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Question Number 58364 Answers: 2 Comments: 0

Let a_n = 10 × ((1/(√2)))^n a_n is a geometrical sequence S_n = a_0 + a_1 + ... + a_(n−1) S_n = 10 × ((1 − ((1/(√2)))^n )/(1 − ((1/(√2))))) Proove that : S_n = ((10(√2))/((√2) − 1)) × (1−((1/(√2)))^n )

$$\mathrm{Let}\:{a}_{{n}} =\:\mathrm{10}\:×\:\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{{n}} \\ $$$$\:\:\:\:{a}_{{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{geometrical}\:\mathrm{sequence} \\ $$$$\:\:{S}_{{n}} \:=\:{a}_{\mathrm{0}} \:+\:{a}_{\mathrm{1}} \:+\:...\:+\:{a}_{{n}−\mathrm{1}} \\ $$$$\:\:\:\:\:{S}_{{n}} =\:\mathrm{10}\:×\:\frac{\mathrm{1}\:−\:\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{{n}} }{\mathrm{1}\:−\:\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)} \\ $$$$\boldsymbol{\mathrm{Proove}}\:\boldsymbol{\mathrm{that}}\:: \\ $$$$ \\ $$$${S}_{{n}} \:=\:\frac{\mathrm{10}\sqrt{\mathrm{2}}}{\sqrt{\mathrm{2}}\:−\:\mathrm{1}}\:×\:\left(\mathrm{1}−\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{{n}} \right) \\ $$$$ \\ $$

Question Number 58362 Answers: 1 Comments: 0

∫((sin x)/(sin 3x))dx

$$\int\frac{\mathrm{sin}\:{x}}{\mathrm{sin}\:\mathrm{3}{x}}{dx} \\ $$

Question Number 58361 Answers: 0 Comments: 0

the molar heat capacity of constant pressure of a gas varies with the temperature according to the equation n c_(p=a+bθ−c/θ^(2 _ ) ) where a.b and c are constant how much heat transfered during an isobaric process in which n makes of gas undergo a temperature rise from θ_i to θ_f ? (b)the molar heat capacity of a metal at low temperature varies with the temperature according to the equation c=bθ+(a/h)θ^3 where a and b are constant .how much heat p_(γ ) make is transfered during a process in which the tempereture change from 0.01(h)to 0.02(h)

$${the}\:{molar}\:{heat}\:{capacity}\:{of}\:{constant} \\ $$$${pressure}\:{of}\:{a}\:{gas}\:{varies}\:{with}\:{the} \\ $$$${temperature}\:{according}\:{to}\:{the}\:{equation} \\ $$$${n}\:{c}_{{p}={a}+{b}\theta−{c}/\theta^{\mathrm{2}\:\:\:\:\:\:\underset{} {\:}} } \\ $$$${where}\:{a}.{b}\:{and}\:{c}\:{are}\:{constant}\:{how}\:{much}\:{heat}\:{transfered}\:{during}\:{an}\:{isobaric}\:{process}\:{in}\:{which}\:{n}\:{makes}\:{of}\:{gas}\:{undergo}\:{a}\:{temperature}\:{rise}\:{from}\:\theta_{{i}} \:\:{to}\:\theta_{{f}} \:? \\ $$$$\left({b}\right){the}\:{molar}\:{heat}\:{capacity}\:{of}\:{a}\:{metal}\:{at}\:{low}\:{temperature}\:{varies}\:{with}\:{the}\:{temperature}\:{according}\:{to}\:{the}\:{equation}\:{c}={b}\theta+\frac{{a}}{{h}}\theta^{\mathrm{3}} \:\:{where}\:{a}\:{and}\:{b}\:{are}\:{constant}\:.{how}\:{much}\:{heat}\:{p}_{\gamma\:\:} \:\:{make}\:{is}\:{transfered}\:{during}\:{a}\:{process}\:{in}\:{which}\:{the}\:{tempereture}\:{change}\:{from}\:\mathrm{0}.\mathrm{01}\left({h}\right){to}\:\mathrm{0}.\mathrm{02}\left({h}\right) \\ $$

Question Number 58358 Answers: 0 Comments: 0

knowing that: cos(x)=Σ_(n=1) ^∞ (((−1)^n x^(2n) )/((2n)!)) sin(x)=Σ_(n=1) ^∞ (((−1)^n x^(2n+1) )/((2n +1)!)) prof that: cos(x+y)= cos(x)cos(y)−sin(x)sin(y)

$${knowing}\:{that}:\: \\ $$$${cos}\left({x}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\left(\mathrm{2}{n}\right)!}\:\:\:\: \\ $$$${sin}\left({x}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}\:+\mathrm{1}\right)!} \\ $$$${prof}\:{that}:\:{cos}\left({x}+{y}\right)=\:{cos}\left({x}\right){cos}\left({y}\right)−{sin}\left({x}\right){sin}\left({y}\right) \\ $$

Question Number 58350 Answers: 0 Comments: 2

Question Number 58346 Answers: 2 Comments: 0

∫((cos(x))/(cos(2016°+cos(x)))×dx

$$\int\frac{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}{\boldsymbol{\mathrm{cos}}\left(\mathrm{2016}°+\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\right.}×\boldsymbol{\mathrm{dx}} \\ $$

Question Number 58354 Answers: 1 Comments: 0

let U_n =((1^2 +2^2 +3^2 +....+n^2 )/(1^4 +2^4 +3^4 +....+n^4 )) 1)find lim_(n→+∞) U_n 2) calculate Σ_(n=1) ^∞ U_n

$${let}\:{U}_{{n}} =\frac{\mathrm{1}^{\mathrm{2}} \:+\mathrm{2}^{\mathrm{2}} \:+\mathrm{3}^{\mathrm{2}} \:+....+{n}^{\mathrm{2}} }{\mathrm{1}^{\mathrm{4}} \:+\mathrm{2}^{\mathrm{4}} \:+\mathrm{3}^{\mathrm{4}} \:+....+{n}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right){find}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{U}_{{n}} \\ $$

Question Number 58319 Answers: 1 Comments: 0

∫x^n (lnx)^n dx

$$\int{x}^{{n}} \left(\mathrm{ln}{x}\right)^{{n}} {dx} \\ $$

Question Number 58309 Answers: 2 Comments: 3

Prove that lim_(x→0) ((1−cos(x)cos(x/2)cos(x/3)...)/x^2 )=(π^2 /(12))

$${Prove}\:{that}\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{\mathrm{1}−{cos}\left({x}\right){cos}\left({x}/\mathrm{2}\right){cos}\left({x}/\mathrm{3}\right)...}{{x}^{\mathrm{2}} }=\frac{\pi^{\mathrm{2}} }{\mathrm{12}} \\ $$

Question Number 58299 Answers: 1 Comments: 1

find ∫ (dx/((x^2 +x+1)^(3/2) ))

$${find}\:\int\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$

Question Number 58286 Answers: 0 Comments: 1

A small body with temperature θ and absorbtivity τ is placed in a large evaluated capacity whose interior walls are at a temperature θ_w . when θ_w −θ is small, show that the rate of heat transfer by radiation is Q^• = 4θ_w ^3 Aτδ(θ_w −θ).

$$\mathrm{A}\:\mathrm{small}\:\mathrm{body}\:\mathrm{with}\:\mathrm{temperature}\:\theta\:\mathrm{and} \\ $$$$\mathrm{absorbtivity}\:\tau\:\mathrm{is}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{a}\:\mathrm{large}\: \\ $$$$\mathrm{evaluated}\:\mathrm{capacity}\:\mathrm{whose}\:\mathrm{interior}\: \\ $$$$\mathrm{walls}\:\mathrm{are}\:\mathrm{at}\:\mathrm{a}\:\mathrm{temperature}\:\theta_{\mathrm{w}} . \\ $$$$\mathrm{when}\:\:\:\:\theta_{\mathrm{w}} −\theta\:\:\:\:\mathrm{is}\:\:\mathrm{small},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{rate}\:\mathrm{of}\:\mathrm{heat}\:\mathrm{transfer}\:\mathrm{by}\:\mathrm{radiation}\:\mathrm{is} \\ $$$$\:\:\:\:\:\overset{\bullet} {\mathrm{Q}}=\:\mathrm{4}\theta_{\mathrm{w}} ^{\mathrm{3}} \mathrm{A}\tau\delta\left(\theta_{\mathrm{w}} −\theta\right). \\ $$

Question Number 58282 Answers: 1 Comments: 0

A circle tangents to :x and y axes and x^(1/2) +y^(1/2) =a^(1/2) .find its radious.

Question Number 58276 Answers: 0 Comments: 0

Question Number 58267 Answers: 0 Comments: 13

lim_(x→0) [((1−cos (x)cos (2x)cos (3x)cos (4x))/x^2 )]

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{\mathrm{1}−\mathrm{cos}\:\left({x}\right)\mathrm{cos}\:\left(\mathrm{2}{x}\right)\mathrm{cos}\:\left(\mathrm{3}{x}\right)\mathrm{cos}\:\left(\mathrm{4}{x}\right)}{{x}^{\mathrm{2}} }\right] \\ $$

Question Number 58259 Answers: 4 Comments: 4

a .∫ (dx/(2sin^2 x+3tg^2 x))=? b .∫(( 1+(x)^(1/3) )/(1+(√x)+(x)^(1/3) +(x)^(1/6) ))dx=? c .∫ ((cosx)/(1+cos2x))dx=? d .∫ ((sin^2 x)/((√2)+(√3).cos^2 x))dx=?

Question Number 58257 Answers: 1 Comments: 1

Question Number 58254 Answers: 1 Comments: 0

The line 3x−2y−5=0 is parallel to a diameter of a circle x^2 +y^2 −4x+2y−4=0. find the equation of the diameter.

$$\mathrm{The}\:\mathrm{line}\:\mathrm{3x}−\mathrm{2y}−\mathrm{5}=\mathrm{0}\:\mathrm{is}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{a}\:\mathrm{diameter} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} −\mathrm{4x}+\mathrm{2y}−\mathrm{4}=\mathrm{0}.\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{diameter}. \\ $$

Question Number 58253 Answers: 1 Comments: 0

There are 100, 150 and 250 students in forms one, two and three, respectively in a school. If the mean ages of tbe students in the forms are 15.6 years, 16.8 years and 18years respectively, find i. the total number of students in the forms ii. correct to one decimal place, the mean age of all the students

$$\mathrm{There}\:\mathrm{are}\:\mathrm{100},\:\mathrm{150}\:\mathrm{and}\:\mathrm{250}\:\mathrm{students}\:\mathrm{in} \\ $$$$\mathrm{forms}\:\mathrm{one},\:\mathrm{two}\:\mathrm{and}\:\mathrm{three},\:\mathrm{respectively} \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{school}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{ages}\:\mathrm{of}\:\mathrm{tbe}\:\mathrm{students}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{forms}\:\mathrm{are}\:\mathrm{15}.\mathrm{6}\:\mathrm{years},\:\mathrm{16}.\mathrm{8}\:\mathrm{years}\:\mathrm{and} \\ $$$$\mathrm{18years}\:\mathrm{respectively},\:\mathrm{find} \\ $$$$\mathrm{i}.\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{students}\:\mathrm{in}\:\mathrm{the}\:\mathrm{forms} \\ $$$$\mathrm{ii}.\:\mathrm{correct}\:\mathrm{to}\:\mathrm{one}\:\mathrm{decimal}\:\mathrm{place},\:\mathrm{the}\:\mathrm{mean} \\ $$$$\mathrm{age}\:\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{students} \\ $$

Question Number 58250 Answers: 1 Comments: 0

∫_( 0) ^( 1) (((3x^3 − x^2 + 2x − 4)/(√(x^2 − 3x + 2)))) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\:\left(\frac{\mathrm{3x}^{\mathrm{3}} \:−\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{2x}\:−\:\mathrm{4}}{\sqrt{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{3x}\:+\:\mathrm{2}}}\right)\:\mathrm{dx} \\ $$

Question Number 58249 Answers: 0 Comments: 0

I_n ^ =∫_0 ^(π/2) cos^n xcos(nx)dx then show that I_1 ,I_2 ,I_3 ....are in G.P

$$\overset{} {{I}}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {cos}^{{n}} {xcos}\left({nx}\right){dx} \\ $$$${then}\:{show}\:{that}\:{I}_{\mathrm{1}} ,{I}_{\mathrm{2}} ,{I}_{\mathrm{3}} ....{are}\:{in}\:{G}.{P} \\ $$

Question Number 58248 Answers: 1 Comments: 0

leg A_1 ,A_2 ,...A_n and H_1 ,H_2 ,...H_n are n A.M′S and H.M′S respectively between a and b prove that A_r H_(n−r+1) =ab n≥r≥1

$${leg}\:{A}_{\mathrm{1}} ,{A}_{\mathrm{2}} ,...{A}_{{n}} \:{and}\:{H}_{\mathrm{1}} ,{H}_{\mathrm{2}} ,...{H}_{{n}} \:{are}\:{n}\:{A}.{M}'{S}\: \\ $$$${and}\:{H}.{M}'{S}\:{respectively}\:{between}\:{a}\:{and}\:{b} \\ $$$${prove}\:{that}\:{A}_{{r}} {H}_{{n}−{r}+\mathrm{1}} ={ab} \\ $$$$\:{n}\geqslant{r}\geqslant\mathrm{1} \\ $$

Question Number 58247 Answers: 0 Comments: 0

lim_(n→∞) ((S_1 S_n +S_2 S_(n−1) +S_3 S_(n−2) +...+S_n S_1 )/(S_1 ^2 +S_2 ^2 +...+S_n ^2 )) when S_n is sum of infinite series whose first term=n and common ratio (1/(n+1)) find the value of limit

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{S}_{\mathrm{1}} {S}_{{n}} +{S}_{\mathrm{2}} {S}_{{n}−\mathrm{1}} +{S}_{\mathrm{3}} {S}_{{n}−\mathrm{2}} +...+{S}_{{n}} {S}_{\mathrm{1}} }{{S}_{\mathrm{1}} ^{\mathrm{2}} +{S}_{\mathrm{2}} ^{\mathrm{2}} +...+{S}_{{n}} ^{\mathrm{2}} } \\ $$$${when}\:{S}_{{n}} \:{is}\:{sum}\:{of}\:{infinite}\:{series}\:{whose} \\ $$$${first}\:{term}={n}\:\:\:{and}\:{common}\:{ratio}\:\frac{\mathrm{1}}{{n}+\mathrm{1}} \\ $$$${find}\:{the}\:{value}\:{of}\:{limit} \\ $$

Question Number 58246 Answers: 1 Comments: 0

show that P=x^(9999) +x^(8888) +x^(7777) +x^(6666) +x^(5555) +x^(4444) +x^(3333) +x^(2222) +x^(1111) +1 Q=x^9 +x^8 +x^7 +x^6 +x^5 +x^4 +x^3 +x^2 +x+1 prove P is divisible by Q

$${show}\:{that} \\ $$$${P}={x}^{\mathrm{9999}} +{x}^{\mathrm{8888}} +{x}^{\mathrm{7777}} +{x}^{\mathrm{6666}} +{x}^{\mathrm{5555}} +{x}^{\mathrm{4444}} +{x}^{\mathrm{3333}} +{x}^{\mathrm{2222}} +{x}^{\mathrm{1111}} +\mathrm{1} \\ $$$${Q}={x}^{\mathrm{9}} +{x}^{\mathrm{8}} +{x}^{\mathrm{7}} +{x}^{\mathrm{6}} +{x}^{\mathrm{5}} +{x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1} \\ $$$${prove}\:\:{P}\:\:{is}\:{divisible}\:{by}\:{Q} \\ $$

Question Number 58245 Answers: 1 Comments: 0

if log(a+b+c)=loga+logb+logc prove log(((2a)/(1−a^2 ))+((2b)/(1−b^2 ))+((2c)/(1−c^2 )))=log(((2a)/(1−a^2 )))+log(((2b)/(1−b^2 )))+log(((2c)/(1−c^2 )))

$${if}\:{log}\left({a}+{b}+{c}\right)={loga}+{logb}+{logc} \\ $$$${prove} \\ $$$${log}\left(\frac{\mathrm{2}{a}}{\mathrm{1}−{a}^{\mathrm{2}} }+\frac{\mathrm{2}{b}}{\mathrm{1}−{b}^{\mathrm{2}} }+\frac{\mathrm{2}{c}}{\mathrm{1}−{c}^{\mathrm{2}} }\right)={log}\left(\frac{\mathrm{2}{a}}{\mathrm{1}−{a}^{\mathrm{2}} }\right)+{log}\left(\frac{\mathrm{2}{b}}{\mathrm{1}−{b}^{\mathrm{2}} }\right)+{log}\left(\frac{\mathrm{2}{c}}{\mathrm{1}−{c}^{\mathrm{2}} }\right) \\ $$

Question Number 58240 Answers: 1 Comments: 2

i=∫dx/(ax^2 +bx+c)^(3/2)

$$\mathrm{i}=\int\mathrm{dx}/\left(\mathrm{ax}^{\mathrm{2}} +\mathrm{bx}+\mathrm{c}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$

Question Number 58239 Answers: 1 Comments: 0

lim_(x→0^+ ) (x^(1/x) )

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left({x}^{\frac{\mathrm{1}}{{x}}} \right) \\ $$

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