Question Number 128012 by mathocean1 last updated on 03/Jan/21 $${show}\:{that}\:\forall\:{n}\:\in\:\mathbb{N},\:\mathrm{12}\:{divise}\:{n}^{\mathrm{2}} \left({n}^{\mathrm{4}} −\mathrm{1}\right) \\ $$ Answered by MJS_new last updated on 03/Jan/21 $${n}=\mathrm{2}{k}\:\Rightarrow\:\mathrm{2}\mid{n}\:\Rightarrow\:\mathrm{4}\mid{n}^{\mathrm{2}} \\ $$$${n}=\mathrm{2}{k}+\mathrm{1}\:\Rightarrow\:{n}^{\mathrm{4}} −\mathrm{1}=\mathrm{8}{k}\left({k}−\mathrm{1}\right)\left(\mathrm{2}{k}^{\mathrm{2}}…
Question Number 128010 by Study last updated on 03/Jan/21 Commented by Study last updated on 03/Jan/21 $${help}\:{me}\:{and}\:{show}\:{practice} \\ $$ Commented by som(math1967) last updated on…
Question Number 128008 by AgnibhoMukhopadhyay last updated on 03/Jan/21 $$\mathrm{If}\:\mathrm{347}.\mathrm{9823}\:=\:\frac{\mathrm{3}}{{P}}\:+\:\mathrm{4}{Q}\:+\:\mathrm{7}{R}\:+\:\frac{\mathrm{9}}{\mathrm{10}}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\:\frac{\mathrm{8}}{\mathrm{100}}\:+\:\frac{\mathrm{2}}{{S}}\:+\:\frac{\mathrm{3}}{{T}} \\ $$$${Then}\:{find}\:{the}\:{value}\:{of}\: \\ $$$${P}\:+\:{Q}\:+\:{R}\:+\:{S}\:+\:{T} \\ $$ Answered by MJS_new last updated on 03/Jan/21…
Question Number 128007 by ajfour last updated on 03/Jan/21 $${Can}\:{anyone}\:{find}\:{any}\:{error}\:{in} \\ $$$${my}\:{attempt}\:{to}\:{improve}\:{upon} \\ $$$${the}\:{Cardano}'{s}\:{cubic}\:{formula}.. \\ $$$${or}\:{as}\:{an}\:{alternate}\:{to}\:{the} \\ $$$${trigonometric}\:{solution}… \\ $$$${here}\:{it}\:{goes}: \\ $$$$\:\:\:\:{X}^{\mathrm{3}} −{pX}−{q}=\mathrm{0} \\ $$$${let}\:\:\frac{{X}}{\:\sqrt{{p}}}\:=\:{x}\:;\:\:{and}\:{with}\:\frac{{q}}{{p}\sqrt{{p}}}\:=\:{c}\:,…
Question Number 128005 by mnjuly1970 last updated on 03/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:….{nice}\:\:\:{calculus}… \\ $$$$\:\:{if}\:\:{b}>{a}>\mathrm{0}\:{then}\:{prove}\::: \\ $$$$\:\:\:\:\:\:\:\:\frac{{a}+{b}}{\mathrm{2}}>\:\frac{{b}−{a}}{{ln}\left({b}\right)−{ln}\left({a}\right)}\:>\:\sqrt{{ab}}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:………………………. \\ $$ Answered by mindispower last updated on 06/Jan/21…
Question Number 62468 by bshahid010@gmail.com last updated on 21/Jun/19 Commented by mathmax by abdo last updated on 21/Jun/19 $${let}\:\mathrm{3}^{{x}} \:={t}\:\:\:\Rightarrow\left({k}−\mathrm{2}\right){t}^{\mathrm{2}} −\mathrm{2}{t}\:+\mathrm{3}>\mathrm{0}\:\:\:\:\forall{t}>\mathrm{0}\:\Rightarrow\:\Delta^{'} <\mathrm{0}\:\Rightarrow \\ $$$$\left(−\mathrm{1}\right)^{\mathrm{2}} −\mathrm{3}\left({k}−\mathrm{2}\right)<\mathrm{0}\:\Rightarrow\mathrm{1}−\mathrm{3}{k}\:+\mathrm{6}\:<\mathrm{0}\:\Rightarrow\mathrm{7}−\mathrm{3}{k}\:<\mathrm{0}\:\Rightarrow{k}>\frac{\mathrm{7}}{\mathrm{3}}…
Question Number 128001 by Agnibhoo last updated on 03/Jan/21 $$\left(\mathrm{x}\:−\:\mathrm{a}\right)\:\left(\mathrm{x}\:−\:\mathrm{b}\right)\:\left(\mathrm{x}\:−\:\mathrm{c}\right)\:…..\:\left(\mathrm{x}\:−\:\mathrm{z}\right)\:=\:? \\ $$ Answered by prakash1956 last updated on 03/Jan/21 $$\left(\mathrm{x}\:−\:\mathrm{a}\right)\:\left(\mathrm{x}\:−\:\mathrm{b}\right)\:\left(\mathrm{x}\:−\:\mathrm{c}\right)\:…..\:\left(\mathrm{x}\:−\:\mathrm{z}\right)\:=\:? \\ $$$$=\mathrm{0}\:\:\:\left(\mathrm{x}−\mathrm{x}\right)\:\mathrm{term}\:\mathrm{involved}\:\mathrm{in}\:\mathrm{this} \\ $$$$ \\…
Question Number 62462 by naka3546 last updated on 21/Jun/19 Answered by MJS last updated on 21/Jun/19 $$\mathrm{let} \\ $$$${a}=\alpha−\sqrt{\beta} \\ $$$${b}=\alpha+\sqrt{\beta} \\ $$$${c}=\gamma−\sqrt{\delta} \\ $$$${d}=\gamma+\sqrt{\delta}…
Question Number 127997 by mr W last updated on 05/Jan/21 Commented by mr W last updated on 05/Jan/21 $${a}\:{paraboloid}\:{bowl}\:{has}\:{a}\:{depth}\:{of}\:{H} \\ $$$${and}\:{an}\:{opening}\:{with}\:{diameter}\:{L}. \\ $$$${a}\:{small}\:{block}\:{of}\:{mass}\:{m}\:{is}\:{released} \\ $$$${from}\:{rest}\:{at}\:{the}\:{rim}\:{of}\:{the}\:{bowl}.…
Question Number 62456 by ajfour last updated on 21/Jun/19 Commented by ajfour last updated on 21/Jun/19 $${If}\:{the}\:{regular}\:{tetrahedron}\:{with} \\ $$$${edge}\:{length}\:{unity}\:{inscribes}\:{a} \\ $$$${hemisphere}\:{such}\:{that}\:{three}\:{faces} \\ $$$${are}\:{tangent}\:{to}\:{the}\:{spherical} \\ $$$${surface}\:{while}\:{fourth}\:{serves}\:{to}…