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Question Number 145229 Answers: 0 Comments: 3

Riddle (clue) 1. I have different types 2. I may be considered natural, whole, positive or negative 3. I am the basic building block of mathematics 4. I am often considered reasonable or rational as well as crazy or irrational 5. I may be terminating or repeating 6. You can locate me on a line bearing my name. who am i?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Riddle} \\ $$$$\:\:\:\:\left(\mathrm{clue}\right) \\ $$$$\mathrm{1}.\:\mathrm{I}\:\mathrm{have}\:\mathrm{different}\:\mathrm{types} \\ $$$$\mathrm{2}.\:\mathrm{I}\:\mathrm{may}\:\mathrm{be}\:\mathrm{considered}\:\mathrm{natural},\:\mathrm{whole}, \\ $$$$\mathrm{positive}\:\mathrm{or}\:\mathrm{negative} \\ $$$$\mathrm{3}.\:\mathrm{I}\:\mathrm{am}\:\mathrm{the}\:\mathrm{basic}\:\mathrm{building}\:\mathrm{block}\:\mathrm{of} \\ $$$$\mathrm{mathematics} \\ $$$$\mathrm{4}.\:\mathrm{I}\:\mathrm{am}\:\mathrm{often}\:\mathrm{considered}\:\mathrm{reasonable}\:\mathrm{or} \\ $$$$\:\:\:\:\:\:\:\mathrm{rational}\:\mathrm{as}\:\mathrm{well}\:\mathrm{as}\:\mathrm{crazy}\:\mathrm{or}\:\mathrm{irrational} \\ $$$$\mathrm{5}.\:\mathrm{I}\:\:\mathrm{may}\:\mathrm{be}\:\mathrm{terminating}\:\mathrm{or}\:\mathrm{repeating} \\ $$$$\mathrm{6}.\:\mathrm{You}\:\mathrm{can}\:\mathrm{locate}\:\mathrm{me}\:\mathrm{on}\:\mathrm{a}\:\mathrm{line}\:\mathrm{bearing} \\ $$$$\mathrm{my}\:\mathrm{name}. \\ $$$$\:\:\:\:\:\boldsymbol{\mathrm{who}}\:\boldsymbol{\mathrm{am}}\:\boldsymbol{\mathrm{i}}? \\ $$$$ \\ $$

Question Number 145227 Answers: 1 Comments: 1

Question Number 145224 Answers: 1 Comments: 0

(√(2(√3) + 2)) - (√((√3) - (√2))) = ?

$$\sqrt{\mathrm{2}\sqrt{\mathrm{3}}\:+\:\mathrm{2}}\:-\:\sqrt{\sqrt{\mathrm{3}}\:-\:\sqrt{\mathrm{2}}}\:=\:? \\ $$

Question Number 145208 Answers: 1 Comments: 0

If P : Q = tan 2A : cos A Q : R = cos 2A : sin 2A then P : R =?

$$\:\mathrm{If}\:\mathrm{P}\::\:\mathrm{Q}\:=\:\mathrm{tan}\:\mathrm{2A}\::\:\mathrm{cos}\:\mathrm{A} \\ $$$$\:\:\:\:\:\mathrm{Q}\::\:\mathrm{R}\:=\:\mathrm{cos}\:\mathrm{2A}\::\:\mathrm{sin}\:\mathrm{2A} \\ $$$$\:\:\:\:\:\mathrm{then}\:\mathrm{P}\::\:\mathrm{R}\:=? \\ $$

Question Number 145205 Answers: 2 Comments: 0

lim_(x→0) ((1−cos 2(sin (sin x)))/x^2 )=?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{2}\left(\mathrm{sin}\:\left(\mathrm{sin}\:\mathrm{x}\right)\right)}{\mathrm{x}^{\mathrm{2}} }=? \\ $$

Question Number 145202 Answers: 2 Comments: 1

Question Number 145200 Answers: 1 Comments: 0

evaluate:: Σ_(n=0) ^∞ (1/(n!(n^4 +n^2 +1)))=(e/2)

$$\mathrm{evaluate}::\:\:\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{n}!\left(\mathrm{n}^{\mathrm{4}} +\mathrm{n}^{\mathrm{2}} +\mathrm{1}\right)}=\frac{\mathrm{e}}{\mathrm{2}} \\ $$

Question Number 145198 Answers: 1 Comments: 0

Let a≥b≥c≥0 , c^3 +(a+b)^3 ≠0 and a^2 +b^2 +c^2 = 3. Prove that (1/2) ≤ ((a^3 +(b+c)^3 +b^3 +(c+a)^3 )/(c^3 +(a+b)^3 )) ≤ 2 Determine when equality holds.

$$\mathrm{Let}\:{a}\geqslant{b}\geqslant{c}\geqslant\mathrm{0}\:,\:{c}^{\mathrm{3}} +\left({a}+{b}\right)^{\mathrm{3}} \neq\mathrm{0}\:\mathrm{and}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \:=\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\leqslant\:\frac{{a}^{\mathrm{3}} +\left({b}+{c}\right)^{\mathrm{3}} +{b}^{\mathrm{3}} +\left({c}+{a}\right)^{\mathrm{3}} }{{c}^{\mathrm{3}} +\left({a}+{b}\right)^{\mathrm{3}} }\:\leqslant\:\mathrm{2} \\ $$$$\mathrm{Determine}\:\mathrm{when}\:\mathrm{equality}\:\mathrm{holds}. \\ $$

Question Number 145197 Answers: 1 Comments: 1

d/dx of x!=?

$${d}/{dx}\:{of}\:{x}!=? \\ $$

Question Number 145190 Answers: 1 Comments: 0

f(x)+f(1−(1/x))=tan^(−1) x,(x≠0) Find f(x)=?

$$\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}}\right)=\mathrm{tan}^{−\mathrm{1}} \mathrm{x},\left(\mathrm{x}\neq\mathrm{0}\right) \\ $$$$\mathrm{Find}\:\mathrm{f}\left(\mathrm{x}\right)=? \\ $$

Question Number 145187 Answers: 0 Comments: 0

((3)^(1/3) + (5)^(1/5) )^(120) How many single units are there at the opening of the binomial

$$\left(\sqrt[{\mathrm{3}}]{\mathrm{3}}\:+\:\sqrt[{\mathrm{5}}]{\mathrm{5}}\right)^{\mathrm{120}} \\ $$$${How}\:{many}\:{single}\:{units}\:{are}\:{there}\:{at} \\ $$$${the}\:{opening}\:{of}\:{the}\:{binomial} \\ $$

Question Number 145186 Answers: 0 Comments: 3

f:R→R f(x-1)+f(x+1)=(√3)∙f(x) ; ∀x∈R find f(x-1)+f(x+5)=?

$${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${f}\left({x}-\mathrm{1}\right)+{f}\left({x}+\mathrm{1}\right)=\sqrt{\mathrm{3}}\centerdot{f}\left({x}\right)\:;\:\forall{x}\in\mathbb{R} \\ $$$${find}\:\:{f}\left({x}-\mathrm{1}\right)+{f}\left({x}+\mathrm{5}\right)=? \\ $$

Question Number 145184 Answers: 0 Comments: 0

find lim_(x→0) ((sin(sh(2x))−sh(sin(3x)))/x^2 )

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\:\frac{\mathrm{sin}\left(\mathrm{sh}\left(\mathrm{2x}\right)\right)−\mathrm{sh}\left(\mathrm{sin}\left(\mathrm{3x}\right)\right)}{\mathrm{x}^{\mathrm{2}} } \\ $$

Question Number 145183 Answers: 2 Comments: 1

find ∫_0 ^1 (dx/(((√x)+(√(x+1)))^3 ))

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{dx}}{\left(\sqrt{\mathrm{x}}+\sqrt{\mathrm{x}+\mathrm{1}}\right)^{\mathrm{3}} } \\ $$

Question Number 145166 Answers: 1 Comments: 1

Question Number 145165 Answers: 1 Comments: 0

calculate Σ_(n=0) ^∞ arctan(((2n+1)/(n^4 +2n^3 +n^2 +1)))

$$\mathrm{calculate}\:\:\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \mathrm{arctan}\left(\frac{\mathrm{2n}+\mathrm{1}}{\mathrm{n}^{\mathrm{4}} \:+\mathrm{2n}^{\mathrm{3}} \:+\mathrm{n}^{\mathrm{2}} \:+\mathrm{1}}\right) \\ $$

Question Number 145164 Answers: 3 Comments: 0

Question Number 145163 Answers: 2 Comments: 0

Question Number 145162 Answers: 0 Comments: 1

we have z = e^(((2pi)/7)i) a = z+ z^2 + z^4 and b = z^3 + z^5 +z^6 we know a + b = −1 and 1−b=a find S = cos(((2pi)/7))+ cos(((4pi)/7)) + cos(((8pi)/7)) thanks for help

$${we}\:{have}\:{z}\:=\:{e}^{\frac{\mathrm{2}{pi}}{\mathrm{7}}{i}} \: \\ $$$${a}\:=\:{z}+\:{z}^{\mathrm{2}} \:+\:{z}^{\mathrm{4}} \:\:{and}\:{b}\:=\:{z}^{\mathrm{3}} \:+\:{z}^{\mathrm{5}} \:+{z}^{\mathrm{6}} \\ $$$${we}\:{know}\:\:{a}\:+\:{b}\:=\:−\mathrm{1}\:{and}\:\mathrm{1}−{b}={a} \\ $$$${find}\:{S}\:=\:{cos}\left(\frac{\mathrm{2}{pi}}{\mathrm{7}}\right)+\:{cos}\left(\frac{\mathrm{4}{pi}}{\mathrm{7}}\right)\:+\:{cos}\left(\frac{\mathrm{8}{pi}}{\mathrm{7}}\right) \\ $$$${thanks}\:{for}\:{help} \\ $$

Question Number 145174 Answers: 1 Comments: 0

Question Number 145193 Answers: 1 Comments: 1

Question Number 145191 Answers: 1 Comments: 0

if q≥1 and x>−1 then: (1+x)^q ≥ (1+x)^(q−1) + x ≥ 1+qx

$${if}\:\:{q}\geqslant\mathrm{1}\:\:{and}\:\:{x}>−\mathrm{1}\:\:{then}: \\ $$$$\left(\mathrm{1}+{x}\right)^{\boldsymbol{{q}}} \:\geqslant\:\left(\mathrm{1}+{x}\right)^{\boldsymbol{{q}}−\mathrm{1}} \:+\:{x}\:\geqslant\:\mathrm{1}+{qx} \\ $$

Question Number 145154 Answers: 0 Comments: 0

Question Number 145244 Answers: 0 Comments: 0

consider the circle (x−1)^2 +(y−1)^2 =2, A(1,4), B(1,−5). if P is a point on the circle such that PA+PB is maximum then prove that P,A,B are collinear points.

$$\mathrm{consider}\:\mathrm{the}\:\mathrm{circle}\: \\ $$$$\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{y}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{2}, \\ $$$$\mathrm{A}\left(\mathrm{1},\mathrm{4}\right),\:\mathrm{B}\left(\mathrm{1},−\mathrm{5}\right).\:\mathrm{if}\:\mathrm{P}\:\mathrm{is}\: \\ $$$$\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{PA}+\mathrm{PB}\:\mathrm{is}\:\mathrm{maximum}\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{P},\mathrm{A},\mathrm{B}\:\mathrm{are}\:\mathrm{collinear}\: \\ $$$$\mathrm{points}. \\ $$

Question Number 145136 Answers: 1 Comments: 0

Soit X une variable aleatoire de loi geometrique de parametre p∈]0.1[ calculer P({X≥4})

$${Soit}\:{X}\:{une}\:{variable}\:{aleatoire}\:{de}\:{loi} \\ $$$$\left.{geometrique}\:{de}\:{parametre}\:{p}\in\right]\mathrm{0}.\mathrm{1}\left[\right. \\ $$$${calculer}\:{P}\left(\left\{{X}\geqslant\mathrm{4}\right\}\right) \\ $$

Question Number 145134 Answers: 0 Comments: 1

On dispose de N+1 urnes.l′urne U_k contient k boules blanches et N−k boules noires.on tire successivement sans remise n boules de l′urne et on note An l′evenement ′′choisir n boules noires lors des n premiers tirages′′. Determiner P(An). on notera U_k =′′choisir l′urne k′′

$${On}\:{dispose}\:{de}\:{N}+\mathrm{1}\:{urnes}.{l}'{urne}\:{U}_{{k}} \\ $$$${contient}\:{k}\:{boules}\:{blanches}\:{et}\:{N}−{k}\:{boules} \\ $$$${noires}.{on}\:{tire}\:{successivement}\:{sans}\: \\ $$$${remise}\:{n}\:{boules}\:{de}\:{l}'{urne}\:{et}\:{on}\:{note}\: \\ $$$${An}\:{l}'{evenement}\:''{choisir}\:{n}\:{boules}\:{noires} \\ $$$${lors}\:{des}\:{n}\:{premiers}\:{tirages}''.\:{Determiner} \\ $$$${P}\left({An}\right).\:{on}\:{notera}\:{U}_{{k}} =''{choisir}\:{l}'{urne}\:{k}'' \\ $$

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