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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.

$$\boldsymbol{\mathrm{A}}\:\boldsymbol{\mathrm{circle}}\:\boldsymbol{\mathrm{tangents}}\:\boldsymbol{\mathrm{to}}\::\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{y}}\:\boldsymbol{\mathrm{axes}}\:\boldsymbol{\mathrm{and}} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{2}}} +\boldsymbol{\mathrm{y}}^{\frac{\mathrm{1}}{\mathrm{2}}} =\boldsymbol{\mathrm{a}}^{\frac{\mathrm{1}}{\mathrm{2}}} .\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{its}}\:\boldsymbol{\mathrm{radious}}. \\ $$

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=?

$$\boldsymbol{\mathrm{a}}\:\:.\int\:\:\frac{\boldsymbol{\mathrm{dx}}}{\mathrm{2}\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}+\mathrm{3}\boldsymbol{\mathrm{tg}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}}=? \\ $$$$\boldsymbol{\mathrm{b}}\:\:\:.\int\frac{\:\:\mathrm{1}+\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{x}}}}{\mathrm{1}+\sqrt{\boldsymbol{\mathrm{x}}}+\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{x}}}+\sqrt[{\mathrm{6}}]{\boldsymbol{\mathrm{x}}}}\boldsymbol{\mathrm{dx}}=? \\ $$$$\boldsymbol{\mathrm{c}}\:\:\:\:\:.\int\:\:\frac{\boldsymbol{\mathrm{cosx}}}{\mathrm{1}+\boldsymbol{\mathrm{cos}}\mathrm{2}\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{dx}}=? \\ $$$$\boldsymbol{\mathrm{d}}\:\:\:\:\:.\int\:\:\:\frac{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}}{\sqrt{\mathrm{2}}+\sqrt{\mathrm{3}}.\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{dx}}=? \\ $$

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) )

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

Question Number 58238    Answers: 0   Comments: 0

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

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

Question Number 58222    Answers: 2   Comments: 4

∫_0 ^1 x^x dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{x}} {dx} \\ $$

Question Number 58220    Answers: 1   Comments: 0

find ∫ (dx/((x^2 +x)(√(−x^2 +2x +3))))

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

Question Number 58216    Answers: 1   Comments: 1

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, b and H are constant. How much heat per mole is transfered during the process in which the temperature change from 0.01H to 0.02H ?

$$\mathrm{The}\:\mathrm{molar}\:\mathrm{heat}\:\mathrm{capacity}\:\mathrm{of}\:\mathrm{a}\:\mathrm{metal}\:\:\mathrm{at} \\ $$$$\mathrm{low}\:\mathrm{temperature}\:\mathrm{varies}\:\mathrm{with}\:\mathrm{the}\: \\ $$$$\mathrm{temperature}\:\mathrm{according}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\:\:\:\:\:\mathrm{C}\:=\:\mathrm{b}\theta\:+\:\frac{\mathrm{a}}{\mathrm{H}}\theta^{\mathrm{3}} \\ $$$$\mathrm{where}\:\mathrm{a},\:\mathrm{b}\:\mathrm{and}\:\mathrm{H}\:\mathrm{are}\:\mathrm{constant}. \\ $$$$\mathrm{How}\:\mathrm{much}\:\mathrm{heat}\:\mathrm{per}\:\mathrm{mole}\:\mathrm{is}\:\mathrm{transfered} \\ $$$$\mathrm{during}\:\mathrm{the}\:\mathrm{process}\:\mathrm{in}\:\mathrm{which}\:\mathrm{the}\: \\ $$$$\mathrm{temperature}\:\mathrm{change}\:\mathrm{from}\:\mathrm{0}.\mathrm{01H}\: \\ $$$$\mathrm{to}\:\mathrm{0}.\mathrm{02H}\:? \\ $$

Question Number 58212    Answers: 0   Comments: 0

let f(x) =∫_0 ^∞ e^(−x[t]) sin(xt)dt with x>0 1) find a explicit form for f(x) 2) let U_n =nf(n) find lim_(n→+∞) U_n and study the convergence of ΣU_n

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}\left[{t}\right]} \:{sin}\left({xt}\right){dt}\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{let}\:{U}_{{n}} ={nf}\left({n}\right)\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \:\:\:{and}\:{study}\:{the}\:{convergence}\:{of}\:\Sigma{U}_{{n}} \\ $$

Question Number 58211    Answers: 0   Comments: 2

∫_( 0) ^(2π) ∣ cos x−sin x ∣dx =

$$\underset{\:\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}\:\mid\:\mathrm{cos}\:{x}−\mathrm{sin}\:{x}\:\mid{dx}\:= \\ $$

Question Number 58210    Answers: 0   Comments: 5

find two possible number such that 1) xy=(x/y)=x−y 2)xy=((2x)/y)=3(x−y) 3) xy=(x/y)=2(x−y).

$$\mathrm{find}\:\mathrm{two}\:\mathrm{possible}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left.\mathrm{1}\right)\:\:\mathrm{xy}=\frac{\mathrm{x}}{\mathrm{y}}=\mathrm{x}−\mathrm{y} \\ $$$$\left.\mathrm{2}\right)\mathrm{xy}=\frac{\mathrm{2x}}{\mathrm{y}}=\mathrm{3}\left(\mathrm{x}−\mathrm{y}\right) \\ $$$$\left.\mathrm{3}\right)\:\:\mathrm{xy}=\frac{\mathrm{x}}{\mathrm{y}}=\mathrm{2}\left(\mathrm{x}−\mathrm{y}\right). \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 58203    Answers: 1   Comments: 0

A battery can supply current of 1.2A and 0.4A through 4Ω and 14Ω respectively. Calculate the internal resistance of the battery

$$\mathrm{A}\:\mathrm{battery}\:\mathrm{can}\:\mathrm{supply}\:\mathrm{current}\:\mathrm{of}\:\mathrm{1}.\mathrm{2A}\: \\ $$$$\mathrm{and}\:\mathrm{0}.\mathrm{4A}\:\mathrm{through}\:\mathrm{4}\Omega\:\mathrm{and}\:\mathrm{14}\Omega\:\:\mathrm{respectively}. \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{internal}\:\mathrm{resistance}\:\mathrm{of}\:\mathrm{the}\:\mathrm{battery} \\ $$

Question Number 58196    Answers: 1   Comments: 0

The molar heat capacity of constant presure of a gas varies with the temperature according to the equation C_p = a + bθ −(C/θ^2 ) where a,b and C are constants. How much heat is transfered during an isobaric process in which n mole of gas undergo a temperature rise from θ_(i ) to θ_f ?

$$\mathrm{The}\:\mathrm{molar}\:\mathrm{heat}\:\mathrm{capacity}\:\mathrm{of}\:\mathrm{constant} \\ $$$$\mathrm{presure}\:\mathrm{of}\:\mathrm{a}\:\mathrm{gas}\:\mathrm{varies}\:\mathrm{with}\:\mathrm{the}\:\mathrm{temperature} \\ $$$$\mathrm{according}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{C}_{\mathrm{p}} \:=\:\:\mathrm{a}\:+\:\mathrm{b}\theta\:−\frac{\mathrm{C}}{\theta^{\mathrm{2}} } \\ $$$$\mathrm{where}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{C}\:\mathrm{are}\:\mathrm{constants}. \\ $$$$\:\:\mathrm{How}\:\mathrm{much}\:\mathrm{heat}\:\mathrm{is}\:\mathrm{transfered}\:\mathrm{during} \\ $$$$\:\:\:\mathrm{an}\:\mathrm{isobaric}\:\mathrm{process}\:\mathrm{in}\:\mathrm{which}\:\mathrm{n}\:\mathrm{mole} \\ $$$$\:\:\:\mathrm{of}\:\mathrm{gas}\:\mathrm{undergo}\:\mathrm{a}\:\mathrm{temperature}\:\mathrm{rise} \\ $$$$\:\:\:\:\mathrm{from}\:\theta_{{i}\:} \mathrm{to}\:\theta_{{f}} \:? \\ $$

Question Number 58195    Answers: 2   Comments: 0

(1/x)+(1/y)=(3/4) (x^2 /y)+(y^2 /x)=9 find the value of x and y

$$\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{y}}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{y}}+\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{x}}=\mathrm{9} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$

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