Question Number 19505 by Tinkutara last updated on 12/Aug/17 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{two}\:\mathrm{straight}\:\mathrm{lines}\:\mathrm{with} \\ $$$$\mathrm{complex}\:\mathrm{slopes}\:\mu_{\mathrm{1}} \:\mathrm{and}\:\mu_{\mathrm{2}} \:\mathrm{are}\:\mathrm{parallel} \\ $$$$\mathrm{and}\:\mathrm{perpendicular}\:\mathrm{according}\:\mathrm{as}\:\mu_{\mathrm{1}} \:=\:\mu_{\mathrm{2}} \\ $$$$\mathrm{and}\:\mu_{\mathrm{1}} \:+\:\mu_{\mathrm{2}} \:=\:\mathrm{0}.\:\mathrm{Hence}\:\mathrm{if}\:\mathrm{the}\:\mathrm{straight} \\ $$$$\mathrm{lines}\:\bar {\alpha}{z}\:+\:\alpha\bar {{z}}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{and}\:\bar…
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Question Number 85036 by gopikrishnan last updated on 18/Mar/20 $${If}\:{A}=\begin{bmatrix}{{x}\:\:\:\:{x}\:\:\:{x}\:\:}\\{\underset{\mathrm{2}\:\:} {\mathrm{4}}\:−\underset{\mathrm{3}} {\mathrm{2}}\:\:\:\underset{\mathrm{4}} {\mathrm{1}}}\end{bmatrix}{findX}\:{if}\:{p}\left({A}\right)=\mathrm{3} \\ $$$$ \\ $$$$ \\ $$ Commented by jagoll last updated on…
Question Number 19499 by tawa tawa last updated on 12/Aug/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{of}\:\:\mathrm{2}^{\mathrm{253}} \\ $$ Answered by Tinkutara last updated on 12/Aug/17 $$\mathrm{Cyclicity}\:\mathrm{of}\:\mathrm{2}\:=\:\mathrm{4}\:\left(\mathrm{2},\:\mathrm{4},\:\mathrm{8},\:\mathrm{6}\right) \\ $$$$\mathrm{253}\:\equiv\:\mathrm{1}\:\left(\mathrm{mod}\:\mathrm{4}\right) \\ $$$$\therefore\:\mathrm{2}^{\mathrm{253}}…
Question Number 150571 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Find}\:\boldsymbol{\mathrm{A}}\:\mathrm{and}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{2021}\in\boldsymbol{\mathrm{A}}\:\mathrm{if} \\ $$$$\overline {\mathrm{abcd}}\in\boldsymbol{\mathrm{A}},\:\:\frac{\mathrm{a}}{\mathrm{d}\:+\:\mathrm{1}}\:=\:\frac{\mathrm{c}\:-\:\mathrm{b}}{\mathrm{c}}\:=\:\frac{\mathrm{a}\:+\:\mathrm{b}}{\mathrm{b}\:+\:\mathrm{c}} \\ $$ Answered by Rasheed.Sindhi last updated on 15/Aug/21 $$\overline {\mathrm{abcd}}\in\boldsymbol{\mathrm{A}};\:\:\frac{\mathrm{a}}{\mathrm{d}\:+\:\mathrm{1}}\:=\:\frac{\mathrm{c}\:-\:\mathrm{b}}{\mathrm{c}}\:=\:\frac{\mathrm{a}\:+\:\mathrm{b}}{\mathrm{b}\:+\:\mathrm{c}};\mathrm{A}=? \\…
Question Number 150567 by DELETED last updated on 13/Aug/21 Answered by DELETED last updated on 13/Aug/21 $$\left.\mathrm{1}\right).\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\:\frac{\mathrm{3x}^{\mathrm{2}} −\mathrm{4}}{\mathrm{2x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\:=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\:\frac{\mathrm{3x}^{\mathrm{2}} /\mathrm{x}^{\mathrm{2}} −\mathrm{4}/\mathrm{x}^{\mathrm{2}} }{\mathrm{2x}^{\mathrm{2}} /\mathrm{x}^{\mathrm{2}}…
Question Number 150560 by DELETED last updated on 13/Aug/21 Answered by DELETED last updated on 13/Aug/21 $$\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\frac{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{7x}+\mathrm{12}\right)\mathrm{sin}\:\left(\mathrm{x}−\mathrm{3}\right)}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}−\mathrm{6}\right)^{\mathrm{2}} }\:\:\:\:\:\:\: \\ $$$$\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\frac{\:\left(\mathrm{x}−\mathrm{4}\right)\left(\mathrm{x}−\mathrm{3}\right)\mathrm{sin}\:\left(\mathrm{x}−\mathrm{3}\right)}{\left\{\left(\mathrm{x}+\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)\right\}^{\mathrm{2}} }…
Question Number 150559 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Solve}\:\mathrm{for}\:\mathrm{natural}\:\mathrm{numbers}: \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{4}} \:=\:\mathrm{2022} \\ $$ Answered by MJS_new last updated on 13/Aug/21 $$\mathrm{if}\:{x}=\mathrm{0}:\:{y}^{\mathrm{4}} =\mathrm{2022}\:\Rightarrow\:{y}\approx\mathrm{6}.\mathrm{7}…
Question Number 85020 by M±th+et£s last updated on 18/Mar/20 $${solve}\:{integration} \\ $$$$\int_{\mathrm{1}} ^{\mathrm{2}} {x}\:{d}\lfloor{x}^{\mathrm{2}} \rfloor \\ $$ Commented by mr W last updated on 18/Mar/20…
Question Number 150558 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Solve}\:\mathrm{for}\:\mathrm{integers}: \\ $$$$\left(\mathrm{6x}\:+\:\mathrm{5y}^{\mathrm{2}} \right)\left(\mathrm{4z}\:+\:\mathrm{x}\right)\left(\mathrm{2y}^{\mathrm{2}} \:+\:\mathrm{3z}\right)\:=\:\mathrm{2021} \\ $$ Answered by MJS_new last updated on 13/Aug/21 $$\mathrm{2021}=\mathrm{43}×\mathrm{47} \\…