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Question Number 207225    Answers: 0   Comments: 1

6×5=30

$$\mathrm{6}×\mathrm{5}=\mathrm{30} \\ $$

Question Number 207220    Answers: 2   Comments: 0

Find: (i − 1)^(−100) = ?

$$\mathrm{Find}:\:\:\:\left(\boldsymbol{\mathrm{i}}\:−\:\mathrm{1}\right)^{−\mathrm{100}} \:\:=\:\:? \\ $$

Question Number 207219    Answers: 1   Comments: 0

Find: ((tg 20)/(1 + tg^2 20)) + ((tg 21)/(1 + tg^2 21)) +...+ ((tg 70)/(1 + tg^2 70))

$$\mathrm{Find}: \\ $$$$\frac{\mathrm{tg}\:\mathrm{20}}{\mathrm{1}\:+\:\mathrm{tg}^{\mathrm{2}} \:\mathrm{20}}\:+\:\frac{\mathrm{tg}\:\mathrm{21}}{\mathrm{1}\:+\:\mathrm{tg}^{\mathrm{2}} \:\mathrm{21}}\:+...+\:\frac{\mathrm{tg}\:\mathrm{70}}{\mathrm{1}\:+\:\mathrm{tg}^{\mathrm{2}} \:\mathrm{70}} \\ $$

Question Number 207218    Answers: 1   Comments: 0

2^x + ((6 ∙ 2^x − 10)/(2^x − 2)) = 4 ∙ 2^x + (1/(2^x − 2)) Find: x = ?

$$\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:\:+\:\:\frac{\mathrm{6}\:\centerdot\:\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:−\:\mathrm{10}}{\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:−\:\mathrm{2}}\:\:=\:\:\mathrm{4}\:\centerdot\:\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:\:+\:\:\frac{\mathrm{1}}{\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:−\:\mathrm{2}} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{x}\:=\:? \\ $$

Question Number 207212    Answers: 1   Comments: 1

arcsin (sin 6) = ?

$$\mathrm{arcsin}\:\left(\mathrm{sin}\:\mathrm{6}\right)\:=\:? \\ $$

Question Number 207207    Answers: 1   Comments: 0

100 x^(lg x) = x^3 Find: x = ?

$$\mathrm{100}\:\mathrm{x}^{\boldsymbol{\mathrm{lg}}\:\boldsymbol{\mathrm{x}}} \:\:=\:\:\mathrm{x}^{\mathrm{3}} \\ $$$$\mathrm{Find}:\:\:\mathrm{x}\:=\:? \\ $$

Question Number 207206    Answers: 2   Comments: 0

(√(x−3 + 2 (√(x−4)))) − (√(x + 5−6 (√(x−2)))) = 2 Find: x = ?

$$\sqrt{\mathrm{x}−\mathrm{3}\:+\:\mathrm{2}\:\sqrt{\mathrm{x}−\mathrm{4}}}\:−\:\sqrt{\mathrm{x}\:+\:\mathrm{5}−\mathrm{6}\:\sqrt{\mathrm{x}−\mathrm{2}}}\:=\:\mathrm{2} \\ $$$$\mathrm{Find}:\:\:\mathrm{x}\:=\:? \\ $$

Question Number 207205    Answers: 1   Comments: 0

{ ((x^2 + xy = 4x)),((y^2 + xy = 4y)) :} Find: log_(16) (x_1 + y_1 + x_2 + y_2 ) = ?

$$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:\:+\:\:\mathrm{xy}\:\:=\:\:\mathrm{4x}}\\{\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{xy}\:\:=\:\:\mathrm{4y}}\end{cases} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{log}_{\mathrm{16}} \:\left(\mathrm{x}_{\mathrm{1}} \:+\:\mathrm{y}_{\mathrm{1}} \:+\:\mathrm{x}_{\mathrm{2}} \:+\:\mathrm{y}_{\mathrm{2}} \right)\:=\:? \\ $$

Question Number 207197    Answers: 2   Comments: 0

If xy = (1/3) , xz = (3/8) and yz = (1/2) For x , y and z compare.

$$\mathrm{If}\:\:\:\mathrm{xy}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\:\:,\:\:\:\mathrm{xz}\:=\:\frac{\mathrm{3}}{\mathrm{8}}\:\:\:\mathrm{and}\:\:\:\mathrm{yz}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{For}\:\:\:\mathrm{x}\:,\:\mathrm{y}\:\mathrm{and}\:\mathrm{z}\:\:\:\mathrm{compare}. \\ $$

Question Number 207196    Answers: 0   Comments: 0

if 2(a^2 −b^2 )^(5 ) it is a term of notable quotient of ((a+b)^n −(a−b)^n )/(ab+b^2 ) find n.

$${if}\:\mathrm{2}\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)^{\mathrm{5}\:} {it}\:{is}\:{a}\:{term}\:{of}\:{notable}\:{quotient}\:{of}\:\:\left(\left({a}+{b}\right)^{{n}} −\left({a}−{b}\right)^{{n}} \right)/\left({ab}+{b}^{\mathrm{2}} \right)\:\:{find}\:\:{n}. \\ $$

Question Number 207194    Answers: 1   Comments: 0

Let x = cos(π/9) Show that 8x^3 −6x−1=0 Deduce x is not rational

$${Let}\:\:{x}\:=\:{cos}\frac{\pi}{\mathrm{9}}\: \\ $$$${Show}\:{that}\:\mathrm{8}{x}^{\mathrm{3}} −\mathrm{6}{x}−\mathrm{1}=\mathrm{0} \\ $$$${Deduce}\:{x}\:{is}\:{not}\:\:{rational}\: \\ $$

Question Number 207184    Answers: 2   Comments: 0

Question Number 207179    Answers: 2   Comments: 0

Question Number 207175    Answers: 2   Comments: 0

If x^m .y^n = (x + y)^(m + n) then (d^2 y/dx^2 ) = ?

$$\mathrm{If}\:{x}^{{m}} .{y}^{{n}} \:=\:\left({x}\:+\:{y}\right)^{{m}\:+\:{n}} \:\mathrm{then}\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:=\:? \\ $$

Question Number 207170    Answers: 1   Comments: 4

Find: ∫_0 ^( 4) (√(16 − x^2 )) dx = ?

$$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{4}} \:\sqrt{\mathrm{16}\:−\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 207169    Answers: 1   Comments: 0

Find: ∫_0 ^( 3) (√(9 − x^2 )) dx = ?

$$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{3}} \:\sqrt{\mathrm{9}\:−\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 207157    Answers: 1   Comments: 0

Find: ((5 sin (π/6))/((1/(tg 75°)) − tg 75°)) = ?

$$\mathrm{Find}:\:\:\:\frac{\mathrm{5}\:\mathrm{sin}\:\frac{\pi}{\mathrm{6}}}{\frac{\mathrm{1}}{\mathrm{tg}\:\mathrm{75}°}\:\:−\:\:\mathrm{tg}\:\mathrm{75}°}\:\:=\:\:? \\ $$

Question Number 207156    Answers: 1   Comments: 0

Find: (1/(sin 10°)) − ((√3)/(cos 10°)) = ?

$$\mathrm{Find}:\:\:\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{10}°}\:−\:\frac{\sqrt{\mathrm{3}}}{\mathrm{cos}\:\mathrm{10}°}\:\:=\:\:? \\ $$

Question Number 207154    Answers: 1   Comments: 0

we have a system made up of two cells x and y. Both of the cell types are dividing and dying. X type cells also differentiate into Y type cells. The dynamics of this system interms of size of X and Y population is given below. calculate the steady state of this system (dx/dt)=−3x (dy/dt)=2x−2y

$$\mathrm{we}\:\mathrm{have}\:\mathrm{a}\:\mathrm{system}\:\mathrm{made}\:\mathrm{up}\:\mathrm{of}\:\mathrm{two}\:\mathrm{cells} \\ $$$$\mathrm{x}\:\mathrm{and}\:\mathrm{y}.\:\mathrm{Both}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cell}\:\mathrm{types}\:\mathrm{are} \\ $$$$\mathrm{dividing}\:\mathrm{and}\:\mathrm{dying}.\:\mathrm{X}\:\mathrm{type}\:\mathrm{cells}\:\mathrm{also} \\ $$$$\mathrm{differentiate}\:\mathrm{into}\:\mathrm{Y}\:\mathrm{type}\:\mathrm{cells}.\:\mathrm{The} \\ $$$$\mathrm{dynamics}\:\mathrm{of}\:\mathrm{this}\:\mathrm{system}\:\mathrm{interms}\:\mathrm{of}\:\:\mathrm{size} \\ $$$$\mathrm{of}\:\mathrm{X}\:\mathrm{and}\:\mathrm{Y}\:\mathrm{population}\:\mathrm{is}\:\mathrm{given}\:\mathrm{below}. \\ $$$$\mathrm{calculate}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{of}\:\mathrm{this}\:\mathrm{system} \\ $$$$\frac{\mathrm{dx}}{\mathrm{dt}}=−\mathrm{3x}\:\:\:\frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{2x}−\mathrm{2y} \\ $$

Question Number 207150    Answers: 0   Comments: 0

Expression of protein is controlled by an external S. the protein also controls its own exprexxion by a negative feedback. The fol<lwing system of ODEs represents the dynamics of the system, with m and p representing mRNA and protein respectively, prove that for any value of S≥0, the steady state of the system is spiral sink (dm/dt)=(S/(1+S))+(1/(1+P))−m (dp/dt)=m−p

$$\mathrm{Expression}\:\mathrm{of}\:\mathrm{protein}\:\mathrm{is}\:\mathrm{controlled} \\ $$$$\mathrm{by}\:\mathrm{an}\:\mathrm{external}\:\mathrm{S}.\:\mathrm{the}\:\mathrm{protein}\:\mathrm{also}\:\mathrm{controls} \\ $$$$\mathrm{its}\:\mathrm{own}\:\mathrm{exprexxion}\:\mathrm{by}\:\mathrm{a}\:\mathrm{negative}\:\mathrm{feedback}. \\ $$$$\mathrm{The}\:\mathrm{fol}<\mathrm{lwing}\:\mathrm{system}\:\mathrm{of}\:\mathrm{ODEs}\:\mathrm{represents} \\ $$$$\mathrm{the}\:\mathrm{dynamics}\:\mathrm{of}\:\mathrm{the}\:\mathrm{system},\:\mathrm{with} \\ $$$$\mathrm{m}\:\mathrm{and}\:\mathrm{p}\:\mathrm{representing}\:\mathrm{mRNA}\:\mathrm{and} \\ $$$$\mathrm{protein}\:\mathrm{respectively},\:\mathrm{prove}\:\mathrm{that}\:\mathrm{for} \\ $$$$\mathrm{any}\:\mathrm{value}\:\mathrm{of}\:\mathrm{S}\geqslant\mathrm{0},\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{system}\:\mathrm{is}\:\mathrm{spiral}\:\mathrm{sink} \\ $$$$\frac{\mathrm{dm}}{\mathrm{dt}}=\frac{\mathrm{S}}{\mathrm{1}+\mathrm{S}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{P}}−\mathrm{m}\:\:\frac{\mathrm{dp}}{\mathrm{dt}}=\mathrm{m}−\mathrm{p} \\ $$

Question Number 207151    Answers: 1   Comments: 0

Two proteins (x and y) control each other through mutual repression. The dynamic model for the system consists of the following system of ODEs. calculate the steady state values of these two proteins. comment on the stability of the steady state and the type of phase portrait expected for this system. consider x,y≥0 (dx/dt)=(y/(1+y))−2 (dy/dt)=(x/(1+x))−y

$$\mathrm{Two}\:\mathrm{proteins}\:\left(\mathrm{x}\:\mathrm{and}\:\mathrm{y}\right)\:\mathrm{control}\:\mathrm{each} \\ $$$$\mathrm{other}\:\mathrm{through}\:\mathrm{mutual}\:\mathrm{repression}.\:\mathrm{The} \\ $$$$\mathrm{dynamic}\:\mathrm{model}\:\mathrm{for}\:\mathrm{the}\:\mathrm{system}\:\mathrm{consists} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{system}\:\mathrm{of}\:\mathrm{ODEs}.\: \\ $$$$\mathrm{calculate}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{values}\:\mathrm{of}\: \\ $$$$\mathrm{these}\:\mathrm{two}\:\mathrm{proteins}.\:\mathrm{comment}\:\mathrm{on}\:\mathrm{the}\: \\ $$$$\mathrm{stability}\:\mathrm{of}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{and}\:\mathrm{the}\:\mathrm{type}\:\mathrm{of}\: \\ $$$$\mathrm{phase}\:\mathrm{portrait}\:\mathrm{expected}\:\mathrm{for}\:\mathrm{this}\:\mathrm{system}. \\ $$$$\mathrm{consider}\:\mathrm{x},\mathrm{y}\geqslant\mathrm{0} \\ $$$$\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{y}}{\mathrm{1}+\mathrm{y}}−\mathrm{2}\:\:\:\:\frac{\mathrm{dy}}{\mathrm{dt}}=\frac{\mathrm{x}}{\mathrm{1}+\mathrm{x}}−\mathrm{y} \\ $$

Question Number 207139    Answers: 1   Comments: 0

Two proteins (x&y) control each other through mutual repression. the dynamic model for the system consists of the following system of ODEs. calculate the steady state values of these two peoteins. comment of the stabilityof the steady state and the type of phase of portrait expected foe this system. consider x,y≥0 (dx/dt)=(y/(1+y))−2x (dy/dt)=(x/(1+x))−y

$$ \\ $$$$\mathrm{Two}\:\mathrm{proteins}\:\left(\mathrm{x\&y}\right)\:\mathrm{control}\:\mathrm{each} \\ $$$$\mathrm{other}\:\mathrm{through}\:\mathrm{mutual}\:\mathrm{repression}.\:\mathrm{the} \\ $$$$\mathrm{dynamic}\:\mathrm{model}\:\mathrm{for}\:\mathrm{the}\:\mathrm{system}\:\mathrm{consists}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{system}\:\mathrm{of}\:\mathrm{ODEs}. \\ $$$$\mathrm{calculate}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{values}\:\mathrm{of}\:\mathrm{these}\:\mathrm{two}\:\mathrm{peoteins}.\: \\ $$$$\mathrm{comment}\:\mathrm{of}\:\mathrm{the}\:\mathrm{stabilityof}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{type}\:\mathrm{of}\:\mathrm{phase}\:\mathrm{of}\:\mathrm{portrait}\:\mathrm{expected}\:\mathrm{foe}\:\mathrm{this} \\ $$$$\mathrm{system}.\:\mathrm{consider}\:\mathrm{x},\mathrm{y}\geqslant\mathrm{0} \\ $$$$\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{y}}{\mathrm{1}+\mathrm{y}}−\mathrm{2x}\:\:\:\frac{\mathrm{dy}}{\mathrm{dt}}=\frac{\mathrm{x}}{\mathrm{1}+\mathrm{x}}−\mathrm{y} \\ $$

Question Number 207130    Answers: 1   Comments: 0

Find: ∫ ((x − 1)/(x − 2)) dx = ?

$$\mathrm{Find}:\:\:\:\int\:\frac{\mathrm{x}\:−\:\mathrm{1}}{\mathrm{x}\:−\:\mathrm{2}}\:\mathrm{dx}\:\:=\:\:? \\ $$

Question Number 207129    Answers: 1   Comments: 0

Find: ∫ ((x + 4)/(x + 3)) dx = ?

$$\mathrm{Find}:\:\:\:\:\:\int\:\frac{\mathrm{x}\:+\:\mathrm{4}}{\mathrm{x}\:+\:\mathrm{3}}\:\mathrm{dx}\:=\:? \\ $$

Question Number 207128    Answers: 1   Comments: 0

log_(tan x) sinx − (1/(log_(cos x) tanx)) = ?

$$\mathrm{log}_{\boldsymbol{\mathrm{tan}}\:\boldsymbol{\mathrm{x}}} \:\:\mathrm{sinx}\:\:−\:\:\frac{\mathrm{1}}{\mathrm{log}_{\boldsymbol{\mathrm{cos}}\:\boldsymbol{\mathrm{x}}} \:\:\mathrm{tanx}}\:\:=\:\:? \\ $$

Question Number 207127    Answers: 2   Comments: 0

If f(x) = sin^4 2x Find f^′ ((π/(12))) = ?

$$\mathrm{If}\:\:\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{sin}^{\mathrm{4}} \:\mathrm{2x} \\ $$$$\mathrm{Find}\:\:\:\:\:\mathrm{f}\:^{'} \:\left(\frac{\pi}{\mathrm{12}}\right)\:=\:? \\ $$

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