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Question Number 179775    Answers: 4   Comments: 0

(122−a)^2 = (123−a)^2

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

Question Number 179773    Answers: 1   Comments: 1

What′s 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 = (π/( (√( 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 )−8x−((12)/x)+25)) +(√(x^2 +(4/x^2 )−16x−((16)/x)+80)) −−−−−−−−−−−−−−−−−− f(x)=(√((x−4)^2 +((2/x)−3)^2 )) +(√((x−8)^2 +((2/x)−4)^2 )) Df=R−{0} f(x)≥0 Min(f(x))=x / f(x)=0 (√((x−4)^2 +((2/x)−3)^2 )) +(√((x−8)^2 +((2/x)−4)^2 )) =0 (x−4)^2 +((2/x)−3)^2 =(x−8)^2 +((2/x)−4)^2 (1) x−8=(x−4)−4 and ((2/x)−4)=((2/x)−3)−1 x^2 −((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)

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

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

Question Number 179927    Answers: 1   Comments: 3

Question Number 179679    Answers: 1   Comments: 0

∫_2 ^4 tan (x)

$$\int_{\mathrm{2}} ^{\mathrm{4}} \mathrm{tan}\:\left({x}\right) \\ $$

Question Number 181567    Answers: 0   Comments: 0

Question Number 181565    Answers: 0   Comments: 0

Question Number 179665    Answers: 0   Comments: 0

Q179420

$${Q}\mathrm{179420} \\ $$

Question Number 179660    Answers: 3   Comments: 0

((a+b)/( (√(ab))))=4 then a:b=?

$$\frac{{a}+{b}}{\:\sqrt{{ab}}}=\mathrm{4}\:\:\:\:{then}\:{a}:{b}=? \\ $$

Question Number 179659    Answers: 1   Comments: 0

cos((2π)/7)=((ab)/c) cos((4π)/7)=((bc)/a) cos((6π)/7)=((ac)/b) faind a^2 +b^2 +c^2 =?

$${cos}\frac{\mathrm{2}\pi}{\mathrm{7}}=\frac{{ab}}{{c}} \\ $$$${cos}\frac{\mathrm{4}\pi}{\mathrm{7}}=\frac{{bc}}{{a}} \\ $$$${cos}\frac{\mathrm{6}\pi}{\mathrm{7}}=\frac{{ac}}{{b}}\:\:\:\:\:\:\: \\ $$$${faind}\:\:\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =? \\ $$

Question Number 179653    Answers: 1   Comments: 0

Question Number 179657    Answers: 1   Comments: 2

prove that lim_(n→∞) ((√((1/2)×(√((1/2)+(1/2)(√(1/2))))))×(√((1/2)+(1/2)(√((1/2)+(1/2)(√(1/2))))))×∙∙∙∙∙(√((1/2)+(1/2)(√((1/2)+(1/2)(√((1/(2+)).....+(1/2)(√(1/2))))))))_(n term) )=(2/π)

$${prove}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\sqrt{\frac{\mathrm{1}}{\mathrm{2}}×\sqrt{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{1}}{\mathrm{2}}}}}×\sqrt{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{1}}{\mathrm{2}}}}}×\centerdot\centerdot\centerdot\centerdot\centerdot\underset{{n}\:{term}} {\underbrace{\sqrt{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{1}}{\mathrm{2}+}.....+\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{1}}{\mathrm{2}}}}}}}}\right)=\frac{\mathrm{2}}{\pi} \\ $$

Question Number 179655    Answers: 2   Comments: 0

Evaluate ∫6 arctan (8/w) dw

$${Evaluate}\:\int\mathrm{6}\:\mathrm{arctan}\:\frac{\mathrm{8}}{{w}}\:{dw} \\ $$

Question Number 179654    Answers: 2   Comments: 1

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