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Question Number 208533    Answers: 2   Comments: 0

z′ = (1/2)(z+(1/z)) z and z′ are complex numbers show that z = 2e^(iθ) show that M′ describes a conic section

$${z}'\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left({z}+\frac{\mathrm{1}}{{z}}\right) \\ $$$${z}\:{and}\:{z}'\:{are}\:{complex}\:{numbers} \\ $$$${show}\:{that}\:{z}\:=\:\mathrm{2}{e}^{{i}\theta} \\ $$$${show}\:{that}\:{M}'\:{describes}\:{a}\:{conic}\:{section} \\ $$

Question Number 208526    Answers: 1   Comments: 2

Question Number 208520    Answers: 1   Comments: 0

Question Number 208519    Answers: 1   Comments: 0

If a^x = b^y (a/x) + (b/y) = 1 then, a^x + b^y = ?

$$\mathrm{If}\:\:\:\:\:\mathrm{a}^{\mathrm{x}} \:\:=\:\:\mathrm{b}^{\mathrm{y}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{a}}{\mathrm{x}}\:\:+\:\:\frac{\mathrm{b}}{\mathrm{y}}\:\:=\:\:\mathrm{1} \\ $$$$\mathrm{then},\:\:\:\:\mathrm{a}^{\mathrm{x}} \:\:+\:\:\mathrm{b}^{\mathrm{y}} \:\:=\:\:? \\ $$

Question Number 208515    Answers: 0   Comments: 0

∫_0 ^( ∞) ((ln^2 x)/(1+x^4 )) dx

$$\int_{\mathrm{0}} ^{\:\infty} \:\frac{{ln}^{\mathrm{2}} {x}}{\mathrm{1}+{x}^{\mathrm{4}} }\:{dx} \\ $$

Question Number 208513    Answers: 0   Comments: 0

a, b>0. 2a+b=1, prove that ((a^2 +b^2 )/(2a))+((2a)/b^2 ) > 2.

$${a},\:{b}>\mathrm{0}.\:\mathrm{2}{a}+{b}=\mathrm{1},\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{\mathrm{2}{a}}+\frac{\mathrm{2}{a}}{{b}^{\mathrm{2}} }\:>\:\mathrm{2}. \\ $$

Question Number 208508    Answers: 1   Comments: 0

Evaluate and leave your answer in exponent form. 5^(2024) − 5^(2023) − 5^(2022) − 5^(2021) − 5^(2020) − 5^(2019) = ?

$$\mathrm{Evaluate}\:\mathrm{and}\:\mathrm{leave}\:\mathrm{your}\:\mathrm{answer}\:\mathrm{in}\:\mathrm{exponent}\:\mathrm{form}. \\ $$$$\mathrm{5}^{\mathrm{2024}} \:−\:\:\mathrm{5}^{\mathrm{2023}} \:\:−\:\:\mathrm{5}^{\mathrm{2022}} \:\:−\:\:\mathrm{5}^{\mathrm{2021}} \:\:−\:\:\mathrm{5}^{\mathrm{2020}} \:\:−\:\:\mathrm{5}^{\mathrm{2019}} \:\:=\:\:? \\ $$

Question Number 208507    Answers: 0   Comments: 0

An aeroplane flies 25 km in the direction 60° North of East 35 km Straight East, then 15 km straight North. What direction is the pane from the straight point?

An aeroplane flies 25 km in the direction 60° North of East 35 km Straight East, then 15 km straight North. What direction is the pane from the straight point?

Question Number 208506    Answers: 2   Comments: 0

What is the smallest number which must added to 9454351626 so that it will become divisible by 11?

What is the smallest number which must added to 9454351626 so that it will become divisible by 11?

Question Number 208503    Answers: 0   Comments: 0

Question Number 208502    Answers: 1   Comments: 0

Question Number 208499    Answers: 0   Comments: 0

Question Number 208493    Answers: 2   Comments: 0

Question Number 208489    Answers: 1   Comments: 0

Question Number 208482    Answers: 0   Comments: 0

Question Number 208693    Answers: 2   Comments: 0

Calculate the area enclosed by the curve ((1/x)−2)^2 +((1/y)−2)^2 =1

$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{area}\:\mathrm{enclosed}\:\mathrm{by}\:\mathrm{the}\:\mathrm{curve} \\ $$$$\left(\frac{\mathrm{1}}{{x}}−\mathrm{2}\right)^{\mathrm{2}} +\left(\frac{\mathrm{1}}{{y}}−\mathrm{2}\right)^{\mathrm{2}} =\mathrm{1} \\ $$

Question Number 208469    Answers: 1   Comments: 0

Find: ((61^3 + 24^3 )/(61^3 + 37^3 )) = ?

$$\mathrm{Find}:\:\:\:\:\:\frac{\mathrm{61}^{\mathrm{3}} \:\:+\:\:\mathrm{24}^{\mathrm{3}} }{\mathrm{61}^{\mathrm{3}} \:\:+\:\:\mathrm{37}^{\mathrm{3}} }\:\:=\:\:? \\ $$

Question Number 208468    Answers: 2   Comments: 0

Question Number 208467    Answers: 3   Comments: 0

Question Number 208466    Answers: 1   Comments: 0

Question Number 208465    Answers: 1   Comments: 0

Question Number 208463    Answers: 1   Comments: 0

Compare it: a = log_3 4 b = log_5 6 c = log_6 2

$$\mathrm{Compare}\:\mathrm{it}: \\ $$$$\mathrm{a}\:=\:\mathrm{log}_{\mathrm{3}} \:\mathrm{4} \\ $$$$\mathrm{b}\:=\:\mathrm{log}_{\mathrm{5}} \:\mathrm{6} \\ $$$$\mathrm{c}\:=\:\mathrm{log}_{\mathrm{6}} \:\mathrm{2} \\ $$

Question Number 208455    Answers: 1   Comments: 2

Question Number 208453    Answers: 2   Comments: 0

If f(x) = (((2a + 1)∙x + 1)/(x − a)) and f(x) = f^(−1) (x) Find: a^2 + 3 = ?

$$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\left(\mathrm{2a}\:+\:\mathrm{1}\right)\centerdot\mathrm{x}\:+\:\mathrm{1}}{\mathrm{x}\:−\:\mathrm{a}}\:\:\:\mathrm{and}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right) \\ $$$$\mathrm{Find}:\:\:\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{3}\:=\:? \\ $$

Question Number 208447    Answers: 0   Comments: 0

$$\:\:\underbrace{\:} \\ $$

Question Number 208445    Answers: 1   Comments: 0

u_(n+1) = u_n −u_n ^3 , u_0 ∈]0,1[ v_n = (1/u_(n+1) ^2 )−(1/u_n ^2 ) = f(u_n ^2 ) ; f(x) = ((2−x)/((1−x)^2 )) v_n converges to 2, v_n is decreasing . show that v_n ≥ 2 x_n =(1/(n+1))Σ_(m=0) ^m (v_m ) . show that x_0 ≥x_n ≥v_n . show that x_n is decreasing and lim_(n→∞) x_n = l ≥2 . show that 2x_(n+1) −x_n ≤v_(n+1) and deduce l . express x_(n+1) −x_n interms of u_n . deduce lim_(n→∞) nu_n ^2

$$\left.{u}_{{n}+\mathrm{1}} \:=\:{u}_{{n}} −{u}_{{n}} ^{\mathrm{3}} ,\:{u}_{\mathrm{0}} \in\right]\mathrm{0},\mathrm{1}\left[\right. \\ $$$${v}_{{n}} \:=\:\frac{\mathrm{1}}{{u}_{{n}+\mathrm{1}} ^{\mathrm{2}} }−\frac{\mathrm{1}}{{u}_{{n}} ^{\mathrm{2}} }\:=\:{f}\left({u}_{{n}} ^{\mathrm{2}} \right)\:;\:{f}\left({x}\right)\:=\:\frac{\mathrm{2}−{x}}{\left(\mathrm{1}−{x}\right)^{\mathrm{2}} } \\ $$$${v}_{{n}} \:{converges}\:{to}\:\mathrm{2},\:{v}_{{n}} \:{is}\:{decreasing} \\ $$$$.\:{show}\:{that}\:{v}_{{n}} \:\geqslant\:\mathrm{2} \\ $$$${x}_{{n}} =\frac{\mathrm{1}}{{n}+\mathrm{1}}\underset{{m}=\mathrm{0}} {\overset{{m}} {\sum}}\left({v}_{{m}} \right) \\ $$$$.\:{show}\:{that}\:{x}_{\mathrm{0}} \geqslant{x}_{{n}} \geqslant{v}_{{n}} \\ $$$$.\:{show}\:{that}\:{x}_{{n}} \:{is}\:{decreasing}\:{and}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}{x}_{{n}} =\:{l}\:\geqslant\mathrm{2} \\ $$$$.\:{show}\:{that}\:\mathrm{2}{x}_{{n}+\mathrm{1}} −{x}_{{n}} \leqslant{v}_{{n}+\mathrm{1}} \:{and}\:{deduce}\:{l} \\ $$$$.\:{express}\:{x}_{{n}+\mathrm{1}} −{x}_{{n}} \:{interms}\:{of}\:{u}_{{n}} \\ $$$$.\:{deduce}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}{nu}_{{n}} ^{\mathrm{2}} \\ $$

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