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Question Number 45291    Answers: 1   Comments: 0

How many possible triple of (a,b,c) ∈ integers so that : ∣ a + b ∣ + c = 19 ab + ∣ c ∣ = 97

$${How}\:\:{many}\:\:{possible}\:\:{triple}\:\:{of}\:\:\left({a},{b},{c}\right)\:\:\in\:\:{integers}\:\:{so}\:\:{that}\:\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mid\:{a}\:+\:{b}\:\mid\:+\:{c}\:\:=\:\:\mathrm{19} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{ab}\:+\:\mid\:{c}\:\mid\:\:=\:\:\mathrm{97} \\ $$

Question Number 45284    Answers: 1   Comments: 2

Question Number 45281    Answers: 1   Comments: 2

Question Number 45283    Answers: 1   Comments: 0

if x=1+a+a^2 +a^3 +……… y=1+b+b^2 +b^3 +……… prove that 1+ab+a^2 b^2 +a^3 b^3 +………=((xy)/(x+y−1))

$$\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{x}}=\mathrm{1}+\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{a}}^{\mathrm{3}} +\ldots\ldots\ldots \\ $$$$\:\:\:\:\boldsymbol{\mathrm{y}}=\mathrm{1}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{3}} +\ldots\ldots\ldots \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}} \\ $$$$\mathrm{1}+\boldsymbol{\mathrm{ab}}+\boldsymbol{\mathrm{a}}^{\mathrm{2}} \boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{a}}^{\mathrm{3}} \boldsymbol{\mathrm{b}}^{\mathrm{3}} +\ldots\ldots\ldots=\frac{\boldsymbol{\mathrm{xy}}}{\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}−\mathrm{1}} \\ $$

Question Number 45280    Answers: 2   Comments: 0

solve the simultaneous equation a)sin(x+y)=(1/((√2) )) cos2x=((-1 )/2) for x and y ranging from 0 to 360 inclusive b)if siny+cosx=x show that (d^2 y/dx^(2 ) ) =(x/((2−x^2 )^(3/2) ))

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{simultaneous}}\:\boldsymbol{\mathrm{equation}} \\ $$$$\left.\boldsymbol{\mathrm{a}}\right)\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:\:\:} \\ $$$$\boldsymbol{\mathrm{cos}}\mathrm{2}\boldsymbol{\mathrm{x}}=\frac{-\mathrm{1}\:\:}{\mathrm{2}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{y}}\:\boldsymbol{\mathrm{ranging}}\:\boldsymbol{\mathrm{from}}\:\mathrm{0}\:\boldsymbol{\mathrm{to}}\:\mathrm{360}\:\boldsymbol{\mathrm{inclusive}} \\ $$$$\left.\boldsymbol{\mathrm{b}}\right)\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{siny}}+\boldsymbol{\mathrm{cosx}}=\boldsymbol{\mathrm{x}}\:\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{2}\:} }\:=\frac{\boldsymbol{\mathrm{x}}}{\left(\mathrm{2}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$

Question Number 45270    Answers: 2   Comments: 1

Question Number 45268    Answers: 0   Comments: 0

Question Number 45259    Answers: 2   Comments: 1

Find distance of point (1,1,1) from the line passing through (2,3,4) & (−1,2,3) ?

$${Find}\:\:{distance}\:{of}\:{point}\:\left(\mathrm{1},\mathrm{1},\mathrm{1}\right)\:{from} \\ $$$${the}\:{line}\:{passing}\:{through}\:\left(\mathrm{2},\mathrm{3},\mathrm{4}\right)\:\& \\ $$$$\left(−\mathrm{1},\mathrm{2},\mathrm{3}\right)\:? \\ $$

Question Number 45264    Answers: 0   Comments: 4

Question Number 45239    Answers: 0   Comments: 0

find the range of the following (i) f(x)=arcsin(x) (ii)f(x)=arccos(x) (iii)f(x)=arctan(x)

$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{range}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{following}} \\ $$$$\left(\boldsymbol{\mathrm{i}}\right)\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{{x}}\right)=\boldsymbol{\mathrm{arcsin}}\left(\boldsymbol{{x}}\right) \\ $$$$\left(\boldsymbol{\mathrm{ii}}\right)\boldsymbol{\mathrm{f}}\left(\boldsymbol{{x}}\right)=\boldsymbol{\mathrm{arccos}}\left(\boldsymbol{{x}}\right) \\ $$$$\left(\boldsymbol{\mathrm{iii}}\right)\boldsymbol{\mathrm{f}}\left(\boldsymbol{{x}}\right)=\boldsymbol{\mathrm{arctan}}\left(\boldsymbol{{x}}\right) \\ $$

Question Number 45240    Answers: 2   Comments: 1

calculate Σ_(n=2) ^∞ ((2n+1)/(n^4 −n^2 ))

$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\:\frac{\mathrm{2}{n}+\mathrm{1}}{{n}^{\mathrm{4}} −{n}^{\mathrm{2}} } \\ $$

Question Number 45237    Answers: 1   Comments: 1

calculate Σ_(n=2) ^∞ (1/(n^3 −n))

$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{3}} −{n}} \\ $$

Question Number 45235    Answers: 0   Comments: 2

let ∣a∣<1 and f(a)=∫_0 ^1 ln(x)ln(1+ax)dx 1) find a explicit form of f(a) 2) calculate g(a) =∫_0 ^1 ((xln(x))/(1+ax))dx 3) calculate ∫_0 ^1 ln(x)ln(2+x)dx 4) calculate ∫_0 ^1 ((xln(x))/(2+x))dx 5) calculate u_n =∫_0 ^1 ((xln(x))/(n+x))dx with n integr and n>1 find nature of the serie Σ u_n

$${let}\:\mid{a}\mid<\mathrm{1}\:{and}\:\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}+{ax}\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{xln}\left({x}\right)}{\mathrm{1}+{ax}}{dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{2}+{x}\right){dx} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{xln}\left({x}\right)}{\mathrm{2}+{x}}{dx}\: \\ $$$$\left.\mathrm{5}\right)\:{calculate}\:{u}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{xln}\left({x}\right)}{{n}+{x}}{dx}\:{with}\:{n}\:{integr}\:{and}\:{n}>\mathrm{1} \\ $$$${find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{u}_{{n}} \\ $$

Question Number 45213    Answers: 3   Comments: 0

Find ∫(√(a^2 −tan^2 x)) dx (with a>0) (related to Q45187)

$${Find}\:\int\sqrt{{a}^{\mathrm{2}} −\mathrm{tan}^{\mathrm{2}} \:{x}}\:{dx}\:\left({with}\:{a}>\mathrm{0}\right) \\ $$$$\left({related}\:{to}\:{Q}\mathrm{45187}\right) \\ $$

Question Number 45211    Answers: 1   Comments: 0

Let A= (((1 3)),((2 5)) ). Find an expression for the enteries of A^n .

$$\mathrm{Let}\:\mathrm{A}=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\mathrm{3}}\\{\mathrm{2}\:\:\:\:\:\mathrm{5}}\end{pmatrix}.\:\mathrm{Find}\:\mathrm{an}\:\mathrm{expression}\: \\ $$$$\mathrm{for}\:\mathrm{the}\:\mathrm{enteries}\:\mathrm{of}\:\mathrm{A}^{\mathrm{n}} . \\ $$

Question Number 45208    Answers: 0   Comments: 1

Question Number 45204    Answers: 0   Comments: 2

Question Number 45201    Answers: 0   Comments: 6

find ∫ (dx/(√(1+x^3 )))

$${find}\:\int\:\:\frac{{dx}}{\sqrt{\mathrm{1}+{x}^{\mathrm{3}} }} \\ $$

Question Number 45206    Answers: 0   Comments: 1

Question Number 45205    Answers: 2   Comments: 0

Question Number 45192    Answers: 0   Comments: 0

To the developer, of tinkutara; Sir, since yesterday the app crashes when i go for any page other than the main ′sort by question id page′, please help, my phone is 2G compatible android old version, but the app had been running all nice before yesterday night.

$${To}\:{the}\:{developer},\:{of}\:{tinkutara}; \\ $$$${Sir},\:{since}\:{yesterday}\:{the} \\ $$$${app}\:{crashes}\:{when}\:{i}\:{go}\:{for} \\ $$$${any}\:{page}\:{other}\:{than}\:{the} \\ $$$${main}\:'{sort}\:{by}\:{question}\:{id}\:{page}', \\ $$$${please}\:{help},\:{my}\:{phone}\:{is}\:\mathrm{2}{G} \\ $$$${compatible}\:{android}\:{old}\:{version}, \\ $$$${but}\:{the}\:{app}\:{had}\:{been}\:{running}\:{all} \\ $$$${nice}\:{before}\:{yesterday}\:{night}. \\ $$

Question Number 45187    Answers: 1   Comments: 6

Question Number 45173    Answers: 0   Comments: 3

Question Number 45232    Answers: 1   Comments: 1

calculate ∫_0 ^1 ln(x)ln(1+x)dx .

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}+{x}\right){dx}\:. \\ $$

Question Number 45231    Answers: 2   Comments: 1

find ∫ (√((x−1)(3−x)))dx

$${find}\:\int\:\sqrt{\left({x}−\mathrm{1}\right)\left(\mathrm{3}−{x}\right)}{dx} \\ $$

Question Number 45225    Answers: 1   Comments: 0

solve for x 9^(x+1) +3^(2x+1) =36

$$\mathrm{solve}\:\mathrm{for}\:{x} \\ $$$$\mathrm{9}^{{x}+\mathrm{1}} +\mathrm{3}^{\mathrm{2}{x}+\mathrm{1}} =\mathrm{36} \\ $$

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