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Question Number 117082 Answers: 1 Comments: 0

If 6−3x is the geometric mean between the integer x^2 +2 and 2^ what are the values of x ?

$$\mathrm{If}\:\mathrm{6}−\mathrm{3x}\:\mathrm{is}\:\mathrm{the}\:\mathrm{geometric}\:\mathrm{mean}\:\mathrm{between} \\ $$$$\mathrm{the}\:\mathrm{integer}\:\mathrm{x}^{\mathrm{2}} +\mathrm{2}\:\mathrm{and}\:\bar {\mathrm{2}}\:\mathrm{what}\:\mathrm{are}\:\mathrm{the}\: \\ $$$$\mathrm{values}\:\mathrm{of}\:\mathrm{x}\:? \\ $$

Question Number 117078 Answers: 1 Comments: 0

lim_(n→∞) [ tan ((π/(2n))).tan (((2π)/(2n))).tan (((3π)/(2n)))...tan (((nπ)/(2n)))]^(1/n) =?

$$\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left[\:\mathrm{tan}\:\left(\frac{\pi}{\mathrm{2n}}\right).\mathrm{tan}\:\left(\frac{\mathrm{2}\pi}{\mathrm{2n}}\right).\mathrm{tan}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2n}}\right)...\mathrm{tan}\:\left(\frac{\mathrm{n}\pi}{\mathrm{2n}}\right)\right]^{\frac{\mathrm{1}}{\mathrm{n}}} =?\: \\ $$

Question Number 117066 Answers: 1 Comments: 0

Given f(x)= ((x−1)/(x+1)) . If f^2 (x)=f(f(x)), f^3 (x)=f(f(f(x))) , f^(1998) (x) = g(x) then ∫_(1/e) ^1 g(x) dx = _?

$$\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)=\:\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}+\mathrm{1}}\:.\:\mathrm{If}\:\mathrm{f}^{\mathrm{2}} \left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right),\: \\ $$$$\mathrm{f}^{\mathrm{3}} \left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\right)\:,\:\mathrm{f}^{\mathrm{1998}} \left(\mathrm{x}\right)\:=\:\mathrm{g}\left(\mathrm{x}\right) \\ $$$$\mathrm{then}\:\int_{\frac{\mathrm{1}}{\mathrm{e}}} ^{\mathrm{1}} \mathrm{g}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\_? \\ $$

Question Number 117061 Answers: 2 Comments: 3

How many 5-digits positive integers x_1 x_2 x_3 x_4 x_5 are there such that x_1 ≤x_2 ≤x_3 ≤x_4 ≤x_5 ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{5}-\mathrm{digits}\:\mathrm{positive}\:\mathrm{integers} \\ $$$${x}_{\mathrm{1}} {x}_{\mathrm{2}} {x}_{\mathrm{3}} {x}_{\mathrm{4}} {x}_{\mathrm{5}} \:\mathrm{are}\:\mathrm{there}\:\mathrm{such}\:\mathrm{that}\: \\ $$$${x}_{\mathrm{1}} \leqslant{x}_{\mathrm{2}} \leqslant{x}_{\mathrm{3}} \leqslant{x}_{\mathrm{4}} \leqslant{x}_{\mathrm{5}} ? \\ $$

Question Number 117053 Answers: 4 Comments: 0

∫ (x/( (√(2−x^4 )))) dx =?

$$\:\:\:\int\:\frac{\mathrm{x}}{\:\sqrt{\mathrm{2}−\mathrm{x}^{\mathrm{4}} }}\:\mathrm{dx}\:=? \\ $$

Question Number 117052 Answers: 1 Comments: 0

lim_(x→∞) (√(4x+(√(4x+(√(4x+(√(4x)))))))) − (√(4x)) =?

$$\:\:\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{4x}+\sqrt{\mathrm{4x}+\sqrt{\mathrm{4x}+\sqrt{\mathrm{4x}}}}}\:−\:\sqrt{\mathrm{4x}}\:=? \\ $$

Question Number 117121 Answers: 2 Comments: 0

...nice mathematics.. please evaluate... Ω =∫_(−∞) ^( +∞) (((x^2 −4)/(x^2 +4))∗ ((sin(2x))/x)) dx =??? m.n.1970

$$\:\:\:\:\:\:\:\:...{nice}\:\:{mathematics}.. \\ $$$$ \\ $$$$\:\:{please}\:\:{evaluate}... \\ $$$$\: \\ $$$$\:\Omega\:=\int_{−\infty} ^{\:+\infty} \left(\frac{{x}^{\mathrm{2}} −\mathrm{4}}{{x}^{\mathrm{2}} +\mathrm{4}}\ast\:\frac{{sin}\left(\mathrm{2}{x}\right)}{{x}}\right)\:{dx}\:=???\:\: \\ $$$$\:{m}.{n}.\mathrm{1970} \\ $$$$ \\ $$

Question Number 117046 Answers: 3 Comments: 0

Calculate without using caculator: a)−2(√2)sin10°(2sin35°−((sec5°)/2)−((cos40°)/(sin5°))) b)sin6°−sin42°−sin66°+sin78°

$$\mathrm{Calculate}\:\mathrm{without}\:\mathrm{using}\:\mathrm{caculator}: \\ $$$$\left.\mathrm{a}\right)−\mathrm{2}\sqrt{\mathrm{2}}\mathrm{sin10}°\left(\mathrm{2sin35}°−\frac{\mathrm{sec5}°}{\mathrm{2}}−\frac{\mathrm{cos40}°}{\mathrm{sin5}°}\right) \\ $$$$\left.\mathrm{b}\right)\mathrm{sin6}°−\mathrm{sin42}°−\mathrm{sin66}°+\mathrm{sin78}° \\ $$

Question Number 117045 Answers: 4 Comments: 0

If x is a complex number satisfying x^2 +x+1 = 0 , what is the value of x^(53) +x^(52) +x^(51) +x^(50) +x^(49) ?

$$\mathrm{If}\:\mathrm{x}\:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{number}\:\mathrm{satisfying}\: \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\:=\:\mathrm{0}\:,\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{x}^{\mathrm{53}} +\mathrm{x}^{\mathrm{52}} +\mathrm{x}^{\mathrm{51}} +\mathrm{x}^{\mathrm{50}} +\mathrm{x}^{\mathrm{49}} \:? \\ $$

Question Number 117037 Answers: 3 Comments: 0

lim_(x→∞) (√(x+(√(x+(√x))))) − (√x) =?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}}}}\:−\:\sqrt{\mathrm{x}}\:=?\: \\ $$

Question Number 117034 Answers: 1 Comments: 0

∫(dx/( ((1+x^3 ))^(1/3) ))

$$\int\frac{\boldsymbol{\mathrm{dx}}}{\:\sqrt[{\mathrm{3}}]{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{3}} }} \\ $$

Question Number 117029 Answers: 1 Comments: 0

(x^2 +2y^2 ) dx + (4xy−y^2 ) dy = 0

$$\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2y}^{\mathrm{2}} \right)\:\mathrm{dx}\:+\:\left(\mathrm{4xy}−\mathrm{y}^{\mathrm{2}} \right)\:\mathrm{dy}\:=\:\mathrm{0} \\ $$

Question Number 117027 Answers: 2 Comments: 0

solve: ((x − 4)/(x − 3)) < ((2x − 1)/2)

$$\mathrm{solve}:\:\:\:\:\frac{\mathrm{x}\:\:−\:\:\mathrm{4}}{\mathrm{x}\:\:−\:\:\mathrm{3}}\:\:\:<\:\:\:\frac{\mathrm{2x}\:\:−\:\:\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 117026 Answers: 2 Comments: 0

((√(1−x))/( (√x))) + ((√x)/( (√(1−x)))) = ((13)/6)

$$\:\frac{\sqrt{\mathrm{1}−\mathrm{x}}}{\:\sqrt{\mathrm{x}}}\:+\:\frac{\sqrt{\mathrm{x}}}{\:\sqrt{\mathrm{1}−\mathrm{x}}}\:=\:\frac{\mathrm{13}}{\mathrm{6}} \\ $$

Question Number 117010 Answers: 0 Comments: 0

Question Number 117007 Answers: 1 Comments: 0

find unit digit of 1!+2!+3!+4!+5!+....+100!

$${find}\:{unit}\:{digit}\:{of} \\ $$$$\mathrm{1}!+\mathrm{2}!+\mathrm{3}!+\mathrm{4}!+\mathrm{5}!+....+\mathrm{100}! \\ $$

Question Number 117006 Answers: 2 Comments: 1

∫_0 ^(100π) ∣sinx∣ dx

$$\int_{\mathrm{0}} ^{\mathrm{100}\pi} \mid{sinx}\mid\:{dx} \\ $$

Question Number 117002 Answers: 1 Comments: 0

∫_0 ^(nπ+v) ∣sinx∣ dx

$$\int_{\mathrm{0}} ^{{n}\pi+{v}} \mid{sinx}\mid\:{dx} \\ $$

Question Number 116999 Answers: 3 Comments: 0

∫(dx/((a+bcosx)^2 ))

$$\int\frac{{dx}}{\left({a}+{bcosx}\right)^{\mathrm{2}} } \\ $$

Question Number 116998 Answers: 2 Comments: 2

∫_((−π)/2) ^(π/2) ((sin^2 x)/(1+2^x ))dx

$$\int_{\frac{−\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{sin}^{\mathrm{2}} {x}}{\mathrm{1}+\mathrm{2}^{{x}} }{dx} \\ $$

Question Number 116997 Answers: 1 Comments: 0

Question Number 116996 Answers: 2 Comments: 0

Question Number 116987 Answers: 1 Comments: 0

Find the largest integer smaller than (7+4(√3))^3 .

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{integer}\:\mathrm{smaller}\:\mathrm{than} \\ $$$$\left(\mathrm{7}+\mathrm{4}\sqrt{\mathrm{3}}\right)^{\mathrm{3}} . \\ $$

Question Number 116983 Answers: 1 Comments: 5

Luigi, an Italian cook, drops a spaghetti which breaks into three pieces. What is the probability of making a triangle with the three pieces ?

$$\mathrm{Luigi},\:\mathrm{an}\:\mathrm{Italian}\:\mathrm{cook},\:\mathrm{drops}\:\mathrm{a}\:\mathrm{spaghetti} \\ $$$$\mathrm{which}\:\mathrm{breaks}\:\mathrm{into}\:\mathrm{three}\:\mathrm{pieces}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{making}\:\mathrm{a} \\ $$$$\mathrm{triangle}\:\mathrm{with}\:\mathrm{the}\:\mathrm{three}\:\mathrm{pieces}\:? \\ $$

Question Number 116976 Answers: 2 Comments: 0

use the formula P=Ie^(kt) ,where P is resulting population ,I is the initial population and t is measured in hours. A bacterial culture has an initial population of 10,000. If its declines to 5000 in 6 hours , what will it be at the end of 8 hours? (a) 1985 (b) 3969 (c) 2500 (d) 4353

$$\mathrm{use}\:\mathrm{the}\:\mathrm{formula}\:\mathrm{P}=\mathrm{Ie}^{\mathrm{kt}} \:,\mathrm{where}\:\mathrm{P}\:\mathrm{is}\:\mathrm{resulting} \\ $$$$\mathrm{population}\:,\mathrm{I}\:\mathrm{is}\:\mathrm{the}\:\mathrm{initial}\:\mathrm{population}\:\mathrm{and}\:\mathrm{t}\:\mathrm{is} \\ $$$$\mathrm{measured}\:\mathrm{in}\:\mathrm{hours}.\:\mathrm{A}\:\mathrm{bacterial}\:\mathrm{culture} \\ $$$$\mathrm{has}\:\mathrm{an}\:\mathrm{initial}\:\mathrm{population}\:\mathrm{of}\:\mathrm{10},\mathrm{000}.\:\mathrm{If} \\ $$$$\mathrm{its}\:\mathrm{declines}\:\mathrm{to}\:\mathrm{5000}\:\mathrm{in}\:\mathrm{6}\:\mathrm{hours}\:,\:\mathrm{what}\: \\ $$$$\mathrm{will}\:\mathrm{it}\:\mathrm{be}\:\mathrm{at}\:\mathrm{the}\:\mathrm{end}\:\mathrm{of}\:\mathrm{8}\:\mathrm{hours}? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{1985}\:\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{3969}\:\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{2500}\:\:\:\:\left(\mathrm{d}\right)\:\mathrm{4353} \\ $$

Question Number 116975 Answers: 0 Comments: 1

Prove ((cos(A−B))/(sin(A+B)))=((1+tan Atan B)/(tan Atan B))

$$\mathrm{Prove}\:\frac{\mathrm{cos}\left({A}−{B}\right)}{\mathrm{sin}\left({A}+{B}\right)}=\frac{\mathrm{1}+\mathrm{tan}\:{A}\mathrm{tan}\:{B}}{\mathrm{tan}\:{A}\mathrm{tan}\:{B}} \\ $$

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