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AllQuestion and Answers: Page 1936

Question Number 4817    Answers: 0   Comments: 1

f(αx)=αf(x−α) f(x)=?

$${f}\left(\alpha{x}\right)=\alpha{f}\left({x}−\alpha\right) \\ $$$${f}\left({x}\right)=? \\ $$

Question Number 4816    Answers: 0   Comments: 1

Question Number 4812    Answers: 0   Comments: 6

Question Number 4809    Answers: 1   Comments: 0

Show that ((x^2 +a^2 )/(x^2 −a^2 )) > ((x+a)/(x−a)).

$${Show}\:{that}\:\frac{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }{{x}^{\mathrm{2}} −{a}^{\mathrm{2}} }\:>\:\frac{{x}+{a}}{{x}−{a}}. \\ $$

Question Number 4807    Answers: 1   Comments: 2

Find X if... ∫_0 ^e x≠0 and ((xr)/r^e )=(√(e+n)) or Σ_(n!) e=0 and ((℧x≠y)/(℧y≠x))=−1

$${Find}\:\mathbb{X}\:{if}... \\ $$$$\underset{\mathrm{0}} {\overset{{e}} {\int}}{x}\neq\mathrm{0}\:{and}\:\frac{{xr}}{{r}^{{e}} }=\sqrt{{e}+{n}} \\ $$$${or} \\ $$$$\underset{{n}!} {\sum}{e}=\mathrm{0}\:{and}\:\frac{\mho{x}\neq{y}}{\mho{y}\neq{x}}=−\mathrm{1} \\ $$

Question Number 4800    Answers: 2   Comments: 0

m(d^2 x/dt^(2 ) )=f−k(dx/dt) x(0)=x_0 x′(0)=v_0 x(t)=?

$${m}\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}\:} }={f}−{k}\frac{{dx}}{{dt}} \\ $$$${x}\left(\mathrm{0}\right)={x}_{\mathrm{0}} \\ $$$${x}'\left(\mathrm{0}\right)={v}_{\mathrm{0}} \\ $$$${x}\left({t}\right)=? \\ $$

Question Number 4793    Answers: 0   Comments: 1

please explain. V(x)=−∫F(x)dx

$${please}\:{explain}. \\ $$$${V}\left({x}\right)=−\int{F}\left({x}\right){dx} \\ $$$$ \\ $$$$ \\ $$

Question Number 4795    Answers: 1   Comments: 0

a,b,c∈C x_0 =a y_0 =b z_0 =c x_(n+1) =x_n −y_n y_(n+1) =y_n −z_n z_(n+1) =z_n −x_n x_n +y_n +z_n =?,n≥1

$${a},{b},{c}\in\mathbb{C} \\ $$$${x}_{\mathrm{0}} ={a} \\ $$$${y}_{\mathrm{0}} ={b} \\ $$$${z}_{\mathrm{0}} ={c} \\ $$$${x}_{{n}+\mathrm{1}} ={x}_{{n}} −{y}_{{n}} \\ $$$${y}_{{n}+\mathrm{1}} ={y}_{{n}} −{z}_{{n}} \\ $$$${z}_{{n}+\mathrm{1}} ={z}_{{n}} −{x}_{{n}} \\ $$$${x}_{{n}} +{y}_{{n}} +{z}_{{n}} =?,{n}\geqslant\mathrm{1} \\ $$

Question Number 4790    Answers: 0   Comments: 0

Function Γ is a+b a+b=AB a≠b−4 b−4=4+b b=a−1 a−1=2 b=2 a=3 Function Γ is (a/b)+sin a+b Γ=(a/b)+sin a+b a=b−1 b=5 a=4 9−a=b 9−b=a Funcion Γ is ((sin a+sin b)/(sin^(−1) a+sin^(−1) b))×(a+b) sin a+sin b<c_1 c_1 =a×3 a=b+3 b=2 b+3=2+3 2+3=5 a=5 a×3=15 c_1 =15 c_2 =c_1 ÷5 c_1 ÷5=3 c_2 =3 c_1 +c_2 =c_3 c_3 =18 c_3 ≈sin a+b a×b×2=20 sin 20=sin a+b c_3 ≈20

$${Function}\:\Gamma\:{is}\:{a}+{b} \\ $$$${a}+{b}={AB} \\ $$$${a}\neq{b}−\mathrm{4} \\ $$$${b}−\mathrm{4}=\mathrm{4}+{b} \\ $$$${b}={a}−\mathrm{1} \\ $$$${a}−\mathrm{1}=\mathrm{2} \\ $$$${b}=\mathrm{2} \\ $$$${a}=\mathrm{3} \\ $$$${Function}\:\Gamma\:{is}\:\frac{{a}}{{b}}+\mathrm{sin}\:{a}+{b} \\ $$$$\Gamma=\frac{{a}}{{b}}+\mathrm{sin}\:{a}+{b} \\ $$$${a}={b}−\mathrm{1} \\ $$$${b}=\mathrm{5} \\ $$$${a}=\mathrm{4} \\ $$$$\mathrm{9}−{a}={b} \\ $$$$\mathrm{9}−{b}={a} \\ $$$${Funcion}\:\Gamma\:{is}\:\frac{\mathrm{sin}\:{a}+\mathrm{sin}\:{b}}{\mathrm{sin}^{−\mathrm{1}} {a}+\mathrm{sin}^{−\mathrm{1}} {b}}×\left({a}+{b}\right) \\ $$$$\mathrm{sin}\:{a}+\mathrm{sin}\:{b}<{c}_{\mathrm{1}} \\ $$$${c}_{\mathrm{1}} ={a}×\mathrm{3} \\ $$$${a}={b}+\mathrm{3} \\ $$$${b}=\mathrm{2} \\ $$$${b}+\mathrm{3}=\mathrm{2}+\mathrm{3} \\ $$$$\mathrm{2}+\mathrm{3}=\mathrm{5} \\ $$$${a}=\mathrm{5} \\ $$$${a}×\mathrm{3}=\mathrm{15} \\ $$$${c}_{\mathrm{1}} =\mathrm{15} \\ $$$${c}_{\mathrm{2}} ={c}_{\mathrm{1}} \boldsymbol{\div}\mathrm{5} \\ $$$${c}_{\mathrm{1}} \boldsymbol{\div}\mathrm{5}=\mathrm{3} \\ $$$${c}_{\mathrm{2}} =\mathrm{3} \\ $$$${c}_{\mathrm{1}} +{c}_{\mathrm{2}} ={c}_{\mathrm{3}} \\ $$$${c}_{\mathrm{3}} =\mathrm{18} \\ $$$${c}_{\mathrm{3}} \approx\mathrm{sin}\:{a}+{b} \\ $$$${a}×{b}×\mathrm{2}=\mathrm{20} \\ $$$$\mathrm{sin}\:\mathrm{20}=\mathrm{sin}\:{a}+{b} \\ $$$${c}_{\mathrm{3}} \approx\mathrm{20} \\ $$

Question Number 4785    Answers: 0   Comments: 2

n^i (n/(i×i))≈sin n^i +α

$${n}^{{i}} \frac{{n}}{{i}×{i}}\approx\mathrm{sin}\:{n}^{{i}} +\alpha \\ $$

Question Number 4783    Answers: 0   Comments: 1

cos α+β ≈(((cos α+cos β)/(cos^(−1) α+cos^(−1) β)))^(α+β) sin a+b≈(((sin a+sin b)/(sin^(−1) a+sin^(−1) b)))^(a+b) tan (a+(a/b))^k ≈(((tan (a+b)×k)/(tan^(−1) (a+b)×k)))

$$\mathrm{cos}\:\alpha+\beta\:\approx\left(\frac{\mathrm{cos}\:\alpha+\mathrm{cos}\:\beta}{\mathrm{cos}^{−\mathrm{1}} \alpha+\mathrm{cos}^{−\mathrm{1}} \beta}\right)^{\alpha+\beta} \\ $$$$\mathrm{sin}\:{a}+{b}\approx\left(\frac{\mathrm{sin}\:{a}+\mathrm{sin}\:{b}}{\mathrm{sin}^{−\mathrm{1}} {a}+\mathrm{sin}^{−\mathrm{1}} {b}}\right)^{{a}+{b}} \\ $$$$\mathrm{tan}\:\left({a}+\frac{{a}}{{b}}\right)^{{k}} \approx\left(\frac{\mathrm{tan}\:\left({a}+{b}\right)×{k}}{\mathrm{tan}^{−\mathrm{1}} \left({a}+{b}\right)×{k}}\right) \\ $$

Question Number 4781    Answers: 0   Comments: 0

((√(δy))/(√(δx)))×f_1 = determinant (((δy)),((δx)))^(x/z)

$$\frac{\sqrt{\delta{y}}}{\sqrt{\delta{x}}}×{f}_{\mathrm{1}} =\begin{vmatrix}{\delta{y}}\\{\delta{x}}\end{vmatrix}^{{x}/{z}} \\ $$

Question Number 4780    Answers: 0   Comments: 0

D_1 ρ=∣(√(f(α+β)))×c^2 ∣

$${D}_{\mathrm{1}} \rho=\mid\sqrt{{f}\left(\alpha+\beta\right)}×{c}^{\mathrm{2}} \mid \\ $$

Question Number 4775    Answers: 0   Comments: 2

Let ∗ be a binary operation on Z defined by x∗y=(1/2)(x+y+1+(1/2)(1+(−1)^(x+y) )). Is ∗ associative?

$${Let}\:\ast\:{be}\:{a}\:{binary}\:{operation}\:{on}\:\mathbb{Z} \\ $$$${defined}\:{by}\: \\ $$$${x}\ast{y}=\frac{\mathrm{1}}{\mathrm{2}}\left({x}+{y}+\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\left(−\mathrm{1}\right)^{{x}+{y}} \right)\right). \\ $$$${Is}\:\ast\:{associative}? \\ $$

Question Number 4772    Answers: 0   Comments: 4

Let z=Ax^2 +Bxy+Cy^2 . Find conditions on the constants A,B,C that ensure that the point (0,0,0) is a (i) local minimum, (ii) local maximum, (ii) saddle point.

$${Let}\:{z}={Ax}^{\mathrm{2}} +{Bxy}+{Cy}^{\mathrm{2}} .\:{Find}\:{conditions} \\ $$$${on}\:{the}\:{constants}\:{A},{B},{C}\:{that}\:{ensure} \\ $$$${that}\:{the}\:{point}\:\left(\mathrm{0},\mathrm{0},\mathrm{0}\right)\:{is}\:{a}\: \\ $$$$\left({i}\right)\:{local}\:{minimum}, \\ $$$$\left({ii}\right)\:{local}\:{maximum}, \\ $$$$\left({ii}\right)\:{saddle}\:{point}. \\ $$$$ \\ $$$$ \\ $$

Question Number 4773    Answers: 1   Comments: 1

lim_(x→0 ) ((sin(x))/x)= 1 how is this so?

$${lim}_{{x}\rightarrow\mathrm{0}\:} \:\frac{{sin}\left({x}\right)}{{x}}=\:\mathrm{1} \\ $$$${how}\:{is}\:{this}\:{so}? \\ $$$$ \\ $$

Question Number 4760    Answers: 0   Comments: 2

the number 27000001 has 4 prime factors. find thier sum

$${the}\:{number}\:\mathrm{27000001} \\ $$$${has}\:\mathrm{4}\:{prime}\:{factors}. \\ $$$${find}\:{thier}\:{sum} \\ $$

Question Number 4758    Answers: 1   Comments: 0

∫_(−∞) ^∞ e^(−x^(2 ) ) dx = (√(π )) is this true, if so how?

$$\int_{−\infty} ^{\infty} {e}^{−{x}^{\mathrm{2}\:} } \:{dx}\:=\:\sqrt{\pi\:} \\ $$$${is}\:{this}\:{true},\:{if}\:{so}\:{how}? \\ $$

Question Number 4753    Answers: 0   Comments: 2

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Selecting a random star is a (1/(15)) chance at random. Lets say you have to pick a second random star that is next to it. Either above, below, or to the side. What this the probability of selecting the two stars correctly? −−−−−− As far as i have worked out, if the first star is not an edge or corner, the second has a (1/4) chance if the first is an edge, the second had a (1/3) chance if the first is a corner, the second is a (1/2) chance. what is the overall probability???

$$\ast\:\:\ast\:\:\ast\:\:\ast\:\:\ast \\ $$$$\ast\:\:\ast\:\:\ast\:\:\ast\:\:\ast \\ $$$$\ast\:\:\ast\:\:\ast\:\:\ast\:\:\ast \\ $$$$ \\ $$$$\mathrm{Selecting}\:\mathrm{a}\:\mathrm{random}\:\mathrm{star}\:\mathrm{is}\:\mathrm{a}\:\:\frac{\mathrm{1}}{\mathrm{15}}\:\mathrm{chance} \\ $$$$\mathrm{at}\:\mathrm{random}.\:\mathrm{Lets}\:\mathrm{say}\:\mathrm{you}\:\mathrm{have}\:\mathrm{to}\:\mathrm{pick} \\ $$$$\mathrm{a}\:\mathrm{second}\:\mathrm{random}\:\mathrm{star}\:\mathrm{that}\:\mathrm{is}\:\mathrm{next}\:\mathrm{to}\:\mathrm{it}. \\ $$$${Either}\:{above},\:{below},\:{or}\:{to}\:{the}\:{side}. \\ $$$$\mathrm{What}\:\mathrm{this}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{selecting} \\ $$$$\mathrm{the}\:\mathrm{two}\:\mathrm{stars}\:\mathrm{correctly}? \\ $$$$ \\ $$$$−−−−−− \\ $$$$ \\ $$$$\mathrm{As}\:\mathrm{far}\:\mathrm{as}\:\mathrm{i}\:\mathrm{have}\:\mathrm{worked}\:\mathrm{out},\:\mathrm{if}\:\mathrm{the}\:\mathrm{first} \\ $$$$\mathrm{star}\:\mathrm{is}\:\mathrm{not}\:\mathrm{an}\:\mathrm{edge}\:\mathrm{or}\:\mathrm{corner},\:\mathrm{the}\:\mathrm{second} \\ $$$$\mathrm{has}\:\mathrm{a}\:\frac{\mathrm{1}}{\mathrm{4}}\:\mathrm{chance} \\ $$$$ \\ $$$$\mathrm{if}\:\mathrm{the}\:\mathrm{first}\:\mathrm{is}\:\mathrm{an}\:\mathrm{edge},\:\mathrm{the}\:\mathrm{second}\:\mathrm{had}\:\mathrm{a} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{chance} \\ $$$$ \\ $$$$\mathrm{if}\:\mathrm{the}\:\mathrm{first}\:\mathrm{is}\:\mathrm{a}\:\mathrm{corner},\:\mathrm{the}\:\mathrm{second}\:\mathrm{is}\:\mathrm{a}\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{chance}. \\ $$$$ \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{overall}\:\mathrm{probability}??? \\ $$$$ \\ $$

Question Number 4750    Answers: 0   Comments: 1

f_n (x)= { ((1 n=1)),(((((1−x^2 )...(1−x^n ))/((1−2x)...(1−nx))) n∈N,n>1)) :} lim_(n→∞) f_n (x)=? n>1,f(x)=0,x=?

$${f}_{{n}} \left({x}\right)=\begin{cases}{\mathrm{1}\:\:\:\:{n}=\mathrm{1}}\\{\frac{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)...\left(\mathrm{1}−{x}^{{n}} \right)}{\left(\mathrm{1}−\mathrm{2}{x}\right)...\left(\mathrm{1}−{nx}\right)}\:\:\:\:{n}\in\mathbb{N},{n}>\mathrm{1}}\end{cases} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}{f}_{{n}} \left({x}\right)=? \\ $$$${n}>\mathrm{1},{f}\left({x}\right)=\mathrm{0},{x}=? \\ $$

Question Number 4752    Answers: 0   Comments: 1

For 0≤x,y,z≤1 solve the equation (x/(1+y+zx))+(y/(1+z+xy))+(z/(1+x+yz))=(3/(x+y+z)).

$${For}\:\mathrm{0}\leqslant{x},{y},{z}\leqslant\mathrm{1}\:{solve}\:{the}\:{equation} \\ $$$$\frac{{x}}{\mathrm{1}+{y}+{zx}}+\frac{{y}}{\mathrm{1}+{z}+{xy}}+\frac{{z}}{\mathrm{1}+{x}+{yz}}=\frac{\mathrm{3}}{{x}+{y}+{z}}. \\ $$

Question Number 4751    Answers: 0   Comments: 6

Find all functions h:Z→Z such that h(x+y)+h(xy)=h(x)h(y)+1 for all x,y∈Z.

$${Find}\:{all}\:{functions}\:{h}:\mathbb{Z}\rightarrow\mathbb{Z}\:{such}\:{that} \\ $$$${h}\left({x}+{y}\right)+{h}\left({xy}\right)={h}\left({x}\right){h}\left({y}\right)+\mathrm{1} \\ $$$${for}\:{all}\:{x},{y}\in\mathbb{Z}. \\ $$

Question Number 4748    Answers: 0   Comments: 0

Find all real a such that f(x)={ax+sinx} is periodic. {u} is the fractional−part function of the real number u.

$${Find}\:{all}\:{real}\:\boldsymbol{{a}}\:{such}\:{that}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{f}\left({x}\right)=\left\{\boldsymbol{{a}}{x}+{sinx}\right\}\: \\ $$$${is}\:{periodic}.\:\left\{{u}\right\}\:{is}\:{the}\:{fractional}−{part} \\ $$$${function}\:{of}\:{the}\:{real}\:{number}\:{u}. \\ $$

Question Number 4739    Answers: 1   Comments: 0

Question Number 4733    Answers: 0   Comments: 10

Question Number 4730    Answers: 0   Comments: 1

∫_0 ^(π/2) (dx/(sinx^(cosx ) + cosx^(sinx ) ))

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{dx}}{{sinx}^{{cosx}\:} \:\:+\:\:\:\:{cosx}^{{sinx}\:} } \\ $$

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