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Question Number 79266 Answers: 0 Comments: 3

let ABC be a escalene triangle of area 7. Let A_1 be a point on the side BC, and let B_1 and C_1 be points on the sides AC and AB, such that AA_1 , BB_1 and CC_1 are parallel. Find the area of triangle A_1 B_1 C_1 .

$${let}\:{ABC}\:{be}\:{a}\:{escalene}\:{triangle}\:{of} \\ $$$${area}\:\mathrm{7}.\:{Let}\:{A}_{\mathrm{1}} \:{be}\:{a}\:{point}\:{on}\:{the}\:{side} \\ $$$${BC},\:{and}\:{let}\:{B}_{\mathrm{1}} \:{and}\:{C}_{\mathrm{1}} \:{be}\:{points}\:{on} \\ $$$${the}\:{sides}\:{AC}\:{and}\:{AB},\:{such}\:{that} \\ $$$${AA}_{\mathrm{1}} ,\:{BB}_{\mathrm{1}} \:{and}\:{CC}_{\mathrm{1}} \:{are}\:{parallel}.\:{Find} \\ $$$${the}\:{area}\:{of}\:{triangle}\:{A}_{\mathrm{1}} {B}_{\mathrm{1}} {C}_{\mathrm{1}} . \\ $$

Question Number 79264 Answers: 0 Comments: 0

Question Number 79263 Answers: 0 Comments: 4

4^(2x−1) +(1/4)^2 log^2 (2x)>^2 log(x) {^2 log((1/x))−2^(2x) }

$$\mathrm{4}^{\mathrm{2x}−\mathrm{1}} +\frac{\mathrm{1}}{\mathrm{4}}\:^{\mathrm{2}} \mathrm{log}^{\mathrm{2}} \left(\mathrm{2x}\right)>\:^{\mathrm{2}} \mathrm{log}\left(\mathrm{x}\right) \\ $$$$\left\{^{\mathrm{2}} \mathrm{log}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)−\mathrm{2}^{\mathrm{2x}} \right\} \\ $$

Question Number 79256 Answers: 1 Comments: 0

3s^2 −2ps−3cp−1=0 and 3s−2p−sp^2 −3cp^2 =0 find s and p both real in terms of c ∈R.

$$\mathrm{3}{s}^{\mathrm{2}} −\mathrm{2}{ps}−\mathrm{3}{cp}−\mathrm{1}=\mathrm{0}\:\:\:{and} \\ $$$$\mathrm{3}{s}−\mathrm{2}{p}−{sp}^{\mathrm{2}} −\mathrm{3}{cp}^{\mathrm{2}} =\mathrm{0} \\ $$$${find}\:{s}\:{and}\:{p}\:{both}\:{real}\:{in}\:{terms} \\ $$$${of}\:{c}\:\in\mathbb{R}. \\ $$

Question Number 79249 Answers: 1 Comments: 3

Question Number 79254 Answers: 4 Comments: 2

Question Number 79236 Answers: 1 Comments: 3

lim_(x→+∞) x{e−(1+(1/x))^x }=?

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

Question Number 79233 Answers: 1 Comments: 3

(1/(x(x+1)))+(1/((x+1)(x+2)))+ (1/((x+2)(x+3)))≤(3/4)

$$\frac{\mathrm{1}}{\mathrm{x}\left(\mathrm{x}+\mathrm{1}\right)}+\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{x}+\mathrm{2}\right)}+ \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{2}\right)\left(\mathrm{x}+\mathrm{3}\right)}\leqslant\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Question Number 79222 Answers: 0 Comments: 3

∫_0 ^1 (x^n /(Σ_(k=0) ^(n−1) x^k ))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}{x}^{{k}} }{dx}=? \\ $$

Question Number 79190 Answers: 4 Comments: 13

if x^2 +y^2 =50, find the minimum and maximum of (x+y)^2 −8(x+y)+20

$${if}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{50}, \\ $$$${find}\:{the}\:{minimum}\:{and}\:{maximum}\:{of} \\ $$$$\left({x}+{y}\right)^{\mathrm{2}} −\mathrm{8}\left({x}+{y}\right)+\mathrm{20} \\ $$

Question Number 79187 Answers: 0 Comments: 0

∫_0 ^π ((cos (nx)−cos (nα))/(cos (x)−cos (α))) dx

$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{cos}\:\left({nx}\right)−\mathrm{cos}\:\left({n}\alpha\right)}{\mathrm{cos}\:\left({x}\right)−\mathrm{cos}\:\left(\alpha\right)}\:{dx} \\ $$

Question Number 79186 Answers: 1 Comments: 0

∫_(−1) ^1 ((cos (x))/(1+e^(1/x) )) dx ?

$$\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{cos}\:\left({x}\right)}{\mathrm{1}+{e}^{\frac{\mathrm{1}}{{x}}} }\:{dx}\:? \\ $$

Question Number 79181 Answers: 1 Comments: 1

lim_(x→ 0^+ ) (x^2 + 1)^(ln x) = ...

$$\underset{{x}\rightarrow\:\mathrm{0}^{+} } {\mathrm{lim}}\:\:\left({x}^{\mathrm{2}} \:+\:\mathrm{1}\right)^{\mathrm{ln}\:{x}} \:\:=\:\:... \\ $$

Question Number 79177 Answers: 1 Comments: 0

Question Number 79667 Answers: 1 Comments: 2

find the equation of the tangent and normal to the curve xy=9 at x=4

$${find}\:{the}\:{equation}\:{of}\:{the}\:{tangent}\:{and} \\ $$$${normal}\:{to}\:{the}\:{curve}\:{xy}=\mathrm{9}\:{at}\:{x}=\mathrm{4} \\ $$

Question Number 79147 Answers: 1 Comments: 0

(√(x+(1/x^2 )))+(√(x−(1/x^2 ) ))≤(2/x)

$$\sqrt{\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }}+\sqrt{\mathrm{x}−\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\:}\leqslant\frac{\mathrm{2}}{\mathrm{x}} \\ $$

Question Number 79131 Answers: 1 Comments: 2

Determine the set of points M such as ∣∣MA^→ +MB^→ +2MC^→ ∣∣=6(√3) AB=BC=AC=6 ABC is triangle.

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:\mathrm{M}\: \\ $$$$\mathrm{such}\:\mathrm{as}\:\mid\mid\mathrm{M}\overset{\rightarrow} {\mathrm{A}}+\mathrm{M}\overset{\rightarrow} {\mathrm{B}}+\mathrm{2M}\overset{\rightarrow} {\mathrm{C}}\mid\mid=\mathrm{6}\sqrt{\mathrm{3}} \\ $$$$\mathrm{AB}=\mathrm{BC}=\mathrm{AC}=\mathrm{6} \\ $$$$\mathrm{ABC}\:\mathrm{is}\:\mathrm{triangle}. \\ $$

Question Number 79128 Answers: 1 Comments: 4

Find out ∫_0 ^1 ln(1−t+t^2 )dt Then deduce the value of A=Σ_(n=1) ^∞ (1/(n(n+1) (((2n+1)),(n) )))

$${Find}\:{out}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{t}+{t}^{\mathrm{2}} \right){dt} \\ $$$${Then}\:{deduce}\:{the}\:{value}\:{of}\:\:\:{A}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)\begin{pmatrix}{\mathrm{2}{n}+\mathrm{1}}\\{{n}}\end{pmatrix}} \\ $$

Question Number 79127 Answers: 2 Comments: 10

Solve on R∗R the following system {_(9^A +9^B +9^C =1) ^(3^A +3^B +3^C =(√3))

$$\:{Solve}\:\:{on}\:\mathbb{R}\ast\mathbb{R}\:\:{the}\:{following}\:{system} \\ $$$$\left\{_{\mathrm{9}^{{A}} +\mathrm{9}^{{B}} +\mathrm{9}^{{C}} =\mathrm{1}} ^{\mathrm{3}^{{A}} +\mathrm{3}^{{B}} +\mathrm{3}^{{C}} =\sqrt{\mathrm{3}}} \:\:\:\right. \\ $$

Question Number 79126 Answers: 1 Comments: 2

Study f(x)=Σ_(n=1) ^∞ ((x^n sin(nx))/n) Find out Σ_(n=1) ^∞ (−1)^n ((sin(n))/n) and Σ_(n=1) ^∞ ((sin(n))/n)

$${Study}\:\:\:{f}\left({x}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{x}^{{n}} {sin}\left({nx}\right)}{{n}} \\ $$$${Find}\:{out}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \:\frac{{sin}\left({n}\right)}{{n}}\:\:\:{and}\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}\left({n}\right)}{{n}}\: \\ $$

Question Number 79124 Answers: 1 Comments: 1

Prove that 16arctan((1/5))−4arctan((1/(239)))=π

$${Prove}\:{that}\: \\ $$$$\:\mathrm{16}{arctan}\left(\frac{\mathrm{1}}{\mathrm{5}}\right)−\mathrm{4}{arctan}\left(\frac{\mathrm{1}}{\mathrm{239}}\right)=\pi \\ $$$$ \\ $$

Question Number 79121 Answers: 1 Comments: 0

Question Number 79210 Answers: 0 Comments: 6

Question Number 79111 Answers: 0 Comments: 3

Show that E={(x,y,z) ∈ R^3 / x−2y+z=0} is a subspace vector of which we will determine one base. please help sirs...

$$\mathrm{Show}\:\mathrm{that} \\ $$$$\mathrm{E}=\left\{\left({x},\mathrm{y},{z}\right)\:\in\:\mathbb{R}^{\mathrm{3}} \:\:/\:\:{x}−\mathrm{2}{y}+{z}=\mathrm{0}\right\} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{subspace}\:\mathrm{vector}\:\mathrm{of}\:\mathrm{which}\:\mathrm{we} \\ $$$$\mathrm{will}\:\mathrm{determine}\:\mathrm{one}\:\mathrm{base}. \\ $$$$\mathrm{please}\:\mathrm{help}\:\mathrm{sirs}... \\ $$

Question Number 79108 Answers: 1 Comments: 1

decompose F(x)=((nx^n )/(x^(2n) +1)) inside C(x) and R(x) (n≥2) and determine ∫_0 ^(+∞) F(x)dx

$${decompose}\:{F}\left({x}\right)=\frac{{nx}^{{n}} }{{x}^{\mathrm{2}{n}} \:+\mathrm{1}}\:\:{inside}\:{C}\left({x}\right)\:{and}\:{R}\left({x}\right)\:\:\left({n}\geqslant\mathrm{2}\right) \\ $$$${and}\:{determine}\:\int_{\mathrm{0}} ^{+\infty} {F}\left({x}\right){dx} \\ $$

Question Number 79107 Answers: 2 Comments: 1

calculate f(a) =∫_0 ^∞ e^(−(x^2 +(a/x^2 ))) dx with a>0

$${calculate}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left({x}^{\mathrm{2}} \:+\frac{{a}}{{x}^{\mathrm{2}} }\right)} {dx}\:{with}\:{a}>\mathrm{0} \\ $$

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