Question Number 183036 by Mastermind last updated on 18/Dec/22 $$\mathrm{A}\:\mathrm{bullet}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{1kg}\:\mathrm{is}\:\mathrm{fired}\:\mathrm{and}\:\mathrm{get} \\ $$$$\mathrm{embedded}\:\mathrm{into}\:\mathrm{a}\:\mathrm{block}\:\mathrm{of}\:\mathrm{wood}\:\mathrm{of} \\ $$$$\mathrm{mass}\:\mathrm{1kg}\:\mathrm{initially}\:\mathrm{at}\:\mathrm{rest}\:\mathrm{the}\:\mathrm{velocity} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{bullet}\:\mathrm{before}\:\mathrm{collision}\:\mathrm{is}\:\mathrm{90m}/\mathrm{s} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{the}\:\mathrm{system} \\ $$$$\mathrm{after}\:\mathrm{collision}? \\ $$$$\left.\mathrm{2}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{kinetic}\:\mathrm{energy}\:\mathrm{before} \\ $$$$\mathrm{and}\:\mathrm{after}\:\mathrm{the}\:\mathrm{collision}. \\…
Question Number 183039 by Mastermind last updated on 18/Dec/22 $$\mathrm{A}\:\mathrm{Golfer}\:\mathrm{practising}\:\mathrm{on}\:\mathrm{a}\:\mathrm{range} \\ $$$$\mathrm{with}\:\mathrm{an}\:\mathrm{accelerated}\:\mathrm{tree}\:\mathrm{4}.\mathrm{9m}\:\mathrm{above} \\ $$$$\mathrm{the}\:\mathrm{fairway}\:\mathrm{is}\:\mathrm{able}\:\mathrm{to}\:\mathrm{strike}\:\mathrm{a}\:\mathrm{ball} \\ $$$$\mathrm{so}\:\mathrm{that}\:\mathrm{it}\:\mathrm{leaves}\:\mathrm{the}\:\mathrm{club}\:\mathrm{with}\:\mathrm{a}\: \\ $$$$\mathrm{horizontal}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{20m}/\mathrm{s}. \\ $$$$\left(\mathrm{Assume}\:\mathrm{the}\:\mathrm{acceleration}\:\mathrm{due}\:\mathrm{to}\:\right. \\ $$$$\mathrm{gravity}\:\mathrm{is}\:\mathrm{9}.\mathrm{8m}/\mathrm{s}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{the}\:\mathrm{effect}\:\mathrm{of}\:\mathrm{air} \\ $$$$\mathrm{resistance}\:\mathrm{maybe}\:\mathrm{ignored}\:\mathrm{unless}…
Question Number 183038 by Mastermind last updated on 18/Dec/22 $$\mathrm{Solve} \\ $$$$\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{xy}=\mathrm{x}^{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$ Answered by TheSupreme last updated on 21/Dec/22…
Question Number 117475 by Dwaipayan Shikari last updated on 11/Oct/20 $$\frac{\mathrm{6}}{\mathrm{5}}.\frac{\mathrm{24}}{\mathrm{23}}.\frac{\mathrm{54}}{\mathrm{53}}.\frac{\mathrm{96}}{\mathrm{95}}.\frac{\mathrm{150}}{\mathrm{149}}.\frac{\mathrm{216}}{\mathrm{215}}.\frac{\mathrm{294}}{\mathrm{293}}…. \\ $$ Answered by frc2crc last updated on 12/Oct/20 $$\frac{\mathrm{sin}\:\pi{x}}{\pi{x}}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{{n}^{\mathrm{2}} }\right)…
Question Number 51933 by Tawa1 last updated on 01/Jan/19 $$\mathrm{If}\:\:\mathrm{p}\:=\:\mathrm{cos}\:\theta\:+\:\mathrm{i}\:\mathrm{sin}\:\theta\:\:\:\:\:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\:\:\:\mathrm{q}\:\:=\:\:\mathrm{cos}\:\phi\:+\:\mathrm{i}\:\mathrm{sin}\:\phi \\ $$$$\mathrm{Show}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\frac{\left(\mathrm{p}\:+\:\mathrm{q}\right)\left(\mathrm{pq}\:−\:\mathrm{1}\right)}{\left(\mathrm{p}\:−\:\mathrm{q}\right)\left(\mathrm{pq}\:+\:\mathrm{1}\right)}\:\:=\:\:\frac{\mathrm{sin}\:\theta\:+\:\mathrm{sin}\:\phi}{\mathrm{sin}\:\theta\:−\:\mathrm{sin}\:\phi} \\ $$ Commented by peter frank last updated on 01/Jan/19 Answered by tanmay.chaudhury50@gmail.com…
Question Number 117460 by Dwaipayan Shikari last updated on 11/Oct/20 $$\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{9801}}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{4}{n}\right)!\left(\mathrm{1103}+\mathrm{26390}{n}\right)}{\left({n}!\right)^{\mathrm{4}} \mathrm{396}^{\mathrm{4}{n}} }=\frac{\mathrm{1}}{\pi}\:\:\:\left({Prove}\:{that}\right) \\ $$ Commented by Dwaipayan Shikari last updated on 11/Oct/20…
Question Number 117458 by Dwaipayan Shikari last updated on 11/Oct/20 $$\frac{\mathrm{4}}{\mathrm{3}}.\frac{\mathrm{16}}{\mathrm{15}}.\frac{\mathrm{36}}{\mathrm{35}}.\frac{\mathrm{64}}{\mathrm{63}}.\frac{\mathrm{100}}{\mathrm{99}}.\frac{\mathrm{144}}{\mathrm{143}}.\frac{\mathrm{196}}{\mathrm{195}}.\frac{\mathrm{256}}{\mathrm{255}}.\frac{\mathrm{324}}{\mathrm{323}}……\infty \\ $$ Answered by Olaf last updated on 11/Oct/20 $$\mathrm{P}_{{n}} \:=\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\frac{\left(\mathrm{2}{k}\right)^{\mathrm{2}} }{\left(\mathrm{2}{k}−\mathrm{1}\right)\left(\mathrm{2}{k}+\mathrm{1}\right)}…
Question Number 182984 by Mastermind last updated on 18/Dec/22 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{for}\:\mathrm{the}\:\mathrm{plane} \\ $$$$\mathrm{through}\:\mathrm{the}\:\mathrm{point}\:\mathrm{A}\left(\mathrm{6},\:\mathrm{2},\:−\mathrm{4}\right), \\ $$$$\mathrm{B}\left(−\mathrm{2},\:\mathrm{4},\:\mathrm{8}\right),\:\mathrm{C}\left(\mathrm{4},\:−\mathrm{2},\:\mathrm{2}\right).\:−\mathrm{Vector}\:\mathrm{Analysis} \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$ Answered by cortano1 last updated…
Question Number 51897 by Tawa1 last updated on 31/Dec/18 $$\mathrm{If}\:\:\:\:\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{x}}\:\:=\:\:\mathrm{2cos}\theta\:,\:\:\:\:\:\:\mathrm{y}\:+\:\frac{\mathrm{1}}{\mathrm{y}}\:\:=\:\:\mathrm{2cos}\phi\:,\:\:\:\:\:\:\:\:\:\mathrm{z}\:+\:\frac{\mathrm{1}}{\mathrm{z}}\:\:=\:\:\mathrm{2cos}\psi \\ $$$$\mathrm{Show}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\:\mathrm{xyz}\:+\:\frac{\mathrm{1}}{\mathrm{xyz}}\:\:=\:\:\mathrm{2cos}\left(\theta\:+\:\phi\:+\:\psi\right) \\ $$ Answered by behi83417@gmail.com last updated on 31/Dec/18 $${cos}\theta=\frac{{e}^{{i}\theta} +{e}^{−{i}\theta} }{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}\left({e}^{{i}\theta} +\frac{\mathrm{1}}{{e}^{{i}\theta}…
Question Number 51887 by Tawa1 last updated on 31/Dec/18 $$\mathrm{Prove}\:\mathrm{that};\:\:\:\:\mathrm{tanh}\left(\mathrm{log}\:\sqrt{\mathrm{3}}\right)\:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 01/Jan/19 $${e}^{{ix}} ={cosx}+{isinx} \\ $$$${e}^{−{ix}} ={cosx}−{isinx} \\…