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Question Number 11887    Answers: 1   Comments: 0

(dy/dt) +3t^2 y = t^(2 ) , y(0) = 1 y(t) = ?

$$\frac{{dy}}{{dt}}\:+\mathrm{3}{t}^{\mathrm{2}} {y}\:=\:{t}^{\mathrm{2}\:} \:\:\:\:,\:{y}\left(\mathrm{0}\right)\:=\:\mathrm{1} \\ $$$${y}\left({t}\right)\:=\:? \\ $$

Question Number 11898    Answers: 1   Comments: 0

(a) You are listening to your favourite song on a CD. You note that the sound wave has a pleasant frequency of 12 Hertz. (i) What is the velovity of the sound wave (ii) What wavelenght are the wave moving at (iii) What is the period (b) 60 complete waves pass a particular point in 4 secs, if the distance between 3 successive troughs of the water is 15m. Calculate the speed of the wave.

$$\left(\mathrm{a}\right) \\ $$$$\mathrm{You}\:\mathrm{are}\:\mathrm{listening}\:\mathrm{to}\:\mathrm{your}\:\mathrm{favourite}\:\mathrm{song}\:\mathrm{on}\:\mathrm{a}\:\mathrm{CD}.\:\mathrm{You}\:\mathrm{note}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sound} \\ $$$$\mathrm{wave}\:\mathrm{has}\:\mathrm{a}\:\mathrm{pleasant}\:\mathrm{frequency}\:\mathrm{of}\:\mathrm{12}\:\mathrm{Hertz}. \\ $$$$\left(\mathrm{i}\right)\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{velovity}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sound}\:\mathrm{wave} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{What}\:\mathrm{wavelenght}\:\mathrm{are}\:\mathrm{the}\:\mathrm{wave}\:\mathrm{moving}\:\mathrm{at} \\ $$$$\left(\mathrm{iii}\right)\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{period} \\ $$$$\left(\mathrm{b}\right) \\ $$$$\mathrm{60}\:\mathrm{complete}\:\mathrm{waves}\:\mathrm{pass}\:\mathrm{a}\:\mathrm{particular}\:\mathrm{point}\:\mathrm{in}\:\mathrm{4}\:\mathrm{secs},\:\mathrm{if}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between} \\ $$$$\mathrm{3}\:\mathrm{successive}\:\mathrm{troughs}\:\mathrm{of}\:\mathrm{the}\:\mathrm{water}\:\mathrm{is}\:\mathrm{15m}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wave}. \\ $$

Question Number 11883    Answers: 1   Comments: 0

Draw the structural formula of the compound 2,2,7 - trimethyl - 4 - (1 - methylpropyl) nonane

$$\mathrm{Draw}\:\mathrm{the}\:\mathrm{structural}\:\mathrm{formula}\:\mathrm{of}\:\mathrm{the}\:\mathrm{compound} \\ $$$$\mathrm{2},\mathrm{2},\mathrm{7}\:-\:\mathrm{trimethyl}\:-\:\mathrm{4}\:-\:\left(\mathrm{1}\:-\:\mathrm{methylpropyl}\right)\:\mathrm{nonane} \\ $$

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.

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

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