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Question Number 27769 Answers: 1 Comments: 0

A glass bottle full of mercury has mass 500g. On being heated through 35°C, 2.43g of mercury are expelled. calculate the mass of mercury remaining in the bottle (Cubic expansivity of mercury is 1.8 × 10^(−4) per K. linear expansivity of glass is 8.0 × 10^(−6) per K.

$$\mathrm{A}\:\mathrm{glass}\:\mathrm{bottle}\:\mathrm{full}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{has}\:\mathrm{mass}\:\mathrm{500g}.\:\mathrm{On}\:\mathrm{being}\:\mathrm{heated}\:\mathrm{through}\:\mathrm{35}°\mathrm{C}, \\ $$$$\mathrm{2}.\mathrm{43g}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{are}\:\mathrm{expelled}.\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{remaining}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{bottle}\:\:\left(\mathrm{Cubic}\:\mathrm{expansivity}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{is}\:\mathrm{1}.\mathrm{8}\:×\:\mathrm{10}^{−\mathrm{4}} \:\mathrm{per}\:\mathrm{K}.\right. \\ $$$$\mathrm{linear}\:\mathrm{expansivity}\:\mathrm{of}\:\mathrm{glass}\:\mathrm{is}\:\mathrm{8}.\mathrm{0}\:×\:\mathrm{10}^{−\mathrm{6}} \:\mathrm{per}\:\mathrm{K}. \\ $$

Question Number 27767 Answers: 1 Comments: 0

If the number of divisors of a number is odd,prove that the number is perfect square and vice versa.

$$\mathrm{If}\:\mathrm{the}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{divisors}}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{number}\:\mathrm{is}\:\boldsymbol{\mathrm{odd}},\mathrm{prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{is}\:\boldsymbol{\mathrm{perfect}}\:\boldsymbol{\mathrm{square}}\:\mathrm{and} \\ $$$$\mathrm{vice}\:\mathrm{versa}. \\ $$

Question Number 27764 Answers: 1 Comments: 0

∫(√(tan x))dx

$$\int\sqrt{\mathrm{tan}\:{x}}{dx} \\ $$

Question Number 27761 Answers: 1 Comments: 0

4(2a+b)^2 −(a−b)^2

$$\mathrm{4}\left(\mathrm{2a}+\mathrm{b}\right)^{\mathrm{2}} −\left(\mathrm{a}−\mathrm{b}\right)^{\mathrm{2}} \\ $$

Question Number 27729 Answers: 0 Comments: 0

Question Number 27727 Answers: 1 Comments: 0

What are the conditions whereby the limit of a function does not exist at a poont?

$${What}\:{are}\:{the}\:{conditions}\:{whereby} \\ $$$${the}\:{limit}\:{of}\:{a}\:{function}\:{does}\:{not} \\ $$$${exist}\:{at}\:{a}\:{poont}? \\ $$

Question Number 27723 Answers: 1 Comments: 0

lim_(x→1) ((x^3 −1)/((x−1)^2 ))

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

Question Number 27722 Answers: 1 Comments: 0

find the limit of f(x)= { ((1+x x<1)),((k x=0 c=0)),((1+x , x>0)) :}

$${find}\:{the}\:{limit}\:{of} \\ $$$$ \\ $$$${f}\left({x}\right)=\begin{cases}{\mathrm{1}+{x}\:\:\:\:\:\:\:{x}<\mathrm{1}}\\{{k}\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}=\mathrm{0}\:\:\:{c}=\mathrm{0}}\\{\mathrm{1}+{x}\:\:\:\:\:,\:{x}>\mathrm{0}}\end{cases} \\ $$

Question Number 27718 Answers: 1 Comments: 0

Question Number 27717 Answers: 0 Comments: 0

Question Number 27701 Answers: 1 Comments: 1

If the function f(x) satisfies lim_(x→1) ((f(x)−2)/(x^2 −1)) =π, evaluate lim_(x→1) f(x)

$${If}\:{the}\:{function}\:{f}\left({x}\right)\:{satisfies} \\ $$$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\:\frac{{f}\left({x}\right)−\mathrm{2}}{{x}^{\mathrm{2}} −\mathrm{1}}\:=\pi,\:{evaluate}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}{f}\left({x}\right) \\ $$

Question Number 27700 Answers: 0 Comments: 1

if f(x)= { ((mx^2 +n, x<0)),((nx+m, 0≤x≤1)),((nx^3 +m, x>1)) :} for what integers m and n does both lim_(x→0) f(x) and lim_(x→1) f(x) exist?

$${if}\:{f}\left({x}\right)=\begin{cases}{{mx}^{\mathrm{2}} +{n},\:\:\:\:\:{x}<\mathrm{0}}\\{{nx}+{m},\:\:\:\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}}\\{{nx}^{\mathrm{3}} +{m},\:\:\:{x}>\mathrm{1}}\end{cases} \\ $$$${for}\:{what}\:{integers}\:{m}\:{and}\:{n}\:{does} \\ $$$${both}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}\right)\:{and}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}{f}\left({x}\right)\:{exist}? \\ $$

Question Number 27699 Answers: 1 Comments: 0

suppose f(x)= { ((a+bx, x<1)),((4, x=1)),((b−ax, x>1)) :} and if lim_(x→1) f(x)=f(1) what are possible values of a and b?

$${suppose}\:{f}\left({x}\right)=\begin{cases}{{a}+{bx},\:\:{x}<\mathrm{1}}\\{\mathrm{4},\:\:\:\:\:\:\:{x}=\mathrm{1}}\\{{b}−{ax},\:\:{x}>\mathrm{1}}\end{cases}\:{and} \\ $$$${if}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:{f}\left({x}\right)={f}\left(\mathrm{1}\right)\:{what}\:{are}\:{possible} \\ $$$${values}\:{of}\:{a}\:{and}\:{b}? \\ $$$$ \\ $$

Question Number 27696 Answers: 0 Comments: 2

lim_(x→0) ((ae^x +3e^(2x) −b)/x) =8 find a and b

$$ \\ $$$$ \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ae}^{\mathrm{x}} +\mathrm{3e}^{\mathrm{2x}} −\mathrm{b}}{\mathrm{x}}\:=\mathrm{8} \\ $$$$ \\ $$$$\mathrm{find}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b} \\ $$

Question Number 27693 Answers: 1 Comments: 1

1) calculate ∫∫_(]0,1]×]0,(π/2)]) ((dxdy)/(1+(xtany)^2 )) 2) find the value of ∫_0 ^(π/2) (t/(tant))dt .

$$\left.\mathrm{1}\right)\:{calculate}\:\:\int\int_{\left.\right]\left.\mathrm{0}\left.,\left.\mathrm{1}\right]×\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\right]} \:\:\:\frac{{dxdy}}{\mathrm{1}+\left({xtany}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{t}}{{tant}}{dt}\:. \\ $$

Question Number 27692 Answers: 0 Comments: 1

find by two ways the value of ∫∫_([0,1]) x^y dxdxy then calculate ∫_0 ^1 ((t−1)/(lnt))dt .

$${find}\:{by}\:{two}\:{ways}\:{the}\:{value}\:{of}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]} \:\:{x}^{{y}} \:\:{dxdxy}\:{then} \\ $$$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{t}−\mathrm{1}}{{lnt}}{dt}\:\:. \\ $$

Question Number 27691 Answers: 0 Comments: 1

let give A=∫∫_(0≤y≤x≤1) ((dxdxy)/((1+x^2 )(1+y^2 ))) and B= ∫_0 ^(π/4) ((ln(2cos^2 θ))/(2cos(2θ)))dθ calculate A and prove that B=A.

$${let}\:{give}\:\:{A}=\int\int_{\mathrm{0}\leqslant{y}\leqslant{x}\leqslant\mathrm{1}} \:\:\:\:\:\:\frac{{dxdxy}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)}\:\:{and} \\ $$$${B}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{ln}\left(\mathrm{2}{cos}^{\mathrm{2}} \theta\right)}{\mathrm{2}{cos}\left(\mathrm{2}\theta\right)}{d}\theta\:\:{calculate}\:{A}\:{and}\:{prove}\:{that}\:{B}={A}. \\ $$

Question Number 27690 Answers: 0 Comments: 1

find I= ∫∫_D ln(1+x+y)dxdy with D= {(x,y)∈R^2 / x+y≤1 and x≥0 and y≥0 }.

$${find}\:\:\:{I}=\:\:\int\int_{{D}} {ln}\left(\mathrm{1}+{x}+{y}\right){dxdy}\:\:{with} \\ $$$${D}=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:\:\:/\:\:{x}+{y}\leqslant\mathrm{1}\:{and}\:{x}\geqslant\mathrm{0}\:{and}\:{y}\geqslant\mathrm{0}\:\right\}. \\ $$

Question Number 27757 Answers: 1 Comments: 0

calculate I= ∫_0 ^(π/2) (dx/(1+cosx)) and J= ∫_0^ ^(π/2) ((cosx)/(1+cosx))dx .

$${calculate}\:\:{I}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dx}}{\mathrm{1}+{cosx}}\:{and}\:{J}=\:\int_{\mathrm{0}^{} } ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{cosx}}{\mathrm{1}+{cosx}}{dx}\:. \\ $$

Question Number 27688 Answers: 0 Comments: 0

Write the series,indicating the 5th term,the 5th partial sum 0+1+3+...+(((n^2 +n)/2))+...

$${Write}\:{the}\:{series},{indicating}\:{the} \\ $$$$\mathrm{5}{th}\:{term},{the}\:\mathrm{5}{th}\:{partial}\:{sum} \\ $$$$ \\ $$$$\mathrm{0}+\mathrm{1}+\mathrm{3}+...+\left(\frac{{n}^{\mathrm{2}} +{n}}{\mathrm{2}}\right)+... \\ $$

Question Number 28038 Answers: 0 Comments: 0

let give f(x)=(√(x+y)) +1 and D={(x,y)∈R^2 / 0≤x≤1 and −1≤y≤1} find the value of ∫∫ f(x,y)dxdy .

$${let}\:{give}\:{f}\left({x}\right)=\sqrt{{x}+{y}}\:+\mathrm{1}\:\:{and}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:\right. \\ $$$$\left.{and}\:−\mathrm{1}\leqslant{y}\leqslant\mathrm{1}\right\}\:\:{find}\:{the}\:{value}\:{of}\:\:\int\int\:{f}\left({x},{y}\right){dxdy}\:. \\ $$

Question Number 27685 Answers: 0 Comments: 2

Write the first five series indicating the 5th term,5th partial sum Σ_(n=1) ^∞ t_n , where t_n = { ((1 for n=1)),(((1/2) for n=2)),((1−(1/2)+...+(−1)^(n+1) ((1/n)) for n>2)) :}

$${Write}\:{the}\:{first}\:{five}\:{series}\:{indicating} \\ $$$${the}\:\mathrm{5}{th}\:{term},\mathrm{5}{th}\:{partial}\:{sum} \\ $$$$ \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{t}_{{n}} ,\:{where} \\ $$$${t}_{{n}} =\begin{cases}{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{for}\:{n}=\mathrm{1}}\\{\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{for}\:{n}=\mathrm{2}}\\{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+...+\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} \left(\frac{\mathrm{1}}{{n}}\right)\:\:\:{for}\:\:\:{n}>\mathrm{2}}\end{cases} \\ $$

Question Number 27684 Answers: 0 Comments: 1

1) prove the existence of the integral I=∫_0 ^(π/2) ((ln(1+cosx))/(cosx))dx 2)prove that I= ∫∫_D ((siny)/(1+cosx cosy))dxdy with D=[0,(π/2)]^2 3)find the value of I.

$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{existence}\:{of}\:{the}\:{integral} \\ $$$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{ln}\left(\mathrm{1}+{cosx}\right)}{{cosx}}{dx} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{I}=\:\int\int_{{D}} \:\:\frac{{siny}}{\mathrm{1}+{cosx}\:{cosy}}{dxdy}\:{with}\: \\ $$$${D}=\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\right]^{\mathrm{2}} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:{I}. \\ $$

Question Number 27706 Answers: 0 Comments: 0

the number of xε[0,2π]for which∣(√)2sin^(4 ) x+18cos^2 x−(√(2cos^4 x+18sin^2 x))∣=1

$${the}\:{number}\:{of}\:{x}\epsilon\left[\mathrm{0},\mathrm{2}\pi\right]{for}\:{which}\mid\sqrt{}\mathrm{2}{sin}^{\mathrm{4}\:} {x}+\mathrm{18}{cos}^{\mathrm{2}} {x}−\sqrt{\mathrm{2}{cos}^{\mathrm{4}} {x}+\mathrm{18}{sin}^{\mathrm{2}} {x}}\mid=\mathrm{1} \\ $$

Question Number 27677 Answers: 1 Comments: 4

Question Number 27681 Answers: 0 Comments: 2

Find square root of 7−30(√2)i .

$${Find}\:{square}\:{root}\:{of}\:\mathrm{7}−\mathrm{30}\sqrt{\mathrm{2}}{i}\:. \\ $$

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