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Question Number 112359 Answers: 0 Comments: 0

Suppose a job consists of n tasks each of which takes time t seconds. Thus if there are no failuers the sum over all computed nodes of the time taken to execute tasks at that node is nt. Sppose also that the probability of a task failing is p per job per second and when a task fails the overhead of management of the restart is such that it adds 10t seconds to the total execution time of the job. What is the total expected execution time of the job?

$$\mathrm{Suppose}\:\mathrm{a}\:\mathrm{job}\:\mathrm{consists}\:\mathrm{of}\:\boldsymbol{{n}}\:\mathrm{tasks} \\ $$$$\mathrm{each}\:\mathrm{of}\:\mathrm{which}\:\mathrm{takes}\:\mathrm{time}\:\boldsymbol{{t}} \\ $$$$\mathrm{seconds}.\:\mathrm{Thus}\:\mathrm{if}\:\mathrm{there}\:\mathrm{are}\:\mathrm{no} \\ $$$$\mathrm{failuers}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{over}\:\mathrm{all}\:\mathrm{computed} \\ $$$$\mathrm{nodes}\:\mathrm{of}\:\mathrm{the}\:\mathrm{time}\:\mathrm{taken}\:\mathrm{to}\: \\ $$$$\mathrm{execute}\:\mathrm{tasks}\:\mathrm{at}\:\mathrm{that}\:\mathrm{node}\:\mathrm{is}\:\boldsymbol{{nt}}. \\ $$$$\mathrm{Sppose}\:\mathrm{also}\:\mathrm{that}\:\mathrm{the}\:\mathrm{probability} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{task}\:\mathrm{failing}\:\mathrm{is}\:\boldsymbol{{p}}\:\mathrm{per}\:\mathrm{job}\:\mathrm{per} \\ $$$$\mathrm{second}\:\mathrm{and}\:\mathrm{when}\:\mathrm{a}\:\mathrm{task}\:\mathrm{fails}\:\mathrm{the} \\ $$$$\mathrm{overhead}\:\mathrm{of}\:\mathrm{management}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{restart}\:\mathrm{is}\:\mathrm{such}\:\mathrm{that}\:\mathrm{it}\:\mathrm{adds}\:\mathrm{10}\boldsymbol{{t}}\: \\ $$$$\mathrm{seconds}\:\mathrm{to}\:\mathrm{the}\:\mathrm{total}\:\mathrm{execution}\: \\ $$$$\mathrm{time}\:\mathrm{of}\:\mathrm{the}\:\mathrm{job}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{total} \\ $$$$\mathrm{expected}\:\mathrm{execution}\:\mathrm{time}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{job}? \\ $$

Question Number 112358 Answers: 2 Comments: 0

The point of the curve 3x^2 −4y^2 =72 which nearest to the line 3x+2y=1 is___ (a) (6,3) (c) (6,6) (b) (6,−3) (d) (6,5)

$${The}\:{point}\:{of}\:{the}\:{curve}\:\mathrm{3}{x}^{\mathrm{2}} −\mathrm{4}{y}^{\mathrm{2}} =\mathrm{72} \\ $$$${which}\:{nearest}\:{to}\:{the}\:{line}\:\mathrm{3}{x}+\mathrm{2}{y}=\mathrm{1} \\ $$$${is\_\_\_} \\ $$$$\left({a}\right)\:\left(\mathrm{6},\mathrm{3}\right)\:\:\:\:\:\:\:\left({c}\right)\:\left(\mathrm{6},\mathrm{6}\right) \\ $$$$\left({b}\right)\:\left(\mathrm{6},−\mathrm{3}\right)\:\:\left({d}\right)\:\left(\mathrm{6},\mathrm{5}\right) \\ $$

Question Number 112356 Answers: 3 Comments: 1

Question Number 112338 Answers: 0 Comments: 0

Question Number 112334 Answers: 0 Comments: 0

Question Number 112331 Answers: 0 Comments: 2

Question Number 112328 Answers: 1 Comments: 0

find the all root (m^3 −6m+9)=0

$${find}\:{the}\:{all}\:{root}\:\left({m}^{\mathrm{3}} −\mathrm{6}{m}+\mathrm{9}\right)=\mathrm{0} \\ $$

Question Number 112313 Answers: 5 Comments: 2

solve ∫_0 ^1 ((lnx)/(√(1+x^2 )))dx

$${solve} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}{x}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 112323 Answers: 0 Comments: 1

_≤^≥ = SOLVE the EQUATION_ ^ _( •) n−⌊(√n)⌋−⌊(n)^(1/3) ⌋+⌊(n)^(1/6) ⌋=2016

$$\:_{\leqslant} ^{\geqslant} =\:\:\mathbb{SOLVE}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathcal{EQUATION}}_{} ^{\:_{\:\:\bullet} } \\ $$$$\:\:\:\boldsymbol{\mathrm{n}}−\lfloor\sqrt{\boldsymbol{\mathrm{n}}}\rfloor−\lfloor\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{n}}}\rfloor+\lfloor\sqrt[{\mathrm{6}}]{\boldsymbol{\mathrm{n}}}\rfloor=\mathrm{2016} \\ $$$$ \\ $$

Question Number 112320 Answers: 2 Comments: 1

Question Number 112311 Answers: 1 Comments: 2

lim_(x→0) (√((3sin^2 x+cos 2x−1)/(x.tan 2x))) ?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt{\frac{\mathrm{3sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{cos}\:\mathrm{2x}−\mathrm{1}}{\mathrm{x}.\mathrm{tan}\:\mathrm{2x}}}\:? \\ $$

Question Number 112310 Answers: 4 Comments: 0

Question Number 112308 Answers: 0 Comments: 0

Question Number 112301 Answers: 0 Comments: 1

Question Number 112280 Answers: 2 Comments: 0

If 15 ≤ ∣10−(1/3)a∣ < 20 find a ∈ R

$$\:\:\mathrm{If}\:\mathrm{15}\:\leqslant\:\mid\mathrm{10}−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{a}\mid\:<\:\mathrm{20}\: \\ $$$$\mathrm{find}\:\mathrm{a}\:\in\:\mathbb{R} \\ $$

Question Number 112271 Answers: 2 Comments: 0

(√(bemath)) ∫ sin x (√(1−sin x)) dx ?

$$\:\:\:\:\sqrt{\mathrm{bemath}} \\ $$$$\:\:\:\int\:\mathrm{sin}\:\mathrm{x}\:\sqrt{\mathrm{1}−\mathrm{sin}\:\mathrm{x}}\:\mathrm{dx}\:? \\ $$

Question Number 112266 Answers: 1 Comments: 0

Let Ω denote the circumcircle of ABC. The tangent to Ω at A meets BC at X. Let the angle bisectors of ∠AXB meet AC and AB at E and F respectively. D is the foot of the angle bisector from ∠BAC on BC. Let AD intersect EF at K and Ω again at L(other than A). Prove that AEDF is a rhombus and further prove that the circle defined by triangle KLX passes through the midpoint of line segment BC.

$$\mathrm{Let}\:\Omega\:\mathrm{denote}\:\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{of}\:\mathrm{ABC}. \\ $$$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{to}\:\Omega\:\mathrm{at}\:\mathrm{A}\:\mathrm{meets}\:\mathrm{BC}\:\mathrm{at}\:\mathrm{X}. \\ $$$$\mathrm{Let}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{bisectors}\:\mathrm{of}\:\angle\mathrm{AXB}\:\mathrm{meet} \\ $$$$\mathrm{AC}\:\mathrm{and}\:\mathrm{AB}\:\mathrm{at}\:\mathrm{E}\:\mathrm{and}\:\mathrm{F} \\ $$$$\mathrm{respectively}.\:\mathrm{D}\:\mathrm{is}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of}\:\mathrm{the}\:\mathrm{angle} \\ $$$$\mathrm{bisector}\:\mathrm{from}\:\angle\mathrm{BAC}\:\mathrm{on}\:\mathrm{BC}.\:\mathrm{Let}\:\mathrm{AD} \\ $$$$\mathrm{intersect}\:\mathrm{EF}\:\mathrm{at}\:\mathrm{K}\:\mathrm{and}\:\Omega\:\mathrm{again}\:\mathrm{at} \\ $$$$\mathrm{L}\left(\mathrm{other}\:\mathrm{than}\:\mathrm{A}\right).\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{AEDF}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{rhombus}\:\mathrm{and}\:\mathrm{further}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{circle}\:\mathrm{defined}\:\mathrm{by}\:\mathrm{triangle}\:\mathrm{KLX}\:\mathrm{passes} \\ $$$$\mathrm{through}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of}\:\mathrm{line}\:\mathrm{segment} \\ $$$$\mathrm{BC}. \\ $$

Question Number 112287 Answers: 3 Comments: 0

{ ((x^2 + 3xy + y^2 = −1)),((x^3 + y^3 = 7)) :}

$$\begin{cases}{{x}^{\mathrm{2}} \:+\:\mathrm{3}{xy}\:+\:{y}^{\mathrm{2}} \:=\:−\mathrm{1}}\\{{x}^{\mathrm{3}} \:+\:{y}^{\mathrm{3}} \:=\:\mathrm{7}}\end{cases} \\ $$

Question Number 112254 Answers: 1 Comments: 0

sin ((π/( 7))).sin (((2π)/7)).sin (((3π)/7)) =?

$$\:\:\:\mathrm{sin}\:\left(\frac{\pi}{\:\mathrm{7}}\right).\mathrm{sin}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right).\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)\:=? \\ $$

Question Number 112251 Answers: 1 Comments: 0

∫ (dx/((x^4 −1)(√(x^2 +1))))

$$\int\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{4}} −\mathrm{1}\right)\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}} \\ $$

Question Number 112249 Answers: 2 Comments: 1

∫(dx/((αx^2 +px+β)(√(αx^2 +qx+β))))=?