Question and Answers Forum

AllQuestion and Answers: Page 1957

Question Number 6099 Answers: 2 Comments: 1

∫_0 ^∞ e^(−st) sinh atdt=? or... −(1/s)[e^(−st) sinh at]_0 ^∞ =? −(a/s^2 )[e^(−st) coth at]_0 ^∞ =?

$$\int_{\mathrm{0}} ^{\infty} {e}^{−{st}} \mathrm{sinh}\:{atdt}=? \\ $$$$\mathrm{or}... \\ $$$$−\frac{\mathrm{1}}{{s}}\left[{e}^{−{st}} \mathrm{sinh}\:{at}\right]_{\mathrm{0}} ^{\infty} =? \\ $$$$−\frac{{a}}{{s}^{\mathrm{2}} }\left[{e}^{−{st}} \mathrm{coth}\:{at}\right]_{\mathrm{0}} ^{\infty} =? \\ $$

Question Number 6098 Answers: 0 Comments: 0

A party of 7 members is to be chosen from a group of 6 ladies and 5 gents. In how many ways can the party be formed if it is to contain (i) exactly 4 ladies (i) at least 4 ladies (i) at most 4 ladies ?

$$\mathrm{A}\:\mathrm{party}\:\mathrm{of}\:\:\mathrm{7}\:\:\:\mathrm{members}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{chosen} \\ $$$$\mathrm{from}\:\mathrm{a}\:\mathrm{group}\:\mathrm{of}\:\:\mathrm{6}\:\:\mathrm{ladies}\:\mathrm{and}\:\:\mathrm{5}\:\:\mathrm{gents}. \\ $$$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{party}\:\mathrm{be} \\ $$$$\mathrm{formed}\:\mathrm{if}\:\mathrm{it}\:\mathrm{is}\:\mathrm{to}\:\mathrm{contain} \\ $$$$\left(\mathrm{i}\right)\:\:\:\mathrm{exactly}\:\mathrm{4}\:\:\mathrm{ladies} \\ $$$$\left(\mathrm{i}\right)\:\:\:\mathrm{at}\:\mathrm{least}\:\mathrm{4}\:\:\mathrm{ladies} \\ $$$$\left(\mathrm{i}\right)\:\:\:\mathrm{at}\:\mathrm{most}\:\mathrm{4}\:\:\mathrm{ladies}\:? \\ $$

Question Number 6111 Answers: 1 Comments: 0

log^2 x+logx^2 −3=0 , x=?

$${log}^{\mathrm{2}} {x}+{logx}^{\mathrm{2}} −\mathrm{3}=\mathrm{0}\:,\:{x}=? \\ $$

Question Number 6146 Answers: 1 Comments: 0

Determine the value of sin ((Π/3)+p)cos ((Π/6)+p)−cos ((Π/3)+p)sin ((Π/6)+p)

$${Determine}\:{the}\:{value}\:{of}\:\mathrm{sin}\:\left(\frac{\Pi}{\mathrm{3}}+{p}\right)\mathrm{cos}\:\left(\frac{\Pi}{\mathrm{6}}+{p}\right)−\mathrm{cos}\:\left(\frac{\Pi}{\mathrm{3}}+{p}\right)\mathrm{sin}\:\left(\frac{\Pi}{\mathrm{6}}+{p}\right) \\ $$

Question Number 6145 Answers: 1 Comments: 0

show that sin 64^(° ) ×cos 26^° + cos 64^(° ) × sin 26^° =1

$${show}\:{that}\:\mathrm{sin}\:\mathrm{64}^{°\:} \:×\mathrm{cos}\:\:\mathrm{26}^{°} \:+\:\mathrm{cos}\:\mathrm{64}^{°\:} ×\:\mathrm{sin}\:\mathrm{26}^{°} \:=\mathrm{1}\: \\ $$$$ \\ $$

Question Number 6090 Answers: 1 Comments: 0

Question Number 6086 Answers: 1 Comments: 0

Find the solution of the differential equation (y − x + 1)dy − (y + x + 2)dx = 0

$${Find}\:{the}\:{solution}\:{of}\:{the}\:{differential}\:{equation}\: \\ $$$$\left({y}\:−\:{x}\:+\:\mathrm{1}\right){dy}\:−\:\left({y}\:+\:{x}\:+\:\mathrm{2}\right){dx}\:=\:\mathrm{0} \\ $$$$ \\ $$

Question Number 6087 Answers: 0 Comments: 2

Given the terms a_k and a_(k + r ) of an A.P find a_1 and a_(k + r) in terms of a_(k ) and a_(k + r)

$${Given}\:{the}\:{terms}\:\:\:\:{a}_{{k}} \:\:\:{and}\:\:{a}_{{k}\:+\:{r}\:} \:\:{of}\:{an}\:{A}.{P} \\ $$$${find}\:\:{a}_{\mathrm{1}} \:\:{and}\:\:{a}_{{k}\:+\:{r}} \:\:{in}\:{terms}\:{of}\:\:{a}_{{k}\:} \:{and}\:\:{a}_{{k}\:+\:{r}} \\ $$

Question Number 6071 Answers: 0 Comments: 2

SHARING: Martin Gardner (famous mathematician) found a way to write his name so that it can also be read upside down. See the comment below

$$\mathcal{SHARING}: \\ $$$$\mathrm{Martin}\:\mathrm{Gardner}\:\left(\mathrm{famous}\:\mathrm{mathematician}\right) \\ $$$$\mathrm{found}\:\mathrm{a}\:\mathrm{way}\:\mathrm{to}\:\mathrm{write}\:\mathrm{his}\:\mathrm{name}\:\mathrm{so}\:\mathrm{that}\:\mathrm{it}\:\mathrm{can} \\ $$$$\mathrm{also}\:\mathrm{be}\:\mathrm{read}\:\mathrm{upside}\:\:\mathrm{down}. \\ $$$$\mathrm{See}\:\mathrm{the}\:\mathrm{comment}\:\mathrm{below} \\ $$

Question Number 6058 Answers: 0 Comments: 1

sin(a)=sin(−a) For what values of a is the statement true?

$$\mathrm{sin}\left({a}\right)=\mathrm{sin}\left(−{a}\right) \\ $$$$\mathrm{For}\:\mathrm{what}\:\mathrm{values}\:\mathrm{of}\:{a}\:\mathrm{is}\:\mathrm{the}\:\mathrm{statement}\:\mathrm{true}? \\ $$

Question Number 6056 Answers: 1 Comments: 3

Question Number 6055 Answers: 1 Comments: 0

ln(x)+x=a x=?

$${ln}\left({x}\right)+{x}={a} \\ $$$${x}=? \\ $$

Question Number 6054 Answers: 1 Comments: 2

Determine distance between opposite corners of a cubic room of dimention x units.

$$\mathcal{D}{etermine}\:{distance}\:{between} \\ $$$${opposite}\:{corners}\:\:{of}\:{a}\:{cubic} \\ $$$${room}\:{of}\:{dimention}\:{x}\:{units}. \\ $$$$ \\ $$

Question Number 6045 Answers: 1 Comments: 1

4^x = 2x find x

$$\mathrm{4}^{{x}} \:=\:\mathrm{2}{x} \\ $$$$ \\ $$$${find}\:{x} \\ $$

Question Number 6044 Answers: 0 Comments: 2

Find the locus in the complex plain such that arg ((z/(z + 2))) = (Π/2) please help.

$${Find}\:{the}\:{locus}\:{in}\:{the}\:{complex}\:{plain}\:{such}\:{that}\: \\ $$$${arg}\:\left(\frac{{z}}{{z}\:+\:\mathrm{2}}\right)\:=\:\frac{\Pi}{\mathrm{2}} \\ $$$$ \\ $$$${please}\:{help}. \\ $$

Question Number 6043 Answers: 1 Comments: 0

The number of positive integral pairs satisfying the equation tan^(−1) a + tan^(−1) b = tan^(−1) 3 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integral}\:\mathrm{pairs} \\ $$$$\mathrm{satisfying}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{tan}^{−\mathrm{1}} {a}\:+\:\mathrm{tan}^{−\mathrm{1}} {b}\:=\:\mathrm{tan}^{−\mathrm{1}} \mathrm{3}\:\:\mathrm{is} \\ $$

Question Number 6042 Answers: 0 Comments: 4

Can you solve the indefinite integral: ∫e^(−u) u^n du

$$\mathrm{Can}\:\mathrm{you}\:\mathrm{solve}\:\mathrm{the}\:\mathrm{indefinite}\:\mathrm{integral}: \\ $$$$\int{e}^{−{u}} {u}^{{n}} {du} \\ $$

Question Number 6028 Answers: 0 Comments: 2

Can you please show me how to solve: L=lim_(n→∞) (x^n /(n!))

$$\mathrm{Can}\:\mathrm{you}\:\mathrm{please}\:\mathrm{show}\:\mathrm{me}\:\mathrm{how}\:\mathrm{to}\:\mathrm{solve}: \\ $$$${L}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}^{{n}} }{{n}!} \\ $$

Question Number 6027 Answers: 0 Comments: 3

x^x =e^(xln x) e^x =1+x+(x^2 /(2!))+(x^3 /(3!))+... ∴ e^(xln x) = 1+xln(x)+(((xln x)^2 )/(2!))+... e^(xln x) = (((xln x)^0 )/(0!))+(((xln x)^1 )/(1!))+(((xln x)^2 )/(2!))+... x^x =e^(xln x) =Σ_(n=0) ^∞ ((x^n (ln x)^n )/(n!)) Question: ∫_0 ^( 1) x^x dx=Σ_(i=0) ^∞ ∫_0 ^1 ((x^n (ln x)^n )/(n!))dx by substituting: x=exp(−(u/(n+1))), 0<u<∞ show that: ∫_0 ^( 1) x^n (ln x)^n =(−1)^n (n+1)^(−(n+1)) ∫_0 ^( ∞) u^n e^(−u) du

$${x}^{{x}} ={e}^{{x}\mathrm{ln}\:{x}} \\ $$$${e}^{{x}} =\mathrm{1}+{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+... \\ $$$$\therefore\:{e}^{{x}\mathrm{ln}\:{x}} \:=\:\mathrm{1}+{x}\mathrm{ln}\left({x}\right)+\frac{\left({x}\mathrm{ln}\:{x}\right)^{\mathrm{2}} }{\mathrm{2}!}+... \\ $$$${e}^{{x}\mathrm{ln}\:{x}} \:=\:\frac{\left({x}\mathrm{ln}\:{x}\right)^{\mathrm{0}} }{\mathrm{0}!}+\frac{\left({x}\mathrm{ln}\:{x}\right)^{\mathrm{1}} }{\mathrm{1}!}+\frac{\left({x}\mathrm{ln}\:{x}\right)^{\mathrm{2}} }{\mathrm{2}!}+... \\ $$$${x}^{{x}} ={e}^{{x}\mathrm{ln}\:{x}} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} \left(\mathrm{ln}\:{x}\right)^{{n}} }{{n}!} \\ $$$$\mathrm{Question}: \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{x}} {dx}=\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{x}^{{n}} \left(\mathrm{ln}\:{x}\right)^{{n}} }{{n}!}{dx} \\ $$$$\mathrm{by}\:\mathrm{substituting}: \\ $$$${x}=\mathrm{exp}\left(−\frac{{u}}{{n}+\mathrm{1}}\right),\:\:\:\:\:\mathrm{0}<{u}<\infty \\ $$$$\mathrm{show}\:\mathrm{that}: \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{n}} \left(\mathrm{ln}\:{x}\right)^{{n}} =\left(−\mathrm{1}\right)^{{n}} \left({n}+\mathrm{1}\right)^{−\left({n}+\mathrm{1}\right)} \int_{\mathrm{0}} ^{\:\infty} {u}^{{n}} {e}^{−{u}} {du} \\ $$

Question Number 6021 Answers: 1 Comments: 3

2^x + 2x = 8 find the value of x

$$\mathrm{2}^{{x}} \:+\:\mathrm{2}{x}\:=\:\mathrm{8}\: \\ $$$$ \\ $$$${find}\:{the}\:{value}\:{of}\:{x} \\ $$

Question Number 6020 Answers: 0 Comments: 0