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AllQuestion and Answers: Page 1350

Question Number 70296    Answers: 1   Comments: 1

Question Number 70225    Answers: 1   Comments: 3

Question Number 70198    Answers: 1   Comments: 0

Question Number 70197    Answers: 0   Comments: 1

Question Number 70196    Answers: 1   Comments: 0

Question Number 70216    Answers: 0   Comments: 0

let A = (((1 −1)),((1 2)) ) 1)calculate A^n 2) find e^A and e^(−A)

$${let}\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:−\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right){calculate}\:{A}^{{n}} \:\:\:\:\: \\ $$$$\left.\mathrm{2}\right)\:{find}\:{e}^{{A}} \:{and}\:{e}^{−{A}} \\ $$

Question Number 70237    Answers: 0   Comments: 3

calculate ∫_0 ^∞ ((xsin(αx))/(1+x^4 ))dx with α real

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{xsin}\left(\alpha{x}\right)}{\mathrm{1}+{x}^{\mathrm{4}} }{dx}\:{with}\:\alpha\:{real} \\ $$

Question Number 70298    Answers: 1   Comments: 0

Question Number 70168    Answers: 1   Comments: 3

Question Number 70167    Answers: 0   Comments: 1

find minima of (x_1 −x_2 )^2 +5+(√(1−(x_1 )^2 ))+(√(4x_2 )) ∀ x_1 ,x_2 ∈R

$${find}\:{minima}\:{of} \\ $$$$\left({x}_{\mathrm{1}} −{x}_{\mathrm{2}} \right)^{\mathrm{2}} +\mathrm{5}+\sqrt{\mathrm{1}−\left({x}_{\mathrm{1}} \right)^{\mathrm{2}} }+\sqrt{\mathrm{4}{x}_{\mathrm{2}} }\:\:\forall\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} \in{R} \\ $$

Question Number 70163    Answers: 1   Comments: 0

Question Number 70162    Answers: 1   Comments: 0

Question Number 70161    Answers: 1   Comments: 0

Question Number 70159    Answers: 1   Comments: 1

Question Number 70150    Answers: 0   Comments: 1

prove that ∫_0 ^(π/2) (√((4−sin^2 x)))dx < ((π(√(14)))/4)

$${prove}\:{that}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\left(\mathrm{4}−{sin}^{\mathrm{2}} {x}\right)}{dx}\:<\:\frac{\pi\sqrt{\mathrm{14}}}{\mathrm{4}} \\ $$

Question Number 70147    Answers: 1   Comments: 4

Consider the functions f(x)=5×4^(−x) and g(x)=(0.25)^(2x) +4 For what values of x do these functions assume equal values?

$${Consider}\:{the}\:{functions}\: \\ $$$${f}\left({x}\right)=\mathrm{5}×\mathrm{4}^{−{x}} \:{and}\:{g}\left({x}\right)=\left(\mathrm{0}.\mathrm{25}\right)^{\mathrm{2}{x}} +\mathrm{4} \\ $$$${For}\:{what}\:{values}\:{of}\:{x}\:{do}\:{these}\: \\ $$$${functions}\:{assume}\:{equal}\:{values}? \\ $$

Question Number 70145    Answers: 1   Comments: 0

prove that ; arg(z1z2)=arg(z1)+arg(z2). arg(z1/z2)=arg(z1)−arg(z2).

$${prove}\:{that}\:;\:{arg}\left(\boldsymbol{{z}}\mathrm{1}\boldsymbol{{z}}\mathrm{2}\right)={arg}\left({z}\mathrm{1}\right)+{arg}\left({z}\mathrm{2}\right). \\ $$$${arg}\left({z}\mathrm{1}/{z}\mathrm{2}\right)={arg}\left({z}\mathrm{1}\right)−{arg}\left({z}\mathrm{2}\right). \\ $$

Question Number 70138    Answers: 1   Comments: 0

prove that e^(iθ) =e^(i(θ+2kΠ)) given that k=0,±1,±2...

$${prove}\:{that}\:\:\:{e}^{{i}\theta} ={e}^{{i}\left(\theta+\mathrm{2}{k}\Pi\right)} \:\:{given}\:{that}\:{k}=\mathrm{0},\pm\mathrm{1},\pm\mathrm{2}... \\ $$

Question Number 70135    Answers: 0   Comments: 1

sophie−Germain identity a^4 +4b^4 =((a+b)^2 +b^2 )((a−b)^2 +b^2 )

$${sophie}−{Germain}\:{identity} \\ $$$${a}^{\mathrm{4}} +\mathrm{4}{b}^{\mathrm{4}} =\left(\left({a}+{b}\right)^{\mathrm{2}} +{b}^{\mathrm{2}} \right)\left(\left({a}−{b}\right)^{\mathrm{2}} +{b}^{\mathrm{2}} \right) \\ $$

Question Number 70132    Answers: 1   Comments: 1

Σ_(n=1) ^(3050) i^n

$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{3050}} {\sum}}\:{i}^{{n}} \\ $$

Question Number 70121    Answers: 1   Comments: 0

Question Number 70108    Answers: 1   Comments: 0

Question Number 70103    Answers: 2   Comments: 0

if m^3 +2p^3 =3mn, a^3 +b^3 =p^3 and a^2 +b^2 =n then prove that a+b=m.

$$\mathrm{if}\:\mathrm{m}^{\mathrm{3}} +\mathrm{2p}^{\mathrm{3}} =\mathrm{3mn},\:\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} =\mathrm{p}^{\mathrm{3}} \:\mathrm{and} \\ $$$$\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} =\mathrm{n}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{a}+\mathrm{b}=\mathrm{m}. \\ $$

Question Number 70075    Answers: 0   Comments: 3

Question Number 70074    Answers: 1   Comments: 1

∫_1 ^2 [3+(1/t^2 )]dt=

$$\int_{\mathrm{1}} ^{\mathrm{2}} \left[\mathrm{3}+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right]{dt}= \\ $$

Question Number 70069    Answers: 1   Comments: 2

Π_(n=1) ^5 (((12n−2)^4 +18^2 )/((12n−8)^4 +18^2 )) =(((10^4 +324)(22^4 +324)(34^4 +324)(46^4 +324)(58^4 +324))/((4^4 +324)(16^4 +324)(28^4 +324)(40^4 +324)(52^4 +324)))

$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{5}} {\prod}}\frac{\left(\mathrm{12}{n}−\mathrm{2}\right)^{\mathrm{4}} +\mathrm{18}^{\mathrm{2}} }{\left(\mathrm{12}{n}−\mathrm{8}\right)^{\mathrm{4}} +\mathrm{18}^{\mathrm{2}} } \\ $$$$=\frac{\left(\mathrm{10}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{22}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{34}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{46}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{58}^{\mathrm{4}} +\mathrm{324}\right)}{\left(\mathrm{4}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{16}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{28}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{40}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{52}^{\mathrm{4}} +\mathrm{324}\right)} \\ $$

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