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Question Number 164538    Answers: 0   Comments: 0

Question Number 164533    Answers: 2   Comments: 0

Prove, that 1) Σ_(k=1) ^α (1/(k(k+1))) = (α/(α+1)) 2) lim_(x→∞) Σ_(k=1) ^∞ (1/(k(k+x))) = 0

$$\mathrm{Prove},\:\mathrm{that} \\ $$$$\left.\mathrm{1}\right)\:\underset{{k}=\mathrm{1}} {\overset{\alpha} {\sum}}\frac{\mathrm{1}}{{k}\left({k}+\mathrm{1}\right)}\:=\:\frac{\alpha}{\alpha+\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{k}\left({k}+{x}\right)}\:=\:\mathrm{0} \\ $$

Question Number 164530    Answers: 1   Comments: 1

Question Number 164524    Answers: 1   Comments: 0

Question Number 164520    Answers: 0   Comments: 0

A circle is drawn with center (0,1) and radius 1. A line OAB is drawn, making an angle θ with the x-axis to cut the circle at A and the tangent to the circle at (0,2) at B. Lines are now drawn through A and B parallel to the x- and y-axes respectively to intersect at P. Prove that (i) OA=2 sin θ and (ii)the coordinates of P are (2 cot θ, 2 sin^2 θ) Hence, find the Cartesian equation of the locus of P.

$$\mathrm{A}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{with}\:\mathrm{center}\:\left(\mathrm{0},\mathrm{1}\right)\:\mathrm{and}\:\mathrm{radius}\:\mathrm{1}. \\ $$$$\mathrm{A}\:\mathrm{line}\:\mathrm{OAB}\:\mathrm{is}\:\mathrm{drawn},\:\mathrm{making}\:\mathrm{an}\:\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the} \\ $$$${x}-\mathrm{axis}\:\mathrm{to}\:\mathrm{cut}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{at}\:{A}\:\mathrm{and}\:\mathrm{the}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{circle}\:\mathrm{at}\:\left(\mathrm{0},\mathrm{2}\right)\:\mathrm{at}\:{B}.\:\mathrm{Lines}\:\mathrm{are}\:\mathrm{now}\:\mathrm{drawn}\:\mathrm{through} \\ $$$${A}\:\mathrm{and}\:{B}\:\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:{x}-\:\mathrm{and}\:{y}-\mathrm{axes}\:\mathrm{respectively} \\ $$$$\mathrm{to}\:\mathrm{intersect}\:\mathrm{at}\:{P}.\:\:\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{OA}=\mathrm{2}\:\mathrm{sin}\:\theta\:\:\:\:\:\mathrm{and}\: \\ $$$$\left(\mathrm{ii}\right)\mathrm{the}\:\mathrm{coordinates}\:\mathrm{of}\:{P}\:\mathrm{are}\:\left(\mathrm{2}\:\mathrm{cot}\:\theta,\:\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \theta\right) \\ $$$$\mathrm{Hence},\:\mathrm{find}\:\mathrm{the}\:\mathrm{Cartesian}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{P}. \\ $$

Question Number 164542    Answers: 0   Comments: 0

((x^2 −x)/(3−2x)) + (1/2) < (5/(−3x+x^2 −2))

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

Question Number 164515    Answers: 0   Comments: 3

Question Number 164511    Answers: 1   Comments: 0

sin∙(π sin x) - cos∙(π sin x) = 1 find x=?

$$\mathrm{sin}\centerdot\left(\pi\:\mathrm{sin}\:\mathrm{x}\right)\:-\:\mathrm{cos}\centerdot\left(\pi\:\mathrm{sin}\:\mathrm{x}\right)\:=\:\mathrm{1} \\ $$$$\mathrm{find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$

Question Number 164582    Answers: 0   Comments: 0

Question Number 164507    Answers: 1   Comments: 2

Question Number 164497    Answers: 1   Comments: 0

Question Number 164483    Answers: 1   Comments: 1

((cos25−sin65)/(sin20+sin10))=?

$$\frac{{cos}\mathrm{25}−{sin}\mathrm{65}}{{sin}\mathrm{20}+{sin}\mathrm{10}}=? \\ $$

Question Number 164488    Answers: 0   Comments: 1

(1/(sin 30))−(cot 30)^4 −(cot 30)^2 =.... a)sin^2 30 b)(1/(sin 30)) c)2sin^2 30 d)(1/(sin^2 30))

$$\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{30}}−\left(\mathrm{cot}\:\mathrm{30}\right)^{\mathrm{4}} −\left(\mathrm{cot}\:\mathrm{30}\right)^{\mathrm{2}} =.... \\ $$$$\left.{a}\left.\right)\mathrm{sin}\:^{\mathrm{2}} \mathrm{30}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{b}\right)\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{30}} \\ $$$$\left.{c}\left.\right)\mathrm{2sin}\:^{\mathrm{2}} \mathrm{30}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{d}\right)\frac{\mathrm{1}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{30}} \\ $$

Question Number 164478    Answers: 1   Comments: 0

Given { ((u_0 =α ∈ C)),((u_(n+1) =((u_n +∣u_n ∣)/2))) :} ; n∈ N where (u_n ) _(n∈N) is a complex sequence. Determinate the sequence (Im(u_n )) _(n∈N) and calculate its limit. NB: Im(u_n ) is the complex part of u_(n.)

$${Given}\:\begin{cases}{{u}_{\mathrm{0}} =\alpha\:\in\:\mathbb{C}}\\{{u}_{{n}+\mathrm{1}} =\frac{{u}_{{n}} +\mid{u}_{{n}} \mid}{\mathrm{2}}}\end{cases}\:;\:{n}\in\:\mathbb{N} \\ $$$${where}\:\left({u}_{{n}} \right)\:_{{n}\in\mathbb{N}} \:{is}\:{a}\:{complex}\:{sequence}. \\ $$$${Determinate}\:{the}\:{sequence}\:\left({Im}\left({u}_{{n}} \right)\right)\:_{{n}\in\mathbb{N}} \\ $$$${and}\:{calculate}\:{its}\:{limit}. \\ $$$${NB}:\:{Im}\left({u}_{{n}} \right)\:{is}\:{the}\:{complex}\:{part}\:{of}\:{u}_{{n}.} \\ $$$$ \\ $$

Question Number 164477    Answers: 0   Comments: 0

Using the definition, show that U_n =Σ_(k=1) ^n ((sin(k))/(k!)) is a sequence of Cauchy.

$${Using}\:{the}\:{definition},\:{show}\:{that}\: \\ $$$${U}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{{sin}\left({k}\right)}{{k}!}\:{is}\:{a}\:{sequence}\:{of}\:{Cauchy}. \\ $$

Question Number 164475    Answers: 1   Comments: 0

Etudiez la convergence de de la suite U_n =((1+cos(n)+2n)/(ni+(√((n+1)(n+2))))) ; n ∈ N. [study the convergence of U_n ]

$${Etudiez}\:\:{la}\:{convergence}\:{de}\:{de}\:{la} \\ $$$${suite}\:{U}_{{n}} =\frac{\mathrm{1}+{cos}\left({n}\right)+\mathrm{2}{n}}{{ni}+\sqrt{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}}\:;\:{n}\:\in\:\mathbb{N}. \\ $$$$\left[{study}\:{the}\:{convergence}\:{of}\:{U}_{{n}} \right] \\ $$

Question Number 164474    Answers: 0   Comments: 0

E={x ∈ Q_+ :x^2 >3}. Show that E has not lower bound in Q. [Montrez que E n′admet pas de borne inferieure dans Q]

$${E}=\left\{{x}\:\in\:\mathbb{Q}_{+} :{x}^{\mathrm{2}} >\mathrm{3}\right\}.\: \\ $$$${Show}\:{that}\:{E}\:{has}\:{not}\:{lower}\:{bound} \\ $$$${in}\:\mathbb{Q}. \\ $$$$\left[{Montrez}\:{que}\:{E}\:{n}'{admet}\:{pas}\:{de}\:{borne}\right. \\ $$$$\left.{inferieure}\:{dans}\:\mathbb{Q}\right] \\ $$

Question Number 164473    Answers: 1   Comments: 0

Show for z_1 ; z_(2 ) ∈ C that: ∣z_1 +z_2 ∣^2 +∣z_1 −z_2 ∣^2 =2(∣z_1 ∣^2 +∣z_2 ∣^2 ).

$${Show}\:{for}\:{z}_{\mathrm{1}} ;\:{z}_{\mathrm{2}\:} \:\in\:\mathbb{C}\:{that}: \\ $$$$\mid{z}_{\mathrm{1}} +{z}_{\mathrm{2}} \mid^{\mathrm{2}} +\mid{z}_{\mathrm{1}} −{z}_{\mathrm{2}} \mid^{\mathrm{2}} =\mathrm{2}\left(\mid{z}_{\mathrm{1}} \mid^{\mathrm{2}} +\mid{z}_{\mathrm{2}} \mid^{\mathrm{2}} \right). \\ $$

Question Number 164462    Answers: 2   Comments: 2

Find x, such that f(x) is minimum. f(x)={((√(c^2 −x^2 ))/(c−x))−(c−x)}^2

$${Find}\:{x},\:{such}\:{that}\:{f}\left({x}\right)\:{is}\:{minimum}. \\ $$$${f}\left({x}\right)=\left\{\frac{\sqrt{{c}^{\mathrm{2}} −{x}^{\mathrm{2}} }}{{c}−{x}}−\left({c}−{x}\right)\right\}^{\mathrm{2}} \\ $$

Question Number 164445    Answers: 0   Comments: 1

Question Number 164447    Answers: 1   Comments: 0

Question Number 164437    Answers: 1   Comments: 0

Find the max. area of Δle witth sides a,b,c such as 0<a≤1;1≤b≤2;2≤c≤3 is a)1 b)1/2 c)2 d)3/2

$${Find}\:{the}\:{max}.\:{area}\:{of}\:\Delta{le} \\ $$$${witth}\:{sides}\:{a},{b},{c}\:{such}\:{as} \\ $$$$\mathrm{0}<{a}\leqslant\mathrm{1};\mathrm{1}\leqslant{b}\leqslant\mathrm{2};\mathrm{2}\leqslant{c}\leqslant\mathrm{3}\:{is} \\ $$$$\left.{a}\left.\right)\left.\mathrm{1}\left.\:\:\:\:\:\:\:\:{b}\right)\mathrm{1}/\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:{c}\right)\mathrm{2}\:\:\:\:\:\:\:\:{d}\right)\mathrm{3}/\mathrm{2} \\ $$

Question Number 164434    Answers: 1   Comments: 0

Question Number 164429    Answers: 1   Comments: 0

If C(2x , x) = 70 prove that x = 4

$${If}\:{C}\left(\mathrm{2}{x}\:,\:{x}\right)\:=\:\mathrm{70} \\ $$$${prove}\:{that}\:{x}\:=\:\mathrm{4} \\ $$

Question Number 164426    Answers: 1   Comments: 1

In ΔABC if { ((cot A.cot C = (1/2))),((cot B.cot C = (1/(18)))) :} then tan C = ?

$$\:\:{In}\:\Delta{ABC}\:{if}\:\begin{cases}{\mathrm{cot}\:{A}.\mathrm{cot}\:{C}\:=\:\frac{\mathrm{1}}{\mathrm{2}}}\\{\mathrm{cot}\:{B}.\mathrm{cot}\:{C}\:=\:\frac{\mathrm{1}}{\mathrm{18}}}\end{cases} \\ $$$$\:{then}\:\mathrm{tan}\:\:{C}\:=\:? \\ $$

Question Number 164425    Answers: 1   Comments: 0

What it′s ?? 5×55×555×5555×55555.............∞=

$$\mathrm{What}\:\mathrm{it}'\mathrm{s}\:?? \\ $$$$\mathrm{5}×\mathrm{55}×\mathrm{555}×\mathrm{5555}×\mathrm{55555}.............\infty= \\ $$

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