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Question Number 42289    Answers: 1   Comments: 1

log(x+y)=1 log_2 x+log_4 y^2 =4

$$\boldsymbol{\mathrm{log}}\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)=\mathrm{1} \\ $$$$\boldsymbol{\mathrm{log}}_{\mathrm{2}} \boldsymbol{{x}}+\boldsymbol{\mathrm{log}}_{\mathrm{4}} \boldsymbol{{y}}^{\mathrm{2}} =\mathrm{4} \\ $$

Question Number 42287    Answers: 1   Comments: 0

Your family will be attending a family reunion at a particular beach resort. To avoid hassle, you consider renting a car that charges a flat rate of P2 000 plus P150 per kilometer. Write a piecewise function that model the situation.

$${Your}\:{family}\:{will}\:{be}\:{attending}\:{a}\: \\ $$$${family}\:{reunion}\:{at}\:{a}\:{particular}\:{beach} \\ $$$${resort}.\:{To}\:{avoid}\:{hassle},\:{you}\:{consider} \\ $$$${renting}\:{a}\:{car}\:{that}\:{charges}\:{a}\:{flat}\:{rate} \\ $$$${of}\:{P}\mathrm{2}\:\mathrm{000}\:{plus}\:{P}\mathrm{150}\:{per}\:{kilometer}.\: \\ $$$${Write}\:{a}\:{piecewise}\:{function}\:{that}\: \\ $$$${model}\:{the}\:{situation}. \\ $$

Question Number 42285    Answers: 0   Comments: 2

let S_p =Σ_(n=0) ^∞ cos(((nπ)/p)) and W_p =Σ_(n=0) ^∞ sin(((nπ)/p)) with p natural integr not0 1) find a simple form of S_p and W_p 2) find the value of Σ_(n=0) ^∞ cos(((nπ)/3)) and Σ_(n=0) ^∞ sin(((nπ)/3)) 3) find the value of Σ_(n=0) ^∞ cos(((nπ)/5)) and Σ_(n=0) ^∞ sin(((nπ)/5)) 4) calculate A =Σ_(n=0) ^∞ cos^2 (((nπ)/3)) and B =Σ_(n=0) ^∞ sin^2 (((nπ)/3)) .

$${let}\:\:{S}_{{p}} =\sum_{{n}=\mathrm{0}} ^{\infty} \:\:{cos}\left(\frac{{n}\pi}{{p}}\right)\:\:{and}\:\:{W}_{{p}} \:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{sin}\left(\frac{{n}\pi}{{p}}\right)\:{with}\:{p}\:{natural}\:{integr}\:{not}\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{S}_{{p}} \:{and}\:{W}_{{p}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{cos}\left(\frac{{n}\pi}{\mathrm{3}}\right)\:{and}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{sin}\left(\frac{{n}\pi}{\mathrm{3}}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{cos}\left(\frac{{n}\pi}{\mathrm{5}}\right)\:{and}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{sin}\left(\frac{{n}\pi}{\mathrm{5}}\right) \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\:{A}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{cos}^{\mathrm{2}} \left(\frac{{n}\pi}{\mathrm{3}}\right)\:{and}\:{B}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{sin}^{\mathrm{2}} \left(\frac{{n}\pi}{\mathrm{3}}\right)\:. \\ $$

Question Number 42304    Answers: 0   Comments: 1

solve in Z^3 x+2y +3z =12 .

$${solve}\:{in}\:{Z}^{\mathrm{3}} \:\:\:\:{x}+\mathrm{2}{y}\:+\mathrm{3}{z}\:=\mathrm{12}\:\:. \\ $$

Question Number 42278    Answers: 0   Comments: 1

Question Number 42266    Answers: 0   Comments: 1

find the value of Σ_(n =0) ^∞ (1/(n^2 +1)) and Σ_(n=0) ^∞ (((−1)^n )/(n^2 +1)) .

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}\:=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:+\mathrm{1}}\:\:{and}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \:+\mathrm{1}}\:. \\ $$

Question Number 42264    Answers: 1   Comments: 0

Thd local barangay recieved a budget of 150000 to provide medical check−ups for the children in the barangay. Write an equation representing the relationship of the alloted money per child.

$${Thd}\:{local}\:{barangay}\:{recieved}\:{a}\:{budget}\:{of}\:\mathrm{150000}\:{to}\:{provide}\:{medical}\:{check}−{ups}\:{for}\:{the}\:{children}\:{in}\:{the}\:{barangay}.\:{Write}\:{an}\:{equation}\:{representing}\:{the}\:{relationship}\:{of}\:{the}\:{alloted}\:{money}\:{per}\:{child}. \\ $$

Question Number 42262    Answers: 1   Comments: 0

Heat is supplied at a rate of 500W to a pressure cooker containing water and fitted with a safety valve.Steam escape such that water is lost at 12g/min.If the heat is supplied at 900W,water is lost at 20g/min.Calculate the specific latent heat of steam.

$${Heat}\:{is}\:{supplied}\:{at}\:{a}\:{rate}\:{of}\:\mathrm{500}{W} \\ $$$${to}\:{a}\:{pressure}\:{cooker}\:{containing} \\ $$$${water}\:{and}\:{fitted}\:{with}\:{a}\:{safety}\: \\ $$$${valve}.{Steam}\:{escape}\:{such}\:{that}\: \\ $$$${water}\:{is}\:{lost}\:{at}\:\mathrm{12}{g}/{min}.{If}\:{the} \\ $$$${heat}\:{is}\:{supplied}\:{at}\:\mathrm{900}{W},{water}\:{is} \\ $$$${lost}\:{at}\:\mathrm{20}{g}/{min}.{Calculate}\:{the} \\ $$$${specific}\:{latent}\:{heat}\:{of}\:{steam}. \\ $$$$ \\ $$

Question Number 42261    Answers: 1   Comments: 1

How much sweat must evaporate from the surface of 150kg of human body to be able to cool the human by 2°C.(assume C=3.35J/g/K for human and L=2.5mJ/kg for water at body temperature)

$${How}\:{much}\:{sweat}\:{must}\:{evaporate} \\ $$$${from}\:{the}\:{surface}\:{of}\:\mathrm{150}{kg}\:{of}\:{human} \\ $$$${body}\:{to}\:{be}\:{able}\:{to}\:{cool}\:{the}\:{human} \\ $$$${by}\:\mathrm{2}°{C}.\left({assume}\:{C}=\mathrm{3}.\mathrm{35}{J}/{g}/{K}\:{for}\right. \\ $$$${human}\:{and}\:{L}=\mathrm{2}.\mathrm{5}{mJ}/{kg}\:{for} \\ $$$$\left.{water}\:{at}\:{body}\:{temperature}\right) \\ $$

Question Number 42260    Answers: 0   Comments: 2

let f(a) = ∫_(−∞) ^(+∞) cos(ax^2 )dx with a>0 1) calculate f(a) interms of a ) calculate ∫_(−∞) ^(+∞) cos(2x^2 )dx 3) find the value of ∫_(−∞) ^(+∞) cos(x^2 +x+1)dx .

$${let}\:{f}\left({a}\right)\:=\:\int_{−\infty} ^{+\infty} \:{cos}\left({ax}^{\mathrm{2}} \right){dx}\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right)\:{interms}\:{of}\:{a} \\ $$$$\left.\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:{cos}\left(\mathrm{2}{x}^{\mathrm{2}} \right){dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:{cos}\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right){dx}\:. \\ $$

Question Number 42305    Answers: 0   Comments: 6

let f(x) = ∫_(−∞) ^(+∞) ((cos(xt))/((t−i)^2 )) dt 1) let R =Re(f(x)) and I =Im(f(x)) extract R and I 2) calculate R and I 3) conclude the value of f(x) 4) calculate ∫_(−∞) ^(+∞) ((cos(2t))/((t−i)^2 ))dt 5) let u_n = ∫_(−∞) ^(+∞) ((cos((t/n)))/((t−i)^2 ))dt (n natral integer not o) find lim_(n→+∞) u_n and study the convergence of Σu_n

$${let}\:{f}\left({x}\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\frac{{cos}\left({xt}\right)}{\left({t}−{i}\right)^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{let}\:{R}\:={Re}\left({f}\left({x}\right)\right)\:{and}\:{I}\:={Im}\left({f}\left({x}\right)\right)\:{extract}\:{R}\:{and}\:{I} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{R}\:{and}\:{I} \\ $$$$\left.\mathrm{3}\right)\:{conclude}\:{the}\:{value}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{t}\right)}{\left({t}−{i}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{5}\right)\:{let}\:{u}_{{n}} =\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\frac{{t}}{{n}}\right)}{\left({t}−{i}\right)^{\mathrm{2}} }{dt}\:\:\:\:\left({n}\:{natral}\:{integer}\:{not}\:{o}\right) \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} \:\:\:\:{and}\:\:{study}\:{the}\:{convergence}\:{of}\:\Sigma{u}_{{n}} \\ $$

Question Number 42228    Answers: 1   Comments: 0

Solve : 2x^2 ydx −2y^4 dx+2x^3 dy+3xy^3 dy=0.

$$\mathrm{Solve}\:: \\ $$$$\mathrm{2}{x}^{\mathrm{2}} {ydx}\:−\mathrm{2}{y}^{\mathrm{4}} {dx}+\mathrm{2}{x}^{\mathrm{3}} {dy}+\mathrm{3}{xy}^{\mathrm{3}} {dy}=\mathrm{0}. \\ $$

Question Number 42224    Answers: 0   Comments: 5

a + b + c = 180 a,b,c ∈ N number of triplets possible (a,b,c) for the above equation are ? ( the order of a,b,c doesn′t matter)

$${a}\:+\:{b}\:+\:{c}\:=\:\mathrm{180}\: \\ $$$${a},{b},{c}\:\in\:\mathbb{N} \\ $$$${number}\:{of}\:{triplets}\:{possible} \\ $$$$\left({a},{b},{c}\right)\:{for}\:{the}\:{above} \\ $$$${equation}\:{are}\:? \\ $$$$\left(\:{the}\:{order}\:{of}\:{a},{b},{c}\:\:{doesn}'{t}\right. \\ $$$$\left.{matter}\right) \\ $$

Question Number 42222    Answers: 1   Comments: 0

let f(x) =e^(−∣x∣) , 2π periodic even developp f at fourier serie .

$${let}\:{f}\left({x}\right)\:={e}^{−\mid{x}\mid} \:,\:\:\mathrm{2}\pi\:{periodic}\:{even}\:\:{developp}\:{f}\:{at}\:{fourier}\:{serie}\:. \\ $$

Question Number 42221    Answers: 1   Comments: 0

Solve: (dt/dx) = (2/(x+t)) .

$$\mathrm{Solve}: \\ $$$$\frac{\mathrm{dt}}{\mathrm{d}{x}}\:=\:\frac{\mathrm{2}}{{x}+\mathrm{t}}\:. \\ $$

Question Number 42215    Answers: 0   Comments: 3

Number of straight lines which satisfy the differential equation (dy/dx) + x((dy/dx))^2 − y =0 is ?

$$\mathrm{Number}\:\mathrm{of}\:\mathrm{straight}\:\mathrm{lines}\:\mathrm{which}\:\mathrm{satisfy} \\ $$$$\mathrm{the}\:\mathrm{differential}\:\mathrm{equation} \\ $$$$\frac{\mathrm{dy}}{{dx}}\:+\:{x}\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} −\:{y}\:=\mathrm{0}\:{is}\:? \\ $$

Question Number 42207    Answers: 1   Comments: 0

Find the equation of tangent and normal to the curve y given by y = x^3 + 3x^2 + 7 .

$${Find}\:{the}\:{equation}\:{of}\:{tangent}\:{and}\:{normal}\:{to}\:\:{the}\:{curve}\:{y} \\ $$$${given}\:{by}\:\:\:{y}\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{7}\:. \\ $$

Question Number 42200    Answers: 1   Comments: 0

Question Number 42199    Answers: 1   Comments: 0

Question Number 42196    Answers: 0   Comments: 4

Let P be an interior point of a triangle ABC and AP,BP,CP meet the sides BC, CA,AB in D,E,F respectively. Show that ((AP)/(PD))= ((AF)/(FB)) + ((AE)/(EC)) .

$$\mathrm{Let}\:\mathrm{P}\:\mathrm{be}\:\mathrm{an}\:\mathrm{interior}\:\mathrm{point}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle} \\ $$$$\mathrm{ABC}\:\mathrm{and}\:\mathrm{AP},\mathrm{BP},\mathrm{CP}\:\mathrm{meet}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{BC}, \\ $$$$\mathrm{CA},\mathrm{AB}\:\mathrm{in}\:\mathrm{D},\mathrm{E},\mathrm{F}\:\mathrm{respectively}.\:\mathrm{Show} \\ $$$$\mathrm{that}\:\frac{\mathrm{AP}}{\mathrm{PD}}=\:\frac{\mathrm{AF}}{\mathrm{FB}}\:+\:\frac{\mathrm{AE}}{\mathrm{EC}}\:. \\ $$

Question Number 42195    Answers: 1   Comments: 0

let p(x)=x^(10) −1 1) find roots of p(x) 2) factorize i nside C[x] p(x{ 3) factorize inside R[x] p(x) .

$${let}\:{p}\left({x}\right)={x}^{\mathrm{10}} −\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{i}\:{nside}\:\:{C}\left[{x}\right]\:{p}\left({x}\left\{\right.\right. \\ $$$$\left.\mathrm{3}\right)\:{factorize}\:{inside}\:{R}\left[{x}\right]\:{p}\left({x}\right)\:. \\ $$

Question Number 42410    Answers: 1   Comments: 1

∫_( 0) ^(2a) ((f(x))/(f(x)+f(2a−x))) dx =

$$\underset{\:\mathrm{0}} {\overset{\mathrm{2}{a}} {\int}}\:\:\frac{{f}\left({x}\right)}{{f}\left({x}\right)+{f}\left(\mathrm{2}{a}−{x}\right)}\:{dx}\:= \\ $$

Question Number 42191    Answers: 0   Comments: 1

let A_p =∫_0 ^∞ ((sin(px))/(e^x −1)) dx with p>0 1)give A_p at form of serie 2) give A_1 at form of serie .

$${let}\:{A}_{{p}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({px}\right)}{{e}^{{x}} −\mathrm{1}}\:{dx}\:\:{with}\:{p}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){give}\:{A}_{{p}} \:\:{at}\:{form}\:{of}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:{give}\:{A}_{\mathrm{1}} \:{at}\:{form}\:{of}\:{serie}\:. \\ $$

Question Number 42190    Answers: 0   Comments: 1

let u_n =Σ_(k=1) ^n (((−1)^k )/(√k)) 1) prove that (u_n )is convergente 2) find a equivalent of u_n when n→+∞

$${let}\:\:\:{u}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\sqrt{{k}}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left({u}_{{n}} \right){is}\:{convergente} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \:{when}\:{n}\rightarrow+\infty \\ $$

Question Number 42189    Answers: 0   Comments: 0

calculate f(x) = ∫_0 ^∞ e^(−t^2 ) arctan(xt^2 )dt

$${calculate}\:\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{t}^{\mathrm{2}} } \:{arctan}\left({xt}^{\mathrm{2}} \right){dt} \\ $$

Question Number 42188    Answers: 0   Comments: 1

let x>0 calculate f(x) =∫_0 ^(+∞) e^(−t) ∣sin(xt)∣ dt

$${let}\:{x}>\mathrm{0}\:\:\:{calculate}\:\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:{e}^{−{t}} \:\mid{sin}\left({xt}\right)\mid\:{dt} \\ $$

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