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The number 1,2,3,4,...,7 are randomly divided into two non −empty subsets. The probability that the sum of the numbers in the two subsets being equal is (r/s) expressed in the lowest term. Find r+s ?

$${The}\:{number}\:\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},...,\mathrm{7}\:{are}\:{randomly} \\ $$$${divided}\:{into}\:{two}\:{non}\:−{empty}\:{subsets}. \\ $$$${The}\:{probability}\:{that}\:{the}\:{sum}\:{of}\:{the} \\ $$$${numbers}\:{in}\:{the}\:{two}\:{subsets}\:{being} \\ $$$${equal}\:{is}\:\frac{{r}}{{s}}\:{expressed}\:{in}\:{the}\:{lowest} \\ $$$${term}.\:{Find}\:{r}+{s}\:? \\ $$

Question Number 116436 Answers: 3 Comments: 2

... advanced calculus... evaluate :: I= ∫_( 0) ^( ∞) (((sin(x).sin(2x))/x)) dx =??? ... m.n.1970...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{advanced}\:\:{calculus}... \\ $$$$\:\:\:\:\:\:{evaluate}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}=\:\int_{\:\mathrm{0}} ^{\:\:\infty} \:\:\left(\frac{{sin}\left({x}\right).{sin}\left(\mathrm{2}{x}\right)}{{x}}\right)\:{dx}\:=???\: \\ $$$$\:\:\:\:\:\:\:\:\:...\:{m}.{n}.\mathrm{1970}... \\ $$$$\: \\ $$$$ \\ $$

Question Number 116391 Answers: 1 Comments: 7

∫ (dx/((x+1)(√x) )) ?

$$\int\:\frac{\mathrm{dx}}{\left(\mathrm{x}+\mathrm{1}\right)\sqrt{\mathrm{x}}\:}\:? \\ $$

Question Number 116385 Answers: 3 Comments: 0

∫ ((xe^x )/( (√(1+e^x )))) dx

$$\:\int\:\frac{\mathrm{xe}^{\mathrm{x}} }{\:\sqrt{\mathrm{1}+\mathrm{e}^{\mathrm{x}} }}\:\mathrm{dx}\: \\ $$

Question Number 116376 Answers: 2 Comments: 0

Let f be a function defined on non zero real numbers such that ((27 f(−x))/x) −x^2 f((1/x)) =−2x^2 for ∀x ≠ 0 . Find → { ((f(x))),((f(3))) :} ?

$$\mathrm{Let}\:\mathrm{f}\:\mathrm{be}\:\mathrm{a}\:\mathrm{function}\:\mathrm{defined}\:\mathrm{on}\:\mathrm{non}\:\mathrm{zero}\:\: \\ $$$$\mathrm{real}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\:\frac{\mathrm{27}\:\mathrm{f}\left(−\mathrm{x}\right)}{\mathrm{x}}\:−\mathrm{x}^{\mathrm{2}} \:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:=−\mathrm{2x}^{\mathrm{2}} \\ $$$$\mathrm{for}\:\forall\mathrm{x}\:\neq\:\mathrm{0}\:.\:\mathrm{Find}\:\rightarrow\begin{cases}{\mathrm{f}\left(\mathrm{x}\right)}\\{\mathrm{f}\left(\mathrm{3}\right)}\end{cases}\:?\: \\ $$

Question Number 116375 Answers: 2 Comments: 0

... nice calculus... ordinary differential equation(o.d.e) y(d^2 y/dx^2 ) −((dy/dx))^2 =y^2 (lny) ... find : general solution ..m.n.1970..

$$\:\:\:\:\:\:\:...\:\:{nice}\:\:{calculus}... \\ $$$$\:{ordinary}\:{differential} \\ $$$${equation}\left({o}.{d}.{e}\right) \\ $$$$\:\:\: \\ $$$$\:\:\:{y}\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:−\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} ={y}^{\mathrm{2}} \left({lny}\right)\:\:... \\ $$$$\:\:\:{find}\::\:\:{general}\:\:{solution} \\ $$$$\:\:\:\:\:..{m}.{n}.\mathrm{1970}.. \\ $$

Question Number 116374 Answers: 2 Comments: 0

Determine the maximum value of ((1+cos x)/(sin x+cos x+2)) where x ranges over all real numbers.

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\frac{\mathrm{1}+\mathrm{cos}\:\mathrm{x}}{\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}+\mathrm{2}}\:\mathrm{where}\:\mathrm{x}\:\mathrm{ranges}\:\mathrm{over}\:\mathrm{all} \\ $$$$\mathrm{real}\:\mathrm{numbers}. \\ $$

Question Number 116367 Answers: 2 Comments: 0

Question Number 116365 Answers: 1 Comments: 0

Question Number 116363 Answers: 0 Comments: 0

Question Number 116361 Answers: 1 Comments: 1

Question Number 116360 Answers: 1 Comments: 0

A five digits number divisible by 3 is to be formed using the number 0,1,2,3,4 and 5 without repetition The total number of ways this can be done is __

$$\mathrm{A}\:\mathrm{five}\:\mathrm{digits}\:\mathrm{number}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{3} \\ $$$$\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{using}\:\mathrm{the}\:\mathrm{number}\: \\ $$$$\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}\:\mathrm{and}\:\mathrm{5}\:\mathrm{without}\:\mathrm{repetition} \\ $$$$\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{this}\:\mathrm{can} \\ $$$$\mathrm{be}\:\mathrm{done}\:\mathrm{is}\:\_\_ \\ $$

Question Number 116359 Answers: 1 Comments: 0

If 0 < θ < (π/4) such that cosec θ−sec θ=((√(13))/6) then cot θ−tan θ equals to __

$$\mathrm{If}\:\mathrm{0}\:<\:\theta\:<\:\frac{\pi}{\mathrm{4}}\:\mathrm{such}\:\mathrm{that}\:\mathrm{cosec}\:\theta−\mathrm{sec}\:\theta=\frac{\sqrt{\mathrm{13}}}{\mathrm{6}} \\ $$$$\mathrm{then}\:\mathrm{cot}\:\theta−\mathrm{tan}\:\theta\:\mathrm{equals}\:\mathrm{to}\:\_\_ \\ $$

Question Number 116358 Answers: 2 Comments: 0

Given that ((17−((27)/4)(√6)))^(1/(3 )) and ((17+((27)/4)(√6)))^(1/(3 )) are the roots of the equation x^2 −ax+b = 0. Find the value of ab.

$$\mathrm{Given}\:\mathrm{that}\:\sqrt[{\mathrm{3}\:}]{\mathrm{17}−\frac{\mathrm{27}}{\mathrm{4}}\sqrt{\mathrm{6}}}\:\mathrm{and}\:\sqrt[{\mathrm{3}\:}]{\mathrm{17}+\frac{\mathrm{27}}{\mathrm{4}}\sqrt{\mathrm{6}}} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{ax}+\mathrm{b}\:=\:\mathrm{0}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{ab}. \\ $$

Question Number 116356 Answers: 0 Comments: 0

Let S ={1,2,3,4,...,48,49} .What is the maximum value of n such that it is possible to select n numbers from S and arrange them in a circle in such a way that the product of any two adjacent numbers in the circle is less than 100?

$$\mathrm{Let}\:\mathrm{S}\:=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},...,\mathrm{48},\mathrm{49}\right\}\:.\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{it}\:\mathrm{is}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{select}\:\mathrm{n}\:\mathrm{numbers}\: \\ $$$$\mathrm{from}\:\mathrm{S}\:\mathrm{and}\:\mathrm{arrange}\:\mathrm{them}\:\mathrm{in}\:\mathrm{a}\:\mathrm{circle}\: \\ $$$$\mathrm{in}\:\mathrm{such}\:\mathrm{a}\:\mathrm{way}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of} \\ $$$$\mathrm{any}\:\mathrm{two}\:\mathrm{adjacent}\:\mathrm{numbers}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{circle}\:\mathrm{is}\:\mathrm{less}\:\mathrm{than}\:\mathrm{100}? \\ $$

Question Number 116355 Answers: 0 Comments: 0

the speed v of a projectile launched at an angle α to the horizontal is thought to be given by : v = [(1/k)(A + ((sin αt)/g)) + e^(−C/t) ] where k,A, α and C are constants, t is time and g is the gravitational acceleration. determine the base units of A, α and C

$$\mathrm{the}\:\mathrm{speed}\:{v}\:\mathrm{of}\:\mathrm{a}\:\mathrm{projectile}\:\mathrm{launched}\:\mathrm{at} \\ $$$$\mathrm{an}\:\mathrm{angle}\:\alpha\:\mathrm{to}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{is}\:\mathrm{thought}\:\mathrm{to} \\ $$$$\mathrm{be}\:\mathrm{given}\:\mathrm{by}\:: \\ $$$$\:\:\:\:\:\:\:{v}\:=\:\left[\frac{\mathrm{1}}{{k}}\left({A}\:+\:\frac{\mathrm{sin}\:\alpha{t}}{\mathrm{g}}\right)\:+\:{e}^{−{C}/{t}} \right] \\ $$$$\mathrm{where}\:{k},{A},\:\:\alpha\:\mathrm{and}\:{C}\:\mathrm{are}\:\mathrm{constants},\:{t}\:\mathrm{is}\:\mathrm{time} \\ $$$$\mathrm{and}\:\mathrm{g}\:\mathrm{is}\:\mathrm{the}\:\mathrm{gravitational}\:\mathrm{acceleration}. \\ $$$$\mathrm{determine}\:\mathrm{the}\:\mathrm{base}\:\mathrm{units}\:\mathrm{of}\:{A},\:\alpha\:\mathrm{and}\:{C} \\ $$

Question Number 116348 Answers: 1 Comments: 2

Find the minimum value of (((5+x)(2+x)^2 )/(x(1+x))) (x≠−1,0)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\frac{\left(\mathrm{5}+\mathrm{x}\right)\left(\mathrm{2}+\mathrm{x}\right)^{\mathrm{2}} }{\mathrm{x}\left(\mathrm{1}+\mathrm{x}\right)}\:\:\left(\mathrm{x}\neq−\mathrm{1},\mathrm{0}\right) \\ $$

Question Number 116347 Answers: 1 Comments: 0

Question Number 116331 Answers: 2 Comments: 0

lim_(x→0) ((3x(cos 7x−cos 3x))/( (√(tan 2x+1))−(√(sin 2x+1)))) ?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3x}\left(\mathrm{cos}\:\mathrm{7x}−\mathrm{cos}\:\mathrm{3x}\right)}{\:\sqrt{\mathrm{tan}\:\mathrm{2x}+\mathrm{1}}−\sqrt{\mathrm{sin}\:\mathrm{2x}+\mathrm{1}}}\:? \\ $$

Question Number 116329 Answers: 1 Comments: 0

solve the diff equation (1)(dy/dx) = ((x+3y−5)/(x−y−1)) (2) (3y−7x−3)dx+(7y−3x−7)dy=0

$$\mathrm{solve}\:\mathrm{the}\:\mathrm{diff}\:\mathrm{equation}\:\: \\ $$$$\left(\mathrm{1}\right)\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\frac{\mathrm{x}+\mathrm{3y}−\mathrm{5}}{\mathrm{x}−\mathrm{y}−\mathrm{1}} \\ $$$$\left(\mathrm{2}\right)\:\left(\mathrm{3y}−\mathrm{7x}−\mathrm{3}\right)\mathrm{dx}+\left(\mathrm{7y}−\mathrm{3x}−\mathrm{7}\right)\mathrm{dy}=\mathrm{0} \\ $$

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