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Question Number 179852    Answers: 0   Comments: 4

prove ∫_0 ^(𝛑/2) log(sinx)dx=(Ο€/2)log(1/2)

$$\boldsymbol{{prove}}\:\:\:\:\:\int_{\mathrm{0}} ^{\frac{\boldsymbol{\pi}}{\mathrm{2}}} \boldsymbol{{log}}\left(\boldsymbol{{sinx}}\right)\boldsymbol{{dx}}=\frac{\pi}{\mathrm{2}}\boldsymbol{{log}}\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 179848    Answers: 0   Comments: 1

In how many years ,Rs 50000 amounts to Rs672720 at compounded annually if the interest rate is 1 paisa per rupee per month?

$$ \\ $$$$\:\:\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{years}\:,\mathrm{Rs}\:\mathrm{50000}\:\mathrm{amounts}\:\mathrm{to}\:\mathrm{Rs672720} \\ $$$$\mathrm{at}\:\mathrm{compounded}\:\mathrm{annually}\:\mathrm{if}\:\mathrm{the}\:\mathrm{interest}\:\mathrm{rate}\:\mathrm{is}\:\mathrm{1}\:\mathrm{paisa}\:\mathrm{per} \\ $$$$\mathrm{rupee}\:\mathrm{per}\:\mathrm{month}? \\ $$

Question Number 179844    Answers: 1   Comments: 0

∫^1 _0 ((xβˆ’1)/((x+1)lnx))dx=?

$$\underset{\mathrm{0}} {\int}^{\mathrm{1}} \:\frac{{x}βˆ’\mathrm{1}}{\left({x}+\mathrm{1}\right){lnx}}{dx}=? \\ $$

Question Number 179829    Answers: 1   Comments: 1

Question Number 179819    Answers: 1   Comments: 0

x+y+z=2 x+2yβˆ’z=4 2x+yβˆ’2z=2

$${x}+{y}+{z}=\mathrm{2} \\ $$$${x}+\mathrm{2}{y}βˆ’{z}=\mathrm{4} \\ $$$$\mathrm{2}{x}+{y}βˆ’\mathrm{2}{z}=\mathrm{2} \\ $$$$ \\ $$

Question Number 179818    Answers: 4   Comments: 0

in how many ways can you distribute 20 different books to 5 students such that each student gets at least 2 books?

$${in}\:{how}\:{many}\:{ways}\:{can}\:{you}\:{distribute} \\ $$$$\mathrm{20}\:{different}\:{books}\:{to}\:\mathrm{5}\:{students}\:{such} \\ $$$${that}\:{each}\:{student}\:{gets}\:{at}\:{least}\:\mathrm{2}\:{books}? \\ $$

Question Number 179811    Answers: 1   Comments: 4

Question Number 179869    Answers: 0   Comments: 5

Question Number 179799    Answers: 2   Comments: 0

Question Number 179781    Answers: 2   Comments: 8

Question Number 179777    Answers: 1   Comments: 0

p(x)=(1βˆ’x)(1+2x)(1βˆ’3x)....(1+14x)(1βˆ’15x) the absolute value of the coefficient of the x^2 in the expression?

$${p}\left({x}\right)=\left(\mathrm{1}βˆ’{x}\right)\left(\mathrm{1}+\mathrm{2}{x}\right)\left(\mathrm{1}βˆ’\mathrm{3}{x}\right)....\left(\mathrm{1}+\mathrm{14}{x}\right)\left(\mathrm{1}βˆ’\mathrm{15}{x}\right) \\ $$$$ \\ $$$${the}\:{absolute}\:{value}\:{of}\:{the}\:{coefficient}\:{of}\:{the}\:{x}^{\mathrm{2}} \:{in}\:{the}\:{expression}? \\ $$

Question Number 179775    Answers: 4   Comments: 0

(122βˆ’a)^2 = (123βˆ’a)^2

$$\:\left(\mathrm{122}βˆ’{a}\right)^{\mathrm{2}} =\:\left(\mathrm{123}βˆ’{a}\right)^{\mathrm{2}} \\ $$

Question Number 179773    Answers: 1   Comments: 1

Whatβ€²s the sum of the odd numbers between 2313 and 4718

$${What}'{s}\:{the}\:{sum}\:{of}\:{the}\:{odd}\:{numbers} \\ $$$$\:{between}\:\mathrm{2313}\:{and}\:\mathrm{4718} \\ $$

Question Number 179948    Answers: 2   Comments: 3

20 students should stand in 5 different rows. each row should have at least 2 students. in how many ways can you arrange them? (an unsolved old question)

$$\mathrm{20}\:{students}\:{should}\:{stand}\:{in}\:\mathrm{5} \\ $$$${different}\:{rows}.\:{each}\:{row}\:{should}\:{have} \\ $$$${at}\:{least}\:\mathrm{2}\:{students}.\:{in}\:{how}\:{many}\:{ways} \\ $$$${can}\:{you}\:{arrange}\:{them}? \\ $$$$\left({an}\:{unsolved}\:{old}\:{question}\right) \\ $$

Question Number 179752    Answers: 2   Comments: 2

Question Number 179749    Answers: 0   Comments: 0

Question Number 179741    Answers: 0   Comments: 0

prove that ∫_0 ^( ∞ ) (( (√x))/(1 +7x^( 2) + x^( 4) )) dx = (Ο€/( (√( 90))))

$$ \\ $$$$\:\:\:\:\:{prove}\:\:{that} \\ $$$$ \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\infty\:} \frac{\:\sqrt{{x}}}{\mathrm{1}\:+\mathrm{7}{x}^{\:\mathrm{2}} \:+\:{x}^{\:\mathrm{4}} }\:{dx}\:=\:\frac{\pi}{\:\sqrt{\:\mathrm{90}}}\:\: \\ $$$$ \\ $$

Question Number 179738    Answers: 2   Comments: 0

Number of distributions of 12 different things be taken to 3different boxes so as 1)5 things in 1st box exactly 2)5 things in any one box?

$${Number}\:{of}\:{distributions}\:{of} \\ $$$$\mathrm{12}\:{different}\:{things}\:{be}\:{taken}\:{to} \\ $$$$\mathrm{3}{different}\:{boxes}\:{so}\:{as} \\ $$$$\left.\mathrm{1}\right)\mathrm{5}\:{things}\:{in}\:\mathrm{1}{st}\:{box}\:{exactly} \\ $$$$\left.\mathrm{2}\right)\mathrm{5}\:{things}\:{in}\:{any}\:{one}\:{box}? \\ $$

Question Number 179737    Answers: 0   Comments: 2

Prouve that R=((L^2 +H^2 )/(2H)) ?

$$\mathrm{P}{rouve}\:{that} \\ $$$$\mathrm{R}=\frac{\mathrm{L}^{\mathrm{2}} +\mathrm{H}^{\mathrm{2}} }{\mathrm{2H}}\:\:\:? \\ $$

Question Number 179726    Answers: 0   Comments: 0

Question Number 179722    Answers: 1   Comments: 2

1000^(500 x) = (√(200))

$$\mathrm{1000}^{\mathrm{500}\:{x}} \:=\:\sqrt{\mathrm{200}} \\ $$

Question Number 179719    Answers: 1   Comments: 3

Question Number 179717    Answers: 1   Comments: 6

Q 179570 (Posted by Infinityaction 30.10.2022) find the minimum of f(x) f(x)=(√(x^2 +(4/x^2 )βˆ’8xβˆ’((12)/x)+25)) +(√(x^2 +(4/x^2 )βˆ’16xβˆ’((16)/x)+80)) βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ f(x)=(√((xβˆ’4)^2 +((2/x)βˆ’3)^2 )) +(√((xβˆ’8)^2 +((2/x)βˆ’4)^2 )) Df=Rβˆ’{0} f(x)β‰₯0 Min(f(x))=x / f(x)=0 (√((xβˆ’4)^2 +((2/x)βˆ’3)^2 )) +(√((xβˆ’8)^2 +((2/x)βˆ’4)^2 )) =0 (xβˆ’4)^2 +((2/x)βˆ’3)^2 =(xβˆ’8)^2 +((2/x)βˆ’4)^2 (1) xβˆ’8=(xβˆ’4)βˆ’4 and ((2/x)βˆ’4)=((2/x)βˆ’3)βˆ’1 x^2 βˆ’((55)/8)x+(1/2)=0 x=0,07351334 and x=6,80148666} x=0,073513334 f(x)=49,044679(rejete) x=6,80148666 f(x)=7,7899055 Donc 7,7899055 est minimum de f(x)

$$\mathrm{Q}\:\mathrm{179570}\:\left({Posted}\:{by}\:\mathrm{Infinityaction}\:\mathrm{30}.\mathrm{10}.\mathrm{2022}\right) \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right) \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\mathrm{x}^{\mathrm{2}} +\frac{\mathrm{4}}{\mathrm{x}^{\mathrm{2}} }βˆ’\mathrm{8x}βˆ’\frac{\mathrm{12}}{\mathrm{x}}+\mathrm{25}}\:+\sqrt{\mathrm{x}^{\mathrm{2}} +\frac{\mathrm{4}}{\mathrm{x}^{\mathrm{2}} }βˆ’\mathrm{16x}βˆ’\frac{\mathrm{16}}{{x}}+\mathrm{80}}\: \\ $$$$βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\left(\mathrm{x}βˆ’\mathrm{4}\right)^{\mathrm{2}} +\left(\frac{\mathrm{2}}{\mathrm{x}}βˆ’\mathrm{3}\right)^{\mathrm{2}} }\:\:+\sqrt{\left(\mathrm{x}βˆ’\mathrm{8}\right)^{\mathrm{2}} +\left(\frac{\mathrm{2}}{\mathrm{x}}βˆ’\mathrm{4}\right)^{\mathrm{2}} }\: \\ $$$$\mathrm{D}{f}=\mathbb{R}βˆ’\left\{\mathrm{0}\right\}\:\:\:\:\mathrm{f}\left(\mathrm{x}\right)\geqslant\mathrm{0}\: \\ $$$$\mathrm{Min}\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{x}\:/\:{f}\left(\mathrm{x}\right)=\mathrm{0} \\ $$$$\sqrt{\left(\mathrm{x}βˆ’\mathrm{4}\right)^{\mathrm{2}} +\left(\frac{\mathrm{2}}{\mathrm{x}}βˆ’\mathrm{3}\right)^{\mathrm{2}} \:}\:+\sqrt{\left(\mathrm{x}βˆ’\mathrm{8}\right)^{\mathrm{2}} +\left(\frac{\mathrm{2}}{\mathrm{x}}βˆ’\mathrm{4}\right)^{\mathrm{2}} \:}\:=\mathrm{0} \\ $$$$\left(\mathrm{x}βˆ’\mathrm{4}\right)^{\mathrm{2}} +\left(\frac{\mathrm{2}}{\mathrm{x}}βˆ’\mathrm{3}\right)^{\mathrm{2}} =\left(\mathrm{x}βˆ’\mathrm{8}\right)^{\mathrm{2}} +\left(\frac{\mathrm{2}}{\mathrm{x}}βˆ’\mathrm{4}\right)^{\mathrm{2}} \:\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{x}βˆ’\mathrm{8}=\left(\mathrm{x}βˆ’\mathrm{4}\right)βˆ’\mathrm{4}\:\:\:\:\mathrm{and}\:\:\left(\frac{\mathrm{2}}{\mathrm{x}}βˆ’\mathrm{4}\right)=\left(\frac{\mathrm{2}}{\mathrm{x}}βˆ’\mathrm{3}\right)βˆ’\mathrm{1} \\ $$$$\left.\:\:\mathrm{x}^{\mathrm{2}} βˆ’\frac{\mathrm{55}}{\mathrm{8}}\mathrm{x}+\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{0}\:\:\:{x}=\mathrm{0},\mathrm{07351334}\:{and}\:{x}=\mathrm{6},\mathrm{80148666}\right\} \\ $$$$\mathrm{x}=\mathrm{0},\mathrm{073513334} \\ $$$${f}\left(\mathrm{x}\right)=\mathrm{49},\mathrm{044679}\left({rejete}\right) \\ $$$$\mathrm{x}=\mathrm{6},\mathrm{80148666}\:\:\:\:\:{f}\left(\mathrm{x}\right)=\mathrm{7},\mathrm{7899055} \\ $$$${Donc}\:\:\mathrm{7},\mathrm{7899055}\:\:\:{est}\:{minimum}\:{de}\:{f}\left({x}\right) \\ $$$$ \\ $$

Question Number 179710    Answers: 0   Comments: 0

Q179229

$${Q}\mathrm{179229} \\ $$$$ \\ $$

Question Number 179699    Answers: 0   Comments: 9

20 students are numbered with the numbers from 1 to 20. 10 students are randomly selected. what is the probability that the sum of their numbers is exactly 100?

$$\mathrm{20}\:{students}\:{are}\:{numbered}\:{with}\:{the} \\ $$$${numbers}\:{from}\:\mathrm{1}\:{to}\:\mathrm{20}.\:\mathrm{10}\:{students}\:{are} \\ $$$${randomly}\:{selected}.\:{what}\:{is}\:{the} \\ $$$${probability}\:{that}\:{the}\:{sum}\:{of}\:{their} \\ $$$${numbers}\:{is}\:{exactly}\:\mathrm{100}? \\ $$

Question Number 179688    Answers: 1   Comments: 5

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