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Question Number 153598 Answers: 0 Comments: 3

lim_(x→∞) cos (nπ (e)^(1/(2n)) )=?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}cos}\:\left({n}\pi\:\sqrt[{\mathrm{2}{n}}]{{e}}\:\right)=? \\ $$

Question Number 153736 Answers: 1 Comments: 0

Given that 7 cos 2θ+24 sin^2 θ=R cos(2θ−α), where R>0 and 0<α<(π/2), find the maximum value of 14 cos^2 θ+48 sin θ cos θ.

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{7}\:\mathrm{cos}\:\mathrm{2}\theta+\mathrm{24}\:\mathrm{sin}^{\mathrm{2}} \theta={R}\:\mathrm{cos}\left(\mathrm{2}\theta−\alpha\right), \\ $$$$\mathrm{where}\:{R}>\mathrm{0}\:\mathrm{and}\:\mathrm{0}<\alpha<\frac{\pi}{\mathrm{2}},\:\mathrm{find}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{14}\:\mathrm{cos}^{\mathrm{2}} \theta+\mathrm{48}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta. \\ $$

Question Number 153593 Answers: 1 Comments: 0

Solve : (sin(2x))! = 2

$$\boldsymbol{{Solve}}\::\:\left(\boldsymbol{{sin}}\left(\mathrm{2}\boldsymbol{{x}}\right)\right)!\:=\:\mathrm{2} \\ $$

Question Number 153611 Answers: 3 Comments: 0

Question Number 153596 Answers: 0 Comments: 5

Question Number 153583 Answers: 4 Comments: 3

If fog(x)=((2x−1)/x) and g(x)=5x+2, find f(x).

$$\mathrm{If}\:{fog}\left({x}\right)=\frac{\mathrm{2}{x}−\mathrm{1}}{{x}}\:\mathrm{and}\:\mathrm{g}\left({x}\right)=\mathrm{5}{x}+\mathrm{2},\:\mathrm{find} \\ $$$${f}\left({x}\right). \\ $$

Question Number 153580 Answers: 1 Comments: 0

Question Number 153574 Answers: 1 Comments: 0

1) the radius of a spherical ballon is increasing at the rate of 3cms^(−1) . find the rate in the surface area of the ballon when its radius is 10cm (2) evaluate ∫(5/((3x−2)^4 ))dx

$$\left.\:\mathrm{1}\right)\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{a}\:\mathrm{spherical}\:\mathrm{ballon}\:\mathrm{is}\: \\ $$$$\mathrm{increasing}\:\mathrm{at}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{3cms}^{−\mathrm{1}} \:.\:\mathrm{find} \\ $$$$\mathrm{the}\:\mathrm{rate}\:\mathrm{in}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ballon} \\ $$$$\mathrm{when}\:\mathrm{its}\:\mathrm{radius}\:\mathrm{is}\:\mathrm{10cm} \\ $$$$ \\ $$$$\left(\mathrm{2}\right)\:\mathrm{evaluate}\:\int\frac{\mathrm{5}}{\left(\mathrm{3x}−\mathrm{2}\right)^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 153573 Answers: 1 Comments: 0

A commitee of 5 men and 3 women is to be selected from 10 men and 8 women. In how many ways can this be done if a particular women must be in the committee

$$\mathrm{A}\:\mathrm{commitee}\:\mathrm{of}\:\mathrm{5}\:\mathrm{men}\:\mathrm{and}\:\mathrm{3}\:\mathrm{women}\:\mathrm{is}\: \\ $$$$\mathrm{to}\:\mathrm{be}\:\mathrm{selected}\:\mathrm{from}\:\mathrm{10}\:\mathrm{men}\:\mathrm{and}\:\mathrm{8}\:\mathrm{women}. \\ $$$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{this}\:\mathrm{be}\:\mathrm{done}\:\mathrm{if} \\ $$$$\mathrm{a}\:\mathrm{particular}\:\mathrm{women}\:\mathrm{must}\:\mathrm{be}\:\mathrm{in}\:\mathrm{the}\: \\ $$$$\mathrm{committee} \\ $$

Question Number 153572 Answers: 2 Comments: 1

In how many ways can 6 players be lined up if 2 particlar players must not stand next to each other

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\:\mathrm{can}\:\mathrm{6}\:\mathrm{players}\:\mathrm{be}\: \\ $$$$\mathrm{lined}\:\mathrm{up}\:\mathrm{if}\:\mathrm{2}\:\mathrm{particlar}\:\mathrm{players}\:\mathrm{must}\: \\ $$$$\mathrm{not}\:\mathrm{stand}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other} \\ $$

Question Number 153571 Answers: 1 Comments: 0

Find lim_(n→∞) n∙((π^2 /4) - a_n ^2 ) = ? where a_n =Σ_(k=1) ^n arctan((1/(k^2 -k+1)))

$$\mathrm{Find}\:\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}n}\centerdot\left(\frac{\pi^{\mathrm{2}} }{\mathrm{4}}\:-\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} ^{\mathrm{2}} \right)\:=\:? \\ $$$$\mathrm{where}\:\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} =\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\mathrm{arctan}\left(\frac{\mathrm{1}}{\mathrm{k}^{\mathrm{2}} -\mathrm{k}+\mathrm{1}}\right) \\ $$

Question Number 153568 Answers: 1 Comments: 0

solve in x∈C sin x=(3/2)

$${solve}\:{in}\:{x}\in\mathbb{C} \\ $$$$\:\mathrm{sin}\:{x}=\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$ \\ $$

Question Number 153565 Answers: 1 Comments: 4

define d(n) to be the sum of the digits of n. i.e d(1000) = 1 , d(999) = 27 find d(d(d(d(d(5^(10^(100) ) )))))

$$\: \\ $$$$\:\mathrm{define}\:{d}\left({n}\right)\:\mathrm{to}\:\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\: \\ $$$$\:\mathrm{of}\:{n}.\:\:\: \\ $$$$\:\mathrm{i}.\mathrm{e}\:\:{d}\left(\mathrm{1000}\right)\:=\:\mathrm{1}\:,\:\:\:\:{d}\left(\mathrm{999}\right)\:=\:\mathrm{27} \\ $$$$\: \\ $$$$\:\mathrm{find}\:\:{d}\left({d}\left({d}\left({d}\left({d}\left(\mathrm{5}^{\mathrm{10}^{\mathrm{100}} } \right)\right)\right)\right)\right) \\ $$$$\: \\ $$

Question Number 153563 Answers: 1 Comments: 0

∫ (1/(1+(√(2x)) )) dx =?

$$\int\:\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}{x}}\:}\:{dx}\:=? \\ $$

Question Number 153555 Answers: 1 Comments: 0

Ω := ∫_0 ^( (π/2)) cos(2x).ln(sin(x))dx=^? −(π/4) solution (1 ) Ω := ∫_0 ^( (π/2)) ( 2cos^( 2) (x)−1)ln(sin(x))dx := 2∫_0 ^( (π/2)) cos^( 2) (x).ln(sin(x))dx−∫_0 ^( (π/2)) ln(sin(x))dx we know that : ∫_0 ^(π/2) ln(sin(x))dx=_(earlier) ^(derived) ((−π)/2) ln(2) ∫_0 ^( (π/2)) cos^( 2) (x).ln(sin(x))dx=_(posts) ^(previous) −(π/4)ln(2)−(π/8) ∴ Ω := −(π/2) ln(2) −(π/4) +(π/2) ln(2) ◂ Ω =− (π/4) ▶ m.n

$$ \\ $$$$\:\:\:\:\:\Omega\::=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {cos}\left(\mathrm{2}{x}\right).{ln}\left({sin}\left({x}\right)\right){dx}\overset{?} {=}\:−\frac{\pi}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:{solution}\:\left(\mathrm{1}\:\right) \\ $$$$\:\:\:\:\:\Omega\::=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \left(\:\mathrm{2}{cos}^{\:\mathrm{2}} \left({x}\right)−\mathrm{1}\right){ln}\left({sin}\left({x}\right)\right){dx} \\ $$$$\:\:\:\:\:\::=\:\mathrm{2}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {cos}^{\:\mathrm{2}} \left({x}\right).{ln}\left({sin}\left({x}\right)\right){dx}−\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {ln}\left({sin}\left({x}\right)\right){dx} \\ $$$$\:\:\:\:\:{we}\:{know}\:{that}\::\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({sin}\left({x}\right)\right){dx}\underset{{earlier}} {\overset{{derived}} {=}}\:\frac{−\pi}{\mathrm{2}}\:{ln}\left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {cos}^{\:\mathrm{2}} \left({x}\right).{ln}\left({sin}\left({x}\right)\right){dx}\underset{{posts}} {\overset{{previous}} {=}}\:−\frac{\pi}{\mathrm{4}}{ln}\left(\mathrm{2}\right)−\frac{\pi}{\mathrm{8}} \\ $$$$\:\:\:\:\:\:\therefore\:\:\:\Omega\::=\:−\frac{\pi}{\mathrm{2}}\:{ln}\left(\mathrm{2}\right)\:−\frac{\pi}{\mathrm{4}}\:+\frac{\pi}{\mathrm{2}}\:{ln}\left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\blacktriangleleft\:\:\:\:\Omega\:=−\:\frac{\pi}{\mathrm{4}}\:\:\blacktriangleright\:\:\:\:\:\:{m}.{n} \\ $$

Question Number 153553 Answers: 1 Comments: 0

sin(9) + sin(21)+sin(39)=^? (ϕ/( (√2))) ϕ:= golden ratio m.n

$$ \\ $$$$\:{sin}\left(\mathrm{9}\right)\:+\:{sin}\left(\mathrm{21}\right)+{sin}\left(\mathrm{39}\right)\overset{?} {=}\frac{\varphi}{\:\sqrt{\mathrm{2}}} \\ $$$$\:\:\:\varphi:=\:{golden}\:{ratio} \\ $$$$\:{m}.{n} \\ $$

Question Number 153542 Answers: 1 Comments: 0

y′′′+y′=sec x

$$\:{y}'''+{y}'=\mathrm{sec}\:{x}\: \\ $$

Question Number 153537 Answers: 1 Comments: 0