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AllQuestion and Answers: Page 447

Question Number 174024    Answers: 2   Comments: 0

Question Number 173859    Answers: 0   Comments: 0

Question Number 173842    Answers: 1   Comments: 0

The probability that an athlete will not win any of the three races is 1/4. If the athlete runs in all the races, what is the probability that the athlete will win: (I) only the second race: (ii) all the three races (iii) only two of the races?

$$ \\ $$The probability that an athlete will not win any of the three races is 1/4. If the athlete runs in all the races, what is the probability that the athlete will win: (I) only the second race: (ii) all the three races (iii) only two of the races?

Question Number 173839    Answers: 3   Comments: 1

x^4 +x^3 +x^2 +x+1=0

$${x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}=\mathrm{0} \\ $$

Question Number 173836    Answers: 2   Comments: 1

Question Number 173834    Answers: 2   Comments: 2

Question Number 173831    Answers: 1   Comments: 0

Solve x+y=4 x^x −y^y =13(x−y)

$$\mathrm{Solve} \\ $$$$\mathrm{x}+\mathrm{y}=\mathrm{4} \\ $$$$\mathrm{x}^{\mathrm{x}} −\mathrm{y}^{\mathrm{y}} =\mathrm{13}\left(\mathrm{x}−\mathrm{y}\right) \\ $$

Question Number 173830    Answers: 0   Comments: 0

How many positive integer multiples of 2431 can be expressed in the form 7^a −7^b where a and b are integer such that 0≤b≤a≤2022

$$ \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{multiples}\:\mathrm{of}\:\mathrm{2431} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{expressed}\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\mathrm{7}^{\mathrm{a}} −\mathrm{7}^{\mathrm{b}} \\ $$$$\mathrm{where}\:\boldsymbol{{a}}\:\mathrm{and}\:\boldsymbol{{b}}\:\mathrm{are}\:\mathrm{integer}\:\mathrm{such}\:\mathrm{that}\:\mathrm{0}\leqslant\boldsymbol{{b}}\leqslant\boldsymbol{{a}}\leqslant\mathrm{2022}\: \\ $$

Question Number 173829    Answers: 1   Comments: 1

If , x∈ [0 , 1] , ∣ (√(1−x^( 2) )) −ax−b ∣≤ (((√2) −1)/2) find the values of ( a , b ) a , b∈ R.

$$ \\ $$$$\:\:\:\:{If}\:\:,\:{x}\in\:\left[\mathrm{0}\:,\:\mathrm{1}\right]\: \\ $$$$\:\:\:\:\:\:,\:\:\:\:\mid\:\sqrt{\mathrm{1}−{x}^{\:\mathrm{2}} }\:−{ax}−{b}\:\mid\leqslant\:\frac{\sqrt{\mathrm{2}}\:−\mathrm{1}}{\mathrm{2}} \\ $$$$\:\:\:\:{find}\:{the}\:{values}\:{of}\:\:\left(\:{a}\:,\:{b}\:\right) \\ $$$$\:\:\:{a}\:,\:{b}\in\:\mathbb{R}. \\ $$$$ \\ $$

Question Number 173827    Answers: 0   Comments: 0

Question Number 173825    Answers: 0   Comments: 2

Question Number 173826    Answers: 0   Comments: 0

Question Number 173820    Answers: 0   Comments: 0

Prove ((cos(π/(19))cos((7π)/(19))cos((8π)/(19))))^(1/3) +((cos((2π)/(19))cos((3π)/(19))cos((5π)/(19))))^(1/3) − −((cos((4π)/(19))cos((6π)/(19))cos((9π)/(19))))^(1/3) =(1/2)((3((19))^(1/3) −1))^(1/3)

$$ \\ $$ Prove $$ \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{cos}\frac{\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{7}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{8}\pi}{\mathrm{19}}}+\sqrt[{\mathrm{3}}]{\mathrm{cos}\frac{\mathrm{2}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{3}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{5}\pi}{\mathrm{19}}}− \\ $$$$−\sqrt[{\mathrm{3}}]{\mathrm{cos}\frac{\mathrm{4}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{6}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{9}\pi}{\mathrm{19}}}=\frac{\mathrm{1}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{19}}−\mathrm{1}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 173815    Answers: 0   Comments: 2

prove (((n+1)/3))^n <n!

$${prove}\: \\ $$$$\left(\frac{{n}+\mathrm{1}}{\mathrm{3}}\right)^{{n}} <{n}! \\ $$

Question Number 173807    Answers: 1   Comments: 0

Question Number 173798    Answers: 3   Comments: 0

( ϕ^( 5) + ϕ^( 4) + 1)^( 2) = mϕ + n m,n=? ........

$$ \\ $$$$\:\:\:\:\:\:\:\left(\:\varphi^{\:\mathrm{5}} +\:\varphi^{\:\mathrm{4}} +\:\mathrm{1}\right)^{\:\mathrm{2}} =\:{m}\varphi\:+\:{n} \\ $$$$\:\:\:\:\:\:\:\:\:{m},{n}=? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:........ \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 173795    Answers: 1   Comments: 1

Soient a et b deux re^ els. Pour tout n ∈ N on pose u_n =(√n)+a(√(n+1))+b(√(n+2)). 1. Ve^ rifier que la suite (u_n ) tend vers 0 si et seulement si a+b=−1. 2. De^ terminer a et b pour que la se^ rie Σu_n soit convergente.

$$\:\:\:\:\mathrm{Soient}\:{a}\:\mathrm{et}\:{b}\:\mathrm{deux}\:\mathrm{r}\acute {\mathrm{e}els}.\:\mathrm{Pour}\:\mathrm{tout}\:{n}\:\in\:\mathbb{N}\:\mathrm{on}\:\mathrm{pose} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{u}_{{n}} =\sqrt{{n}}+{a}\sqrt{{n}+\mathrm{1}}+{b}\sqrt{{n}+\mathrm{2}}. \\ $$$$\:\:\mathrm{1}.\:\:\mathrm{V}\acute {\mathrm{e}rifier}\:\mathrm{que}\:\mathrm{la}\:\mathrm{suite}\:\left({u}_{{n}} \right)\:\mathrm{tend}\:\mathrm{vers}\:\mathrm{0}\:\mathrm{si}\:\mathrm{et}\:\mathrm{seulement}\:\mathrm{si}\:{a}+{b}=−\mathrm{1}. \\ $$$$\:\:\mathrm{2}.\:\:\mathrm{D}\acute {\mathrm{e}terminer}\:{a}\:\mathrm{et}\:{b}\:\mathrm{pour}\:\mathrm{que}\:\mathrm{la}\:\mathrm{s}\acute {\mathrm{e}rie}\:\Sigma{u}_{{n}} \:\mathrm{soit}\:\mathrm{convergente}.\:\: \\ $$

Question Number 173787    Answers: 2   Comments: 11

4sin^( 2) (x)− 4sin(x)=cos^( 2) (x)−2cos(x) , 0<x<(π/2) . Find sin(x)=?

$$\:\mathrm{4}{sin}^{\:\mathrm{2}} \left({x}\right)−\:\mathrm{4}{sin}\left({x}\right)={cos}^{\:\mathrm{2}} \left({x}\right)−\mathrm{2}{cos}\left({x}\right) \\ $$$$\:\:\:,\:\mathrm{0}<{x}<\frac{\pi}{\mathrm{2}}\:.\:\:\:{Find}\:\:\:\:\:\:\:{sin}\left({x}\right)=? \\ $$

Question Number 173776    Answers: 2   Comments: 0

Question Number 173767    Answers: 0   Comments: 2

prouve ,nεN ∫_a ^a (t−a)(t−b)=(((b−a)^(n+2) )/((n+1)(n+2)))

$${prouve}\:,{n}\epsilon{N} \\ $$$$\int_{{a}} ^{{a}} \left({t}−{a}\right)\left({t}−{b}\right)=\frac{\left({b}−{a}\right)^{{n}+\mathrm{2}} }{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)} \\ $$

Question Number 173765    Answers: 4   Comments: 0

Question Number 176804    Answers: 0   Comments: 1

Question Number 173736    Answers: 1   Comments: 0

Show that x^2 + 2xy + 3y^2 = 1, is a solution to the differential equation (x + 3y)^2 (d^2 y/dx^2 ) + 2(x^2 + 2xy + 2y^3 ) = 0

$$\mathrm{Show}\:\mathrm{that}\:\:\:\:\:\mathrm{x}^{\mathrm{2}} \:\:\:+\:\:\:\mathrm{2xy}\:\:\:+\:\:\:\mathrm{3y}^{\mathrm{2}} \:\:\:=\:\:\:\mathrm{1},\:\:\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{solution}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{differential}\:\mathrm{equation}\:\:\:\:\:\left(\mathrm{x}\:\:\:+\:\:\:\mathrm{3y}\right)^{\mathrm{2}} \:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:\:\:\:\:+\:\:\:\mathrm{2}\left(\mathrm{x}^{\mathrm{2}} \:\:\:+\:\:\:\mathrm{2xy}\:\:\:+\:\:\:\mathrm{2y}^{\mathrm{3}} \right)\:\:\:=\:\:\:\mathrm{0} \\ $$

Question Number 173734    Answers: 1   Comments: 0

lim_( n→∞) (n +2) ∫_0 ^( 1) x^( n) ln( 1+x )dx=?

$$ \\ $$$$\:\mathrm{lim}_{\:{n}\rightarrow\infty} \:\left({n}\:+\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:{x}^{\:{n}} \:\mathrm{ln}\left(\:\mathrm{1}+{x}\:\right){dx}=? \\ $$$$ \\ $$

Question Number 173733    Answers: 0   Comments: 3

Show that x^3 + y^3 = 1 is a solution to the differential equation 20x^3 + 3y^2 (d^2 y/dx^2 ) + 6y((dy/dx))^2 = 0

$$\mathrm{Show}\:\mathrm{that}\:\:\:\:\mathrm{x}^{\mathrm{3}} \:\:+\:\:\mathrm{y}^{\mathrm{3}} \:\:=\:\:\mathrm{1}\:\:\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{solution}\:\mathrm{to}\:\mathrm{the}\:\mathrm{differential} \\ $$$$\mathrm{equation}\:\:\:\:\mathrm{20x}^{\mathrm{3}} \:\:\:+\:\:\:\mathrm{3y}^{\mathrm{2}} \:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:\:\:\:+\:\:\:\mathrm{6y}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{\mathrm{2}} \:\:\:=\:\:\:\:\mathrm{0} \\ $$

Question Number 173732    Answers: 4   Comments: 1

solve for x∈R (√(x^2 −4))+2(√(x^2 −1))=x^2

$${solve}\:{for}\:{x}\in{R} \\ $$$$\sqrt{{x}^{\mathrm{2}} −\mathrm{4}}+\mathrm{2}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}={x}^{\mathrm{2}} \\ $$

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