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Question Number 78563 Answers: 0 Comments: 0

Question Number 78549 Answers: 1 Comments: 0

Question Number 78542 Answers: 0 Comments: 0

Question Number 78526 Answers: 0 Comments: 14

lim_(x→0) [((∫_( 0) ^( x^2 ) (√(4 + t^3 )) dt)/x^2 )]

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\left[\frac{\int_{\:\:\mathrm{0}} ^{\:\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\sqrt{\mathrm{4}\:+\:\boldsymbol{\mathrm{t}}^{\mathrm{3}} }\:\:\boldsymbol{\mathrm{dt}}}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right] \\ $$

Question Number 78522 Answers: 0 Comments: 10

what is the line passing through (2,2,1) and parallel to 2i^ − j^ − k^ ?

$${what}\:{is}\:{the}\: \\ $$$${line}\:{passing}\:{through}\:\left(\mathrm{2},\mathrm{2},\mathrm{1}\right) \\ $$$${and}\:{parallel}\:{to}\:\mathrm{2}\hat {{i}}\:−\:\hat {{j}}\:−\:\hat {{k}}\:? \\ $$

Question Number 78503 Answers: 3 Comments: 0

if x,y >1 prove (x^2 /(y−1))+(y^2 /(x−1))≥8

$${if}\:{x},{y}\:>\mathrm{1}\: \\ $$$${prove}\:\frac{{x}^{\mathrm{2}} }{{y}−\mathrm{1}}+\frac{{y}^{\mathrm{2}} }{{x}−\mathrm{1}}\geqslant\mathrm{8} \\ $$

Question Number 78496 Answers: 1 Comments: 4

Question Number 78493 Answers: 2 Comments: 0

the sum to infinity of a Geometric series is S the sum to infinty of the squares of the terms of the series is 2S the sum to infinity of the cubes of the terms of the series is ((64)/(13))S. find the value of S and write iut the first 3 terms if the series.

$${the}\:{sum}\:{to}\:{infinity}\:{of}\:{a}\:{Geometric}\:{series}\:{is}\:{S} \\ $$$${the}\:{sum}\:{to}\:{infinty}\:{of}\:{the}\:{squares}\:{of}\:{the}\:{terms} \\ $$$${of}\:{the}\:{series}\:{is}\:\mathrm{2}{S} \\ $$$${the}\:{sum}\:{to}\:{infinity}\:{of}\:{the}\:{cubes}\:{of}\:{the}\:{terms} \\ $$$${of}\:{the}\:{series}\:{is}\:\frac{\mathrm{64}}{\mathrm{13}}{S}. \\ $$$${find}\:{the}\:{value}\:{of}\:{S}\:{and}\:{write}\:{iut}\:{the}\:{first} \\ $$$$\mathrm{3}\:{terms}\:{if}\:{the}\:{series}. \\ $$

Question Number 78490 Answers: 1 Comments: 0

Find out A =Σ_(n=2) ^∞ ((ζ(n))/(n(−3)^n )) where ζ(p)=Σ_(n=1) ^∞ (1/n^p )

$$\mathrm{Find}\:\mathrm{out}\:\mathrm{A}\:=\underset{\mathrm{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\:\frac{\zeta\left(\mathrm{n}\right)}{\mathrm{n}\left(−\mathrm{3}\right)^{\mathrm{n}} }\:\:\:\:\:\:\:\:\:\:\:\mathrm{where}\:\:\zeta\left(\mathrm{p}\right)=\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{p}} }\: \\ $$

Question Number 78489 Answers: 1 Comments: 0

let P(x)= x^5 −209x+56 Prove that there exist two roots a,b such as ab=1 Find out their sum ( a+b=?) and deduce the decomposition of P(x) in prime factors.

$$\mathrm{let}\:\:\mathrm{P}\left(\mathrm{x}\right)=\:\mathrm{x}^{\mathrm{5}} −\mathrm{209x}+\mathrm{56}\: \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{two}\:\mathrm{roots}\:\:\mathrm{a},\mathrm{b}\:\mathrm{such}\:\mathrm{as}\:\:\:\mathrm{ab}=\mathrm{1} \\ $$$$\mathrm{Find}\:\mathrm{out}\:\mathrm{their}\:\mathrm{sum}\:\left(\:\mathrm{a}+\mathrm{b}=?\right)\:\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{decomposition}\:\mathrm{of}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{in}\:\mathrm{prime}\:\mathrm{factors}. \\ $$

Question Number 78511 Answers: 1 Comments: 2

Question Number 78468 Answers: 0 Comments: 1

If F(x) = ∫_0 ^(arctanx) (√(((t^4 −1)/(t^4 +1))dt,)) find F′(x).

$${If}\:{F}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{{arctanx}} \sqrt{\frac{{t}^{\mathrm{4}} −\mathrm{1}}{{t}^{\mathrm{4}} +\mathrm{1}}{dt},}\:{find}\:{F}'\left({x}\right). \\ $$

Question Number 78465 Answers: 1 Comments: 0

Question Number 78460 Answers: 3 Comments: 3

Σ_(n=1) ^∞ (n^3 /3^n )=?

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{n}^{\mathrm{3}} }{\mathrm{3}^{{n}} }=? \\ $$

Question Number 78456 Answers: 1 Comments: 0

for a>0 and b>a+2 , verify the follwing claim: Σ_(n=1) ^( ∞) n ((a(a+1)(a+2)...(a+n−1))/(b(b+1)(b+2)...(b+n−1))) =((a(b−1))/((b−a−1)(b−a−2)))

$${for}\:{a}>\mathrm{0}\:{and}\:{b}>{a}+\mathrm{2}\:,\:\:{verify}\:{the}\:{follwing}\: \\ $$$${claim}: \\ $$$$\:\:\:\sum_{{n}=\mathrm{1}} ^{\:\:\infty} \:{n}\:\frac{{a}\left({a}+\mathrm{1}\right)\left({a}+\mathrm{2}\right)...\left({a}+{n}−\mathrm{1}\right)}{{b}\left({b}+\mathrm{1}\right)\left({b}+\mathrm{2}\right)...\left({b}+{n}−\mathrm{1}\right)}\:=\frac{{a}\left({b}−\mathrm{1}\right)}{\left({b}−{a}−\mathrm{1}\right)\left({b}−{a}−\mathrm{2}\right)} \\ $$

Question Number 78453 Answers: 0 Comments: 4

ABC is a triangle with points A(−5;−5) B(−5;10) C(15;−5). the cartesian equtions of (AB); (AC) and (BC) are respectively x=−5 y=−5 x+y=5 1) Please help me to determinate the cartesian equations of the interior bisectors of A^ ; B^ ; C^ . 2) Demonstrate that these bisectors meet in some point H(25;25) 3) Give a cartesian equation of inscrited circle in ABC triangle.

$$\mathrm{ABC}\:\mathrm{is}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{with}\:\mathrm{points} \\ $$$$\mathrm{A}\left(−\mathrm{5};−\mathrm{5}\right)\:\mathrm{B}\left(−\mathrm{5};\mathrm{10}\right)\:\mathrm{C}\left(\mathrm{15};−\mathrm{5}\right). \\ $$$$\mathrm{the}\:\mathrm{cartesian}\:\mathrm{equtions}\:\mathrm{of}\:\left(\mathrm{AB}\right);\:\left(\mathrm{AC}\right) \\ $$$$\mathrm{and}\:\left(\mathrm{BC}\right)\:\mathrm{are}\:\mathrm{respectively} \\ $$$$\mathrm{x}=−\mathrm{5} \\ $$$$\mathrm{y}=−\mathrm{5} \\ $$$$\mathrm{x}+\mathrm{y}=\mathrm{5} \\ $$$$ \\ $$$$\left.\mathrm{1}\right)\:\mathrm{Please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{to}\:\mathrm{determinate}\: \\ $$$$\mathrm{the}\:\mathrm{cartesian}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{interior}\:\mathrm{bisectors}\:\mathrm{of}\:\hat {\mathrm{A}}\:;\:\hat {\mathrm{B}}\:;\:\hat {\mathrm{C}}. \\ $$$$\left.\mathrm{2}\right)\:\mathrm{Demonstrate}\:\mathrm{that}\:\mathrm{these}\:\mathrm{bisectors} \\ $$$$\:\mathrm{meet}\:\mathrm{in}\:\mathrm{some}\:\mathrm{point}\:\mathrm{H}\left(\mathrm{25};\mathrm{25}\right) \\ $$$$\left.\mathrm{3}\right)\:\mathrm{Give}\:\mathrm{a}\:\mathrm{cartesian}\:\mathrm{equation}\:\mathrm{of}\: \\ $$$$\mathrm{inscrited}\:\mathrm{circle}\:\:\mathrm{in}\:\mathrm{ABC}\:\mathrm{triangle}. \\ $$$$ \\ $$

Question Number 78449 Answers: 1 Comments: 1

find the domain of definition of f(x)=((−x)/(∣x∣−x))

$$\mathrm{find}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{of}\:\mathrm{definition}\:\mathrm{of}\: \\ $$$$\mathrm{f}\left({x}\right)=\frac{−{x}}{\mid{x}\mid−{x}} \\ $$

Question Number 78447 Answers: 1 Comments: 0

prove that the seq a_n = ((ncos(3n^2 +2n+1))/(n+1)) has convergent subsequence

$${prove}\:{that}\:{the}\:{seq}\:{a}_{{n}} \:=\:\frac{{ncos}\left(\mathrm{3}{n}^{\mathrm{2}} +\mathrm{2}{n}+\mathrm{1}\right)}{{n}+\mathrm{1}} \\ $$$${has}\:{convergent}\:{subsequence} \\ $$

Question Number 78458 Answers: 0 Comments: 2

if I_n =∫_0 ^(π/4) tan^n xdx and I_n =a_n +b_n I_(n−2) then find a_(10)

$${if}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {tan}^{{n}} {xdx}\:{and}\:{I}_{{n}} ={a}_{{n}} +{b}_{{n}} {I}_{{n}−\mathrm{2}} \:{then}\:{find}\:{a}_{\mathrm{10}} \\ $$

Question Number 78444 Answers: 0 Comments: 0

Question Number 78443 Answers: 1 Comments: 0

Question Number 78440 Answers: 0 Comments: 0

ab = p (a−2b)(a+b)(a+2b)=−24q Find ((a+b)/3) in terms of p, q.

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{ab}\:=\:{p} \\ $$$$\:\:\left({a}−\mathrm{2}{b}\right)\left({a}+{b}\right)\left({a}+\mathrm{2}{b}\right)=−\mathrm{24}{q} \\ $$$${Find}\:\:\:\frac{{a}+{b}}{\mathrm{3}}\:\:\:\:{in}\:{terms}\:{of}\:{p},\:{q}. \\ $$

Question Number 78427 Answers: 2 Comments: 0

please i need it urgently show that the midpoint of the hypotenuse of a right triangle is equidistant from its vertices

$${please}\:{i}\:{need}\:{it}\:{urgently} \\ $$$$ \\ $$$${show}\:{that}\:{the}\:{midpoint}\:{of}\:{the}\:{hypotenuse} \\ $$$${of}\:{a}\:{right}\:{triangle}\:{is}\:{equidistant}\:{from}\:{its}\:{vertices} \\ $$

Question Number 78425 Answers: 3 Comments: 0

how to solve find inf and sup of A 1. A={(m/n)+((4n)/m) m,n ∈N} 2. A={((mn)/(4m^2 + n^2 )) m∈Z,n ∈N} 3. A={(m/(∣m∣ + n)) m∈Z,n ∈N}

$${how}\:{to}\:{solve} \\ $$$${find}\:{inf}\:{and}\:{sup}\:{of}\:{A} \\ $$$$ \\ $$$$\mathrm{1}.\:{A}=\left\{\frac{{m}}{{n}}+\frac{\mathrm{4}{n}}{{m}}\:\:\:{m},{n}\:\in{N}\right\} \\ $$$$\mathrm{2}.\:{A}=\left\{\frac{{mn}}{\mathrm{4}{m}^{\mathrm{2}} \:+\:{n}^{\mathrm{2}} }\:\:{m}\in{Z},{n}\:\in{N}\right\} \\ $$$$\mathrm{3}.\:{A}=\left\{\frac{{m}}{\mid{m}\mid\:+\:{n}}\:\:{m}\in{Z},{n}\:\in{N}\right\} \\ $$$$ \\ $$$$ \\ $$

Question Number 78400 Answers: 0 Comments: 0

the convolute function of both f and g is marked f∗g And define by (f∗g)(x)=∫_0 ^x f(x−t)g(t)dt Let noted E=the set of function define on R_+ 0) Prove that there exist a function f_0 ∈E such as for all x>0 , ∫_0 ^∞ f_0 (t)e^(−xt) dt=1 1)Prove that (E,∗) is a semigroup 2)Prove that (E,+,∗) is an integrity domain.Is it a hull? 3) Prove that the sub−set J(x)={f∈E , ∫_0 ^∞ f(t)e^(−xt) dt=0} is a maximal ideal of E for all x>0. 4) let U(E) be the set of units of E Prove that H(x)={f∈U(E), f≡f_0 modJ(x)} is an invariant sub−group of U(E). 5) let be I_n the ideal formed by g_n : x→x^n witb n≥1, let mark I_n =(g_n )=g_n E Prove that , I_n ={∫∫..∫fdx_1 ...dx_n , f∈E} 6) By using the fondamental analysis theorem , can we say that E is a principal ring??

$$\mathrm{the}\:\mathrm{convolute}\:\mathrm{function}\:\mathrm{of}\:\mathrm{both}\:\mathrm{f}\:\mathrm{and}\:\mathrm{g}\:\mathrm{is}\:\mathrm{marked}\:\mathrm{f}\ast\mathrm{g} \\ $$$$\mathrm{And}\:\mathrm{define}\:\mathrm{by}\:\:\left(\mathrm{f}\ast\mathrm{g}\right)\left(\mathrm{x}\right)=\int_{\mathrm{0}} ^{\mathrm{x}} \mathrm{f}\left(\mathrm{x}−\mathrm{t}\right)\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt} \\ $$$$\mathrm{Let}\:\mathrm{noted}\:\mathrm{E}=\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{function}\:\mathrm{define}\:\mathrm{on}\:\mathbb{R}_{+} \\ $$$$\left.\mathrm{0}\right)\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{a}\:\mathrm{function}\:\mathrm{f}_{\mathrm{0}} \in\mathrm{E}\:\mathrm{such}\:\mathrm{as}\:\mathrm{for}\:\mathrm{all}\:\mathrm{x}>\mathrm{0}\:,\:\int_{\mathrm{0}} ^{\infty} \mathrm{f}_{\mathrm{0}} \left(\mathrm{t}\right)\mathrm{e}^{−\mathrm{xt}} \mathrm{dt}=\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\mathrm{Prove}\:\mathrm{that}\:\left(\mathrm{E},\ast\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{semigroup} \\ $$$$\left.\mathrm{2}\right)\mathrm{Prove}\:\mathrm{that}\:\left(\mathrm{E},+,\ast\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{integrity}\:\mathrm{domain}.\mathrm{Is}\:\mathrm{it}\:\mathrm{a}\:\mathrm{hull}?\: \\ $$$$\left.\mathrm{3}\right)\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sub}−\mathrm{set}\:\mathrm{J}\left(\mathrm{x}\right)=\left\{\mathrm{f}\in\mathrm{E}\:,\:\int_{\mathrm{0}} ^{\infty} \mathrm{f}\left(\mathrm{t}\right)\mathrm{e}^{−\mathrm{xt}} \mathrm{dt}=\mathrm{0}\right\}\:\mathrm{is}\:\mathrm{a}\:\mathrm{maximal}\:\mathrm{ideal}\:\mathrm{of}\:\mathrm{E}\:\:\:\mathrm{for}\:\mathrm{all}\:\mathrm{x}>\mathrm{0}. \\ $$$$\left.\mathrm{4}\right)\:\mathrm{let}\:\mathrm{U}\left(\mathrm{E}\right)\:\mathrm{be}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{units}\:\mathrm{of}\:\mathrm{E} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\:\mathrm{H}\left(\mathrm{x}\right)=\left\{\mathrm{f}\in\mathrm{U}\left(\mathrm{E}\right),\:\mathrm{f}\equiv\mathrm{f}_{\mathrm{0}} \mathrm{modJ}\left(\mathrm{x}\right)\right\}\:\mathrm{is}\:\mathrm{an}\:\mathrm{invariant}\:\mathrm{sub}−\mathrm{group}\:\mathrm{of}\:\mathrm{U}\left(\mathrm{E}\right). \\ $$$$\left.\mathrm{5}\right)\:\mathrm{let}\:\mathrm{be}\:\mathrm{I}_{\mathrm{n}} \:\mathrm{the}\:\mathrm{ideal}\:\mathrm{formed}\:\mathrm{by}\:\mathrm{g}_{\mathrm{n}} :\:\mathrm{x}\rightarrow\mathrm{x}^{\mathrm{n}} \:\mathrm{witb}\:\mathrm{n}\geqslant\mathrm{1},\:\mathrm{let}\:\mathrm{mark}\:\:\mathrm{I}_{\mathrm{n}} =\left(\mathrm{g}_{\mathrm{n}} \:\right)=\mathrm{g}_{\mathrm{n}} \mathrm{E} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\:,\:\mathrm{I}_{\mathrm{n}} =\left\{\int\int..\int\mathrm{fdx}_{\mathrm{1}} ...\mathrm{dx}_{\mathrm{n}} \:,\:\:\:\:\:\mathrm{f}\in\mathrm{E}\right\} \\ $$$$\left.\mathrm{6}\right)\:\mathrm{By}\:\mathrm{using}\:\mathrm{the}\:\mathrm{fondamental}\:\mathrm{analysis}\:\mathrm{theorem}\:,\:\mathrm{can}\:\mathrm{we}\:\mathrm{say}\:\mathrm{that}\:\mathrm{E}\:\mathrm{is}\:\mathrm{a}\:\mathrm{principal}\:\mathrm{ring}?? \\ $$

Question Number 78399 Answers: 2 Comments: 14

dear sir W, Mjs the set {1,4,n} have the condition that if two different elements are selected and 2112 is added to the result , then the result is a perfect square if n is a positif number . then the number of possible values of n is (A) 8 (B) 7 (C) 6 (D) 5 (E) 4

$${dear}\:{sir}\:{W},\:{Mjs}\: \\ $$$${the}\:{set}\:\left\{\mathrm{1},\mathrm{4},{n}\right\}\:{have}\:{the}\:{condition}\:{that}\: \\ $$$${if}\:{two}\:{different}\:{elements}\:{are} \\ $$$${selected}\:{and}\:\mathrm{2112}\:{is}\:{added}\:{to} \\ $$$${the}\:{result}\:,\:{then}\:{the}\:{result}\: \\ $$$${is}\:{a}\:{perfect}\:{square}\:{if}\:{n}\:{is}\:{a}\: \\ $$$${positif}\:{number}\:.\:{then}\:{the}\:{number}\: \\ $$$${of}\:{possible}\:{values}\:{of}\:{n}\:{is}\: \\ $$$$\left({A}\right)\:\mathrm{8}\:\:\:\:\left({B}\right)\:\mathrm{7}\:\:\:\:\:\left({C}\right)\:\mathrm{6}\:\:\:\:\:\left({D}\right)\:\mathrm{5} \\ $$$$\left({E}\right)\:\mathrm{4} \\ $$

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