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Question Number 80199    Answers: 2   Comments: 1

Question Number 80180    Answers: 1   Comments: 0

h(x)=((x−x^2 )/(x+1)) we defined this function on R−{−1}→R 1) Study the variations of h then draw up its table of variation. please sirs i need your kind help

$$\mathrm{h}\left({x}\right)=\frac{{x}−{x}^{\mathrm{2}} }{{x}+\mathrm{1}} \\ $$$${we}\:{defined}\:{this}\:{function}\:{on} \\ $$$$\mathbb{R}−\left\{−\mathrm{1}\right\}\rightarrow\mathbb{R} \\ $$$$ \\ $$$$\left.\mathrm{1}\right)\:\mathrm{Study}\:\mathrm{the}\:\mathrm{variations}\:\mathrm{of}\:\mathrm{h}\:\mathrm{then} \\ $$$$\mathrm{draw}\:\mathrm{up}\:\mathrm{its}\:\mathrm{table}\:\mathrm{of}\:\mathrm{variation}. \\ $$$$ \\ $$$$\mathrm{please}\:\mathrm{sirs}\:\mathrm{i}\:\mathrm{need}\:\mathrm{your}\:\mathrm{kind}\:\mathrm{help} \\ $$

Question Number 80178    Answers: 0   Comments: 4

Question Number 80175    Answers: 1   Comments: 1

∫_0 ^1 (dx/(√(x^2 +x+1))) = ?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dx}}{\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}\:=\:? \\ $$

Question Number 80172    Answers: 0   Comments: 2

Given that lim_(x→0) ((√(f(x)+ x))/h) = L then lim_(x→0) ((√(f(x) + 2x))/h) = ?

$${Given}\:{that}\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{{f}\left({x}\right)+\:{x}}}{{h}}\:=\:{L}\:{then} \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{{f}\left({x}\right)\:+\:\mathrm{2}{x}}}{{h}}\:=\:? \\ $$

Question Number 80164    Answers: 0   Comments: 0

Question Number 80161    Answers: 0   Comments: 10

Find the relation between q and r so that x^3 +3px^2 +qx+r is a perfect cube for all value of x

$${Find}\:{the}\:{relation}\:{between} \\ $$$${q}\:{and}\:{r}\:\:{so}\:\:{that} \\ $$$${x}^{\mathrm{3}} +\mathrm{3}{px}^{\mathrm{2}} +{qx}+{r}\:{is}\:{a}\:{perfect} \\ $$$${cube}\:{for}\:{all}\:\:{value}\:{of}\:{x} \\ $$

Question Number 80160    Answers: 0   Comments: 1

∫_0 ^∞ (x^3 /(e^(2x) −e^x ))

$$\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{3}} }{{e}^{\mathrm{2}{x}} −{e}^{{x}} } \\ $$

Question Number 80159    Answers: 1   Comments: 4

IF THE SUM OF p TERMS OF AN A.P. IS EQUAL TO SUM OF ITS q TERMS. PROVE THAT THE SUM OF (p+q) TERMS OF IT IS EQUAL TO 0(ZERO).

$$\boldsymbol{{IF}}\:\:\boldsymbol{{THE}}\:\:\:\boldsymbol{{SUM}}\:\:\:\boldsymbol{{OF}}\:\:\:\boldsymbol{{p}}\:\:\boldsymbol{{TERMS}} \\ $$$$\boldsymbol{{OF}}\:\:\boldsymbol{{AN}}\:\:\:\:\boldsymbol{{A}}.\boldsymbol{{P}}.\:\:\:\boldsymbol{{IS}}\:\:\:\boldsymbol{{EQUAL}}\:\:\boldsymbol{{TO}} \\ $$$$\boldsymbol{{SUM}}\:\:\boldsymbol{{OF}}\:\:\:\boldsymbol{{ITS}}\:\:\:\boldsymbol{{q}}\:\:\:\boldsymbol{{TERMS}}.\:\: \\ $$$$\boldsymbol{{PROVE}}\:\:\boldsymbol{{THAT}}\:\:\boldsymbol{{THE}}\:\:\boldsymbol{{SUM}}\:\:\boldsymbol{{OF}} \\ $$$$\left(\boldsymbol{{p}}+\boldsymbol{{q}}\right)\:\:\boldsymbol{{TERMS}}\:\:\boldsymbol{{OF}}\:\:\:\boldsymbol{{IT}}\:\:\:\boldsymbol{{IS}}\:\:\: \\ $$$$\boldsymbol{{EQUAL}}\:\:\boldsymbol{{TO}}\:\:\mathrm{0}\left(\boldsymbol{{ZERO}}\right). \\ $$

Question Number 80146    Answers: 1   Comments: 0

Two system of rectangular axes have the same origin. If a plane cuts them at distance a, b, c and p, q, r respectively, then prove with the help of an appropriate diagram that : (1/a^2 ) + (1/b^2 ) + (1/c^2 ) = (1/p^2 ) + (1/q^2 ) + (1/r^2 )

$${Two}\:{system}\:{of}\:{rectangular}\:{axes}\:{have} \\ $$$${the}\:{same}\:{origin}.\:{If}\:{a}\:{plane}\:{cuts}\:{them} \\ $$$${at}\:{distance}\:{a},\:{b},\:{c}\:{and}\:{p},\:{q},\:{r} \\ $$$${respectively},\:{then}\:{prove}\:{with}\:{the}\:{help} \\ $$$${of}\:{an}\:{appropriate}\:{diagram}\:{that}\:: \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{b}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{c}^{\mathrm{2}} }\:=\:\frac{\mathrm{1}}{{p}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{q}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{r}^{\mathrm{2}} } \\ $$

Question Number 80139    Answers: 1   Comments: 6

Question Number 80142    Answers: 2   Comments: 0

a. Σ_(k=1) ^∞ ((k^3 /2^k ))=? b. Σ_(k=1) ^∞ (((k^3 +k^2 +k+1)/7^k ))=?

$$\mathrm{a}.\:\:\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\:\left(\frac{\boldsymbol{\mathrm{k}}^{\mathrm{3}} }{\mathrm{2}^{\boldsymbol{\mathrm{k}}} }\right)=? \\ $$$$\boldsymbol{\mathrm{b}}.\:\:\:\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\:\left(\frac{\boldsymbol{\mathrm{k}}^{\mathrm{3}} +\boldsymbol{\mathrm{k}}^{\mathrm{2}} +\boldsymbol{\mathrm{k}}+\mathrm{1}}{\mathrm{7}^{\boldsymbol{\mathrm{k}}} }\right)=? \\ $$

Question Number 80131    Answers: 0   Comments: 2

Question Number 80145    Answers: 1   Comments: 5

{ (((x/a)+(y/b)=a^2 +b^2 )),(( [a,b∈R])),((ab(x^2 −y^2 )=xy(a^2 −b^2 ))) :}

$$\begin{cases}{\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{a}}}+\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{b}}}=\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} }\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}}\in\boldsymbol{\mathrm{R}}\right]}\\{\boldsymbol{\mathrm{ab}}\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{y}}^{\mathrm{2}} \right)=\boldsymbol{\mathrm{xy}}\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{b}}^{\mathrm{2}} \right)}\end{cases} \\ $$

Question Number 80119    Answers: 2   Comments: 0

Question Number 80144    Answers: 0   Comments: 1

solve for x: (((√x)+1)/(√(x+1)))+ax^2 =x(a^2 +1) [a∈R]

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}}: \\ $$$$\frac{\sqrt{\boldsymbol{\mathrm{x}}}+\mathrm{1}}{\sqrt{\boldsymbol{\mathrm{x}}+\mathrm{1}}}+\boldsymbol{\mathrm{ax}}^{\mathrm{2}} =\boldsymbol{\mathrm{x}}\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\mathrm{1}\right)\:\:\:\:\:\:\left[\boldsymbol{\mathrm{a}}\in\boldsymbol{\mathrm{R}}\right] \\ $$

Question Number 80116    Answers: 1   Comments: 1

Find S_m =Σ_(n=0) ^∞ (1/(Π_(k=1) ^m (n+k)))=? (m≥2)

$${Find} \\ $$$${S}_{{m}} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\underset{{k}=\mathrm{1}} {\overset{{m}} {\prod}}\left({n}+{k}\right)}=? \\ $$$$\left({m}\geqslant\mathrm{2}\right) \\ $$

Question Number 80113    Answers: 0   Comments: 1

how do you simply sin (tan^(−1) (3x)+cos^(−1) (x)) ?

$${how}\:{do}\:{you}\:{simply} \\ $$$$\mathrm{sin}\:\left(\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{3}{x}\right)+\mathrm{cos}^{−\mathrm{1}} \left({x}\right)\right)\:? \\ $$

Question Number 80102    Answers: 0   Comments: 0

Question Number 80093    Answers: 3   Comments: 0

Solve for x and y x^(√y) = 64 y^(√x) = 81

$$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\:\:\:\:\:\mathrm{x}^{\sqrt{\mathrm{y}}} \:\:\:=\:\:\mathrm{64} \\ $$$$\:\:\:\:\:\mathrm{y}^{\sqrt{\mathrm{x}}} \:\:\:=\:\mathrm{81} \\ $$

Question Number 80088    Answers: 0   Comments: 0

When the father was son′s age, the son was ten years old; when the son will be father′s age, the father will be seventy. What are their ages ?

$$\:{When}\:\:{the}\:\:{father}\:\:{was}\:{son}'{s}\:\:{age},\:\:{the}\:\:{son} \\ $$$$\:\:{was}\:\:{ten}\:\:{years}\:\:{old};\:\:{when}\:\:{the}\:\:{son}\:\:{will}\:\:{be}\:\:{father}'{s}\:\:{age}, \\ $$$$\:\:{the}\:\:{father}\:\:{will}\:\:{be}\:\:{seventy}. \\ $$$$\:\:{What}\:\:{are}\:\:{their}\:\:{ages}\:\:? \\ $$

Question Number 80084    Answers: 0   Comments: 3

−1=(−1)^1 =(−1)^(2/2) =((−1)^2 )^(1/2) =(1)^(1/2) = =(√1)=1 what do you think about this?

$$\:\:−\mathrm{1}=\left(−\mathrm{1}\right)^{\mathrm{1}} =\left(−\mathrm{1}\right)^{\frac{\mathrm{2}}{\mathrm{2}}} =\left(\left(−\mathrm{1}\right)^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} =\left(\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} = \\ $$$$=\sqrt{\mathrm{1}}=\mathrm{1}\:\: \\ $$$$\mathrm{what}\:\mathrm{do}\:\mathrm{you}\:\mathrm{think}\:\mathrm{about}\:\mathrm{this}? \\ $$

Question Number 80068    Answers: 2   Comments: 3

Question Number 80065    Answers: 0   Comments: 0

Question Number 80064    Answers: 1   Comments: 6

lim_(x→−∞) [(√(1−xe^x ))]

$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\left[\sqrt{\mathrm{1}−{xe}^{{x}} \:}\right] \\ $$

Question Number 80057    Answers: 1   Comments: 2

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