Question Number 76842 by jagoll last updated on 31/Dec/19 $$ \\ $$$${what}\:{is}\:{solution}\:{y}^{''\:} +\:\:{y}\:=\:\mathrm{0}\:. \\ $$ Commented by mathmax by abdo last updated on 31/Dec/19 $${the}\:{caraceristc}\:{equation}\:{is}\:{r}^{\mathrm{2}}…
Question Number 142379 by Rexzie last updated on 30/May/21 $${Show}\:{that}\:\mathrm{1}+\mathrm{3}{n}<{n}^{\mathrm{2}} \:{for}\:{every}\:{positive}\:{integer}\:{n}\geqslant\mathrm{4} \\ $$ Commented by mr W last updated on 30/May/21 $${n}\geqslant\mathrm{4} \\ $$$${n}^{\mathrm{2}} \geqslant\mathrm{4}{n}=\mathrm{3}{n}+{n}\geqslant\mathrm{3}{n}+\mathrm{4}>\mathrm{3}{n}+\mathrm{1}…
Question Number 11303 by tawa last updated on 19/Mar/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{and}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circumference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{formed} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{three}\:\mathrm{lines}. \\ $$$$\mathrm{2y}\:−\:\mathrm{9x}\:+\:\mathrm{26}\:=\:\mathrm{0} \\ $$$$\mathrm{9y}\:+\:\mathrm{2x}\:+\:\mathrm{32}\:=\:\mathrm{0} \\ $$$$\mathrm{11y}\:−\:\mathrm{7x}\:−\:\mathrm{27}\:=\:\mathrm{0} \\ $$ Answered by ajfour last updated…
Question Number 11302 by uni last updated on 19/Mar/17 $$\frac{\mathrm{sin10x}−\mathrm{sin6x}−\mathrm{sin2x}}{\mathrm{sin9x}−\mathrm{sin7x}−\mathrm{sinx}}=? \\ $$ Answered by sandy_suhendra last updated on 20/Mar/17 $$=\frac{\left(\mathrm{sin10x}−\mathrm{sin6x}\right)−\mathrm{sin2x}}{\left(\mathrm{sin9x}−\mathrm{sin7x}\right)−\mathrm{sinx}} \\ $$$$=\frac{\mathrm{2cos8xsin2x}−\mathrm{sin2x}}{\mathrm{2cos8xsinx}−\mathrm{sinx}} \\ $$$$=\frac{\mathrm{sin2x}\left(\mathrm{2cos8x}−\mathrm{1}\right)}{\mathrm{sinx}\left(\mathrm{2cos8x}−\mathrm{1}\right)} \\…
Question Number 11301 by tawa last updated on 19/Mar/17 $$\mathrm{ABCD}\:\mathrm{is}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quad}\:\mathrm{and}\:\mathrm{the}\:\mathrm{diagonal}\:\mathrm{AC}\:\mathrm{and}\:\mathrm{BD}\:\mathrm{intersect}\:\mathrm{at}\:\mathrm{H}.\: \\ $$$$\bigtriangleup\mathrm{DAC}\:=\:\mathrm{41}°\:\mathrm{and}\:\bigtriangleup\mathrm{AHB}\:=\:\mathrm{70}°.\:\mathrm{calulate}\:\:\bigtriangleup\mathrm{ACB} \\ $$ Commented by sandy_suhendra last updated on 20/Mar/17 $$\mathrm{do}\:\mathrm{you}\:\mathrm{mean}\:\angle\mathrm{DAC},\:\angle\mathrm{AHB}\:\mathrm{and}\:\angle\mathrm{ACB}? \\ $$ Answered…
Question Number 11299 by agni5 last updated on 19/Mar/17 $$\mathrm{Define}\:\mathrm{two}\:\mathrm{partition}\:\mathrm{p}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{p}_{\mathrm{2}} \:\mathrm{of}\:\left[\mathrm{2},\mathrm{5}\right]\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{p}_{\mathrm{1}} \subset\mathrm{p}_{\mathrm{2}} \:.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{upper}\:\mathrm{and}\:\mathrm{lower}\:\mathrm{product} \\ $$$$\mathrm{sums}\:\mathrm{with}\:\mathrm{respet}\:\mathrm{to}\:\mathrm{f}\:,?\mathrm{defined}\:\mathrm{by} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}\:,\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}<\mathrm{4} \\ $$$$\:\:\:\:\:\:\:\:=\mathrm{1}−\mathrm{x}^{\mathrm{2}} \:,\:\:\:\:\:\mathrm{x}\:\geqslant\mathrm{4}\:\:. \\ $$$$\mathrm{Also}\:\mathrm{verify}\:\mathrm{the}\:\mathrm{relationship}\:\mathrm{between}\:\mathrm{these}…
Question Number 142365 by mathmax by abdo last updated on 30/May/21 $$\mathrm{calculate}\:\int\:\:\sqrt{\mathrm{1}+\mathrm{e}^{\mathrm{x}} \:+\mathrm{e}^{\mathrm{2x}} }\mathrm{dx} \\ $$ Answered by MJS_new last updated on 05/Jun/21 $$\int\sqrt{\mathrm{e}^{\mathrm{2}{x}} +\mathrm{e}^{{x}}…
Question Number 76830 by peter frank last updated on 30/Dec/19 Commented by mathmax by abdo last updated on 05/Jan/20 $${let}\:{remember}\:{that}\:{arctanz}\:=\frac{\mathrm{1}}{\mathrm{2}{i}}{ln}\left(\frac{\mathrm{1}+{iz}}{\mathrm{1}−{iz}}\right)\:\left({result}\:{proved}\right) \\ $$$${iln}\left(\frac{{a}−{ib}}{{a}+{ib}}\right)\:=−{iln}\left(\frac{{a}+{ib}}{{a}−{ib}}\right)\:=−{iln}\left(\frac{\mathrm{1}+{i}\frac{{b}}{{a}}}{\mathrm{1}−{i}\frac{{b}}{{a}}}\right)\:=−{i}\left(\mathrm{2}{i}\right){arctan}\left(\frac{{b}}{{a}}\right) \\ $$$$=\mathrm{2}\:{arctan}\left(\frac{{b}}{{a}}\right)\:\Rightarrow{tan}\left({iln}\left(\frac{{a}−{ib}}{{a}+{ib}}\right)\right)={tan}\left(\mathrm{2}{arctan}\left(\frac{{b}}{{a}}\right)\right) \\…
Question Number 142361 by BHOOPENDRA last updated on 30/May/21 Answered by mr W last updated on 30/May/21 $$\delta=\mathrm{0}.\mathrm{2}{m} \\ $$$${l}=\mathrm{4}.\mathrm{8}+\mathrm{0}.\mathrm{2}=\mathrm{5}{m} \\ $$$${a}=\mathrm{1}{m} \\ $$$${mgl}\mathrm{sin}\:\theta−\mu{mg}\mathrm{cos}\:\theta{l}=\frac{\mathrm{1}}{\mathrm{2}}{k}\delta^{\mathrm{2}} ={mga}\mathrm{sin}\:\theta+\mu{mg}\mathrm{cos}\:\theta{a}…
Question Number 76826 by Master last updated on 30/Dec/19 Answered by john santu last updated on 31/Dec/19 $$=\:\int\underset{\mathrm{0}} {\overset{\mathrm{2}} {\:}}\:\int\underset{\mathrm{0}} {\overset{\:\mathrm{1}} {\:}}\:\int\underset{\mathrm{0}} {\overset{\:\mathrm{3}} {\:}}\:\left(\mathrm{2}{y}+{z}+{x}\right)\:{dz}\:{dy}\:{dx}\: \\…