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Question Number 215951    Answers: 1   Comments: 0

E_n = 3^E_(n−1) , n≥2 find the unit digit of E_(1000)

$${E}_{{n}} \:=\:\mathrm{3}^{{E}_{{n}−\mathrm{1}} } ,\:{n}\geqslant\mathrm{2} \\ $$$${find}\:{the}\:{unit}\:{digit}\:{of}\:{E}_{\mathrm{1000}} \\ $$

Question Number 215944    Answers: 2   Comments: 0

if m,n,z ∈N so , m<n and z<m then z<n ?

$${if}\:{m},{n},{z}\:\in{N}\:{so}\:,\:{m}<{n}\:{and}\:{z}<{m}\:{then}\:{z}<{n}\:?\: \\ $$

Question Number 215943    Answers: 1   Comments: 0

Question Number 215939    Answers: 1   Comments: 0

for any natural numbers m,n then m=n or m<n or m>n ? prove

$${for}\:{any}\:{natural}\:{numbers}\:{m},{n}\:{then}\:{m}={n}\:{or}\:{m}<{n}\:{or}\:{m}>{n}\:?\:{prove} \\ $$

Question Number 215919    Answers: 0   Comments: 1

Is absolute value of x linear equation ?

$$\:\:\:{Is}\:{absolute}\:{value}\:{of}\:{x}\:{linear}\:{equation}\:?\: \\ $$$$ \\ $$

Question Number 215918    Answers: 3   Comments: 0

why ((d )/dz)e^z =e^z ?? why dose not (de^z /dz)=(de^z /dz)=(e^z /z) ???

$$\mathrm{why}\:\:\frac{\mathrm{d}\:\:}{\mathrm{d}{z}}{e}^{{z}} ={e}^{{z}} \:?? \\ $$$$\mathrm{why}\:\mathrm{dose}\:\mathrm{not}\:\frac{\mathrm{d}{e}^{{z}} }{\mathrm{d}{z}}=\frac{\cancel{\mathrm{d}}{e}^{{z}} }{\cancel{\mathrm{d}}{z}}=\frac{{e}^{{z}} }{{z}}\:??? \\ $$

Question Number 215908    Answers: 1   Comments: 0

K.1 y=(3/2)x+1 determinant ((x,y),((−2),),((−1),),(0,),(1,),(2,))and then plot

$$\mathrm{K}.\mathrm{1} \\ $$$${y}=\frac{\mathrm{3}}{\mathrm{2}}{x}+\mathrm{1} \\ $$$$\begin{array}{|c|c|c|c|c|c|}{{x}}&\hline{{y}}\\{−\mathrm{2}}&\hline{}\\{−\mathrm{1}}&\hline{}\\{\mathrm{0}}&\hline{}\\{\mathrm{1}}&\hline{}\\{\mathrm{2}}&\hline{}\\\hline\end{array}\mathrm{and}\:\mathrm{then}\:\mathrm{plot} \\ $$

Question Number 215898    Answers: 1   Comments: 1

x^n +y^n +z^n = 0 n = ?

$${x}^{{n}} +{y}^{{n}} +{z}^{{n}} \:=\:\mathrm{0} \\ $$$${n}\:=\:? \\ $$

Question Number 215893    Answers: 4   Comments: 0

Find: 1) lim_(x→0) ((1 − cos2x)/x^2 ) = ? 2) Σ_(n=1) ^(n=∞) (n^2 /3^n ) = ?

$$\mathrm{Find}: \\ $$$$\left.\mathrm{1}\right)\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}\:−\:\mathrm{cos2x}}{\mathrm{x}^{\mathrm{2}} }\:=\:? \\ $$$$\left.\mathrm{2}\right)\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}=\infty} {\sum}}\:\frac{\mathrm{n}^{\mathrm{2}} }{\mathrm{3}^{\boldsymbol{\mathrm{n}}} }\:=\:? \\ $$

Question Number 215889    Answers: 1   Comments: 5

Find: lim_(h→0) (((x + h)^3 + x^3 )/h) = ?

$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{h}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{x}\:+\:\mathrm{h}\right)^{\mathrm{3}} \:+\:\mathrm{x}^{\mathrm{3}} }{\mathrm{h}}\:=\:? \\ $$

Question Number 215887    Answers: 0   Comments: 2

Question Number 215885    Answers: 1   Comments: 1

Area of ABC ?

$$\:\mathrm{Area}\:\:\mathrm{of}\:\mathrm{ABC}\:? \\ $$

Question Number 215884    Answers: 1   Comments: 0

lim_(x→0) ((cos 2x−cos 6x)/(1−cos 3x cos 5x)) =?

$$\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{2x}−\mathrm{cos}\:\mathrm{6x}}{\mathrm{1}−\mathrm{cos}\:\mathrm{3x}\:\mathrm{cos}\:\mathrm{5x}}\:=? \\ $$$$\:\:\:\: \\ $$

Question Number 215883    Answers: 1   Comments: 2

Given m>0, n>0, m+n=(√a). Find the range of a such that “(m+(1/m))(n+(1/n)) gets its minimum iff m=n”.

$$\mathrm{Given}\:{m}>\mathrm{0},\:{n}>\mathrm{0},\:{m}+{n}=\sqrt{{a}}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{a}\:\mathrm{such}\:\mathrm{that} \\ $$$$``\left({m}+\frac{\mathrm{1}}{{m}}\right)\left({n}+\frac{\mathrm{1}}{{n}}\right)\:\mathrm{gets}\:\mathrm{its}\:\mathrm{minimum}\:\mathrm{iff}\:{m}={n}''. \\ $$

Question Number 215874    Answers: 3   Comments: 0

If x^2 +3x+2=y^2 +5y+8, Prove that x=((−3±(√(4y^2 +20y+33)))/2).

$$\mathrm{If}\:{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{2}={y}^{\mathrm{2}} +\mathrm{5}{y}+\mathrm{8}, \\ $$$$\mathrm{Prove}\:\mathrm{that}\:{x}=\frac{−\mathrm{3}\pm\sqrt{\mathrm{4}{y}^{\mathrm{2}} +\mathrm{20}{y}+\mathrm{33}}}{\mathrm{2}}. \\ $$

Question Number 215868    Answers: 0   Comments: 2

Does the force of friction increase, decrease, or remain constant with the increase in the number of car tires?

$$ \\ $$Does the force of friction increase, decrease, or remain constant with the increase in the number of car tires?

Question Number 215859    Answers: 0   Comments: 0

Question Number 215845    Answers: 2   Comments: 10

1, 3, − 1, − 3, − 7, − 21, − 25, ___, ___, ___ Next three terms??

$$\mathrm{1},\:\mathrm{3},\:−\:\mathrm{1},\:−\:\mathrm{3},\:−\:\mathrm{7},\:\:−\:\mathrm{21},\:\:−\:\mathrm{25},\:\:\:\:\:\_\_\_,\:\:\:\:\:\_\_\_,\:\:\:\:\:\_\_\_ \\ $$$$ \\ $$$$\mathrm{Next}\:\mathrm{three}\:\mathrm{terms}?? \\ $$

Question Number 215840    Answers: 2   Comments: 2

log _(24) 3= a and log _(24) 6 = (b/6) log _(√8) (b−4a)= ?

$$\:\:\:\mathrm{log}\:_{\mathrm{24}} \:\mathrm{3}=\:{a}\:\mathrm{and}\:\mathrm{log}\:_{\mathrm{24}} \:\mathrm{6}\:=\:\frac{{b}}{\mathrm{6}} \\ $$$$\:\:\:\mathrm{log}\:_{\sqrt{\mathrm{8}}} \:\left({b}−\mathrm{4}{a}\right)=\:? \\ $$

Question Number 215837    Answers: 2   Comments: 0

If b^3 + a^2 c + ac^2 = 3abc then prove that one root of ax^2 + bx + c = 0 is the square of the other one.

$$\mathrm{If}\:{b}^{\mathrm{3}} \:+\:{a}^{\mathrm{2}} {c}\:+\:{ac}^{\mathrm{2}} \:=\:\mathrm{3}{abc}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{one}\:\mathrm{root}\:\mathrm{of}\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{is}\:\mathrm{the}\:\mathrm{square}\: \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{other}\:\mathrm{one}. \\ $$

Question Number 215831    Answers: 2   Comments: 0

lim_(x→∞) (((√((x+1)^3 ))−(√((x−1)^3 )))/( (√x))) =?

$$\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\sqrt{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} }−\sqrt{\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{3}} }}{\:\sqrt{\mathrm{x}}}\:=? \\ $$

Question Number 215828    Answers: 1   Comments: 0

Let Γ be a hyperbola with foci F_1 and F_2 , eccentricity e. M is an arbitrary point on Γ. Let x=∠MF_1 F_2 , y=∠MF_2 F_1 Prove that ((∣ cos x − cos y ∣)/(1 − cos x cos y)) = ((2e)/(e^2 +1)).

$$\mathrm{Let}\:\Gamma\:\mathrm{be}\:\mathrm{a}\:\mathrm{hyperbola}\:\mathrm{with}\:\mathrm{foci}\:{F}_{\mathrm{1}} \:\mathrm{and}\:{F}_{\mathrm{2}} ,\: \\ $$$$\mathrm{eccentricity}\:{e}.\:{M}\:\mathrm{is}\:\mathrm{an}\:\mathrm{arbitrary}\:\mathrm{point}\:\mathrm{on}\:\Gamma. \\ $$$$\mathrm{Let}\:{x}=\angle{MF}_{\mathrm{1}} {F}_{\mathrm{2}} ,\:{y}=\angle{MF}_{\mathrm{2}} {F}_{\mathrm{1}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\frac{\mid\:\mathrm{cos}\:{x}\:−\:\mathrm{cos}\:{y}\:\mid}{\mathrm{1}\:−\:\mathrm{cos}\:{x}\:\mathrm{cos}\:{y}}\:=\:\frac{\mathrm{2}{e}}{{e}^{\mathrm{2}} +\mathrm{1}}. \\ $$

Question Number 215820    Answers: 2   Comments: 1

Question Number 215789    Answers: 1   Comments: 1

Question Number 215811    Answers: 2   Comments: 0

If α, β, γ, δ are the roots of x^4 + x^3 + x^2 + x + 1 = 0 then find α^(2021) + β^(2021) + γ^(2021) + δ^(2021) .

$$\mathrm{If}\:\alpha,\:\beta,\:\gamma,\:\delta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$${x}^{\mathrm{4}} \:+\:{x}^{\mathrm{3}} \:+\:{x}^{\mathrm{2}} \:+\:{x}\:+\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{then}\:\mathrm{find} \\ $$$$\alpha^{\mathrm{2021}} \:+\:\beta^{\mathrm{2021}} \:+\:\gamma^{\mathrm{2021}} \:+\:\delta^{\mathrm{2021}} \:. \\ $$

Question Number 215782    Answers: 1   Comments: 0

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