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Question Number 88610 Answers: 0 Comments: 2

is 1×1×1×..........=1^∞ or 1×1×1×1×1×..........=1

$${is}\:\mathrm{1}×\mathrm{1}×\mathrm{1}×..........=\mathrm{1}^{\infty} \: \\ $$$${or} \\ $$$$\:\mathrm{1}×\mathrm{1}×\mathrm{1}×\mathrm{1}×\mathrm{1}×..........=\mathrm{1} \\ $$$$ \\ $$$$ \\ $$

Question Number 88607 Answers: 0 Comments: 0

chose the right option 1) if x,y ∈Q^c and x<y then ∃a∈Q such that x<a<y in case we say that a)Q^c is an ordered field b)Q dense inQ^c c)not ordered field 2)let Q≠A ∈ Q,A not necessary has least upper bound and greatest lower bound that mean (Q,+,.,≤)is....... a)not complete b)complete c)dense in R 3)the sequence a_n =(n/(n+1)) is convergent to...... a)0 b)1 3)∞ 4) let f:A→B be a real valued function and Q≠S⊆A such that f is not continuous on S then f is a)contiuous on A b) continuous at any points x_0 in A c) continuous on A/S 4)every set of natural numbers has a least element so the order set of natural number is...... a)bounded from above b)bounded from below c)well ordered

$${chose}\:{the}\:{right}\:{option} \\ $$$$\left.\mathrm{1}\right)\:{if}\:{x},{y}\:\in{Q}^{{c}} \:{and}\:{x}<{y}\:\:{then}\:\exists{a}\in{Q} \\ $$$${such}\:{that}\:{x}<{a}<{y}\:{in}\:{case}\:{we}\:{say}\:{that} \\ $$$$\left.{a}\right){Q}^{{c}} \:{is}\:{an}\:{ordered}\:{field} \\ $$$$\left.{b}\right){Q}\:{dense}\:{inQ}^{{c}} \\ $$$$\left.{c}\right){not}\:{ordered}\:{field} \\ $$$$ \\ $$$$\left.\mathrm{2}\right){let}\:{Q}\neq{A}\:\in\:{Q},{A}\:{not}\:{necessary}\:{has} \\ $$$${least}\:{upper}\:{bound}\:{and}\:{greatest}\:{lower} \\ $$$${bound}\:{that}\:{mean}\:\left({Q},+,.,\leqslant\right){is}....... \\ $$$$\left.{a}\right){not}\:{complete} \\ $$$$\left.{b}\right){complete} \\ $$$$\left.{c}\right){dense}\:{in}\:{R} \\ $$$$ \\ $$$$\left.\mathrm{3}\right){the}\:{sequence}\:{a}_{{n}} =\frac{{n}}{{n}+\mathrm{1}}\:{is}\:{convergent} \\ $$$${to}...... \\ $$$$\left.{a}\right)\mathrm{0} \\ $$$$\left.{b}\right)\mathrm{1} \\ $$$$\left.\mathrm{3}\right)\infty \\ $$$$ \\ $$$$\left.\mathrm{4}\right)\:{let}\:{f}:{A}\rightarrow{B}\:{be}\:{a}\:{real}\:{valued}\:{function} \\ $$$${and}\:{Q}\neq{S}\subseteq{A}\:{such}\:{that}\:{f}\:{is}\:{not}\:{continuous} \\ $$$${on}\:{S} \\ $$$${then}\:{f}\:{is} \\ $$$$\left.{a}\right){contiuous}\:{on}\:{A} \\ $$$$\left.{b}\right)\:{continuous}\:{at}\:{any}\:{points}\:{x}_{\mathrm{0}} \:{in}\:\:{A} \\ $$$$\left.{c}\right)\:{continuous}\:{on}\:\:{A}/{S} \\ $$$$ \\ $$$$\left.\mathrm{4}\right){every}\:{set}\:{of}\:{natural}\:{numbers}\:{has} \\ $$$${a}\:{least}\:{element}\:{so}\:{the}\:{order}\:{set}\:{of} \\ $$$${natural}\:{number}\:{is}...... \\ $$$$\left.{a}\right){bounded}\:{from}\:{above} \\ $$$$\left.{b}\right){bounded}\:{from}\:{below} \\ $$$$\left.{c}\right){well}\:{ordered} \\ $$

Question Number 88606 Answers: 2 Comments: 2

Question Number 88603 Answers: 1 Comments: 0

Question Number 88594 Answers: 0 Comments: 3

solve for x∈C cos (x)=a+bi

$${solve}\:{for}\:{x}\in\mathbb{C} \\ $$$$\mathrm{cos}\:\left({x}\right)={a}+{bi} \\ $$

Question Number 88592 Answers: 1 Comments: 0

show that the variance δ^2 of a set of observations x_1 ,x_2 ,...x_n with mean x^_ can be expressed in the form δ^2 = ((Σ_(i=1) ^n x_i ^2 )/n) − x^ ^(2 )

$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{variance}\:\delta^{\mathrm{2}} \:\mathrm{of}\:\mathrm{a}\:\mathrm{set}\:\mathrm{of}\:\mathrm{observations}\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,...{x}_{{n}} \:\mathrm{with}\:\mathrm{mean} \\ $$$$\overset{\_} {{x}}\:\mathrm{can}\:\mathrm{be}\:\mathrm{expressed}\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\:\delta^{\mathrm{2}} \:=\:\frac{\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{x}_{{i}} ^{\mathrm{2}} }{{n}}\:−\:\bar {{x}}\:^{\mathrm{2}\:} \\ $$

Question Number 88586 Answers: 1 Comments: 0

∫((√(cos(2x)+3))/(cos(x)))dx

$$\int\frac{\sqrt{{cos}\left(\mathrm{2}{x}\right)+\mathrm{3}}}{{cos}\left({x}\right)}{dx} \\ $$

Question Number 88590 Answers: 0 Comments: 0

a^a^a^a^3 =5 find−a

$${a}^{{a}^{{a}^{{a}^{\mathrm{3}} } } } =\mathrm{5} \\ $$$${find}−{a} \\ $$

Question Number 88580 Answers: 0 Comments: 4

lim_(x→0) (((√(1+4x)) −((1+6x))^(1/(3 )) )/(1−cos 3x)) =

$$ \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}+\mathrm{4x}}\:−\sqrt[{\mathrm{3}\:\:}]{\mathrm{1}+\mathrm{6x}}}{\mathrm{1}−\mathrm{cos}\:\mathrm{3x}}\:= \\ $$

Question Number 88569 Answers: 1 Comments: 1

Question Number 88567 Answers: 0 Comments: 1

(1+cos (π/(11)))(1+cos ((3π)/(11)))(1+cos ((5π)/(11)))(1+cos ((7π)/(11)))(1+cos ((9π)/(11)))

$$\left(\mathrm{1}+\mathrm{cos}\:\frac{\pi}{\mathrm{11}}\right)\left(\mathrm{1}+\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{11}}\right)\left(\mathrm{1}+\mathrm{cos}\:\frac{\mathrm{5}\pi}{\mathrm{11}}\right)\left(\mathrm{1}+\mathrm{cos}\:\frac{\mathrm{7}\pi}{\mathrm{11}}\right)\left(\mathrm{1}+\mathrm{cos}\:\frac{\mathrm{9}\pi}{\mathrm{11}}\right) \\ $$

Question Number 88559 Answers: 0 Comments: 1

∫_((−π)/6) ^((3π)/4) ((√(tanx))/(1+(√(tanx))))dx

$$\int_{\frac{−\pi}{\mathrm{6}}} ^{\frac{\mathrm{3}\pi}{\mathrm{4}}} \frac{\sqrt{{tanx}}}{\mathrm{1}+\sqrt{{tanx}}}{dx} \\ $$

Question Number 88555 Answers: 1 Comments: 0

slove ⌈(x/a)⌉<a when a>1 ⌈...⌉ is ceil function

$${slove}\: \\ $$$$\lceil\frac{{x}}{{a}}\rceil<{a}\:\:\: \\ $$$${when}\:{a}>\mathrm{1} \\ $$$$\lceil...\rceil\:{is}\:{ceil}\:{function} \\ $$

Question Number 88552 Answers: 0 Comments: 0

let W_1 ,W_2 ,....,W_n be subspaces of a vector space V over a field (F,+,.) prove that: (1) W_1 ∩W_2 ∩....∩W_n a subspace of the vector space V over (F,+,.). (2)W_1 +W_2 +....+W_n is subspace of the vector space V over (F,+,.)

$${let}\:{W}_{\mathrm{1}} ,{W}_{\mathrm{2}} ,....,{W}_{{n}} \:{be}\:{subspaces}\:{of}\:{a}\:{vector} \\ $$$${space}\:{V}\:{over}\:{a}\:{field}\:\left({F},+,.\right) \\ $$$${prove}\:{that}: \\ $$$$\left(\mathrm{1}\right)\:{W}_{\mathrm{1}} \cap{W}_{\mathrm{2}} \cap....\cap{W}_{{n}} \:{a}\:{subspace} \\ $$$${of}\:{the}\:{vector}\:{space}\:{V}\:\:{over}\:\left({F},+,.\right). \\ $$$$\left(\mathrm{2}\right){W}_{\mathrm{1}} +{W}_{\mathrm{2}} +....+{W}_{{n}} \:{is}\:{subspace}\:{of}\:{the} \\ $$$${vector}\:{space}\:{V}\:{over}\:\left({F},+,.\right) \\ $$

Question Number 88547 Answers: 1 Comments: 0

prove for (0<a<2) ∫_0 ^( ∞) ((x^(a−1) dx)/(1+x+x^2 )) = ((2π)/(√3))cos (((2πa+π)/6))cosec πa .

$${prove}\:{for}\:\left(\mathrm{0}<{a}<\mathrm{2}\right) \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \frac{{x}^{{a}−\mathrm{1}} {dx}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\:=\:\frac{\mathrm{2}\pi}{\sqrt{\mathrm{3}}}\mathrm{cos}\:\left(\frac{\mathrm{2}\pi{a}+\pi}{\mathrm{6}}\right)\mathrm{cosec}\:\pi{a}\:. \\ $$

Question Number 88541 Answers: 1 Comments: 5

Question Number 88525 Answers: 1 Comments: 6

Question Number 88507 Answers: 1 Comments: 1

cos^2 12^o +cos^2 24^o +cos^2 48^o +cos^2 84^o

$$\mathrm{cos}\:^{\mathrm{2}} \mathrm{12}^{\mathrm{o}} +\mathrm{cos}\:^{\mathrm{2}} \mathrm{24}^{\mathrm{o}} +\mathrm{cos}\:^{\mathrm{2}} \mathrm{48}^{\mathrm{o}} +\mathrm{cos}\:^{\mathrm{2}} \mathrm{84}^{\mathrm{o}} \\ $$

Question Number 88503 Answers: 0 Comments: 3

(1+cos (π/8))(1+cos ((3π)/8))(1+cos ((5π)/8))(1+cos ((7π)/8))

$$\left(\mathrm{1}+\mathrm{cos}\:\frac{\pi}{\mathrm{8}}\right)\left(\mathrm{1}+\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{8}}\right)\left(\mathrm{1}+\mathrm{cos}\:\frac{\mathrm{5}\pi}{\mathrm{8}}\right)\left(\mathrm{1}+\mathrm{cos}\:\frac{\mathrm{7}\pi}{\mathrm{8}}\right) \\ $$$$ \\ $$

Question Number 88494 Answers: 0 Comments: 2

∫ (√(cos(x))) dx

$$\int\:\sqrt{\mathrm{cos}\left(\mathrm{x}\right)}\:\:\mathrm{dx} \\ $$

Question Number 88492 Answers: 0 Comments: 4

y′′ −4y′+5y = 1+8cos x+e^(2x)

$$\mathrm{y}''\:−\mathrm{4y}'+\mathrm{5y}\:=\:\mathrm{1}+\mathrm{8cos}\:\mathrm{x}+\mathrm{e}^{\mathrm{2x}} \\ $$

Question Number 88491 Answers: 1 Comments: 0

solve cos(x)=k

$$\boldsymbol{{solve}} \\ $$$${cos}\left({x}\right)={k} \\ $$

Question Number 88490 Answers: 0 Comments: 4

∫_1 ^∞ (x^4 /4^x )dx=?

$$\int_{\mathrm{1}} ^{\infty} \:\frac{{x}^{\mathrm{4}} }{\mathrm{4}^{{x}} }{dx}=? \\ $$

Question Number 88487 Answers: 0 Comments: 7

Question Number 88479 Answers: 0 Comments: 0

Consider the transformation f of the plane with all points M wity affix z mapped to the point M ′ with affix z ′ such that z ′=−((√3)+i)z−1+i(1+(√3)) 1) Given M_0 the point z_0 =((√3)/4)+(3/4)i calculate AM_0 and deduce the angle in radians (Taking A as the center of the transformation) 2) Consider the progression with points(M_n )_(n≥0) defined by f(M_n )=M_(n+1) a∙ Show by recurrence that ∀n∈N z_n =2^n e^(ln((7π)/6)) (z_(0 ) −i) Find AM_n then determine the smallest natural number, n, such that AM_n ≥10^2

$${Consider}\:{the}\:{transformation}\:\boldsymbol{{f}}\:{of}\:{the}\:{plane}\:{with}\:{all}\:{points} \\ $$$$\boldsymbol{{M}}\:{wity}\:{affix}\:\boldsymbol{{z}}\:{mapped}\:{to}\:{the}\:{point}\:\boldsymbol{{M}}\:'\:{with}\:{affix}\:\boldsymbol{{z}}\:' \\ $$$${such}\:{that}\:\boldsymbol{{z}}\:'=−\left(\sqrt{\mathrm{3}}+{i}\right){z}−\mathrm{1}+{i}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right) \\ $$$$\left.\mathrm{1}\right)\:{Given}\:\boldsymbol{{M}}_{\mathrm{0}} \:{the}\:{point}\:\boldsymbol{{z}}_{\mathrm{0}} =\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}+\frac{\mathrm{3}}{\mathrm{4}}{i} \\ $$$${calculate}\:\boldsymbol{{AM}}_{\mathrm{0}} \:{and}\:{deduce}\:{the}\:{angle}\:{in}\:{radians} \\ $$$$\left({Taking}\:\boldsymbol{{A}}\:{as}\:{the}\:{center}\:{of}\:{the}\:{transformation}\right) \\ $$$$\left.\mathrm{2}\right)\:{Consider}\:{the}\:{progression}\:{with}\:{points}\left(\boldsymbol{{M}}_{\boldsymbol{{n}}} \right)_{\boldsymbol{{n}}\geqslant\mathrm{0}} \:{defined}\:{by} \\ $$$${f}\left({M}_{{n}} \right)={M}_{{n}+\mathrm{1}} \\ $$$${a}\centerdot\:{Show}\:{by}\:{recurrence}\:{that}\:\forall{n}\in\mathbb{N}\:\boldsymbol{{z}}_{\boldsymbol{{n}}} =\mathrm{2}^{{n}} {e}^{{ln}\frac{\mathrm{7}\pi}{\mathrm{6}}} \:\left({z}_{\mathrm{0}\:} −{i}\right) \\ $$$${Find}\:{AM}_{{n}} \:{then}\:{determine}\:{the}\:{smallest}\:{natural}\:{number},\:{n},\:{such}\:{that} \\ $$$${AM}_{{n}} \geqslant\mathrm{10}^{\mathrm{2}} \\ $$

Question Number 88473 Answers: 0 Comments: 2

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