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Question Number 211632 Answers: 1 Comments: 0

Question Number 211631 Answers: 0 Comments: 0

Question Number 211630 Answers: 0 Comments: 0

Question Number 211627 Answers: 0 Comments: 0

certificate: ∣∣h∣∣_^L^p ^p =∫_0 ^∞ ∣h(x)∣^p dx.

$$ \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mid\mid\boldsymbol{{h}}\mid\mid_{\:^{\boldsymbol{{L}}^{\boldsymbol{{p}}} } } ^{\boldsymbol{{p}}} =\int_{\mathrm{0}} ^{\infty} \mid\boldsymbol{{h}}\left(\boldsymbol{{x}}\right)\mid^{\boldsymbol{{p}}} \boldsymbol{{dx}}. \\ $$

Question Number 211614 Answers: 2 Comments: 1

A group of people are standing in a circle. Every second person is removed from the circle and this process continues until only one person remains in the circle. If there are 100 people in the circle, what will be the number of the last person left?

$$ \\ $$A group of people are standing in a circle. Every second person is removed from the circle and this process continues until only one person remains in the circle. If there are 100 people in the circle, what will be the number of the last person left?

Question Number 211617 Answers: 1 Comments: 0

Ato starts a business with $1,250.00. Ama joins the business later with a capital of 1,875.00. At the end of the first year, profits are shared equally between Ato and Ama. When did Ama join the business?

$$\mathrm{Ato}\:\:\mathrm{starts}\:\mathrm{a}\:\mathrm{business}\:\mathrm{with}\:\$\mathrm{1},\mathrm{250}.\mathrm{00}.\:\mathrm{Ama}\:\mathrm{joins}\:\mathrm{the}\:\mathrm{business} \\ $$$$\mathrm{later}\:\mathrm{with}\:\mathrm{a}\:\mathrm{capital}\:\mathrm{of}\:\mathrm{1},\mathrm{875}.\mathrm{00}.\:\mathrm{At}\:\mathrm{the}\: \\ $$$$\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{year},\:\mathrm{profits}\:\mathrm{are}\:\mathrm{shared}\:\mathrm{equally} \\ $$$$\mathrm{between}\:\mathrm{Ato}\:\mathrm{and}\:\mathrm{Ama}.\:\mathrm{When}\:\mathrm{did}\:\mathrm{Ama} \\ $$$$\mathrm{join}\:\mathrm{the}\:\mathrm{business}? \\ $$

Question Number 211609 Answers: 2 Comments: 0

Prove that, in a triangle the ratios of the sides and the sine of the opposite angles are equal. Also prove that each ratio is equal to the diameter of the circum circle of the triangle.

$$\mathrm{Prove}\:\mathrm{that},\:\mathrm{in}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{the}\:\mathrm{ratios}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{and}\:\mathrm{the}\:\mathrm{sine}\:\mathrm{of}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{angles} \\ $$$$\mathrm{are}\:\mathrm{equal}.\:\mathrm{Also}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{each}\:\mathrm{ratio}\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{diameter}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circum}\:\mathrm{circle} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}. \\ $$

Question Number 211605 Answers: 1 Comments: 1

Let a_1 ,a_2 ,…a_n Is n real numbers. All fall in the interval (−1,1) ________________________ (1)Prove that: Π_(1≤i,j≤n) ((1+a_i a_j )/(1−a_i a_j ))≥1 (2) Determine the necessary andsufficient conditions forl equaity in the inequality.

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Let}\:\boldsymbol{{a}}_{\mathrm{1}} ,\boldsymbol{{a}}_{\mathrm{2}} ,\ldots\boldsymbol{{a}}_{\boldsymbol{{n}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Is}\:\mathrm{n}\:\mathrm{real}\:\mathrm{numbers}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{All}\:\mathrm{fall}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval}\:\left(−\mathrm{1},\mathrm{1}\right) \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$$\left(\mathrm{1}\right)\mathrm{Prove}\:\mathrm{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{1}\leq\boldsymbol{{i}},\boldsymbol{{j}}\leq\boldsymbol{{n}}} {\prod}\frac{\mathrm{1}+\boldsymbol{{a}}_{\boldsymbol{{i}}} \boldsymbol{{a}}_{\boldsymbol{{j}}} }{\mathrm{1}−\boldsymbol{{a}}_{\boldsymbol{{i}}} \boldsymbol{{a}}_{\boldsymbol{{j}}} }\geq\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Determine}\:\mathrm{the}\:\mathrm{necessary}\: \\ $$$$\mathrm{andsufficient}\:\mathrm{conditions}\:\mathrm{forl} \\ $$$$\mathrm{equaity}\:\mathrm{in}\:\mathrm{the}\:\mathrm{inequality}. \\ $$$$ \\ $$

Question Number 211603 Answers: 3 Comments: 0

1.Given a regular tetrahedron ABCD with vertices A(0,0,0)B(a,0,0), C(0,a,0),and D(0,0,a).Calculate the volume V and the surface area S of this tetrahedron.

$$\mathrm{1}.\mathrm{Given}\:\mathrm{a}\:\mathrm{regular}\:\mathrm{tetrahedron}\:\boldsymbol{{ABCD}} \\ $$$$\mathrm{with}\:\mathrm{vertices}\:\boldsymbol{{A}}\left(\mathrm{0},\mathrm{0},\mathrm{0}\right)\boldsymbol{{B}}\left(\boldsymbol{{a}},\mathrm{0},\mathrm{0}\right), \\ $$$$\boldsymbol{{C}}\left(\mathrm{0},\boldsymbol{{a}},\mathrm{0}\right),\boldsymbol{\mathrm{and}}\:\boldsymbol{{D}}\left(\mathrm{0},\mathrm{0},\boldsymbol{{a}}\right).\mathrm{Calculate}\:\mathrm{the} \\ $$$$\:\mathrm{volume}\:\boldsymbol{{V}}\:\:\mathrm{and}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{area}\:\boldsymbol{{S}}\:\boldsymbol{\mathrm{of}} \\ $$$$\mathrm{this}\:\mathrm{tetrahedron}. \\ $$

Question Number 211601 Answers: 0 Comments: 0

set 𝛂(2)^(1/3) ,ask Q(𝛂)Upper irreducible cubic equation: x^3 −3x+1=0 All the roots of

$$ \\ $$$$\:\:\:\:\:\:\:\mathrm{set}\:\boldsymbol{\alpha}\sqrt[{\mathrm{3}}]{\mathrm{2}},\mathrm{ask}\:\mathbb{Q}\left(\boldsymbol{\alpha}\right)\mathrm{Upper}\:\mathrm{irreducible}\:\mathrm{cubic}\:\mathrm{equation}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{x}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{x}}+\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{All}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of} \\ $$$$ \\ $$

Question Number 211598 Answers: 2 Comments: 0

{ ((x^2 −4x+3<0)),((((2x−1)/(x+2))≥1)) :}

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{{x}}+\mathrm{3}<\mathrm{0}}\\{\frac{\mathrm{2}\boldsymbol{{x}}−\mathrm{1}}{\boldsymbol{{x}}+\mathrm{2}}\geq\mathrm{1}}\end{cases} \\ $$$$ \\ $$

Question Number 211597 Answers: 1 Comments: 0

2x^4 −4x^3 −22x^2 +24x+36=0

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\boldsymbol{{x}}^{\mathrm{4}} −\mathrm{4}\boldsymbol{{x}}^{\mathrm{3}} −\mathrm{22}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{24}\boldsymbol{{x}}+\mathrm{36}=\mathrm{0} \\ $$$$ \\ $$

Question Number 211595 Answers: 1 Comments: 0