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Question Number 198363    Answers: 2   Comments: 0

Question Number 198357    Answers: 0   Comments: 0

let a∈R, z∈C resolve (z+1)^n =e^(iπna) deduce that P_n =Π_(k=0) ^(n−1) sin(a+((kπ)/n))

$$\boldsymbol{{let}}\:\:\boldsymbol{{a}}\in\mathbb{R},\:\boldsymbol{{z}}\in\mathbb{C} \\ $$$$\boldsymbol{{resolve}}\: \\ $$$$\left(\boldsymbol{{z}}+\mathrm{1}\right)^{\boldsymbol{{n}}} =\boldsymbol{{e}}^{\boldsymbol{{i}}\pi\boldsymbol{{na}}} \\ $$$$\boldsymbol{{deduce}}\:\boldsymbol{{that}}\:\boldsymbol{{P}}_{\boldsymbol{{n}}} =\underset{{k}=\mathrm{0}} {\overset{\boldsymbol{{n}}−\mathrm{1}} {\prod}}\boldsymbol{{sin}}\left(\boldsymbol{{a}}+\frac{\boldsymbol{{k}}\pi}{\boldsymbol{{n}}}\right) \\ $$

Question Number 198355    Answers: 0   Comments: 2

solve the EDP 1− x(∂f/∂x)−y(∂f/∂y)=0 2− x(∂f/∂x)−(∂f/∂y)=(z^2 /x) 3− x^2 (∂^2 f/∂x^2 )−y^2 (∂^2 f/∂y^2 )=0

$${solve}\:{the}\:{EDP} \\ $$$$\mathrm{1}−\:\:\:{x}\frac{\partial{f}}{\partial{x}}−{y}\frac{\partial{f}}{\partial{y}}=\mathrm{0} \\ $$$$\mathrm{2}−\:\:\:{x}\frac{\partial{f}}{\partial{x}}−\frac{\partial{f}}{\partial{y}}=\frac{{z}^{\mathrm{2}} }{{x}} \\ $$$$ \\ $$$$\mathrm{3}−\:\:{x}^{\mathrm{2}} \frac{\partial^{\mathrm{2}} {f}}{\partial{x}^{\mathrm{2}} }−{y}^{\mathrm{2}} \frac{\partial^{\mathrm{2}} {f}}{\partial{y}^{\mathrm{2}} }=\mathrm{0} \\ $$

Question Number 198352    Answers: 0   Comments: 1

Question Number 198339    Answers: 1   Comments: 0

Question Number 198326    Answers: 0   Comments: 0

express 𝚪(k−n) in pocchammer symbol

$$\boldsymbol{\mathrm{express}}\:\boldsymbol{\Gamma}\left(\boldsymbol{{k}}−\boldsymbol{{n}}\right)\:\boldsymbol{{in}}\:\boldsymbol{{pocchammer}}\:\boldsymbol{{symbol}} \\ $$

Question Number 198321    Answers: 2   Comments: 1

A certain preliminary science class contains 50 students all of whom take Mathematics. 18 study Chemistry, 17 study Biology, 24 study Physics. Of those taking three subjects, 5 study Physics and Chemistry, 7 study Physics and Biology and 6 study Chemistry and Biology while 2 take all four subjects. How many students study only Mathematics?

$${A}\:{certain}\:{preliminary}\:{science}\:{class} \\ $$$${contains}\:\mathrm{50}\:{students}\:{all}\:{of}\:{whom}\:{take} \\ $$$${Mathematics}.\:\mathrm{18}\:{study}\:{Chemistry},\:\mathrm{17} \\ $$$${study}\:{Biology},\:\mathrm{24}\:{study}\:{Physics}.\:{Of} \\ $$$${those}\:{taking}\:{three}\:{subjects},\:\mathrm{5}\:{study} \\ $$$${Physics}\:{and}\:{Chemistry},\:\mathrm{7}\:{study}\:{Physics} \\ $$$${and}\:{Biology}\:{and}\:\mathrm{6}\:{study}\:{Chemistry} \\ $$$${and}\:{Biology}\:{while}\:\mathrm{2}\:{take}\:{all}\:{four}\:{subjects}. \\ $$$${How}\:{many}\:{students}\:{study}\:{only}\:{Mathematics}? \\ $$

Question Number 198319    Answers: 1   Comments: 1

An analysis of 100 personal injury claims made upon a motor insurance company revealed that loss or injury in respect of an eye, an arm or leg occurred in 30, 50 and 70 cases, respectively. Claims involving the loss or injury to two of these number 44. How many claims involved loss or injury to all three. Assume that one or other of the three members was mentioned in each of the 100 claims.

$${An}\:{analysis}\:{of}\:\mathrm{100}\:{personal}\:{injury} \\ $$$${claims}\:{made}\:{upon}\:{a}\:{motor}\:{insurance} \\ $$$${company}\:{revealed}\:{that}\:{loss}\:{or}\:{injury} \\ $$$${in}\:{respect}\:{of}\:{an}\:{eye},\:{an}\:{arm}\:{or}\:{leg} \\ $$$${occurred}\:{in}\:\mathrm{30},\:\mathrm{50}\:{and}\:\mathrm{70}\:{cases},\:{respectively}. \\ $$$${Claims}\:{involving}\:{the}\:{loss}\:{or}\:{injury}\:{to} \\ $$$${two}\:{of}\:{these}\:{number}\:\mathrm{44}.\:{How}\:{many} \\ $$$${claims}\:{involved}\:{loss}\:{or}\:{injury}\:{to}\:{all} \\ $$$${three}.\:{Assume}\:{that}\:{one}\:{or}\:{other}\:{of} \\ $$$${the}\:{three}\:{members}\:{was}\:{mentioned}\:{in} \\ $$$${each}\:{of}\:{the}\:\mathrm{100}\:{claims}. \\ $$

Question Number 198314    Answers: 0   Comments: 0

Solve the EDP x^2 (∂^2 u/∂x^2 )−y^2 (∂^2 u/∂y^2 )=0

$${Solve}\:{the}\:{EDP} \\ $$$${x}^{\mathrm{2}} \frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} }−{y}^{\mathrm{2}} \frac{\partial^{\mathrm{2}} {u}}{\partial{y}^{\mathrm{2}} }=\mathrm{0} \\ $$

Question Number 198313    Answers: 0   Comments: 1

Question Number 198311    Answers: 0   Comments: 0

Let {x_r }_(r=1) ^n be n positive real numbers Show That: (x_1 /(1+x_1 ^2 ))+(x_2 /(1+x_1 ^2 +x_2 ^2 ))+...+(x_n /(1+x_1 ^2 +x_2 ^2 +...+x_n ^2 ))<(√n)

$${Let}\:\left\{{x}_{{r}} \right\}_{{r}=\mathrm{1}} ^{{n}} {be}\:{n}\:{positive}\:{real}\:{numbers}\:{Show}\:{That}: \\ $$$$\frac{{x}_{\mathrm{1}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} }+\frac{{x}_{\mathrm{2}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} }+...+\frac{{x}_{{n}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} +...+{x}_{{n}} ^{\mathrm{2}} }<\sqrt{{n}} \\ $$

Question Number 198309    Answers: 0   Comments: 2

Question Number 198304    Answers: 0   Comments: 7

for {a_n } be a sequence of positive real numbers such that a_1 =1 , a_(n+1) ^2 −2a_n a_(n+1) −a_n = 0 , ∀ n≥ 1 than the sum of series Σ_(n=1) ^∞ (a_n /3^(n ) ) lies in the interval (A) (1,2] (B) (2,3] (C) (3,4] (D) (4,5]

$$\:\:\:\mathrm{for}\:\left\{\mathrm{a}_{\mathrm{n}} \right\}\:\mathrm{be}\:\mathrm{a}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\:\:\:\mathrm{such}\:\mathrm{that}\:\:\mathrm{a}_{\mathrm{1}} =\mathrm{1}\:,\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} ^{\mathrm{2}} −\mathrm{2a}_{\mathrm{n}} \mathrm{a}_{\mathrm{n}+\mathrm{1}} −\mathrm{a}_{\mathrm{n}} \:=\:\mathrm{0}\:,\:\forall\:\mathrm{n}\geqslant\:\mathrm{1} \\ $$$$\:\:\:\mathrm{than}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{series}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{a}_{\mathrm{n}} }{\mathrm{3}^{\mathrm{n}\:} }\:\:\mathrm{lies}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval} \\ $$$$\:\:\left({A}\right)\:\:\left(\mathrm{1},\mathrm{2}\right]\:\:\:\:\left({B}\right)\:\:\left(\mathrm{2},\mathrm{3}\right]\:\:\:\:\left({C}\right)\:\:\left(\mathrm{3},\mathrm{4}\right]\:\:\:\:\left({D}\right)\:\:\left(\mathrm{4},\mathrm{5}\right] \\ $$$$\:\:\:\: \\ $$

Question Number 198302    Answers: 1   Comments: 0

Question Number 198296    Answers: 0   Comments: 0

Let {x_r }_(r=1) ^n be n positive real numbers Show That: (x_1 /(1+x_1 ^2 ))+(x_2 /(1+x_1 ^2 +x_2 ^2 ))+...+(x_n /(1+x_1 ^2 +x_2 ^2 +...+x_n ^2 ))<(√n)

$${Let}\:\left\{{x}_{{r}} \right\}_{{r}=\mathrm{1}} ^{{n}} {be}\:{n}\:{positive}\:{real}\:{numbers}\:{Show}\:{That}: \\ $$$$\frac{{x}_{\mathrm{1}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} }+\frac{{x}_{\mathrm{2}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} }+...+\frac{{x}_{{n}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} +...+{x}_{{n}} ^{\mathrm{2}} }<\sqrt{{n}} \\ $$

Question Number 198295    Answers: 1   Comments: 0

x^3 −((81x−8))^(1/3) = 2x^2 −(4/3)x+2

$$\:\:\:\mathrm{x}^{\mathrm{3}} −\sqrt[{\mathrm{3}}]{\mathrm{81x}−\mathrm{8}}\:=\:\mathrm{2x}^{\mathrm{2}} −\frac{\mathrm{4}}{\mathrm{3}}\mathrm{x}+\mathrm{2}\: \\ $$

Question Number 198293    Answers: 1   Comments: 0

Question Number 198267    Answers: 3   Comments: 0

Find the real values of n: n^6 −n^3 =2

$${Find}\:{the}\:{real}\:{values}\:{of}\:{n}:\:{n}^{\mathrm{6}} −{n}^{\mathrm{3}} =\mathrm{2} \\ $$

Question Number 198266    Answers: 1   Comments: 0

Question Number 198263    Answers: 1   Comments: 0

Question Number 198260    Answers: 0   Comments: 0

help ! (i^→ ,j^→ ,k^→ ) est une base orthonormee. A, B, C et D sont des points de l′espace tels que : AB^(→) =i^→ +j^→ +k^→ AC^(→) =2i^→ +3j^→ +k^→ AD^(→) =i^→ −2j^→ +2k^→ . Determine tous les points P tels que (DP) ⊂ (OAB) et AP^(→) soit un vecteur unitaire orthogonal a AD^(→) . by axel

$$\mathrm{help}\:! \\ $$$$\left(\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}},\overset{\rightarrow} {{k}}\right)\:\mathrm{est}\:\mathrm{une}\:\mathrm{base}\:\mathrm{orthonormee}. \\ $$$$\mathrm{A},\:\mathrm{B},\:\mathrm{C}\:\mathrm{et}\:\mathrm{D}\:\mathrm{sont}\:\mathrm{des}\:\mathrm{points}\:\mathrm{de}\:\mathrm{l}'\mathrm{espace} \\ $$$$\mathrm{tels}\:\mathrm{que}\::\: \\ $$$$\overset{\rightarrow} {\mathrm{AB}}=\overset{\rightarrow} {{i}}+\overset{\rightarrow} {{j}}+\overset{\rightarrow} {{k}} \\ $$$$\overset{\rightarrow} {\mathrm{AC}}=\mathrm{2}\overset{\rightarrow} {{i}}+\mathrm{3}\overset{\rightarrow} {{j}}+\overset{\rightarrow} {{k}} \\ $$$$\overset{\rightarrow} {\mathrm{AD}}=\overset{\rightarrow} {{i}}−\mathrm{2}\overset{\rightarrow} {{j}}+\mathrm{2}\overset{\rightarrow} {{k}}. \\ $$$$\boldsymbol{\mathrm{Determine}}\:\boldsymbol{\mathrm{tous}}\:\boldsymbol{\mathrm{les}}\:\boldsymbol{\mathrm{points}}\:\boldsymbol{\mathrm{P}}\:\boldsymbol{\mathrm{tels}}\:\boldsymbol{\mathrm{que}}\: \\ $$$$\left(\boldsymbol{\mathrm{DP}}\right)\:\subset\:\left(\boldsymbol{\mathrm{OAB}}\right)\:\boldsymbol{\mathrm{et}}\:\overset{\rightarrow} {\boldsymbol{\mathrm{AP}}}\:\boldsymbol{\mathrm{soit}}\:\boldsymbol{\mathrm{un}}\:\boldsymbol{\mathrm{vecteur}}\: \\ $$$$\boldsymbol{\mathrm{unitaire}}\:\boldsymbol{\mathrm{orthogonal}}\:\boldsymbol{\mathrm{a}}\:\overset{\rightarrow} {\boldsymbol{\mathrm{AD}}}. \\ $$$${by}\:{axel} \\ $$

Question Number 198252    Answers: 1   Comments: 2

Question Number 198249    Answers: 1   Comments: 4

if sin(x+ϕ)+cos2x≤(√(3 )) ϕ=?

$${if}\:\:{sin}\left({x}+\varphi\right)+{cos}\mathrm{2}{x}\leqslant\sqrt{\mathrm{3}\:}\: \\ $$$$\varphi=? \\ $$

Question Number 198246    Answers: 0   Comments: 0

prove that Σ_(i=1) ^(2n−1) (((−1)^(i−1) )/i)>ln2+(1/(4n))

$${prove}\:{that} \\ $$$$\:\underset{{i}=\mathrm{1}} {\overset{\mathrm{2}{n}−\mathrm{1}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{i}−\mathrm{1}} }{{i}}>{ln}\mathrm{2}+\frac{\mathrm{1}}{\mathrm{4}{n}} \\ $$

Question Number 198244    Answers: 0   Comments: 0

Question Number 198243    Answers: 3   Comments: 0

find the sum of the first n terms from 1, 2+3, 4+5+6, 7+8+9+10, ...

$${find}\:{the}\:{sum}\:{of}\:{the}\:{first}\:{n}\:{terms}\:{from} \\ $$$$\mathrm{1},\:\mathrm{2}+\mathrm{3},\:\mathrm{4}+\mathrm{5}+\mathrm{6},\:\mathrm{7}+\mathrm{8}+\mathrm{9}+\mathrm{10},\:... \\ $$

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