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AllQuestion and Answers: Page 237

Question Number 198395 Answers: 0 Comments: 4

Question Number 198391 Answers: 2 Comments: 1

Question Number 198426 Answers: 0 Comments: 0

Question Number 198380 Answers: 0 Comments: 2

Question Number 198377 Answers: 1 Comments: 0

Question Number 198373 Answers: 0 Comments: 0

Question Number 198372 Answers: 1 Comments: 1

(1/(1×5)) + (1/(4×8)) + (1/(7×11)) +(1/(10×14)) + …=?

$$\:\:\:\frac{\mathrm{1}}{\mathrm{1}×\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{4}×\mathrm{8}}\:+\:\frac{\mathrm{1}}{\mathrm{7}×\mathrm{11}}\:+\frac{\mathrm{1}}{\mathrm{10}×\mathrm{14}}\:+\:\ldots=? \\ $$

Question Number 198367 Answers: 0 Comments: 7

if f(x) = ((x^2 −(b+1)x+b)/(x^2 −(a+1)x+a)) (a≠b & a,b ∈ R ∼ {1}) can take all values except two values α & β such that α+β = 0 then ∣((a^3 +b^3 −8)/(ab))∣ = ??

$$\:\:\mathrm{if}\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{{x}^{\mathrm{2}} −\left({b}+\mathrm{1}\right){x}+{b}}{{x}^{\mathrm{2}} −\left({a}+\mathrm{1}\right){x}+{a}}\:\:\left({a}\neq{b}\:\&\:{a},{b}\:\in\:\mathbb{R}\:\sim\:\left\{\mathrm{1}\right\}\right) \\ $$$$\:\:\mathrm{can}\:\mathrm{take}\:\mathrm{all}\:\mathrm{values}\:\mathrm{except}\:\mathrm{two}\:\mathrm{values}\:\alpha\:\&\:\beta \\ $$$$\:\:\mathrm{such}\:\mathrm{that}\:\alpha+\beta\:=\:\mathrm{0}\:\:\mathrm{then}\:\mid\frac{\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} −\mathrm{8}}{\mathrm{ab}}\mid\:\:=\:\:?? \\ $$

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)

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

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

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