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Prove using the density of Q in R that every real number x is the limit of a cauchy sequence of rational numbers (r_n )_(n∈N) . Give a sequence of irrational numbers (S_n ) such that S_n → x.

$$\mathrm{Prove}\:\mathrm{using}\:\mathrm{the}\:\mathrm{density}\:\mathrm{of}\:\boldsymbol{\mathrm{Q}}\:\mathrm{in}\:\mathbb{R}\:\mathrm{that}\:\mathrm{every}\:\mathrm{real}\:\mathrm{number}\:\mathrm{x}\:\mathrm{is}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{of} \\ $$$$\mathrm{a}\:\mathrm{cauchy}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{rational}\:\mathrm{numbers}\:\left(\mathrm{r}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathrm{N}} .\:\mathrm{Give}\:\mathrm{a}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{irrational}\: \\ $$$$\mathrm{numbers}\:\left(\mathrm{S}_{\mathrm{n}} \right)\:\mathrm{such}\:\mathrm{that}\:\mathrm{S}_{\mathrm{n}} \:\rightarrow\:\mathrm{x}. \\ $$

Question Number 11770 Answers: 1 Comments: 0

ax^3 +bx^2 +cx+d=0 pls.solve

$${ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0}\:{pls}.{solve} \\ $$

Question Number 11769 Answers: 0 Comments: 0

ax^3 +bx^2 +cx+d=0 solve

$${ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0}\:{solve} \\ $$

Question Number 11768 Answers: 0 Comments: 0

ax^3 +bx^2 +cx+d=0 solve it.

$${ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0}\:{solve}\:{it}. \\ $$

Question Number 11766 Answers: 1 Comments: 0

∫(1/(sin2x))dx=?

$$\int\frac{\mathrm{1}}{\mathrm{sin2x}}\mathrm{dx}=? \\ $$

Question Number 11764 Answers: 1 Comments: 0

if x y and z are solution of ((x + xy)/(x + y + 1)) = 2 ((2x + xz)/(x + z +2)) = 3 ((2 + 2y+ z + yz)/(y + z +3)) = 4 so, the value of (1/x) + (1/(y + 1)) + (1/(z + 2 )) = ....?

$$\mathrm{if}\:\mathrm{x}\:\mathrm{y}\:\mathrm{and}\:\mathrm{z}\:\mathrm{are}\:\mathrm{solution}\:\mathrm{of} \\ $$$$\frac{\mathrm{x}\:+\:\mathrm{xy}}{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{1}}\:=\:\mathrm{2} \\ $$$$\frac{\mathrm{2x}\:+\:\mathrm{xz}}{\mathrm{x}\:+\:\mathrm{z}\:+\mathrm{2}}\:=\:\mathrm{3} \\ $$$$\frac{\mathrm{2}\:+\:\mathrm{2y}+\:\mathrm{z}\:+\:\mathrm{yz}}{\mathrm{y}\:+\:\mathrm{z}\:+\mathrm{3}}\:=\:\mathrm{4} \\ $$$$\mathrm{so},\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{1}}{\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{y}\:+\:\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{z}\:+\:\mathrm{2}\:}\:=\:....? \\ $$$$ \\ $$

Question Number 11759 Answers: 1 Comments: 0

A quadrilateral with consecutive sides of lenght 7, 15, 15, and d is inscribed in a circle with its diameter d. Find the radius of circle

$$\mathrm{A}\:\mathrm{quadrilateral}\:\mathrm{with}\:\mathrm{consecutive}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{lenght}\:\mathrm{7},\:\mathrm{15},\:\mathrm{15},\:\mathrm{and}\:{d} \\ $$$$\mathrm{is}\:\mathrm{inscribed}\:\mathrm{in}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{its}\:\mathrm{diameter}\:{d}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{circle} \\ $$

Question Number 11791 Answers: 1 Comments: 0

What is the velocity of sound at 100°C if the velocity of sound at 0°C is 340 m/s

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{sound}\:\mathrm{at}\:\mathrm{100}°\mathrm{C}\:\:\mathrm{if}\:\mathrm{the}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{sound}\:\mathrm{at}\:\mathrm{0}°\mathrm{C}\:\mathrm{is}\:\mathrm{340}\:\mathrm{m}/\mathrm{s} \\ $$

Question Number 11754 Answers: 1 Comments: 0

How many numbers between 1 − 2017 that aren′t divisible by 5, 6, 7, 8 ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{numbers}\:\mathrm{between}\:\mathrm{1}\:−\:\mathrm{2017} \\ $$$$\mathrm{that}\:\mathrm{aren}'\mathrm{t}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{5},\:\mathrm{6},\:\mathrm{7},\:\mathrm{8}\:? \\ $$

Question Number 11753 Answers: 2 Comments: 0

a, b, c are the roots from equation x^3 − 5x^2 − 9x + 10 = 0 If P(x) = Ax^3 + Bx^2 + Cx − 2015 and P(a) = b + c P(b) = a + c P(c) = a + b What is the value of A + B + C ?

$${a},\:{b},\:{c}\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{from}\:\mathrm{equation}\:{x}^{\mathrm{3}} \:−\:\mathrm{5}{x}^{\mathrm{2}} \:−\:\mathrm{9}{x}\:+\:\mathrm{10}\:=\:\mathrm{0} \\ $$$$\mathrm{If}\:{P}\left({x}\right)\:=\:{Ax}^{\mathrm{3}} \:+\:{Bx}^{\mathrm{2}} \:+\:{Cx}\:−\:\mathrm{2015}\:\mathrm{and} \\ $$$${P}\left({a}\right)\:=\:{b}\:+\:{c}\: \\ $$$${P}\left({b}\right)\:=\:{a}\:+\:{c} \\ $$$${P}\left({c}\right)\:=\:{a}\:+\:{b} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{A}\:+\:{B}\:+\:{C}\:? \\ $$

Question Number 11751 Answers: 1 Comments: 0

Simplify: sin(6θ)

$$\mathrm{Simplify}:\:\mathrm{sin}\left(\mathrm{6}\theta\right) \\ $$

Question Number 11748 Answers: 1 Comments: 0

Solve simultaneously x^2 + 4y^2 + xy = 6 .............. equation (i) 3x^2 + 8y^2 = 14 .............. equation (ii)

$$\mathrm{Solve}\:\mathrm{simultaneously} \\ $$$$\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{4y}^{\mathrm{2}} \:+\:\mathrm{xy}\:=\:\mathrm{6}\:\:\:\:\:..............\:\mathrm{equation}\:\left(\mathrm{i}\right) \\ $$$$\mathrm{3x}^{\mathrm{2}} \:+\:\mathrm{8y}^{\mathrm{2}} \:=\:\mathrm{14}\:\:\:\:\:\:\:\:\:\:\:..............\:\mathrm{equation}\:\left(\mathrm{ii}\right) \\ $$

Question Number 11743 Answers: 1 Comments: 0

9/3(6×4/8)=?

$$\mathrm{9}/\mathrm{3}\left(\mathrm{6}×\mathrm{4}/\mathrm{8}\right)=? \\ $$

Question Number 11744 Answers: 1 Comments: 0

If 5^(n+2) = 625 , find [5(n+3)]^(1/n) .

$$\mathrm{If}\:\:\:\mathrm{5}^{\mathrm{n}+\mathrm{2}} =\:\mathrm{625}\:,\:\mathrm{find}\:\left[\mathrm{5}\left(\mathrm{n}+\mathrm{3}\right)\right]^{\mathrm{1}/{n}} . \\ $$

Question Number 11741 Answers: 1 Comments: 0

∫xtan^2 x dx=?

$$\int\mathrm{xtan}^{\mathrm{2}} \:\mathrm{x}\:\mathrm{dx}=? \\ $$

Question Number 11740 Answers: 0 Comments: 0

Metal pellets of mass 0.1 kg are fired into an iron plate of large mass m and specific heat capacity c at a temperature of 15°C . 25% of the kinetic energy of the pellets is converted thermal energy and the plate temperature rises to 16°C. The average speed of the pellets before hitting the plate was ?

$$\mathrm{Metal}\:\mathrm{pellets}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{0}.\mathrm{1}\:\mathrm{kg}\:\mathrm{are}\:\mathrm{fired}\:\mathrm{into}\:\mathrm{an}\:\mathrm{iron}\:\mathrm{plate}\:\mathrm{of}\:\mathrm{large}\:\mathrm{mass}\:\mathrm{m} \\ $$$$\mathrm{and}\:\mathrm{specific}\:\mathrm{heat}\:\mathrm{capacity}\:\mathrm{c}\:\mathrm{at}\:\mathrm{a}\:\mathrm{temperature}\:\mathrm{of}\:\mathrm{15}°\mathrm{C}\:.\:\mathrm{25\%}\:\mathrm{of}\:\mathrm{the}\:\mathrm{kinetic} \\ $$$$\mathrm{energy}\:\mathrm{of}\:\mathrm{the}\:\mathrm{pellets}\:\mathrm{is}\:\mathrm{converted}\:\mathrm{thermal}\:\mathrm{energy}\:\mathrm{and}\:\mathrm{the}\:\mathrm{plate}\:\mathrm{temperature} \\ $$$$\mathrm{rises}\:\mathrm{to}\:\mathrm{16}°\mathrm{C}.\:\mathrm{The}\:\mathrm{average}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{pellets}\:\mathrm{before}\:\mathrm{hitting}\:\mathrm{the}\:\mathrm{plate}\:\mathrm{was}\:? \\ $$

Question Number 11737 Answers: 1 Comments: 0

∫ (1/(x(x^n +1))) dx =

$$\int\:\frac{\mathrm{1}}{{x}\left({x}^{{n}} +\mathrm{1}\right)}\:{dx}\:= \\ $$

Question Number 11732 Answers: 1 Comments: 0

0.16 ÷ (2/3) of (2/5) ÷ (1/8)

$$\mathrm{0}.\mathrm{16}\:\boldsymbol{\div}\:\frac{\mathrm{2}}{\mathrm{3}}\:\mathrm{of}\:\frac{\mathrm{2}}{\mathrm{5}}\:\boldsymbol{\div}\:\frac{\mathrm{1}}{\mathrm{8}} \\ $$

Question Number 11714 Answers: 0 Comments: 3

Solve the Crazy equation... x(lnx)^2 +xlnx−1=0

$${Solve}\:{the}\:{Crazy}\:{equation}... \\ $$$${x}\left({lnx}\right)^{\mathrm{2}} +{xlnx}−\mathrm{1}=\mathrm{0} \\ $$

Question Number 11707 Answers: 0 Comments: 0

ax^3 +bx^2 +cx+d=0 pls. solve it in a alzebric way , that a O level(ssc) student can understand...

$$ \\ $$$$ \\ $$$${ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0} \\ $$$${pls}.\:{solve}\:{it}\:{in}\:{a}\:{alzebric}\:{way}\:, \\ $$$${that}\:{a}\:{O}\:{level}\left({ssc}\right)\:{student}\:{can} \\ $$$${understand}... \\ $$$$ \\ $$

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