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Question Number 120154 Answers: 1 Comments: 0

solve using LambertW function ((8/7))^x +17=25x

$${solve}\:{using}\:{LambertW}\:{function} \\ $$$$\left(\frac{\mathrm{8}}{\mathrm{7}}\right)^{{x}} +\mathrm{17}=\mathrm{25}{x} \\ $$

Question Number 120152 Answers: 0 Comments: 0

Determinate and construct the set of points M which have as affix z in each case: 1) arg(i−z)=0[π] 2) arg(z+1−i)=(π/6)[2π]

$$\mathrm{Determinate}\:\mathrm{and}\:\mathrm{construct}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:\mathrm{M} \\ $$$$\mathrm{which}\:\mathrm{have}\:\mathrm{as}\:\mathrm{affix}\:\mathrm{z}\:\:\mathrm{in}\:\mathrm{each}\:\mathrm{case}: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{arg}\left(\mathrm{i}−\mathrm{z}\right)=\mathrm{0}\left[\pi\right] \\ $$$$\left.\mathrm{2}\right)\:\mathrm{arg}\left(\mathrm{z}+\mathrm{1}−\mathrm{i}\right)=\frac{\pi}{\mathrm{6}}\left[\mathrm{2}\pi\right] \\ $$$$ \\ $$

Question Number 120151 Answers: 0 Comments: 0

Represent in complex plane the set of points M which have as affix z such that ∣z∣=2 and arg(z+1)=(π/4)[π]

$$\mathrm{Represent}\:\mathrm{in}\:\mathrm{complex}\:\mathrm{plane}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points} \\ $$$$\mathrm{M}\:\mathrm{which}\:\mathrm{have}\:\mathrm{as}\:\mathrm{affix}\:\mathrm{z}\:\mathrm{such}\:\mathrm{that}\:\mid\mathrm{z}\mid=\mathrm{2}\:\mathrm{and} \\ $$$$\mathrm{arg}\left(\mathrm{z}+\mathrm{1}\right)=\frac{\pi}{\mathrm{4}}\left[\pi\right] \\ $$

Question Number 120147 Answers: 1 Comments: 1

Question Number 120133 Answers: 0 Comments: 5

Question Number 120132 Answers: 0 Comments: 0

Suppose you are in a imagnary train which travels at the half of the speed of light. Suppose You have a brother who is 7 year smaller than you. He stands on the platform which you had left. After 1 hour of travelling on the train you come back on the platform. Then you observe something strange. You can see your brother looks older . So what is his age?(He was ten years old)

$${Suppose}\:{you}\:{are}\:{in}\:{a}\:{imagnary}\:{train}\:{which}\:{travels}\:{at}\:{the}\:{half}\: \\ $$$${of}\:{the}\:{speed}\:{of}\:{light}.\:{Suppose}\:{You}\:{have}\:{a}\:{brother}\:{who}\:{is}\:\mathrm{7}\:{year} \\ $$$${smaller}\:{than}\:{you}.\:{He}\:{stands}\:{on}\:{the}\:{platform}\:{which}\:{you}\:{had}\:{left}. \\ $$$${After}\:\mathrm{1}\:{hour}\:{of}\:{travelling}\:{on}\:{the}\:{train}\:{you}\:{come}\:{back}\:{on}\:{the} \\ $$$${platform}.\:{Then}\:{you}\:{observe}\:{something}\:{strange}.\:{You}\:{can}\:{see} \\ $$$${your}\:{brother}\:{looks}\:{older}\:.\:{So}\:{what}\:{is}\:{his}\:{age}?\left({He}\:{was}\:{ten}\:{years}\right. \\ $$$$\left.{old}\right) \\ $$

Question Number 120131 Answers: 1 Comments: 1

Question Number 120127 Answers: 2 Comments: 0

Question Number 120122 Answers: 0 Comments: 3

{ ((s_(12) =30)),((s_8 =4)) :} a_3 =?

$$\begin{cases}{{s}_{\mathrm{12}} =\mathrm{30}}\\{{s}_{\mathrm{8}} =\mathrm{4}}\end{cases}\:\:\:{a}_{\mathrm{3}} =? \\ $$

Question Number 120120 Answers: 2 Comments: 0

{ ((log _a (x) = 8)),((log _b (x) = 3 )),((log _c (x) = 6)) :} ⇒ log _(abc) (x)=?

$$\begin{cases}{\mathrm{log}\:_{{a}} \left({x}\right)\:=\:\mathrm{8}}\\{\mathrm{log}\:_{{b}} \left({x}\right)\:=\:\mathrm{3}\:}\\{\mathrm{log}\:_{{c}} \left({x}\right)\:=\:\mathrm{6}}\end{cases}\:\Rightarrow\:\mathrm{log}\:_{{abc}} \:\left({x}\right)=? \\ $$

Question Number 120118 Answers: 0 Comments: 2

Question Number 120140 Answers: 2 Comments: 0

Question Number 120110 Answers: 2 Comments: 0

{ ((((4x^2 )/(4x^2 +1)) = y)),((((4y^2 )/(4y^2 +1)) = z )),((((4z^2 )/(4z^2 +1)) = x)) :} where x,y,z ≠ 0

$$\begin{cases}{\frac{\mathrm{4}{x}^{\mathrm{2}} }{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{1}}\:=\:{y}}\\{\frac{\mathrm{4}{y}^{\mathrm{2}} }{\mathrm{4}{y}^{\mathrm{2}} +\mathrm{1}}\:=\:{z}\:}\\{\frac{\mathrm{4}{z}^{\mathrm{2}} }{\mathrm{4}{z}^{\mathrm{2}} +\mathrm{1}}\:=\:{x}}\end{cases} \\ $$$${where}\:{x},{y},{z}\:\neq\:\mathrm{0}\: \\ $$

Question Number 120109 Answers: 0 Comments: 0

Question Number 120108 Answers: 0 Comments: 0

(x^4 +ax^2 +a^2 )^2 +x^6 =1 solve for: x,a∈R

$$\left(\boldsymbol{\mathrm{x}}^{\mathrm{4}} +\boldsymbol{\mathrm{ax}}^{\mathrm{2}} +\boldsymbol{\mathrm{a}}^{\mathrm{2}} \right)^{\mathrm{2}} +\boldsymbol{\mathrm{x}}^{\mathrm{6}} =\mathrm{1} \\ $$$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}:\:\:\mathrm{x},\mathrm{a}\in\boldsymbol{\mathrm{R}} \\ $$

Question Number 120102 Answers: 2 Comments: 0

Θ = ∫ ((4x^(−1) +8x^(−3) )/(x^2 (√(x^4 +2x^2 +2)))) dx

$$\:\Theta\:=\:\int\:\frac{\mathrm{4}{x}^{−\mathrm{1}} +\mathrm{8}{x}^{−\mathrm{3}} }{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}}}\:{dx}\: \\ $$

Question Number 120092 Answers: 3 Comments: 1

Question Number 120091 Answers: 1 Comments: 0

When f(x) is divided by (x−1)(x+2), the remainder is (x+3), and when f(x) is divided by (x^2 +2x+5), the remainder is (2x+1). Find the remainder when f(x) is divided by (x−1)(x^2 +2x+5).

$$\mathrm{When}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right),\: \\ $$$$\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\left({x}+\mathrm{3}\right),\:\mathrm{and}\:\mathrm{when}\:{f}\left({x}\right) \\ $$$$\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}\right),\:\mathrm{the}\:\mathrm{remainder} \\ $$$$\mathrm{is}\:\left(\mathrm{2}{x}+\mathrm{1}\right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\left({x}−\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}\right). \\ $$

Question Number 120090 Answers: 0 Comments: 0

Find the number of subsets of { 1,2,3,...,2000 } the sum of whose elements is divisible by 5

$${Find}\:{the}\:{number}\:{of}\:{subsets}\:{of} \\ $$$$\left\{\:\mathrm{1},\mathrm{2},\mathrm{3},...,\mathrm{2000}\:\right\}\:{the}\:{sum}\:{of}\: \\ $$$${whose}\:{elements}\:{is}\:{divisible}\:{by}\:\mathrm{5} \\ $$

Question Number 120089 Answers: 0 Comments: 0

If a continuous function f:R→R satisfies ∫_0 ^1 f(x)dx=∫_0 ^1 xf(x)dx=1 prove that ∫_0 ^1 (f(x))^2 dx≥4

$$\mathrm{If}\:\mathrm{a}\:\mathrm{continuous}\:\mathrm{function}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{satisfies} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}=\int_{\mathrm{0}} ^{\mathrm{1}} {xf}\left({x}\right){dx}=\mathrm{1} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left({f}\left({x}\right)\right)^{\mathrm{2}} {dx}\geqslant\mathrm{4} \\ $$

Question Number 120071 Answers: 1 Comments: 0

Question Number 120068 Answers: 2 Comments: 0

(i) lim_(x→0) ((cos x−1+(x^2 /2))/x^4 ) (ii) lim_(x→0) ((e^x −1−x−(x^2 /2)−(x^3 /6))/x^4 ) (iii) lim_(x→0) ((tan x−x)/(arc sin x−x))

$$\left({i}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:{x}−\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}}{{x}^{\mathrm{4}} }\: \\ $$$$\left({ii}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{e}^{{x}} −\mathrm{1}−{x}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}}{{x}^{\mathrm{4}} } \\ $$$$\left({iii}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:{x}−{x}}{\mathrm{arc}\:\mathrm{sin}\:{x}−{x}} \\ $$

Question Number 120067 Answers: 2 Comments: 0

(i) lim_(x→0) ((x^2 sin (1/x))/(tan x)) (ii) Without L′Hopital rule lim_(x→0^+ ) ((1−cos x−xsin x)/(2−2cos x−sin^2 x))

$$\left({i}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}^{\mathrm{2}} \:\mathrm{sin}\:\left(\mathrm{1}/{x}\right)}{\mathrm{tan}\:{x}} \\ $$$$\left({ii}\right)\:{Without}\:{L}'{Hopital}\:{rule} \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:{x}−{x}\mathrm{sin}\:{x}}{\mathrm{2}−\mathrm{2cos}\:{x}−\mathrm{sin}\:^{\mathrm{2}} {x}} \\ $$

Question Number 120064 Answers: 1 Comments: 0

I = ∫ _0 ^(π/2) ((1/(ln (tan r))) + (1/(1−tan r)) ) dr

$${I}\:=\:\int\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\:}}\left(\frac{\mathrm{1}}{\mathrm{ln}\:\left(\mathrm{tan}\:{r}\right)}\:+\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{tan}\:{r}}\:\right)\:{dr} \\ $$

Question Number 120060 Answers: 2 Comments: 0

(i) ∫_(−2) ^0 (dx/(2x+3)) (ii)∫_3 ^5 (dx/( (((4−x)^2 ))^(1/3) ))

$$\left({i}\right)\:\underset{−\mathrm{2}} {\overset{\mathrm{0}} {\int}}\:\frac{{dx}}{\mathrm{2}{x}+\mathrm{3}} \\ $$$$\left({ii}\right)\underset{\mathrm{3}} {\overset{\mathrm{5}} {\int}}\:\frac{{dx}}{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{4}−{x}\right)^{\mathrm{2}} }}\: \\ $$

Question Number 120059 Answers: 1 Comments: 0

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