Question Number 32792 by Rio Mike last updated on 02/Apr/18 $${the}\:{equation}\: \\ $$$$\mathrm{3}{x}−\:\mathrm{5}=\mathrm{15}\:{represents}\:{a}\:{straight}\:{line}. \\ $$$$\left.{a}\right)\:{find}\:{one}\:{point}\:{on}\:{this}\:{line}. \\ $$$$\left.{b}\right){find}\:{the}\:{coordinates}\:{of}\:{the}\:{points}\:{when} \\ $$$${the}\:{line}\:{cuts}\:{the}\:{x}−{axis}\:{and}\:{the}\:{y}−{axis} \\ $$$$\left.{c}\right){find}\:{the}\:{gradient}\:{of}\:{this}\:{line}. \\ $$$$ \\ $$…
Question Number 98320 by pranesh last updated on 13/Jun/20 Answered by bemath last updated on 13/Jun/20 $$\mathrm{s}\:=\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{y}}+\frac{\mathrm{x}}{\mathrm{1}−\mathrm{y}}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{1}−\mathrm{y}}+\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{1}−\mathrm{y}}+ \\ $$$$…\:+\frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{1}−\mathrm{y}}\:+…=\frac{\frac{\mathrm{1}}{\mathrm{1}−\mathrm{x}}}{\mathrm{1}−\mathrm{y}}\: \\ $$$$=\:\frac{\mathrm{1}}{\left(\mathrm{1}−\mathrm{x}\right)\left(\mathrm{1}−\mathrm{y}\right)}\:;\:\mid\mathrm{x}\mid\:<\mathrm{1}\:\&\mid\mathrm{y}\mid<\mathrm{1} \\…
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Question Number 32760 by Rio Mike last updated on 01/Apr/18 $$\mathrm{if}\:{y}=\mathrm{3}{x}^{\mathrm{4}\:} .{find}\:{the}\:{approximate}\:{percentage} \\ $$$${increase}\:{in}\:{y}\:{when}\:{x}\:{increase}\:{by}\:\:\:\mathrm{2}\frac{\mathrm{1}}{\mathrm{2}}\%. \\ $$ Answered by MJS last updated on 02/Apr/18 $$\frac{{f}\left({x}\right)}{{f}\left({x}\right)}=\frac{\mathrm{3}\left(\mathrm{1}.\mathrm{025}{x}\right)^{\mathrm{4}} }{\mathrm{3}{x}^{\mathrm{4}}…
Question Number 98290 by Rio Michael last updated on 12/Jun/20 Commented by Rio Michael last updated on 13/Jun/20 $$\mathrm{The}\:\mathrm{above}\:\mathrm{figure}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{square}\:\mathrm{lamina}\:\mathrm{of}\:\mathrm{side}\:{a}\:\mathrm{with}\:\mathrm{vertex}\: \\ $$$$\mathrm{inclined}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:{x},\:\mathrm{0}\:\leqslant\:{x}\leqslant\:\frac{\pi}{\mathrm{2}},\:\mathrm{placed}\:\mathrm{on}\:\mathrm{a}\:\mathrm{line}\:{AB}. \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lamina}\:\mathrm{is}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance}\:\frac{{a}\sqrt{\mathrm{2}}}{\mathrm{2}}\:\mathrm{sin}\:\left({x}\:+\:\frac{\pi}{\mathrm{4}}\right)\:\mathrm{from} \\ $$$${AB}.\:\mathrm{The}\:\mathrm{lamina}\:\mathrm{is}\:\mathrm{rotated}\:\mathrm{completely}\:\mathrm{about}\:\mathrm{the}\:\mathrm{line}\:{AB}.…
Question Number 32707 by Rio Mike last updated on 31/Mar/18 $$\left.\mathrm{1}\right)\mathrm{find}\:\mathrm{u}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{u}\:\mathrm{if} \\ $$$$\sum_{\mathrm{n}=\mathrm{1}} ^{\mathrm{4}} \mathrm{2}{u}.\mathrm{2}^{{n}−\mathrm{1}} =\mathrm{64} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{find}\:\mathrm{k}\:\mathrm{if}\: \\ $$$$\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \mathrm{k}.\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{n}−\mathrm{1}} =\frac{\mathrm{2}}{\mathrm{3}} \\ $$…
Question Number 32688 by Rio Mike last updated on 31/Mar/18 $$\mathrm{Evaluate} \\ $$$$\left.\:\mathrm{1}\right)\:\int_{−\mathrm{1}} ^{\mathrm{0}} \left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:\mathrm{1}\right)\:\mathrm{dx} \\ $$$$\left.\mathrm{2}\right)\:\int_{\mathrm{1}} ^{\mathrm{5}} \left(\mathrm{x}\:−\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\right)\mathrm{dx} \\ $$$$ \\ $$…
Question Number 98208 by Rio Michael last updated on 12/Jun/20 $$\mathrm{suppose}\:\mathrm{a}\:\mathrm{force}\:\mathrm{given}\:\mathrm{as}\:{F}_{\mathrm{1}} \:=\:\mathrm{24}\:{N}\:\mathrm{and}\:{F}_{\mathrm{2}} \:=\:\mathrm{50}\:{N}\:\mathrm{act}\:\mathrm{through}\: \\ $$$$\mathrm{points}\:{AB}\:\mathrm{and}\:{AC}\:\mathrm{where}\:\:{OA}\:=\:\mathrm{2}{i}\:+\mathrm{3}{j}\:,\:{OB}\:=\:\mathrm{5}{i}\:+\:\mathrm{6}{j}\:\:\mathrm{and}\: \\ $$$${OC}\:=\:\mathrm{7}{i}\:+\:\mathrm{8}{j} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{find}\:\mathrm{in}\:\mathrm{vector}\:\mathrm{notation}\:{F}_{\mathrm{1}} \:\mathrm{and}\:{F}_{\mathrm{2}} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{thier}\:\mathrm{resultant}. \\ $$ Commented…
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Question Number 98022 by Ar Brandon last updated on 11/Jun/20 $$\mathcal{D}\mathrm{erive}\:\mathrm{the}\:\mathrm{relation}\:\mathrm{between}\:\mathrm{an}\:\mathcal{A}\mathrm{rithmetic}\:\mathcal{M}\mathrm{ean} \\ $$$$\mathrm{and}\:\mathrm{a}\:\mathcal{G}\mathrm{eometric}\:\mathcal{M}\mathrm{ean} \\ $$$$\sqrt[{\mathrm{n}}]{\mathrm{x}_{\mathrm{1}} \mathrm{x}_{\mathrm{2}} …\mathrm{x}_{\mathrm{n}} }\leqslant\frac{\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +\centerdot\centerdot\centerdot+\mathrm{x}_{\mathrm{n}} }{\mathrm{n}}\:\forall\mathrm{n}\in\mathbb{N}^{\ast} ,\:\forall\left(\mathrm{x}_{\mathrm{1}} ,\mathrm{x}_{\mathrm{2}} ,…\mathrm{x}_{\mathrm{n}} \right)\in\left(\mathbb{R}_{+}…