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Question Number 108842 Answers: 2 Comments: 3

Question Number 108841 Answers: 1 Comments: 0

Question Number 108839 Answers: 0 Comments: 1

Question Number 108838 Answers: 0 Comments: 0

Question Number 108836 Answers: 1 Comments: 0

((bemath)/(⊂cooll⊃)) find the particular solution of y′′ + 4y = sin (2x)

$$\:\:\frac{\boldsymbol{{bemath}}}{\subset{cooll}\supset} \\ $$$${find}\:{the}\:{particular}\:{solution}\: \\ $$$${of}\:{y}''\:+\:\mathrm{4}{y}\:=\:\mathrm{sin}\:\left(\mathrm{2}{x}\right) \\ $$

Question Number 108825 Answers: 2 Comments: 1

Σ_(n=1) ^(10) (i^n +i^(n+1) )= ?

$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left({i}^{{n}} +{i}^{{n}+\mathrm{1}} \right)=\:? \\ $$

Question Number 108821 Answers: 1 Comments: 0

find ∫_0 ^∞ ((lnx)/((x^2 +1)^2 ))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{lnx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 108818 Answers: 1 Comments: 0

((Bobhans)/Δ) (1)Let n be a positive integer, and let x and y be positive real number such that x^n + y^n = 1 . Prove that (Σ_(k = 1) ^n ((1+x^(2k) )/(1+x^(4k) )) )(Σ_(k = 1) ^n ((1+y^(2k) )/(1+y^(4k) )) ) < (1/((1−x)(1−y))) (2) All the letters of the word ′EAMCOT ′ are arranged in different possible ways. The number of such arrangement in which no two vowels are adjacent to each other is ___

$$\:\:\frac{\boldsymbol{\mathcal{B}}{ob}\boldsymbol{{hans}}}{\Delta} \\ $$$$\left(\mathrm{1}\right){Let}\:{n}\:{be}\:{a}\:{positive}\:{integer},\:{and}\:{let}\:{x}\:{and}\:{y}\:{be}\:{positive}\:{real}\: \\ $$$${number}\:{such}\:{that}\:{x}^{{n}} \:+\:{y}^{{n}} \:=\:\mathrm{1}\:.\:{Prove}\:{that}\: \\ $$$$\left(\underset{{k}\:=\:\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}+{x}^{\mathrm{2}{k}} }{\mathrm{1}+{x}^{\mathrm{4}{k}} }\:\right)\left(\underset{{k}\:=\:\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}+{y}^{\mathrm{2}{k}} }{\mathrm{1}+{y}^{\mathrm{4}{k}} }\:\right)\:<\:\frac{\mathrm{1}}{\left(\mathrm{1}−{x}\right)\left(\mathrm{1}−{y}\right)} \\ $$$$\left(\mathrm{2}\right)\:{All}\:{the}\:{letters}\:{of}\:{the}\:{word}\:'{EAMCOT}\:'\:{are}\:{arranged}\:{in}\:{different}\:\: \\ $$$${possible}\:{ways}.\:{The}\:{number}\:{of}\:{such}\:{arrangement}\:{in}\:{which}\: \\ $$$${no}\:{two}\:{vowels}\:{are}\:{adjacent}\:{to}\:{each}\:{other}\:{is}\:\_\_\_ \\ $$

Question Number 108815 Answers: 0 Comments: 0

f(x)=(1/( (√(1+x))))+(1/( (√(1+a))))+((√(ax))/( (√(ax+8)))) x>0 , a>0 , x∈R, a∈R prove:1<f(x)<2

$${f}\left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{a}}}+\frac{\sqrt{{ax}}}{\:\sqrt{{ax}+\mathrm{8}}} \\ $$$${x}>\mathrm{0}\:,\:{a}>\mathrm{0}\:,\:{x}\in{R},\:{a}\in{R} \\ $$$${prove}:\mathrm{1}<{f}\left({x}\right)<\mathrm{2} \\ $$

Question Number 108813 Answers: 0 Comments: 0

lim_(x→∞) (−ln 2.1)^(2x) =? ??????

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(−\mathrm{ln}\:\mathrm{2}.\mathrm{1}\right)^{\mathrm{2x}} =? \\ $$$$?????? \\ $$

Question Number 108805 Answers: 0 Comments: 0

Let the first term and the common ratio of a geometric sequence {a_n } be 1 and r. If {a_n } satisfy ∣a_(n−1) −a_1 ∣≤∣a_n −a_1 ∣ for all n≥2, find the range of values of r.

$$\mathrm{Let}\:\mathrm{the}\:\mathrm{first}\:\mathrm{term}\:\mathrm{and}\:\mathrm{the}\:\mathrm{common} \\ $$$$\mathrm{ratio}\:\mathrm{of}\:\mathrm{a}\:\mathrm{geometric}\:\mathrm{sequence}\:\left\{{a}_{\mathrm{n}} \right\}\:\mathrm{be}\:\mathrm{1} \\ $$$$\mathrm{and}\:{r}.\: \\ $$$$\mathrm{If}\:\left\{{a}_{\mathrm{n}} \right\}\:\mathrm{satisfy}\:\mid{a}_{\mathrm{n}−\mathrm{1}} −{a}_{\mathrm{1}} \mid\leqslant\mid{a}_{\mathrm{n}} −{a}_{\mathrm{1}} \mid\:\mathrm{for} \\ $$$$\mathrm{all}\:\mathrm{n}\geqslant\mathrm{2},\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{r}. \\ $$

Question Number 108802 Answers: 1 Comments: 0

Question Number 108790 Answers: 1 Comments: 1

The values of θ lying between 0 and (π/2) and satisfying the equation determinant (((1+sin^2 θ),( cos^2 θ),(4 sin 4θ)),(( sin^2 θ),(1+cos^2 θ),(4 sin 4θ)),(( sin^2 θ),( cos^2 θ),(1+sin^4 θ)))=0 are

$$\mathrm{The}\:\mathrm{values}\:\mathrm{of}\:\theta\:\mathrm{lying}\:\mathrm{between}\:\mathrm{0}\:\mathrm{and} \\ $$$$\frac{\pi}{\mathrm{2}}\:\mathrm{and}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\begin{vmatrix}{\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \theta}&{\:\:\mathrm{cos}^{\mathrm{2}} \theta}&{\mathrm{4}\:\mathrm{sin}\:\mathrm{4}\theta}\\{\:\:\:\mathrm{sin}^{\mathrm{2}} \theta}&{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \theta}&{\mathrm{4}\:\mathrm{sin}\:\mathrm{4}\theta}\\{\:\:\:\mathrm{sin}^{\mathrm{2}} \theta}&{\:\:\:\mathrm{cos}^{\mathrm{2}} \theta}&{\mathrm{1}+\mathrm{sin}^{\mathrm{4}} \theta}\end{vmatrix}=\mathrm{0}\:\:\mathrm{are} \\ $$

Question Number 108789 Answers: 1 Comments: 0

Question Number 108787 Answers: 1 Comments: 0

Question Number 108786 Answers: 1 Comments: 0

Question Number 108781 Answers: 1 Comments: 0

Question Number 108780 Answers: 1 Comments: 0

Question Number 108776 Answers: 1 Comments: 0

⋮^(bobhans) ∫ (dx/( (√(x(√x) −x^2 )))) = ?

$$\:\overset{{bobhans}} {\vdots} \\ $$$$\int\:\frac{{dx}}{\:\sqrt{{x}\sqrt{{x}}\:−{x}^{\mathrm{2}} }}\:=\:? \\ $$

Question Number 108769 Answers: 1 Comments: 0

((⋰BeMath⋱)/★) Given ((2x)/(2x+6)) = ((5y)/(5y+25)) = ((4z)/(4z+16)) and xy + yz + xz = 188 . Find the solution

$$\:\:\:\frac{\iddots\mathcal{B}{e}\mathcal{M}{ath}\ddots}{\bigstar} \\ $$$$\:\mathrm{G}{iven}\:\frac{\mathrm{2}{x}}{\mathrm{2}{x}+\mathrm{6}}\:=\:\frac{\mathrm{5}{y}}{\mathrm{5}{y}+\mathrm{25}}\:=\:\frac{\mathrm{4}{z}}{\mathrm{4}{z}+\mathrm{16}} \\ $$$${and}\:{xy}\:+\:{yz}\:+\:{xz}\:=\:\mathrm{188}\:.\:\mathrm{F}{ind}\:{the} \\ $$$${solution} \\ $$

Question Number 108766 Answers: 5 Comments: 0

((BeMath)/★) (1) find the equation of the tangent line to the graph of the equation sin^(−1) (x)+cos^(−1) (y)=(π/2) at given point (((√2)/2), ((√2)/2)) (2)If f(x)= lim_(t→x) ((sec t−sec x)/(t−x)) , find the value of f ′((π/4)) (3) lim_(x→1) ((tan^(−1) (x)−(π/4))/(x−1))

$$\:\:\:\frac{\mathcal{B}{e}\mathcal{M}{ath}}{\bigstar} \\ $$$$\left(\mathrm{1}\right)\:{find}\:{the}\:{equation}\:{of}\:{the}\:{tangent}\:{line}\:{to} \\ $$$${the}\:{graph}\:{of}\:{the}\:{equation}\:\mathrm{sin}^{−\mathrm{1}} \left({x}\right)+\mathrm{cos}^{−\mathrm{1}} \left({y}\right)=\frac{\pi}{\mathrm{2}} \\ $$$${at}\:{given}\:{point}\:\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}},\:\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{2}\right){If}\:{f}\left({x}\right)=\:\underset{{t}\rightarrow{x}} {\mathrm{lim}}\:\frac{\mathrm{sec}\:{t}−\mathrm{sec}\:{x}}{{t}−{x}}\:,\:{find}\:{the}\:{value}\:{of}\: \\ $$$${f}\:'\left(\frac{\pi}{\mathrm{4}}\right) \\ $$$$\left(\mathrm{3}\right)\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{tan}^{−\mathrm{1}} \left({x}\right)−\frac{\pi}{\mathrm{4}}}{{x}−\mathrm{1}} \\ $$

Question Number 108761 Answers: 2 Comments: 0

((⋮((Be)/(Math))⋮)/★) If ∫_(−1) ^( a) ((x+1)/((x+2)^4 )) = ((10)/(81)) , then the value of a−2 is ___

$$\:\:\:\frac{\vdots\frac{\mathcal{B}{e}}{\mathcal{M}{ath}}\vdots}{\bigstar} \\ $$$${If}\:\int_{−\mathrm{1}} ^{\:\:{a}} \:\frac{{x}+\mathrm{1}}{\left({x}+\mathrm{2}\right)^{\mathrm{4}} }\:=\:\frac{\mathrm{10}}{\mathrm{81}}\:,\:{then}\:{the}\:{value}\:{of} \\ $$$${a}−\mathrm{2}\:{is}\:\_\_\_ \\ $$

Question Number 108753 Answers: 2 Comments: 4

((⋮BeMath⋮)/△) (((√x) +1)/(x(√x) +x+(√x))) : (1/( (√x) −x^2 )) + x = ?

$$\:\:\frac{\vdots\mathcal{B}{e}\mathcal{M}{ath}\vdots}{\bigtriangleup} \\ $$$$\:\frac{\sqrt{{x}}\:+\mathrm{1}}{{x}\sqrt{{x}}\:+{x}+\sqrt{{x}}}\::\:\frac{\mathrm{1}}{\:\sqrt{{x}}\:−{x}^{\mathrm{2}} }\:+\:{x}\:=\:?\: \\ $$

Question Number 108750 Answers: 2 Comments: 0

calculste ∫_0 ^∞ ((ln(x))/(x^2 −x+1))dx

$$\mathrm{calculste}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 108749 Answers: 2 Comments: 0

calculste ∫_0 ^∞ ((ln(x))/((1+x)^4 )) dx

$$\mathrm{calculste}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{ln}\left(\mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{4}} }\:\mathrm{dx} \\ $$

Question Number 108748 Answers: 1 Comments: 0

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