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AllQuestion and Answers: Page 1897

Question Number 11880    Answers: 0   Comments: 0

S_(ABCD) =3+2(√2) ∠BAO=∠MAO=22,5° ∠BCM=∠DCM S_(AOB) =?

$$\boldsymbol{\mathrm{S}}_{\boldsymbol{\mathrm{ABCD}}} =\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}} \\ $$$$\angle\boldsymbol{\mathrm{BAO}}=\angle\boldsymbol{\mathrm{MAO}}=\mathrm{22},\mathrm{5}° \\ $$$$\angle\boldsymbol{\mathrm{BCM}}=\angle\boldsymbol{\mathrm{DCM}} \\ $$$$\boldsymbol{\mathrm{S}}_{\boldsymbol{\mathrm{AOB}}} =? \\ $$

Question Number 11869    Answers: 0   Comments: 0

Why the expansion of (a+b)^(−n) follows newton′s expansion rule?

$${Why}\:{the}\:{expansion}\:{of}\:\:\left({a}+{b}\right)^{−{n}} \:{follows} \\ $$$${newton}'{s}\:{expansion}\:{rule}? \\ $$

Question Number 11868    Answers: 1   Comments: 0

Question Number 11867    Answers: 0   Comments: 1

Question Number 11865    Answers: 1   Comments: 0

∫_2 ^4 (√(16−x^2 ))dx/x^4 =

$$\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\sqrt{\mathrm{16}−{x}^{\mathrm{2}} }{dx}/{x}^{\mathrm{4}} = \\ $$

Question Number 11864    Answers: 2   Comments: 0

∫_7 ^(10) x^2 dx/x^2 −3x+2=

$$\underset{\mathrm{7}} {\overset{\mathrm{10}} {\int}}{x}^{\mathrm{2}} {dx}/{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}= \\ $$

Question Number 11862    Answers: 2   Comments: 0

Lesson1. AM−GM ′ s inequality (Cauchy) form : ((a_1 +a_2 +...+a_n )/n) ≥ ((a_1 a_2 ...a_n ))^(1/n) where a_1 ,a_2 ,....,a_n >0 Equal at a_1 =a_2 =.....=a_n e.g. 1. Given a,b,c>0, prove that (a+b)(b+c)(c+a)≥8abc Solu. by AM−GM a+b ≥ 2(√(ab)) (1) b+c ≥ 2(√(bc)) (2) c+a ≥ 2(√(ca)) (3) (1)×(2)×(3) ⇒ (a+b)(b+c)(c+a)≥8(√(a^2 b^2 c^2 ))=8abc Now practice. . Given a,b,c>0 prove that 1. a^2 +b^2 +c^2 ≥ab+bc+ca 2. (a+(1/b))(b+(1/c))(c+(1/a))≥8 3. 4(a^3 +b^3 )≥(a+b)^3 4. 9(a^3 +b^3 +c^3 )≥(a+b+c)^3 let′s try, I will post my solution for which one that you can′t do ;)

$$\boldsymbol{{Lesson}}\mathrm{1}.\:\boldsymbol{\mathrm{AM}}−\boldsymbol{\mathrm{GM}}\:'\:\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{inequality}}\:\left(\boldsymbol{\mathrm{Cauchy}}\right) \\ $$$$\boldsymbol{\mathrm{form}}\::\:\frac{\boldsymbol{{a}}_{\mathrm{1}} +\boldsymbol{{a}}_{\mathrm{2}} +...+\boldsymbol{{a}}_{\boldsymbol{{n}}} }{\boldsymbol{{n}}}\:\geqslant\:\sqrt[{\boldsymbol{{n}}}]{\boldsymbol{{a}}_{\mathrm{1}} \boldsymbol{{a}}_{\mathrm{2}} ...\boldsymbol{{a}}_{\boldsymbol{{n}}} } \\ $$$$\boldsymbol{{where}}\:\boldsymbol{{a}}_{\mathrm{1}} ,\boldsymbol{{a}}_{\mathrm{2}} ,....,\boldsymbol{{a}}_{\boldsymbol{{n}}} >\mathrm{0} \\ $$$$\boldsymbol{{E}\mathrm{qual}}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{{a}}_{\mathrm{1}} =\boldsymbol{{a}}_{\mathrm{2}} =.....=\boldsymbol{{a}}_{\boldsymbol{{n}}} \\ $$$$\boldsymbol{{e}}.\boldsymbol{{g}}.\:\mathrm{1}.\:{Given}\:{a},{b},{c}>\mathrm{0},\:{prove}\:{that}\: \\ $$$$\:\:\:\:\:\:\left({a}+{b}\right)\left({b}+{c}\right)\left({c}+{a}\right)\geqslant\mathrm{8}{abc} \\ $$$$\boldsymbol{{Solu}}.\:{by}\:{AM}−{GM} \\ $$$$\:\:\:\:\:\:\:\:{a}+{b}\:\geqslant\:\mathrm{2}\sqrt{{ab}}\:\:\:\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:{b}+{c}\:\geqslant\:\mathrm{2}\sqrt{{bc}}\:\:\:\:\:\:\:\:\left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:{c}+{a}\:\geqslant\:\mathrm{2}\sqrt{{ca}}\:\:\:\:\:\:\:\left(\mathrm{3}\right) \\ $$$$\:\left(\mathrm{1}\right)×\left(\mathrm{2}\right)×\left(\mathrm{3}\right)\:\Rightarrow\:\left({a}+{b}\right)\left({b}+{c}\right)\left({c}+{a}\right)\geqslant\mathrm{8}\sqrt{{a}^{\mathrm{2}} {b}^{\mathrm{2}} {c}^{\mathrm{2}} }=\mathrm{8}{abc} \\ $$$$\boldsymbol{{Now}}\:\boldsymbol{{practice}}. \\ $$$$\:.\:{Given}\:{a},{b},{c}>\mathrm{0}\:{prove}\:{that} \\ $$$$\:\:\:\:\:\mathrm{1}.\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \geqslant{ab}+{bc}+{ca} \\ $$$$\:\:\:\:\:\mathrm{2}.\:\left({a}+\frac{\mathrm{1}}{{b}}\right)\left({b}+\frac{\mathrm{1}}{{c}}\right)\left({c}+\frac{\mathrm{1}}{{a}}\right)\geqslant\mathrm{8} \\ $$$$\:\:\:\:\:\mathrm{3}.\:\mathrm{4}\left({a}^{\mathrm{3}} +{b}^{\mathrm{3}} \right)\geqslant\left({a}+{b}\right)^{\mathrm{3}} \\ $$$$\:\:\:\:\:\mathrm{4}.\:\:\mathrm{9}\left({a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} \right)\geqslant\left({a}+{b}+{c}\right)^{\mathrm{3}} \\ $$$$\mathrm{let}'\mathrm{s}\:\mathrm{try},\:\mathrm{I}\:\mathrm{will}\:\mathrm{post}\:\mathrm{my}\:\mathrm{solution}\:\mathrm{for}\:\mathrm{which}\:\mathrm{one} \\ $$$$\left.\mathrm{that}\:\mathrm{you}\:\mathrm{can}'\mathrm{t}\:\mathrm{do}\:;\right) \\ $$

Question Number 11855    Answers: 1   Comments: 0

Ten men are present at a club. In how many ways can four be chosen to play bridge if two men refuse to sit at the same table.

$$\mathrm{Ten}\:\mathrm{men}\:\mathrm{are}\:\mathrm{present}\:\mathrm{at}\:\mathrm{a}\:\mathrm{club}.\:\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{four}\:\mathrm{be}\:\mathrm{chosen}\:\mathrm{to} \\ $$$$\mathrm{play}\:\mathrm{bridge}\:\mathrm{if}\:\mathrm{two}\:\mathrm{men}\:\mathrm{refuse}\:\mathrm{to}\:\mathrm{sit}\:\mathrm{at}\:\mathrm{the}\:\mathrm{same}\:\mathrm{table}. \\ $$

Question Number 11854    Answers: 1   Comments: 0

Solve by mathematical induction that 1 + (1/(1 + 2)) + (1/(1 + 2 + 3)) + ... + (1/(1 + 2 + 3 + ... + n)) = ((2n)/(n + 1))

$$\mathrm{Solve}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that} \\ $$$$\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}}\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:...\:+\:\mathrm{n}}\:=\:\frac{\mathrm{2n}}{\mathrm{n}\:+\:\mathrm{1}} \\ $$

Question Number 11844    Answers: 2   Comments: 0

cotα+cosecα=k then find cosecα−cotα and also find cotα

$$\mathrm{cot}\alpha+\mathrm{cosec}\alpha={k}\:{then}\:{find}\:\mathrm{cosec}\alpha−{cot}\alpha\:{and}\:{also}\:{find}\:{cot}\alpha \\ $$

Question Number 11843    Answers: 1   Comments: 0

Differentiate, ln(cosx) from the first principle.

$$\mathrm{Differentiate},\:\:\mathrm{ln}\left(\mathrm{cosx}\right)\:\:\:\mathrm{from}\:\mathrm{the}\:\mathrm{first}\:\mathrm{principle}. \\ $$

Question Number 11838    Answers: 1   Comments: 0

((2^2 +1)/(2^2 −1)) + ((3^2 +1)/(3^2 −1)) + ((4^2 +1)/(4^2 −1)) + .... + ((20^2 +1)/(20^2 −1)) = ....?

$$\frac{\mathrm{2}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}^{\mathrm{2}} −\mathrm{1}}\:+\:\frac{\mathrm{3}^{\mathrm{2}} +\mathrm{1}}{\mathrm{3}^{\mathrm{2}} −\mathrm{1}}\:+\:\frac{\mathrm{4}^{\mathrm{2}} +\mathrm{1}}{\mathrm{4}^{\mathrm{2}} −\mathrm{1}}\:+\:....\:+\:\frac{\mathrm{20}^{\mathrm{2}} +\mathrm{1}}{\mathrm{20}^{\mathrm{2}} −\mathrm{1}}\:=\:....? \\ $$

Question Number 11834    Answers: 3   Comments: 0

Prove that ∀x,y∈R ⇒7x^2 −6xy+2y^2 +x+3 > 0

$$\boldsymbol{{Prove}}\:\boldsymbol{{that}}\:\forall\boldsymbol{{x}},\boldsymbol{{y}}\in\boldsymbol{{R}} \\ $$$$\Rightarrow\mathrm{7}\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{6}\boldsymbol{{xy}}+\mathrm{2}\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{x}}+\mathrm{3}\:>\:\mathrm{0} \\ $$

Question Number 11831    Answers: 0   Comments: 0

the system of equation a − (√(c^2 −(1/(16)) ))= (√(b^2 − (1/(16)))) b − (√(a^2 − (1/(25))))= (√(c^2 − (1/(25)))) c − (√(b^2 − (1/(36))))= (√(a^2 − (1/(36)))) given that a, b, c are real numbers that satisfy they system of equation ... if a + b + c = (x/(√(y ))) where x, y are positive integers and y is square free find the value of x + y

$$\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equation} \\ $$$$ \\ $$$$\mathrm{a}\:−\:\sqrt{\mathrm{c}^{\mathrm{2}} \:−\frac{\mathrm{1}}{\mathrm{16}}\:}=\:\sqrt{\mathrm{b}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{16}}} \\ $$$$\mathrm{b}\:−\:\sqrt{\mathrm{a}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{25}}}=\:\sqrt{\mathrm{c}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{25}}} \\ $$$$\mathrm{c}\:−\:\sqrt{\mathrm{b}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{36}}}=\:\sqrt{\mathrm{a}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{36}}} \\ $$$$ \\ $$$$\mathrm{given}\:\mathrm{that}\:\mathrm{a},\:\mathrm{b},\:\mathrm{c}\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers}\: \\ $$$$\mathrm{that}\:\mathrm{satisfy}\:\mathrm{they}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equation}\:... \\ $$$$\mathrm{if}\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:=\:\frac{\mathrm{x}}{\sqrt{\mathrm{y}\:}}\:\mathrm{where}\:\mathrm{x},\:\mathrm{y}\:\mathrm{are}\: \\ $$$$\mathrm{positive}\:\mathrm{integers}\:\mathrm{and}\:\mathrm{y}\:\mathrm{is}\:\mathrm{square}\:\mathrm{free} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:+\:\mathrm{y}\: \\ $$$$ \\ $$

Question Number 11828    Answers: 0   Comments: 0

A sample of steam at 140 bar is states to have enthalpy of 3009.1 kJ/kg, Calculate the internal energy and entropy.

$$\mathrm{A}\:\:\mathrm{sample}\:\mathrm{of}\:\mathrm{steam}\:\mathrm{at}\:\mathrm{140}\:\mathrm{bar}\:\mathrm{is}\:\mathrm{states}\:\mathrm{to}\:\mathrm{have}\:\mathrm{enthalpy}\:\mathrm{of}\:\:\mathrm{3009}.\mathrm{1}\:\mathrm{kJ}/\mathrm{kg}, \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{internal}\:\mathrm{energy}\:\mathrm{and}\:\mathrm{entropy}. \\ $$

Question Number 11827    Answers: 1   Comments: 0

∫x^2 dcosx=?

$$\int\mathrm{x}^{\mathrm{2}} \mathrm{dcosx}=? \\ $$

Question Number 11823    Answers: 1   Comments: 0

3 + 2(3^2 ) + 3(3^3 ) + ... + 10(3^(10) ) = ?

$$\mathrm{3}\:+\:\mathrm{2}\left(\mathrm{3}^{\mathrm{2}} \right)\:+\:\mathrm{3}\left(\mathrm{3}^{\mathrm{3}} \right)\:+\:...\:+\:\mathrm{10}\left(\mathrm{3}^{\mathrm{10}} \right)\:=\:? \\ $$

Question Number 11822    Answers: 1   Comments: 0

∫ ((tan x)/(1 + sin x)) dx

$$\int\:\frac{\mathrm{tan}\:{x}}{\mathrm{1}\:+\:\mathrm{sin}\:{x}}\:{dx} \\ $$

Question Number 11813    Answers: 1   Comments: 3

Question Number 11805    Answers: 2   Comments: 0

x^y = y^x x^2 = y^3 find x and y

$$\mathrm{x}^{\mathrm{y}} \:=\:\mathrm{y}^{\mathrm{x}} \:\:\:\: \\ $$$$\mathrm{x}^{\mathrm{2}} \:=\:\mathrm{y}^{\mathrm{3}} \\ $$$$\mathrm{find}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$

Question Number 11801    Answers: 0   Comments: 1

pl show me app?

$$\mathrm{pl}\:\mathrm{show}\:\mathrm{me}\:\mathrm{app}? \\ $$

Question Number 11800    Answers: 1   Comments: 0

Find the sum Σ_(n=1) ^∞ (1/((n + 1)(√n) + n(√(n + 1))))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum} \\ $$$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\left(\mathrm{n}\:+\:\mathrm{1}\right)\sqrt{\mathrm{n}}\:\:+\:\:\mathrm{n}\sqrt{\mathrm{n}\:+\:\mathrm{1}}} \\ $$

Question Number 11799    Answers: 0   Comments: 0

Prove using the density of Q in R that every real number x is the limit of a cauchy sequence of rational numbers (r_n )_(n∈N) . Give a sequence of irrational numbers (S_n ) such that S_n → x.

$$\mathrm{Prove}\:\mathrm{using}\:\mathrm{the}\:\mathrm{density}\:\mathrm{of}\:\boldsymbol{\mathrm{Q}}\:\mathrm{in}\:\mathbb{R}\:\mathrm{that}\:\mathrm{every}\:\mathrm{real}\:\mathrm{number}\:\mathrm{x}\:\mathrm{is}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{of} \\ $$$$\mathrm{a}\:\mathrm{cauchy}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{rational}\:\mathrm{numbers}\:\left(\mathrm{r}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathrm{N}} .\:\mathrm{Give}\:\mathrm{a}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{irrational}\: \\ $$$$\mathrm{numbers}\:\left(\mathrm{S}_{\mathrm{n}} \right)\:\mathrm{such}\:\mathrm{that}\:\mathrm{S}_{\mathrm{n}} \:\rightarrow\:\mathrm{x}. \\ $$

Question Number 11770    Answers: 1   Comments: 0

ax^3 +bx^2 +cx+d=0 pls.solve

$${ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0}\:{pls}.{solve} \\ $$

Question Number 11769    Answers: 0   Comments: 0

ax^3 +bx^2 +cx+d=0 solve

$${ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0}\:{solve} \\ $$

Question Number 11768    Answers: 0   Comments: 0

ax^3 +bx^2 +cx+d=0 solve it.

$${ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0}\:{solve}\:{it}. \\ $$

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