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Question Number 61840    Answers: 1   Comments: 0

consider the triple of real numbers (x,y,z) defined by the addittion (x,y,z)+(x′,y′,z′)=(x+x′,y+y′,z+z′) and scalar multiplication by 𝛂(x,y,z)=(0,0,0). Show that all axioms for a vector space are satisfied except axiom 8.

$$\boldsymbol{{consider}}\:\boldsymbol{{the}}\:\boldsymbol{{triple}}\:\boldsymbol{{of}}\:\boldsymbol{{real}}\:\boldsymbol{{numbers}}\:\left(\boldsymbol{{x}},{y},{z}\right) \\ $$$${defined}\:{by}\:{the}\:{addittion}\:\left(\boldsymbol{{x}},{y},{z}\right)+\left({x}',{y}',{z}'\right)=\left({x}+{x}',{y}+{y}',{z}+{z}'\right) \\ $$$$\boldsymbol{{and}}\:\boldsymbol{{scalar}}\:\boldsymbol{{multiplication}}\:\boldsymbol{{by}}\:\:\:\boldsymbol{\alpha}\left({x},{y},{z}\right)=\left(\mathrm{0},\mathrm{0},\mathrm{0}\right).\: \\ $$$$\boldsymbol{{S}}{how}\:{that}\:{all}\:{axioms}\:{for}\:{a}\:{vector}\:{space}\:{are}\:{satisfied}\:{except}\:{axiom}\:\mathrm{8}. \\ $$

Question Number 61855    Answers: 1   Comments: 0

∫_(0 ) ^1 ((3x^3 −x^2 +2x−4)/(√(x^2 −3x+2))) dx

$$\int_{\mathrm{0}\:} ^{\mathrm{1}} \frac{\mathrm{3}{x}^{\mathrm{3}} −{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{4}}{\sqrt{{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}}}\:{dx} \\ $$

Question Number 61835    Answers: 0   Comments: 1

If α = Cis(2π/7) and f(x) = A_0 + Σ_(n=1) ^(14) A_n x^n Then prove that Σ_(α=0) ^6 f(α^n x)= 7(A_0 +A_7 x^7 +A_(14) x^(14) ) where Cisθ = Cosθ + iSinθ

$$\mathrm{If}\:\alpha\:=\:\mathrm{Cis}\left(\mathrm{2}\pi/\mathrm{7}\right)\:\mathrm{and}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{A}_{\mathrm{0}} \:+\:\sum_{\mathrm{n}=\mathrm{1}} ^{\mathrm{14}} \mathrm{A}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}} \: \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}\:\sum_{\alpha=\mathrm{0}} ^{\mathrm{6}} \mathrm{f}\left(\alpha^{\mathrm{n}} \mathrm{x}\right)=\:\mathrm{7}\left(\mathrm{A}_{\mathrm{0}} +\mathrm{A}_{\mathrm{7}} \mathrm{x}^{\mathrm{7}} +\mathrm{A}_{\mathrm{14}} \mathrm{x}^{\mathrm{14}} \right) \\ $$$$\mathrm{where}\:\mathrm{Cis}\theta\:=\:\mathrm{Cos}\theta\:+\:\mathrm{iSin}\theta \\ $$

Question Number 61834    Answers: 0   Comments: 3

Question Number 61825    Answers: 1   Comments: 0

Question Number 61823    Answers: 1   Comments: 0

If D,E and F are midpoints of the sides BC,CA and AB respectively of the △ABC and O be any point.Prove that OA^→ + OB^→ +OC^→ =OD^→ +OE^→ +OF^→

$${If}\:{D},{E}\:{and}\:{F}\:{are}\:{midpoints}\:{of}\:{the}\:{sides} \\ $$$${BC},{CA}\:{and}\:{AB}\:{respectively}\:{of}\:{the}\:\bigtriangleup{ABC} \\ $$$${and}\:{O}\:{be}\:{any}\:{point}.{Prove}\:{that} \\ $$$${O}\overset{\rightarrow} {{A}}\:+\:{O}\overset{\rightarrow} {{B}}\:+{O}\overset{\rightarrow} {{C}}={O}\overset{\rightarrow} {{D}}+{O}\overset{\rightarrow} {{E}}+{O}\overset{\rightarrow} {{F}} \\ $$

Question Number 61818    Answers: 0   Comments: 3

Find (dy/dx) y = sin^(−1) (((1−x^2 )/(1+x^2 ))), 0<x<1

$$\mathrm{Find}\:\:\:\frac{{dy}}{{dx}}\:\: \\ $$$${y}\:=\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{1}−{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }\right),\:\mathrm{0}<{x}<\mathrm{1} \\ $$

Question Number 61815    Answers: 0   Comments: 0

2H_2 S+SO_2 =3S+H_2 O is this a disproportionation reaction?

$$\mathrm{2}{H}_{\mathrm{2}} {S}+{SO}_{\mathrm{2}} =\mathrm{3}{S}+{H}_{\mathrm{2}} {O} \\ $$$${is}\:{this}\:{a}\:{disproportionation}\:{reaction}? \\ $$

Question Number 61811    Answers: 0   Comments: 0

Question Number 61810    Answers: 0   Comments: 0

Question Number 61809    Answers: 1   Comments: 1

Question Number 61807    Answers: 0   Comments: 0

Question Number 61804    Answers: 1   Comments: 1

calculate Σ_(n=2) ^∞ Σ_(k=2) ^∞ (1/(k^n k!))

$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\sum_{{k}=\mathrm{2}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{k}^{{n}} \:\:{k}!} \\ $$

Question Number 61803    Answers: 0   Comments: 3

find ∫_0 ^∞ (x^2 /(e^x^2 −1))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{\mathrm{2}} }{{e}^{{x}^{\mathrm{2}} } −\mathrm{1}}{dx} \\ $$

Question Number 61801    Answers: 0   Comments: 3

∫_(2π) ^(4π) (√(1−cos(x))) dx

$$\underset{\mathrm{2}\pi} {\overset{\mathrm{4}\pi} {\int}}\sqrt{\mathrm{1}−{cos}\left({x}\right)}\:{dx} \\ $$

Question Number 61799    Answers: 1   Comments: 0

If a = Cosα −iSinα and b = Cosβ −iSinβ Prove that (((a+b)(1−ab))/((a−b)(1+ab))) = ((Sinα+Sinβ)/(Sinα−Sinβ))

$$\mathrm{If}\:\mathrm{a}\:=\:\mathrm{Cos}\alpha\:−\mathrm{iSin}\alpha\:\mathrm{and}\:\mathrm{b}\:=\:\mathrm{Cos}\beta\:−\mathrm{iSin}\beta \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\frac{\left(\mathrm{a}+\mathrm{b}\right)\left(\mathrm{1}−\mathrm{ab}\right)}{\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{1}+\mathrm{ab}\right)}\:=\:\frac{\mathrm{Sin}\alpha+\mathrm{Sin}\beta}{\mathrm{Sin}\alpha−\mathrm{Sin}\beta} \\ $$

Question Number 61796    Answers: 0   Comments: 0

Question Number 61791    Answers: 1   Comments: 1

lim_(x → ∞) (6^x /(2^x + 4^x )) = ∞ ?

$$\underset{{x}\:\rightarrow\:\infty} {\mathrm{lim}}\:\:\:\frac{\mathrm{6}^{{x}} }{\mathrm{2}^{{x}} \:+\:\mathrm{4}^{{x}} }\:\:\:=\:\:\:\infty\:\:\:\:\:? \\ $$

Question Number 61785    Answers: 1   Comments: 0

∫_0 ^(√(3−x^2 )) ((xy(4−x^2 −y^2 )(√(4−x^2 −y^2 ))−xy)/3) dy

$$\underset{\mathrm{0}} {\overset{\sqrt{\mathrm{3}−{x}^{\mathrm{2}} }} {\int}}\frac{{xy}\left(\mathrm{4}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} }−{xy}}{\mathrm{3}}\:{dy} \\ $$

Question Number 61782    Answers: 0   Comments: 5

Any integer(s) which fulfill n^5 − 5n^3 + 5n + 1 ∣ n! ?

$${Any}\:\:{integer}\left({s}\right)\:\:{which}\:\:{fulfill} \\ $$$$\:\:\:\:\:\:\:\:\:\:{n}^{\mathrm{5}} \:−\:\mathrm{5}{n}^{\mathrm{3}} \:+\:\mathrm{5}{n}\:+\:\mathrm{1}\:\:\mid\:\:{n}!\:\:\:? \\ $$

Question Number 61778    Answers: 1   Comments: 0

Any integer(s) which fulfill n^3 − 5n^2 + 5n + 1 ∣ n! ?

$${Any}\:\:{integer}\left({s}\right)\:\:{which}\:\:{fulfill}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{n}^{\mathrm{3}} \:−\:\mathrm{5}{n}^{\mathrm{2}} \:+\:\mathrm{5}{n}\:+\:\mathrm{1}\:\:\mid\:\:{n}!\:\:\:\:? \\ $$

Question Number 61775    Answers: 0   Comments: 1

Question Number 61770    Answers: 0   Comments: 0

Question Number 61767    Answers: 0   Comments: 0

Question Number 61762    Answers: 2   Comments: 1

Question Number 61755    Answers: 1   Comments: 2

if f(z)=Σ_(k=1) ^n a_k z^k ,a_k ,z∈C.Prove a_k =(1/(2πi))∫_(∣z∣=r ) ((f(z))/z^(k+1) )dz

$$\mathrm{if}\:{f}\left({z}\right)=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{a}_{{k}} {z}^{{k}} ,{a}_{{k}} ,{z}\in\mathbb{C}.\mathrm{Prove} \\ $$$$ \\ $$$${a}_{{k}} =\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\underset{\mid{z}\mid={r}\:} {\int}\frac{{f}\left({z}\right)}{{z}^{{k}+\mathrm{1}} }{dz} \\ $$$$ \\ $$

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