Решение задания для выпускного экзамена по алгебре и началам анализа (11-класс 28-задание)

28-nji iş. Çep tarap
Aňlatmany ýönekeýleşdiriň:
32-m2+2m2-4∙m-22-5mm+2 =32-m2+2(m-2)(m+2)∙m-22- - 5mm+2 = 3m+6+2m-42-m2(m+2) · m-22 - 5mm+2 = 5m+2m+2 - 5mm+2 = 2m+2 ;
Deňlemäni çözüň:
2sin3xcosx=3-2cos3xsinx ; 2sin3xcosx+2cos3xsinx=3 ;2sin3x+x = 3 ; sin4x = 32 ; 4x = (-1)k · π3 +πk , k€Z.
x = (-1)k · π12 + πk4 ; , k€Z.
Deňsizligi çözüň:
log13x-1+2log3(x-1)>1; log3(x-1)log313 +2log3(x-1)>1;
x-1 > 0; x > 1; -log3x-1+2log3(x-1)>1; log3(x-1)>1;
x-1 > 3; x > 4; x€(4; +∞);
4. Uzynlygy 60 m bolan töwerek boýunça iki nokat şol bir ugra deňölçegli hereket edýär. Doly aýlawy olaryň biri beýlekisinden 5 sekunt tiz geçýär. Şunlukda, her minutda biri beýlekisiniň yzyndan ýetýär. Her bir nokadyň tizligini tapyň.
Goý, nokalaryň tizlikleri x m/sek we y m/sek bolsun. Onda,
60x = 60y – 5; 50x – 60y = 60; x = 1 + y; 601+y = 60-5yy ; 60y – (60 – 5y)(1+y) = 0 ; 60y + (5y2 – 55y – 60) = 0; 5y2 +y – 12= 0;
y1 = -4; y2 = 3; y=y2 = 3 ; x = 1+y = 1+3 = 4; Jogaby: 4m/sek we 3 m/sek;
5. Gutuda 6 sany ak şar we 4 sany gara şar bar. Gutudan yzly-yzyna 2 sany şar çykarýarlar. Şol şarlaryň ikisiniň hem ak bolmagynyň ähtimalygyny tapyň.
P = C62C102 = 5!2!4!10!2!8! = 5 ·39 ·5 = 13 ; Jogaby: 136071968139948 y=2x00 y=2x6. Berlen çyzyklar bilen çäklenen figuranyň meýdanyny hasaplaň:
6000277114935003283903205584y=12x00y=12x406256811176000359568047994
00
y=, y=2x, y=4;
S=-20(4-12x)dx +-20(4-2x)dx =
= (4x- 12xln⁡(12) ) │0-2 +(4x- 2xln⁡2 ) │20 =
=( - 1-ln2 + 8 + 4-ln2 )+( 8 – 4-ln2 +1-ln2)== 16 - 6- ln⁡2 ;
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7. Apofemasy 43 dm deň bolan dogry üçburçly piramidanyň iň uly göwrümini tapyň.
HD=43 dm; SABC = 34 BC2 ;
AH= 32 BC ; OH=123 BC ;
OD=(43)2-(123 BC)2 =
= 242- BC223 ; BC= x ; x€(0; 24);
V(x) = 13 SABC · OD=
= 13 · 34 · x2 · 123 242- x2 =
=x224 576- x2 ;
Vˊ(x) = 124 (2x576- x2 - x3576- x2 )= x(1152-3x2)24576- x2 = 0; Vˊ(x) = 0;
X(1152-3x2)= 0; x1 = 0; x2 = 11523 =2423 ; x3 = - 86 ;
Vmax(x) = V(x2) = V3(x) = 124 · 11523 ·576- 11523 = 24·23 · 5763 =
= 16 · 243 = 1283 ; Jogaby: 1283 ;
28-nji iş. Sag tarap
Aňlatmany ýönekeýleşdiriň:
23-n2+3n2-9∙n-32-5nn+3 = 23-n2+3(n-3)(n+3)∙n-32- 5nn+3 = 2(n+3)+3(n-3)n-32(n+3) · m-32 - 5nn+3 = 5n-3n+3 - 5nn+3 = 3n+3 ;
Deňlemäni çözüň:
2cos3x∙cosx=2sinx∙sin3x-1; 2cos3x∙cosx-2sinx∙sin3x=1;
2cos3x+x = 1 ; cos4x = - 12 ; 4x = ±2π3 +2πk , k€Z.
x = ±π6 + πk2 ; , k€Z. Jogaby: x = ±π6 + πk2 ;
Deňsizligi çözüň:
2log2x+2+log12x+2>1; 2log2x+2-log2x+2>1; log2x+2>1; x+2>2; x>0; x+2>0; x > - 2; x€(0; +∞);
4. Motosikletli we tigirli ýoly durman 2 sagatda geçdiler. Şunlukda, motosikletli tigirliden her bir kilometri 4 minut çalt geçdi. Eger olaryň 2 sagatda geçen ýolunyň tapawudy 40 km bolsa, olaryň her biriniň tizligini tapyň.
Goý, Motosikletiň tizligi x m/sek , tigirliniň tizligi y m/sek bolsun. Onda,
2x – 2y = 40; 1x = 1y – 115; x = 20 + y; 120+y = 15-y15y ; 15y – (y – 15)(20+y) = 0 ; -15y + y2 + 5y – 300 = 0; y2 - 10y – 300= 0;
(y - 5)2 = 325; y = 5 ± 325 = 5 ± 18,02; y1 = 23,02; y2 = - 13,02; y=y1 = 23,02 ; x = 20+y = 20+23,02 = 43,02;
Jogaby: 43,02m/sek we 23,02 m/sek;
5. Gutuda 3 sany ak şar we 7 sany gara şar bar. Tötänden çykarylan iki şaryň gara bolmagynyň ähtimallygyny tapyň.
P = C72C102 = 7!2!5!10!2!8! = 7 ·39 ·5 = 2145 = 715; Jogaby: 7156. Berlen çyzyklar bilen çäklenen figuranyň meýdanyny hasaplaň:
327914092075
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y=13x, y=3x, y=9;
3255963173037y=12x00y=12xS=-20(9-13x)dx +02(9-3x)dx =
503237579375 y=2x00 y=2x= (9x - 3-xln⁡3 ) │0-2 +(9x- 3xln⁡3 ) │20 =
=( 1ln3 + 18 - 93 )+( 18 – 9ln3 +1ln3)== 36 - 16 ln⁡3 ; Jogaby: 36 - 16 ln⁡3 ;
7. Gapdal gapyrgasy 12 sm deň bolan dogry üçburcly piramidanyň iň uly göwrümini tapyň.
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AD=BD=CD=12sm.
AC=x , x€[0; 123 ];
SABC = 34 x2 ; DH2= 122 – ( x2 )2 ;
BH=32 x; OH = 123x ;
OD=(12)2-(x2)2-x212 =
= (12)2-x23 = 13 432-x2 ;
V(x) = 13 SABC · OD=
=13 · 34 x2 · 13 432-x2 =
= 112 · x2432- x2 ;
Vˊ(x) = 112 (2x432- x2 - x3432- x2 )= x(864-3x2)12432- x2 = 0; Vˊ(x) = 0;
x(864-3x2)= 0; x1 = 0; 964- 3x2 =0; x2 = 288 ; x2=122 ; x3 = - 122 ;
Vmax(x) = V(x2) = V3(x) = 112 · 288 ·432- 288 = 24144 = 24·12=288;
Jogaby: 288 ;