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Question Number 176168 Answers: 1 Comments: 0

Two fair dice are rolled once. Let X be the random variable representing the sum of the numbers that show up on the two dice. Find X.

$$\mathrm{Two}\:\mathrm{fair}\:\mathrm{dice}\:\mathrm{are}\:\mathrm{rolled}\:\mathrm{once}.\:\mathrm{Let}\:{X} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{random}\:\mathrm{variable}\:\mathrm{representing} \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{that}\:\mathrm{show} \\ $$$$\mathrm{up}\:\mathrm{on}\:\mathrm{the}\:\mathrm{two}\:\mathrm{dice}.\:\mathrm{Find}\:{X}. \\ $$

Question Number 176164 Answers: 1 Comments: 0

Proof that : (√((1 − cos x)/(1 + cos x))) + (√((1 + cos x)/(1 − cos x))) = 2 ∙ cosec x

$$\:{Proof}\:{that}\:: \\ $$$$\: \\ $$$$\:\sqrt{\frac{\mathrm{1}\:−\:\mathrm{cos}\:{x}}{\mathrm{1}\:+\:\mathrm{cos}\:{x}}}\:+\:\sqrt{\frac{\mathrm{1}\:+\:\mathrm{cos}\:{x}}{\mathrm{1}\:−\:\mathrm{cos}\:{x}}}\:=\:\mathrm{2}\:\centerdot\:\mathrm{cosec}\:{x} \\ $$$$\: \\ $$

Question Number 176163 Answers: 0 Comments: 0

Question Number 176160 Answers: 1 Comments: 0

∫(x^3 /(x^4 +x^2 +1))dx

$$\int\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\mathrm{dx} \\ $$

Question Number 176159 Answers: 1 Comments: 0

Question Number 176155 Answers: 2 Comments: 3

Find how many distinct integers are there in this sequence: ⌊((1^2 +1)/(100))⌋, ⌊((2^2 +2)/(100))⌋, ⌊((3^2 +3)/(100))⌋, ..., ⌊((100^2 +100)/(100))⌋ where ⌊x⌋ is the greatest integer that is less than or equal to x

$$\mathrm{Find}\:\mathrm{how}\:\mathrm{many}\:\mathrm{distinct}\:\mathrm{integers}\:\mathrm{are}\:\mathrm{there}\:\mathrm{in}\:\mathrm{this}\:\mathrm{sequence}: \\ $$$$\lfloor\frac{\mathrm{1}^{\mathrm{2}} +\mathrm{1}}{\mathrm{100}}\rfloor,\:\lfloor\frac{\mathrm{2}^{\mathrm{2}} +\mathrm{2}}{\mathrm{100}}\rfloor,\:\lfloor\frac{\mathrm{3}^{\mathrm{2}} +\mathrm{3}}{\mathrm{100}}\rfloor,\:...,\:\lfloor\frac{\mathrm{100}^{\mathrm{2}} +\mathrm{100}}{\mathrm{100}}\rfloor \\ $$$$\mathrm{where}\:\lfloor{x}\rfloor\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{that}\:\mathrm{is}\:\mathrm{less}\:\mathrm{than}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to}\:{x} \\ $$

Question Number 176154 Answers: 1 Comments: 0

lim_(x→0) ((tan x−x)/(x−sin x)) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\mathrm{x}−\mathrm{x}}{\mathrm{x}−\mathrm{sin}\:\mathrm{x}}\:=? \\ $$

Question Number 176147 Answers: 1 Comments: 0

Question Number 176146 Answers: 1 Comments: 0

Question Number 176141 Answers: 1 Comments: 1

which is greater ? 4^(√2) or 3^(√3)

$${which}\:{is}\:{greater}\:? \\ $$$$\mathrm{4}^{\sqrt{\mathrm{2}}} \:{or}\:\mathrm{3}^{\sqrt{\mathrm{3}}} \\ $$

Question Number 176140 Answers: 0 Comments: 0

in AB^Δ C , prove that: sin(2A)+sin(2B)+sin(2C) ≤cos ((A/2)) + cos ((B/2))+cos((C/2))

$$ \\ $$$$\:\:\:{in}\:{A}\overset{\Delta} {{B}C}\:,\:{prove}\:{that}: \\ $$$$\:\:{sin}\left(\mathrm{2}{A}\right)+{sin}\left(\mathrm{2}{B}\right)+{sin}\left(\mathrm{2}{C}\right)\: \\ $$$$\leqslant{cos}\:\left(\frac{{A}}{\mathrm{2}}\right)\:+\:{cos}\:\left(\frac{{B}}{\mathrm{2}}\right)+{cos}\left(\frac{{C}}{\mathrm{2}}\right) \\ $$$$ \\ $$

Question Number 176134 Answers: 0 Comments: 0

Question Number 176133 Answers: 1 Comments: 0

Ω = ∫ ((−7x^2 +3x−8)/(4x^3 +8x^2 −20x−24)) dx

$$\:\:\Omega\:=\:\int\:\frac{−\mathrm{7x}^{\mathrm{2}} +\mathrm{3x}−\mathrm{8}}{\mathrm{4x}^{\mathrm{3}} +\mathrm{8x}^{\mathrm{2}} −\mathrm{20x}−\mathrm{24}}\:\mathrm{dx} \\ $$

Question Number 176132 Answers: 0 Comments: 0

Question Number 176128 Answers: 0 Comments: 1

In △ABC , cot (A/2) , cot (B/2) , cot (C/2) ∈ Q Prove that: (Π_(cyc) sin (A/2))^n + (Π_(cyc) cos (A/2))^n ∈ Q , ∀n∈N

$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:,\:\mathrm{cot}\:\frac{\mathrm{A}}{\mathrm{2}}\:,\:\mathrm{cot}\:\frac{\mathrm{B}}{\mathrm{2}}\:,\:\mathrm{cot}\:\frac{\mathrm{C}}{\mathrm{2}}\:\in\:\mathrm{Q} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\left(\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}\right)^{\boldsymbol{\mathrm{n}}} +\:\left(\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\mathrm{cos}\:\frac{\mathrm{A}}{\mathrm{2}}\right)^{\boldsymbol{\mathrm{n}}} \in\:\mathrm{Q}\:,\:\forall\mathrm{n}\in\mathbb{N} \\ $$

Question Number 176123 Answers: 0 Comments: 0

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Question Number 176121 Answers: 0 Comments: 0

Question Number 176118 Answers: 0 Comments: 1

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Question Number 176102 Answers: 2 Comments: 0

Prove that 4n<2^n for all n≥5

$$\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\: \\ $$$$\:\:\:\:\:\mathrm{4}\boldsymbol{\mathrm{n}}<\mathrm{2}^{\boldsymbol{\mathrm{n}}} \:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{all}}\:\boldsymbol{\mathrm{n}}\geqslant\mathrm{5} \\ $$

Question Number 176101 Answers: 0 Comments: 6

x^2 − 4y^2 = 3z^2 [ x, y, z ∈ N ] HCF(x, y, z) = 1 like (4, 1, 2) x, y, z < 100 solve by computer programing

$${x}^{\mathrm{2}} \:−\:\mathrm{4}{y}^{\mathrm{2}} \:=\:\mathrm{3}{z}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\left[\:{x},\:{y},\:{z}\:\in\:\mathbb{N}\:\right]\: \\ $$$$\:\:\:\:\:\:\:\:\:\:{HCF}\left({x},\:{y},\:{z}\right)\:=\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{like}\:\left(\mathrm{4},\:\mathrm{1},\:\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x},\:{y},\:{z}\:<\:\mathrm{100} \\ $$$${solve}\:{by}\:{computer}\:{programing} \\ $$

Question Number 176100 Answers: 2 Comments: 0

Q : in AB^Δ C : A = 90^o and , m_( b) ^2 + m_( c) ^( 2) = 100. find the value of ” a ” . note : ⟨ m_( a) := the median corresponding to ” A ” . −−− m.n−−−

$$ \\ $$$$\:\:\:\:\:{Q}\:: \\ $$$$\:\:\:\:\:{in}\:\:{A}\overset{\Delta} {{B}C}\::\:{A}\:=\:\mathrm{90}^{\mathrm{o}} \:\:\:{and}\:,\:{m}_{\:{b}} ^{\mathrm{2}} \:+\:{m}_{\:{c}} ^{\:\mathrm{2}} \:=\:\mathrm{100}.\:\:\:\:\:\:\:\:\: \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{find}\:\:{the}\:\:{value}\:{of}\:\:\:''\:\:\:{a}\:\:\:''\:. \\ $$$$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:{note}\::\:\:\langle\:{m}_{\:{a}} \::=\:{the}\:{median}\:{corresponding}\:\:{to}\:''\:{A}\:''\:. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:−−−\:\:\boldsymbol{{m}}.\boldsymbol{{n}}−−− \\ $$

Question Number 176098 Answers: 1 Comments: 0

The deviations of a set of numbers from 12 are 3, −2, 1, 0, −1, 4, 0, 1 and 2. Calculate the mean and standard deviation of the numbers.

$$\mathrm{The}\:\mathrm{deviations}\:\mathrm{of}\:\mathrm{a}\:\mathrm{set}\:\mathrm{of}\:\mathrm{numbers} \\ $$$$\mathrm{from}\:\mathrm{12}\:\mathrm{are}\: \\ $$$$\:\:\:\:\:\mathrm{3},\:−\mathrm{2},\:\mathrm{1},\:\mathrm{0},\:−\mathrm{1},\:\mathrm{4},\:\mathrm{0},\:\mathrm{1}\:\mathrm{and}\:\mathrm{2}. \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{and}\:\mathrm{standard}\: \\ $$$$\mathrm{deviation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{numbers}. \\ $$

Question Number 176095 Answers: 0 Comments: 1

In △ABC the following relationship holds: an_a ≥ (s (b + c) − 2bc) cos (A/2)

$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:\:\mathrm{the}\:\mathrm{following}\:\mathrm{relationship}\:\mathrm{holds}: \\ $$$$\mathrm{an}_{\boldsymbol{\mathrm{a}}} \:\geqslant\:\left(\mathrm{s}\:\left(\mathrm{b}\:+\:\mathrm{c}\right)\:−\:\mathrm{2bc}\right)\:\mathrm{cos}\:\frac{\mathrm{A}}{\mathrm{2}} \\ $$

Question Number 176094 Answers: 0 Comments: 0

Question Number 176092 Answers: 0 Comments: 3

∠AOB triangle equilareral de cote a (A;B) : centres de cercles de rayon r 𝛉 =∡ GOH EF=a_0 Determiner l aire de l espace delimite par AFOEBHG comme il est marque sur l image ci−joint en fonction de: a,r,a_0 et 𝛉

$$\angle{AOB}\:\:\:{triangle}\:{equilareral}\:{de}\:{cote}\:\boldsymbol{{a}} \\ $$$$\left({A};\mathrm{B}\right)\::\:\mathrm{c}{e}\mathrm{ntres}\:\mathrm{de}\:\mathrm{cercles}\:\mathrm{de}\:\mathrm{rayon}\:\boldsymbol{\mathrm{r}} \\ $$$$\boldsymbol{\theta}\:=\measuredangle\:\:{GOH}\:\:\:\mathrm{EF}=\boldsymbol{{a}}_{\mathrm{0}} \\ $$$${Determiner}\:{l}\:{aire}\:{de}\:{l}\:{espace}\:{delimite}\:{par} \\ $$$$\:\:\boldsymbol{\mathrm{AFOEBHG}}\:{comme}\:{il}\:{est}\:{marque}\:{sur}\:{l}\:{image}\:{ci}−{joint}\: \\ $$$${en}\:{fonction}\:{de}:\:\boldsymbol{{a}},\boldsymbol{{r}},\boldsymbol{{a}}_{\mathrm{0}} \:{et}\:\boldsymbol{\theta} \\ $$

Question Number 176086 Answers: 0 Comments: 1

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