x200 ; => 200x+530 · 200 = 253x ; 530 · 200 = 53x;
x = 530 ·20053 = 10600053 = 2000; Jogaby: 2000;
5. Toždestwony subut ediň:
2sinα-ctgα2=tg α2.
Subudy: 2sinα-ctgα2= 22sina2 cosa2-cosa2sina2-= 1-cos2a2sina2 cosa2 =
= sin2a2sina2 cosa2 = tg α2 ; Subut edildi.
6. Berlen çyzyklar bilen çäklenen figuranyň meýdanyny hasaplaň:
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y = (x – 3)2, y = 9 – 2x.
S = 04(y1x- y2x)dx =
= 04(9 – 2x- (x – 3)2))dx =
=04(9 – 2x-x2 +6x –9))dx =
= 04(4x-x2))dx = (2x2 - x33 )│40 =
= 2 · 16 - 643 = 96- 643 = 323 ;
Jogaby: 323;
7. Funksiýanyň grafigi abssissa oky bilen haýsy burç astynda kesişýär:
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y=x-1x+2 ;
yˊ = x+2 + x-12x+2 ;
tga = yˊ(x0) = 3 ; a = π3 ;
Jogaby: π3 ;
19-njy iş. Sag tarap
Hasaplaň:
3+2∙320-1+53-1-252∙sin5π4 = 3+2-1 + 36 + 252 · 12 =
= 45 + 15 = 1; Jogaby: 1;
Deňlemäni çözüň:
6x-8x-2=4x-3x2x ; x(6x-8) = x(x-2)(4-3x); 2x(3x-4) + x(x-2)(3x-4) = 0;
x(3x-4)(2+x-1) = 0; x2(3x-4) = 0 ; x1 = 0; x2 = 43 ; Jogaby: 43 ;
Deňsizligi çözüň:
2∙82x2+1x-43x<0 ; 2∙26x2+3x<26x ; 6x2+3+xx < 6x;
6x2+3+x-6x2x < 0; 3+xx < 0; Jogaby: (-3; 0);

4. Sany ilki 22% kiçeltdiler, soňra alnan sany 25% artdyrdylar. Netijede ilkibaşdaky sandan 5,5 san kiçi bolan san alyndy. Başdaky sany tapmaly?
(x+22%x)+(x+22%x)·25% = x-5.5;
(x+ 1150x)(1+ 14 ) = x – 5.5; x(1+ 1150)· 54 = x – 5.5; x· 6150 · 54 = x – 5.5;
x· 6140 - x = - 5.5; x· ( 6140 - 1) = - 5.5; x· 2140 = - 5.5; x= - 10.5;
5. Toždestwony subut ediň:
2sinα-tgα2=ctg α2 ; 2sinα-tgα2=1+tg2a2tgα2 - tgα2 = 1+tg2a2 - tg2a2 tgα2 = 1 tgα2 = ctg α2 ;
6. Berlen çyzyklar bilen çäklenen figuranyň meýdanyny hasaplaň:
y = (x+2)2, y =4–x; 4–x =(x+2)2 ; x2+4x+4 = - x+4; x2+5x=0; x1=0; x2=-5;
-50[x+2)3-4+xdx = ( (x+2)33 – 4x+ x22 ) │-50 = 333 +20+252- 233 =
= 11 +252- 83 = 66+75-166 = 1256 ;
7. Funksiýanyň grafigi abssissa oky bilen haýsy burç astynda kesişýär:
y=x3-x ; yˊ(x)= - x23-x ; y=0;y=x3-x ; x1=0; 3-x =0; x2=3;
a) x0 = 0; bolanda yˊ(0)= 3-x -0 = 3 ; tga = 3 ; a = π3 ;
b) x0 = 3; bolanda yˊ(3)= 3-3 -323-3 = 0- ∞ = - ∞ ;
tga = - ∞; a = 3π2 ; Jogaby: π3 we 3π2 ;