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Question Number 190848    Answers: 0   Comments: 0

Question Number 190845    Answers: 1   Comments: 0

Question Number 190841    Answers: 1   Comments: 0

Question Number 190851    Answers: 1   Comments: 0

Question Number 190824    Answers: 0   Comments: 2

lim_(x→(π/4)) ((sec^2 x−2(√(tan x)))/(2cos x−2sin x)) =?

$$\:\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{4}}} {\mathrm{lim}}\:\frac{\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}−\mathrm{2}\sqrt{\mathrm{tan}\:\mathrm{x}}}{\mathrm{2cos}\:\mathrm{x}−\mathrm{2sin}\:\mathrm{x}}\:=? \\ $$

Question Number 190823    Answers: 0   Comments: 0

Question Number 190820    Answers: 1   Comments: 0

Question Number 190819    Answers: 1   Comments: 0

Question Number 190818    Answers: 0   Comments: 5

It is known that the set A={1 , 2, 3, ..., 100} The numbers of subsets of A which when added together are divisible by 4

$$\:\:\mathrm{It}\:\mathrm{is}\:\mathrm{known}\:\mathrm{that}\:\mathrm{the}\:\mathrm{set}\:\mathrm{A}=\left\{\mathrm{1}\:,\:\mathrm{2},\:\mathrm{3},\:...,\:\mathrm{100}\right\} \\ $$$$\:\mathrm{The}\:\mathrm{numbers}\:\mathrm{of}\:\mathrm{subsets}\:\mathrm{of}\:\mathrm{A}\:\mathrm{which}\:\mathrm{when}\: \\ $$$$\:\mathrm{added}\:\mathrm{together}\:\mathrm{are}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{4} \\ $$

Question Number 190814    Answers: 1   Comments: 0

Question Number 190813    Answers: 1   Comments: 0

Question Number 190812    Answers: 1   Comments: 1

Question Number 190811    Answers: 1   Comments: 0

If the angle between the vectors c =ai+2j and d=3i+j is 45° , find the two possible values of a

$$\:{If}\:{the}\:{angle}\:{between}\:{the}\:{vectors} \\ $$$$\:{c}\:={ai}+\mathrm{2}{j}\:{and}\:\:{d}=\mathrm{3}{i}+{j}\:{is}\:\mathrm{45}°\:,\:{find} \\ $$$$\:{the}\:{two}\:{possible}\:{values}\:{of}\:{a} \\ $$

Question Number 190809    Answers: 0   Comments: 2

Question Number 190802    Answers: 1   Comments: 0

log_x x=x^(5x−10) x=?

$${log}_{{x}} {x}={x}^{\mathrm{5}{x}−\mathrm{10}} \:\:\:\:\:{x}=? \\ $$

Question Number 190800    Answers: 0   Comments: 0

(d^n /da^n )𝚪(a+1)=?

$$\frac{\boldsymbol{\mathrm{d}}^{\boldsymbol{\mathrm{n}}} }{\boldsymbol{\mathrm{da}}^{\boldsymbol{\mathrm{n}}} }\boldsymbol{\Gamma}\left(\boldsymbol{{a}}+\mathrm{1}\right)=? \\ $$

Question Number 190793    Answers: 1   Comments: 1

A projectile of mass M explodes at thee highst point of its trajectory when it hase vlocity . The horizontal distance travelede btween launch and explosion is x_0 . Two fragments are produced with initiale velocitis parallel to the ground. They thenfollow their trajectories until they hitt he ground. The fragment of mass m_1 retuns exactly to the launch point of thei orginal projectile (of mass M) while thee othr fragment of mass m_2 hits the grounda t a distance D from this point. Disregardn iteraction with air and assume that massa ws conserved in the explosion (m_1 +m_2 =M) Determine the magnitude of the velocity of fragment 2 just before it hits theground. (a) ((gx_0 )/v) (b)(√((25)/9))v (c) (√(((25)/9)v^2 +(((gx_0 )/5))2)) (d)(√((5/3)x_0 v^2 +(((gx_0 )/v))2))

$$ \\ $$$$\mathrm{A}\:\mathrm{projectile}\:\mathrm{of}\:\mathrm{mass}\:\boldsymbol{{M}}\:\mathrm{explodes}\:\mathrm{at}\:\mathrm{thee} \\ $$$$\mathrm{highst}\:\mathrm{point}\:\mathrm{of}\:\mathrm{its}\:\mathrm{trajectory}\:\mathrm{when}\:\mathrm{it}\:\mathrm{hase} \\ $$$$\mathrm{vlocity}\:.\:\mathrm{The}\:\mathrm{horizontal}\:\mathrm{distance}\:\mathrm{travelede} \\ $$$$\mathrm{btween}\:\mathrm{launch}\:\mathrm{and}\:\mathrm{explosion}\:\mathrm{is}\:\boldsymbol{{x}}_{\mathrm{0}} \:.\:\mathrm{Two} \\ $$$$\mathrm{fragments}\:\mathrm{are}\:\mathrm{produced}\:\mathrm{with}\:\mathrm{initiale} \\ $$$$\mathrm{velocitis}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:\mathrm{ground}.\:\mathrm{They}\: \\ $$$$\mathrm{thenfollow}\:\mathrm{their}\:\mathrm{trajectories}\:\mathrm{until}\:\mathrm{they}\:\mathrm{hitt} \\ $$$$\mathrm{he}\:\mathrm{ground}.\:\mathrm{The}\:\mathrm{fragment}\:\mathrm{of}\:\mathrm{mass}\:\boldsymbol{{m}}_{\mathrm{1}} \:\mathrm{retuns}\:\mathrm{exactly}\:\mathrm{to}\:\mathrm{the}\:\mathrm{launch}\:\mathrm{point}\:\mathrm{of}\:\mathrm{thei} \\ $$$$\mathrm{orginal}\:\mathrm{projectile}\:\left(\mathrm{of}\:\mathrm{mass}\:\mathrm{M}\right)\:\mathrm{while}\:\mathrm{thee} \\ $$$$\mathrm{othr}\:\mathrm{fragment}\:\mathrm{of}\:\mathrm{mass}\:\boldsymbol{{m}}_{\mathrm{2}} \:\mathrm{hits}\:\mathrm{the}\:\mathrm{grounda} \\ $$$$\mathrm{t}\:\mathrm{a}\:\mathrm{distance}\:\boldsymbol{{D}}\:\mathrm{from}\:\mathrm{this}\:\mathrm{point}.\:\mathrm{Disregardn} \\ $$$$\mathrm{iteraction}\:\mathrm{with}\:\mathrm{air}\:\mathrm{and}\:\mathrm{assume}\:\mathrm{that}\:\mathrm{massa} \\ $$$$\mathrm{ws}\:\mathrm{conserved}\:\mathrm{in}\:\mathrm{the}\:\mathrm{explosion}\:\left(\boldsymbol{{m}}_{\mathrm{1}} +\boldsymbol{{m}}_{\mathrm{2}} =\mathrm{M}\right)\:\mathrm{Determine}\:\mathrm{the}\:\mathrm{magnitude}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{velocity}\:\mathrm{of}\:\mathrm{fragment}\:\mathrm{2}\:\mathrm{just}\:\mathrm{before}\:\mathrm{it}\:\mathrm{hits}\:\mathrm{theground}. \\ $$$$\left(\mathrm{a}\right)\:\frac{\mathrm{g}{x}_{\mathrm{0}} }{{v}} \\ $$$$\left(\mathrm{b}\right)\sqrt{\frac{\mathrm{25}}{\mathrm{9}}}{v} \\ $$$$\left({c}\right)\:\sqrt{\frac{\mathrm{25}}{\mathrm{9}}{v}^{\mathrm{2}} +\left(\frac{{gx}_{\mathrm{0}} }{\mathrm{5}}\right)\mathrm{2}} \\ $$$$\left({d}\right)\sqrt{\frac{\mathrm{5}}{\mathrm{3}}{x}_{\mathrm{0}} {v}^{\mathrm{2}} +\left(\frac{{gx}_{\mathrm{0}} }{{v}}\right)\mathrm{2}} \\ $$

Question Number 190790    Answers: 0   Comments: 1

Question Number 190788    Answers: 1   Comments: 0

calculate Ω= Σ_(k=0) ^n (( 1)/((n−k)!.(n+k )!))

$$ \\ $$$$\:\:\:\:\mathrm{calculate}\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\Omega=\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\:\mathrm{1}}{\left({n}−{k}\right)!.\left({n}+{k}\:\right)!} \\ $$$$ \\ $$

Question Number 190784    Answers: 0   Comments: 0

If , 0 ⇢ M′ ⇢^f M⇢^g M′′⇢0 is a short exact sequence and M′ , M′′ are two finitely generated R −modules then prove M is finitely generated. Hint: f , g are two R − homomorphism.

$$ \\ $$$$\:\:\mathrm{If}\:,\:\mathrm{0}\:\dashrightarrow\:\mathrm{M}'\:\overset{{f}} {\dashrightarrow}\mathrm{M}\overset{{g}} {\dashrightarrow}\mathrm{M}''\dashrightarrow\mathrm{0}\:\mathrm{is} \\ $$$$\:\:\:\:\mathrm{a}\:\mathrm{short}\:\mathrm{exact}\:\mathrm{sequence}\:\:\mathrm{and}\:\:\mathrm{M}'\:,\:\mathrm{M}''\:\mathrm{are} \\ $$$$\:\:\:\mathrm{two}\:\:\mathrm{finitely}\:\mathrm{generated}\:\:\mathrm{R}\:−\mathrm{modules} \\ $$$$\:\:\:\mathrm{then}\:\:\mathrm{prove}\:\:\:\mathrm{M}\:\:\mathrm{is}\:\:\mathrm{finitely}\:\mathrm{generated}.\: \\ $$$$\:\:\:\mathrm{Hint}:\:\:{f}\:\:,\:\:{g}\:\:\:{are}\:\:{two}\:\:\:{R}\:−\:{homomorphism}. \\ $$

Question Number 190781    Answers: 2   Comments: 1

Question Number 190779    Answers: 1   Comments: 0

Re((1/(1−a)))^(a−1)

$$\boldsymbol{\mathrm{R}{e}}\left(\frac{\mathrm{1}}{\mathrm{1}−\boldsymbol{{a}}}\right)^{\boldsymbol{{a}}−\mathrm{1}} \\ $$

Question Number 190774    Answers: 0   Comments: 0

Question Number 190769    Answers: 0   Comments: 0

Question Number 190765    Answers: 1   Comments: 0

Question Number 190760    Answers: 1   Comments: 0

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