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Question Number 115882 Answers: 1 Comments: 0

Question Number 115876 Answers: 1 Comments: 0

Prove that f(x)=ax^2 +bx+c has no real roots if and only if a∙[f(−(b/(2a)))]>0

$$\mathrm{Prove}\:\mathrm{that}\:{f}\left({x}\right)={ax}^{\mathrm{2}} +{bx}+{c}\:\mathrm{has}\: \\ $$$$\mathrm{no}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:{a}\centerdot\left[{f}\left(−\frac{{b}}{\mathrm{2}{a}}\right)\right]>\mathrm{0} \\ $$

Question Number 115869 Answers: 0 Comments: 0

(dy/dx) +2y = x^3 .e^(−2y)

$$\:\frac{{dy}}{{dx}}\:+\mathrm{2}{y}\:=\:{x}^{\mathrm{3}} .{e}^{−\mathrm{2}{y}} \\ $$

Question Number 115859 Answers: 2 Comments: 0

Determine, in simplest form the smallest of the three numbers x, y and z which satisfy the system { ((log _9 (x)+log _9 (y)+log _3 (z)=2)),((log _(16) (x)+log _4 (y)+log _(16) (z)=1)),((log _5 (x)+log _(25) (y)+log _(25) (z)=0)) :}

Question Number 115858 Answers: 2 Comments: 0

What are all ordered pairs of real number (x,y) for which 5^(y−x) (x+y) = 1 and (x+y)^(x−y) = 5

$${What}\:{are}\:{all}\:{ordered}\:{pairs}\:{of}\:{real} \\ $$$${number}\:\left({x},{y}\right)\:{for}\:{which}\: \\ $$$$\mathrm{5}^{{y}−{x}} \:\left({x}+{y}\right)\:=\:\mathrm{1}\:{and}\:\left({x}+{y}\right)^{{x}−{y}} \:=\:\mathrm{5} \\ $$

Question Number 115857 Answers: 1 Comments: 0

What are all real values of p for which the inequality −3<((x^2 +px−2)/(x^2 −x+1))<2 is satisfied by all real values of x

$${What}\:{are}\:{all}\:{real}\:{values}\:{of}\:{p}\:{for} \\ $$$${which}\:{the}\:{inequality}\: \\ $$$$−\mathrm{3}<\frac{{x}^{\mathrm{2}} +{px}−\mathrm{2}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}<\mathrm{2}\:{is}\:{satisfied}\: \\ $$$${by}\:{all}\:{real}\:{values}\:{of}\:{x} \\ $$

Question Number 115856 Answers: 2 Comments: 0

Which is greater P = (1983)(1+2+3+...+1984) , or Q = (1984)(1+2+3+...+1983)

$${Which}\:{is}\:{greater} \\ $$$${P}\:=\:\left(\mathrm{1983}\right)\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+...+\mathrm{1984}\right)\:,\:{or} \\ $$$${Q}\:=\:\left(\mathrm{1984}\right)\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+...+\mathrm{1983}\right) \\ $$

Question Number 115854 Answers: 1 Comments: 0

If f(2x)= x^2 +4x+1 , what all values of t for which f((t/2)) = −((11)/4) where f represents a function

$${If}\:{f}\left(\mathrm{2}{x}\right)=\:{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{1}\:,\:{what}\:{all} \\ $$$${values}\:{of}\:{t}\:{for}\:{which}\:{f}\left(\frac{{t}}{\mathrm{2}}\right)\:=\:−\frac{\mathrm{11}}{\mathrm{4}} \\ $$$${where}\:{f}\:{represents}\:{a}\:{function} \\ $$

Question Number 115871 Answers: 1 Comments: 1

Question Number 115846 Answers: 2 Comments: 1

given vertex parabola at point (1,−2) and focus at (−1,−2). find the equation of parabola

$${given}\:{vertex}\:{parabola}\:{at}\:{point} \\ $$$$\left(\mathrm{1},−\mathrm{2}\right)\:{and}\:{focus}\:{at}\:\left(−\mathrm{1},−\mathrm{2}\right). \\ $$$${find}\:{the}\:{equation}\:{of}\:{parabola} \\ $$

Question Number 115817 Answers: 1 Comments: 9

Question Number 115815 Answers: 1 Comments: 0

Montrer que ∀(a,b,c)∈(R_+ ^∗ )^3 (1/(a^2 +bc))+(1/(b^2 +ac))+(1/(c^2 +ab))≤(1/2)((1/(ab))+(1/(bc))+(1/(ac)))

$$\mathrm{Montrer}\:\mathrm{que}\:\:\forall\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)\in\left(\mathbb{R}_{+} ^{\ast} \right)^{\mathrm{3}} \\ $$$$\frac{\mathrm{1}}{\mathrm{a}^{\mathrm{2}} +\mathrm{bc}}+\frac{\mathrm{1}}{\mathrm{b}^{\mathrm{2}} +\mathrm{ac}}+\frac{\mathrm{1}}{\mathrm{c}^{\mathrm{2}} +\mathrm{ab}}\leqslant\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{ab}}+\frac{\mathrm{1}}{\mathrm{bc}}+\frac{\mathrm{1}}{\mathrm{ac}}\right) \\ $$

Question Number 115812 Answers: 2 Comments: 4

solve lim_(x→∞) (ζ(x)−1)^(1/x)

$${solve} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\zeta\left({x}\right)−\mathrm{1}\right)^{\frac{\mathrm{1}}{{x}}} \\ $$

Question Number 115798 Answers: 1 Comments: 1

Question Number 115793 Answers: 1 Comments: 0

Given that p_∽ = ((( 2)),((−3)) ) , q_∽ = (((−4)),(( m)) ) and r_∽ = ((n),(4) ) If p_∽ +q_∽ −r_∽ is a unit vector, find the value of m and n.

$$\mathrm{Given}\:\mathrm{that}\:\underset{\backsim} {{p}}=\begin{pmatrix}{\:\:\mathrm{2}}\\{−\mathrm{3}}\end{pmatrix}\:,\:\underset{\backsim} {{q}}=\begin{pmatrix}{−\mathrm{4}}\\{\:\:\:{m}}\end{pmatrix}\:\mathrm{and}\:\underset{\backsim} {{r}}=\begin{pmatrix}{{n}}\\{\mathrm{4}}\end{pmatrix} \\ $$$$\mathrm{If}\:\underset{\backsim} {{p}}+\underset{\backsim} {{q}}−\underset{\backsim} {{r}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{unit}\:\mathrm{vector},\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:{m}\:\mathrm{and}\:{n}. \\ $$

Question Number 115781 Answers: 4 Comments: 0

∫_0 ^(1/( (√2))) ((x sin^(−1) (x^2 ))/( (√(1−x^4 )))) dx =? ∫2^(−x) tanh (2^(1−x) ) dx =?

$$\underset{\mathrm{0}} {\overset{\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}} {\int}}\:\frac{{x}\:\mathrm{sin}^{−\mathrm{1}} \left({x}^{\mathrm{2}} \right)}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }}\:{dx}\:=? \\ $$$$\int\mathrm{2}^{−{x}} \:\mathrm{tanh}\:\left(\mathrm{2}^{\mathrm{1}−{x}} \right)\:{dx}\:=? \\ $$

Question Number 115780 Answers: 10 Comments: 1

lim_(x→0) (cos x)^(1/x^2 ) = ? lim_(x→∞) (e^(3x) −5x)^(1/x) =? lim_(x→0) ((e^(2x) −2e^x +1)/(cos 3x−2cos 2x+cos x))=? lim_(x→0) (1/x^2 )−cot^2 x =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{cos}\:{x}\right)^{\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} \:=\:? \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left({e}^{\mathrm{3}{x}} −\mathrm{5}{x}\right)^{\frac{\mathrm{1}}{{x}}} \:=? \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{e}^{\mathrm{2}{x}} −\mathrm{2}{e}^{{x}} +\mathrm{1}}{\mathrm{cos}\:\mathrm{3}{x}−\mathrm{2cos}\:\mathrm{2}{x}+\mathrm{cos}\:{x}}=? \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }−\mathrm{cot}\:^{\mathrm{2}} {x}\:=? \\ $$

Question Number 115777 Answers: 3 Comments: 0

lim_(x→0) ((sin x−tan^(−1) x)/(x^2 ln (1+x)))

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:{x}−\mathrm{tan}^{−\mathrm{1}} {x}}{{x}^{\mathrm{2}} \:\mathrm{ln}\:\left(\mathrm{1}+{x}\right)} \\ $$

Question Number 115775 Answers: 2 Comments: 0

lim_(x→∞) ((((3−(√x))((√x)+2))/(8x−4)))^(1/(3 ))

$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt[{\mathrm{3}\:}]{\frac{\left(\mathrm{3}−\sqrt{{x}}\right)\left(\sqrt{{x}}+\mathrm{2}\right)}{\mathrm{8}{x}−\mathrm{4}}} \\ $$

Question Number 121180 Answers: 1 Comments: 1

Question Number 115773 Answers: 0 Comments: 0

((1−x)/( (√(x^2 −2x+5)))) = (3/5)

$$\:\frac{\mathrm{1}−{x}}{\:\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{5}}}\:=\:\frac{\mathrm{3}}{\mathrm{5}} \\ $$

Question Number 115771 Answers: 2 Comments: 0

y′′−y′−2y=e^(2x) .cos^2 x

$${y}''−{y}'−\mathrm{2}{y}={e}^{\mathrm{2}{x}} .\mathrm{cos}\:^{\mathrm{2}} {x} \\ $$

Question Number 115769 Answers: 1 Comments: 0

There are 3 teachers and 6 students who will sit on the 9 available seats. many arrangements they sit if each teacher is flanked by 2 students

$${There}\:{are}\:\mathrm{3}\:{teachers}\:{and}\:\mathrm{6}\:{students} \\ $$$${who}\:{will}\:{sit}\:{on}\:{the}\:\mathrm{9}\:{available}\:{seats}.\:{many} \\ $$$${arrangements}\:{they}\:{sit}\:{if}\:{each}\: \\ $$$${teacher}\:{is}\:{flanked}\:{by}\:\mathrm{2}\:{students} \\ $$

Question Number 115765 Answers: 1 Comments: 3

Show that sin (α + β) = sin α cos β + cos α sin β.

$$\: \\ $$$$\:\:\:\mathrm{Show}\:\mathrm{that}\:\mathrm{sin}\:\left(\alpha\:+\:\beta\right)\:=\:\mathrm{sin}\:\alpha\:\mathrm{cos}\:\beta\:+\:\mathrm{cos}\:\alpha\:\mathrm{sin}\:\beta. \\ $$

Question Number 115761 Answers: 2 Comments: 1

.... advanced calculus... ... evaluate ... Ψ= ∫_(−∞) ^( +∞) ((x/(2+2^(−x) +2^x )))^2 dx =??? m.n.july 70

$$\:\:\:\:\:\:....\:\:\:{advanced}\:\:{calculus}...\: \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:\:\:...\:\:\:{evaluate}\:...\: \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Psi=\:\int_{−\infty} ^{\:+\infty} \left(\frac{{x}}{\mathrm{2}+\mathrm{2}^{−{x}} +\mathrm{2}^{{x}} }\right)^{\mathrm{2}} {dx}\:=??? \\ $$$$\:{m}.{n}.{july}\:\mathrm{70} \\ $$$$ \\ $$

Question Number 115830 Answers: 0 Comments: 1

create the differention equation from x^2 +y^2 +2ax+2by+c=0

$${create}\:{the}\:{differention}\:{equation}\:{from} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{2}{ax}+\mathrm{2}{by}+{c}=\mathrm{0} \\ $$

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