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AllQuestion and Answers: Page 1487

Question Number 61202 Answers: 1 Comments: 1

Question Number 61205 Answers: 0 Comments: 4

A cubical block of ice of mass m and edge L is placed in a large tray of mass M.If the ice block melts,how far does the centre of mass of the system “ice + tray” come down ? a)((ml)/(m+M)) b)((2ml)/(m+M)) c)((ml)/(2(m+M))) d)none

$${A}\:{cubical}\:{block}\:{of}\:{ice}\:{of}\:{mass}\:{m}\:{and} \\ $$$${edge}\:{L}\:{is}\:{placed}\:{in}\:{a}\:{large}\:{tray}\:{of}\:{mass} \\ $$$${M}.{If}\:{the}\:{ice}\:{block}\:{melts},{how}\:{far}\:{does} \\ $$$${the}\:{centre}\:{of}\:{mass}\:{of}\:{the}\:{system}\:``{ice}\:+\:{tray}'' \\ $$$${come}\:{down}\:? \\ $$$$ \\ $$$$\left.{a}\left.\right)\left.\frac{{ml}}{{m}+{M}}\left.\:\:{b}\right)\frac{\mathrm{2}{ml}}{{m}+{M}}\:\:{c}\right)\frac{{ml}}{\mathrm{2}\left({m}+{M}\right)}\:\:{d}\right){none} \\ $$

Question Number 60987 Answers: 1 Comments: 1

Question Number 60984 Answers: 2 Comments: 9

(a/(a−b)) + (b/(b−c)) + (c/(c−a)) = 4 ab^2 + bc^2 + abc + ca^2 = a^2 b + b^2 c + c^2 a ((a/(a−b)))^3 + ((b/(b−c)))^3 + ((c/(c−a)))^3 = ?

$$\frac{{a}}{{a}−{b}}\:\:+\:\:\frac{{b}}{{b}−{c}}\:\:+\:\:\frac{{c}}{{c}−{a}}\:\:=\:\:\mathrm{4} \\ $$$${ab}^{\mathrm{2}} \:+\:{bc}^{\mathrm{2}} \:+\:{abc}\:+\:{ca}^{\mathrm{2}} \:\:=\:\:{a}^{\mathrm{2}} {b}\:+\:{b}^{\mathrm{2}} {c}\:+\:{c}^{\mathrm{2}} {a} \\ $$$$\left(\frac{{a}}{{a}−{b}}\right)^{\mathrm{3}} \:\:+\:\:\left(\frac{{b}}{{b}−{c}}\right)^{\mathrm{3}} \:\:+\:\:\left(\frac{{c}}{{c}−{a}}\right)^{\mathrm{3}} \:\:=\:\:? \\ $$$$ \\ $$

Question Number 60981 Answers: 2 Comments: 6

Question Number 60976 Answers: 0 Comments: 1

find Σ_(n=1) ^∞ (1/n^2 ) by use of integral ∫_0 ^(π/2) ln(2cosθ)dθ .

$${find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:{by}\:{use}\:{of}\:{integral}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{2}{cos}\theta\right){d}\theta\:\:. \\ $$

Question Number 60980 Answers: 1 Comments: 2

Solve for x, y, z x(y + z) = 33 ..... (i) y(z + x) = 35 ..... (ii) z(x + y) = 14 ..... (iii)

$$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x},\:\mathrm{y},\:\mathrm{z} \\ $$$$\:\:\:\:\:\mathrm{x}\left(\mathrm{y}\:+\:\mathrm{z}\right)\:=\:\mathrm{33}\:\:\:\:\:\:\:\:.....\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\mathrm{y}\left(\mathrm{z}\:+\:\mathrm{x}\right)\:=\:\mathrm{35}\:\:\:\:\:\:\:\:.....\:\left(\mathrm{ii}\right) \\ $$$$\:\:\:\:\:\mathrm{z}\left(\mathrm{x}\:+\:\mathrm{y}\right)\:=\:\mathrm{14}\:\:\:\:\:\:\:\:.....\:\left(\mathrm{iii}\right) \\ $$

Question Number 60967 Answers: 0 Comments: 4

study the integral ∫_(−∞) ^(+∞) (1−cos((2/(x^2 +1))))dx

$${study}\:{the}\:{integral}\:\int_{−\infty} ^{+\infty} \left(\mathrm{1}−{cos}\left(\frac{\mathrm{2}}{{x}^{\mathrm{2}} \:+\mathrm{1}}\right)\right){dx}\: \\ $$

Question Number 60960 Answers: 0 Comments: 2

find ∫_(−∞) ^(+∞) tan((1/(1+x^2 )))dx

$${find}\:\int_{−\infty} ^{+\infty} \:\:{tan}\left(\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\right){dx}\: \\ $$

Question Number 60955 Answers: 1 Comments: 4

Question Number 60948 Answers: 0 Comments: 1

Question Number 60946 Answers: 3 Comments: 4

Find x: x^x = 2x

$$\mathrm{Find}\:\mathrm{x}:\:\:\:\:\:\:\mathrm{x}^{\mathrm{x}} \:\:=\:\:\mathrm{2x} \\ $$

Question Number 60944 Answers: 1 Comments: 1

∫(dx/((1+x^2 )^(3/2) )) solve this pls

$$\int\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$${solve}\:{this}\:{pls} \\ $$

Question Number 60938 Answers: 0 Comments: 1

∫((csc^(2019) (x))/(sec^5 (x))) tan^2 (x) dx

$$\int\frac{{csc}^{\mathrm{2019}} \left({x}\right)}{{sec}^{\mathrm{5}} \left({x}\right)}\:{tan}^{\mathrm{2}} \left({x}\right)\:{dx} \\ $$

Question Number 60921 Answers: 1 Comments: 0

x^2 (d^2 y/dx^2 ) + x(dy/dx) + y=0 please solve this Euler equation

$${x}^{\mathrm{2}} \frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:{x}\frac{{dy}}{{dx}}\:+\:{y}=\mathrm{0} \\ $$$${please}\:{solve}\:{this}\:{Euler}\:{equation} \\ $$

Question Number 60915 Answers: 0 Comments: 0

Let Fibonacci sequence (F_n ) _(n≥0) where F_0 = 0, F_1 = 1, and F_(n+2) = F_(n+1) + F_n , ∀ n ≥ 0 . Find the least of natural numbers n so that F_n and F_(n+1) − 1 can be divided by F_(2019) .

$${Let}\:\:{Fibonacci}\:\:{sequence}\:\:\left({F}_{{n}} \right)\:_{{n}\geqslant\mathrm{0}} \\ $$$${where}\:\:{F}_{\mathrm{0}} \:=\:\mathrm{0},\:{F}_{\mathrm{1}} \:=\:\mathrm{1},\:\:{and}\:\:{F}_{{n}+\mathrm{2}} \:\:=\:\:{F}_{{n}+\mathrm{1}} \:+\:{F}_{{n}} \:\:\:\:,\:\:\forall\:{n}\:\:\geqslant\:\:\mathrm{0}\:. \\ $$$${Find}\:\:{the}\:\:{least}\:\:{of}\:\:{natural}\:\:{numbers}\:\:{n}\:\:{so}\:\:{that} \\ $$$${F}_{{n}} \:\:\:{and}\:\:\:{F}_{{n}+\mathrm{1}} \:−\:\mathrm{1}\:\:\:{can}\:\:{be}\:\:{divided}\:\:{by}\:\:\:{F}_{\mathrm{2019}} \:. \\ $$

Question Number 60906 Answers: 0 Comments: 1