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Question Number 55501 Answers: 0 Comments: 5

Roots of x^3 +px+q=0 are x = u+v u^3 , v^3 are roots of z^2 −α^3 z+β^6 =0 ((d^2 (y/x))/dx^2 )∣_(x=α) =0 , (dy/dx)∣_(x=β) =0 . I have noticed long way back, please hunt why (how come) ?

$${Roots}\:{of}\:{x}^{\mathrm{3}} +{px}+{q}=\mathrm{0} \\ $$$${are}\:\:{x}\:=\:{u}+{v} \\ $$$${u}^{\mathrm{3}} ,\:{v}^{\mathrm{3}} \:{are}\:{roots}\:{of} \\ $$$$\:\:\:\:\:\:\:{z}^{\mathrm{2}} −\alpha^{\mathrm{3}} {z}+\beta^{\mathrm{6}} =\mathrm{0} \\ $$$$\:\:\:\frac{{d}^{\mathrm{2}} \left({y}/{x}\right)}{{dx}^{\mathrm{2}} }\mid_{{x}=\alpha} =\mathrm{0}\:\:,\:\frac{{dy}}{{dx}}\mid_{{x}=\beta} =\mathrm{0}\:. \\ $$$${I}\:{have}\:{noticed}\:{long}\:{way}\:{back}, \\ $$$${please}\:{hunt}\:{why}\:\left({how}\:{come}\right)\:? \\ $$

Question Number 55493 Answers: 0 Comments: 6

Find intgral of tan x by parts ∫tan x dx pls help me (i am a newbie in calculus)

$${Find}\:{intgral}\:{of}\:\mathrm{tan}\:{x}\:{by}\:{parts} \\ $$$$\int{tan}\:{x}\:{dx}\:\:\:{pls}\:{help}\:{me}\:\:\left({i}\:{am}\:{a}\:{newbie}\:{in}\:{calculus}\right) \\ $$

Question Number 55491 Answers: 1 Comments: 0

Question Number 55482 Answers: 1 Comments: 0

A group of n students are numbered continously from first sdtudent as 1,2,3,...... If 1101 digits had to be used in all,what is the number of students in the group

$${A}\:{group}\:{of}\:{n}\:{students}\:{are}\:{numbered} \\ $$$${continously}\:{from}\:{first}\:{sdtudent}\:{as}\:\mathrm{1},\mathrm{2},\mathrm{3},...... \\ $$$${If}\:\mathrm{1101}\:{digits}\:{had}\:{to}\:{be}\:{used}\:{in}\:{all},{what}\:{is}\:{the}\: \\ $$$${number}\:{of}\:{students}\:{in}\:{the}\:{group} \\ $$

Question Number 55476 Answers: 1 Comments: 0

4 metal rods of length 78 cm,104 cm,117cm, a.nd 169 cm are to be cut into parts of equal length Each length must be as long as possible What is the maximum number of pieces that can be cut?

$$\mathrm{4}\:{metal}\:{rods}\:{of}\:{length}\:\mathrm{78}\:{cm},\mathrm{104}\:{cm},\mathrm{117}{cm}, \\ $$$${a}.{nd}\:\mathrm{169}\:{cm}\:{are}\:{to}\:{be}\:{cut}\:{into}\:{parts}\:{of}\:{equal}\:{length} \\ $$$${Each}\:{length}\:{must}\:{be}\:{as}\:{long}\:{as}\:{possible} \\ $$$${What}\:{is}\:{the}\:{maximum}\:{number}\:{of}\:{pieces} \\ $$$${that}\:{can}\:{be}\:{cut}? \\ $$

Question Number 55475 Answers: 1 Comments: 0

Find the equation of the plane passing through the points (1,0,0) and (0,1,0) and makes an angle of (π/4) with the plane x+y = 3

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{plane} \\ $$$$\mathrm{passing}\:\mathrm{through}\:\mathrm{the}\:\mathrm{points} \\ $$$$\left(\mathrm{1},\mathrm{0},\mathrm{0}\right)\:\mathrm{and}\:\left(\mathrm{0},\mathrm{1},\mathrm{0}\right)\:\mathrm{and}\:\mathrm{makes} \\ $$$$\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\:\frac{\pi}{\mathrm{4}}\:\mathrm{with}\:\mathrm{the}\:\mathrm{plane} \\ $$$$\mathrm{x}+\mathrm{y}\:=\:\mathrm{3} \\ $$

Question Number 55473 Answers: 0 Comments: 1

Question Number 55468 Answers: 1 Comments: 0

Question Number 55467 Answers: 0 Comments: 3

Question Number 55458 Answers: 1 Comments: 1

Question Number 55457 Answers: 0 Comments: 2

calculate ∫_0 ^∞ ((ln(x))/(x^2 +x+1))dx .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} +{x}+\mathrm{1}}{dx}\:. \\ $$

Question Number 55449 Answers: 1 Comments: 0

Question Number 55444 Answers: 0 Comments: 0

Question Number 55433 Answers: 1 Comments: 0

Question Number 55431 Answers: 1 Comments: 1

2 plane parallel conducting plates are held horizontal,one above the other in a vacuum.Electrons having a speed of 6×10^6 m/s and moves normally to the plates enter the region between them through a hole in the lower plate which is earthed.What potential must be applied to the other plate so that the electrons just fails to reach it?What is the subsequent motion of these electrons? (ratio of charge to mass of electron= 1.8×10^(11) Ckg^(−1) )

$$\mathrm{2}\:{plane}\:{parallel}\:{conducting}\:{plates}\:{are} \\ $$$${held}\:{horizontal},{one}\:{above}\:{the}\:{other}\:{in} \\ $$$${a}\:{vacuum}.{Electrons}\:{having}\:{a}\:{speed}\:{of} \\ $$$$\mathrm{6}×\mathrm{10}^{\mathrm{6}} {m}/{s}\:{and}\:{moves}\:{normally}\:{to}\:{the} \\ $$$${plates}\:{enter}\:{the}\:{region}\:{between}\:{them} \\ $$$${through}\:{a}\:{hole}\:{in}\:{the}\:{lower}\:{plate}\:{which} \\ $$$${is}\:{earthed}.{What}\:{potential}\:{must}\:{be} \\ $$$${applied}\:{to}\:{the}\:{other}\:{plate}\:{so}\:{that}\:{the} \\ $$$${electrons}\:{just}\:{fails}\:{to}\:{reach}\:{it}?{What} \\ $$$${is}\:{the}\:{subsequent}\:{motion}\:{of}\:{these} \\ $$$${electrons}? \\ $$$$\left({ratio}\:{of}\:{charge}\:{to}\:{mass}\:{of}\:{electron}=\right. \\ $$$$\left.\mathrm{1}.\mathrm{8}×\mathrm{10}^{\mathrm{11}} {Ckg}^{−\mathrm{1}} \right) \\ $$

Question Number 55422 Answers: 0 Comments: 1

Please see these questions:

$${Please}\:{see}\:{these}\:{questions}: \\ $$

Question Number 55418 Answers: 3 Comments: 1