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Question Number 46482 Answers: 1 Comments: 1

Question Number 46481 Answers: 0 Comments: 3

Four spheres with radii a,b,c and d touch each other. Find the radii of their circumscribed sphere (R) and their inscribed sphere (r) in terms of a,b,c and d.

$${Four}\:{spheres}\:{with}\:{radii}\:{a},{b},{c}\:{and}\:{d} \\ $$$${touch}\:{each}\:{other}.\:{Find}\:{the}\:{radii}\:{of} \\ $$$${their}\:{circumscribed}\:{sphere}\:\left({R}\right)\:{and} \\ $$$${their}\:{inscribed}\:{sphere}\:\left({r}\right)\:{in}\:{terms}\:{of}\: \\ $$$${a},{b},{c}\:{and}\:{d}. \\ $$

Question Number 46478 Answers: 1 Comments: 1

If x^2 +y^2 +xy=1. Find range of E = x^3 y+xy^3 +4 .

$${If}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{xy}=\mathrm{1}.\:{Find}\:{range}\:{of}\: \\ $$$${E}\:=\:{x}^{\mathrm{3}} {y}+{xy}^{\mathrm{3}} +\mathrm{4}\:. \\ $$

Question Number 46475 Answers: 1 Comments: 2

Question Number 46474 Answers: 2 Comments: 2

a)[If tan^(−1) a+tan^(−1) b+tan^(−1) c=π show that ((a+b+c)/(abc))=1 b)[ If tanx=((nsinycosy)/(1−nsin^2 y)) show that tan(y−x)=(1−n)tany

$$\left.\mathrm{a}\right)\left[\mathrm{If}\:\mathrm{tan}^{−\mathrm{1}} \mathrm{a}+\mathrm{tan}^{−\mathrm{1}} \mathrm{b}+\mathrm{tan}^{−\mathrm{1}} \mathrm{c}=\pi\:\:\mathrm{show}\:\mathrm{that}\:\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{abc}}=\mathrm{1}\right. \\ $$$$\left.\mathrm{b}\right)\left[\:\:\mathrm{If}\:\:\mathrm{tanx}=\frac{\mathrm{nsinycosy}}{\mathrm{1}−\mathrm{nsin}^{\mathrm{2}} \mathrm{y}}\right. \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{tan}\left(\mathrm{y}−\mathrm{x}\right)=\left(\mathrm{1}−\mathrm{n}\right)\mathrm{tany} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 46473 Answers: 1 Comments: 0

Find the value of θ which satisfy the equation cosθx+cos(x+2)θ=cosθ

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\theta \\ $$$$\mathrm{which}\:\mathrm{satisfy}\:\mathrm{the}\: \\ $$$$\mathrm{equation} \\ $$$$\mathrm{cos}\theta\mathrm{x}+\mathrm{cos}\left(\mathrm{x}+\mathrm{2}\right)\theta=\mathrm{cos}\theta \\ $$

Question Number 46472 Answers: 1 Comments: 0

Question Number 46467 Answers: 0 Comments: 2

Question Number 46465 Answers: 1 Comments: 0

If I=(1/2) ∫_0 ^∞ t^n e^(−t) dt = 360. Find n?

$${If}\:{I}=\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\infty} {t}^{{n}} {e}^{−{t}} {dt}\:\:=\:\mathrm{360}. \\ $$$${Find}\:{n}? \\ $$

Question Number 46464 Answers: 0 Comments: 0

To this formula find the value of S when t=1 u=2 a=3 b=4 c=5 S=((tu)/((a+b+c))) [Answer]========================= S=0.1666666666666... ≈0.17 [Solution]======================== [S=((tu)/((a+b+c)))] =((1×2)/((3+4+5))) =(2/((3+4+5))) =(2/((7+5))) =(2/(12)) =0.1666666666666... ≈0.17 S=0.17

$${To}\:{this}\:{formula}\:{find}\:{the}\:{value}\:{of}\:{S}\:{when}\:{t}=\mathrm{1}\:{u}=\mathrm{2}\:{a}=\mathrm{3}\:{b}=\mathrm{4}\:{c}=\mathrm{5} \\ $$$${S}=\frac{{tu}}{\left({a}+{b}+{c}\right)} \\ $$$$\left[{Answer}\right]========================= \\ $$$$ \\ $$$$\:\:\:\:{S}=\mathrm{0}.\mathrm{1666666666666}... \\ $$$$\:\:\:\:\:\:\:\approx\mathrm{0}.\mathrm{17} \\ $$$$ \\ $$$$\left[{Solution}\right]======================== \\ $$$$ \\ $$$$\left[{S}=\frac{{tu}}{\left({a}+{b}+{c}\right)}\right] \\ $$$$\:\:\:\:\:=\frac{\mathrm{1}×\mathrm{2}}{\left(\mathrm{3}+\mathrm{4}+\mathrm{5}\right)} \\ $$$$\:\:\:\:\:=\frac{\mathrm{2}}{\left(\mathrm{3}+\mathrm{4}+\mathrm{5}\right)} \\ $$$$\:\:\:\:\:=\frac{\mathrm{2}}{\left(\mathrm{7}+\mathrm{5}\right)} \\ $$$$\:\:\:\:\:=\frac{\mathrm{2}}{\mathrm{12}} \\ $$$$\:\:\:\:\:=\mathrm{0}.\mathrm{1666666666666}... \\ $$$$\:\:\:\:\:\approx\mathrm{0}.\mathrm{17} \\ $$$$ \\ $$$${S}=\mathrm{0}.\mathrm{17} \\ $$

Question Number 46461 Answers: 1 Comments: 0

Find the sum of the nth term of the series: (1/2) + (3/4) + (7/8) + ((15)/(16)) + ...

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{nth}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series}:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\mathrm{3}}{\mathrm{4}}\:+\:\frac{\mathrm{7}}{\mathrm{8}}\:+\:\frac{\mathrm{15}}{\mathrm{16}}\:+\:... \\ $$

Question Number 46460 Answers: 0 Comments: 0

If k is odd, then show that 1^k + 2^k + 3^k + ... + n^k is divisible by 1 + 2 + 3 + ... + n, for every n ∈ N

$$\mathrm{If}\:\:\mathrm{k}\:\mathrm{is}\:\mathrm{odd},\:\mathrm{then}\:\mathrm{show}\:\mathrm{that}\:\:\:\:\mathrm{1}^{\mathrm{k}} \:+\:\mathrm{2}^{\mathrm{k}} \:+\:\mathrm{3}^{\mathrm{k}} \:+\:...\:+\:\mathrm{n}^{\mathrm{k}} \:\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\:\: \\ $$$$\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:...\:+\:\mathrm{n},\:\:\:\:\:\mathrm{for}\:\mathrm{every}\:\:\:\mathrm{n}\:\in\:\mathrm{N} \\ $$

Question Number 46432 Answers: 0 Comments: 2

Question Number 46430 Answers: 0 Comments: 5

Question Number 46427 Answers: 1 Comments: 0

Give a proof for the following. ∀ a ∈ R^∗ , (b,c) ∈ R , x ∈ C : ax^2 + bx + c = a(x−((−b − (√(b^2 −4ac)))/(2a)))(x−((−b + (√(b^2 −4ac)))/(2a))) To answer, I should not expand from the second equation but I should start from the first one. How may I do ? Thank you.

$$\mathrm{Give}\:\mathrm{a}\:\mathrm{proof}\:\mathrm{for}\:\mathrm{the}\:\mathrm{following}. \\ $$$$ \\ $$$$ \\ $$$$\forall\:{a}\:\in\:\mathbb{R}^{\ast} \:,\:\left({b},{c}\right)\:\in\:\mathbb{R}\:,\:{x}\:\in\:\mathbb{C}\:: \\ $$$$ \\ $$$$ \\ $$$${ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c} \\ $$$$=\:{a}\left({x}−\frac{−{b}\:−\:\sqrt{{b}^{\mathrm{2}} −\mathrm{4}{ac}}}{\mathrm{2}{a}}\right)\left({x}−\frac{−{b}\:+\:\sqrt{{b}^{\mathrm{2}} −\mathrm{4}{ac}}}{\mathrm{2}{a}}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\mathrm{To}\:\mathrm{answer},\:\mathrm{I}\:\mathrm{should}\:\mathrm{not}\:\mathrm{expand}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{second}\:\mathrm{equation}\:\mathrm{but}\:\mathrm{I}\:\mathrm{should}\:\mathrm{start}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{one}. \\ $$$$ \\ $$$$\mathrm{How}\:\mathrm{may}\:\mathrm{I}\:\mathrm{do}\:?\:\mathrm{Thank}\:\mathrm{you}. \\ $$$$ \\ $$

Question Number 46425 Answers: 0 Comments: 2

let p(x)=(x+i)^n −(x−i)^n with i^2 =−1 1) find p(x) at form Σ a_k x^k 2) find the roots of p(x) 3) factorize inside C[x] p(x) 4) factorize inside R[x] the polynom p(x) 5) decompose the fraction F(x)=(1/(p(x)))

$${let}\:{p}\left({x}\right)=\left({x}+{i}\right)^{{n}} −\left({x}−{i}\right)^{{n}} \:\:\:{with}\:{i}^{\mathrm{2}} =−\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{p}\left({x}\right)\:{at}\:{form}\:\Sigma\:{a}_{{k}} {x}^{{k}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{factorize}\:{inside}\:{C}\left[{x}\right]\:{p}\left({x}\right) \\ $$$$\left.\mathrm{4}\right)\:{factorize}\:{inside}\:{R}\left[{x}\right]\:{the}\:{polynom}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{5}\right)\:{decompose}\:{the}\:{fraction}\:{F}\left({x}\right)=\frac{\mathrm{1}}{{p}\left({x}\right)} \\ $$

Question Number 46424 Answers: 0 Comments: 3

let f(x)=(e^(−x) /(x^2 +4)) 1) calculate f^((n)) (x) and f^((n)) (0) 2) developp f at integr serie .

$${let}\:{f}\left({x}\right)=\frac{{e}^{−{x}} }{{x}^{\mathrm{2}} +\mathrm{4}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 46423 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(x^2 ))/(1+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 46422 Answers: 0 Comments: 0

let f_n (x)=((√(1+x^2 ))−x)^(n ) with n integr natural 1) find ∫ f_n (x)dx 2) calculate f_n ^(−1) (x) 3) find lim_(x→+∞) f_n (x) 4) calculate f_n ^′ (x) and f_n ^′ (0) 5) give the equation of tangent at point E(0,f(0))

$${let}\:{f}_{{n}} \left({x}\right)=\left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }−{x}\right)^{{n}\:} \:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:\int\:\:{f}_{{n}} \left({x}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}_{{n}} ^{−\mathrm{1}} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{lim}_{{x}\rightarrow+\infty} \:{f}_{{n}} \left({x}\right) \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\:{f}_{{n}} ^{'} \left({x}\right)\:\:{and}\:\:{f}_{{n}} ^{'} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{5}\right)\:{give}\:{the}\:{equation}\:{of}\:{tangent}\:{at}\:{point}\:{E}\left(\mathrm{0},{f}\left(\mathrm{0}\right)\right) \\ $$

Question Number 46421 Answers: 0 Comments: 3

let f(x)=(x+(1/x))^n −(x−(1/x))^n with n integr natural and x from R (x≠0) 1) simplify f(x) 2) calculate lim_(x→+∞) f(x) 3) calculate ∫_1 ^3 f(x)dx

$${let}\:{f}\left({x}\right)=\left({x}+\frac{\mathrm{1}}{{x}}\right)^{{n}} −\left({x}−\frac{\mathrm{1}}{{x}}\right)^{{n}} \:{with}\:{n}\:{integr}\:{natural}\:{and}\:{x}\:{from}\:{R}\:\left({x}\neq\mathrm{0}\right) \\ $$$$\left.\mathrm{1}\right)\:{simplify}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{f}\left({x}\right){dx} \\ $$

Question Number 46420 Answers: 0 Comments: 0

let p(x)=(1+ix)^5 −1 with i^2 =−1 1) solve inside C[x] the equation p(x)=0 2)factorize inside C[x] the polynom p(x)

$${let}\:{p}\left({x}\right)=\left(\mathrm{1}+{ix}\right)^{\mathrm{5}} −\mathrm{1}\:\:{with}\:{i}^{\mathrm{2}} =−\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{solve}\:{inside}\:{C}\left[{x}\right]\:{the}\:{equation}\:{p}\left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{2}\right){factorize}\:{inside}\:{C}\left[{x}\right]\:{the}\:{polynom}\:{p}\left({x}\right) \\ $$$$ \\ $$$$ \\ $$

Question Number 46419 Answers: 0 Comments: 4

let f(x)= (√(x+1−(√(x−1)))) 1) find D_f 2) determine f^(−1) (x) 3) find ∫ f(x)dx 4) dtetrmine ∫ f^(−1) (x) 5) let g(x)= (ch(x))^2 calculate fog(x) and (fog)^′ (x) .

$${let}\:{f}\left({x}\right)=\:\sqrt{{x}+\mathrm{1}−\sqrt{{x}−\mathrm{1}}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{f}^{−\mathrm{1}} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:\int\:{f}\left({x}\right){dx} \\ $$$$\left.\mathrm{4}\right)\:{dtetrmine}\:\int\:\:{f}^{−\mathrm{1}} \left({x}\right) \\ $$$$\left.\mathrm{5}\right)\:{let}\:{g}\left({x}\right)=\:\left({ch}\left({x}\right)\right)^{\mathrm{2}} \:\:\:{calculate}\:{fog}\left({x}\right)\:{and}\:\left({fog}\right)^{'} \left({x}\right)\:. \\ $$

Question Number 46418 Answers: 0 Comments: 0

find ∫∫_([0,1]×[0,1]) ((√(x+(√y)))−(√(y+(√x))))dxdy

$${find}\:\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]×\left[\mathrm{0},\mathrm{1}\right]} \left(\sqrt{{x}+\sqrt{{y}}}−\sqrt{{y}+\sqrt{{x}}}\right){dxdy} \\ $$

Question Number 46411 Answers: 1 Comments: 0

Question Number 46410 Answers: 0 Comments: 0

Question Number 46414 Answers: 2 Comments: 1

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