Question Number 150585 by bekzodjumayev last updated on 13/Aug/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\underset{\mathrm{0}} {\overset{\pi} {\int}}{cosx}^{\mathrm{2}} {dx}}{{x}}=?? \\ $$$${Help} \\ $$ Commented by mr W last updated on…
Question Number 19507 by Tinkutara last updated on 12/Aug/17 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{three}\:\mathrm{points}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{are} \\ $$$$\mathrm{collinear}\:\mathrm{if}\:\begin{vmatrix}{{z}_{\mathrm{1}} }&{\bar {{z}}_{\mathrm{1}} }&{\mathrm{1}}\\{{z}_{\mathrm{2}} }&{\bar {{z}}_{\mathrm{2}} }&{\mathrm{1}}\\{{z}_{\mathrm{3}} }&{\bar {{z}}_{\mathrm{3}} }&{\mathrm{1}}\end{vmatrix}=\:\mathrm{0} \\…
Question Number 19508 by Tinkutara last updated on 12/Aug/17 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{perpendicular} \\ $$$$\mathrm{drawn}\:\mathrm{from}\:\mathrm{the}\:\mathrm{point}\:{z}_{\mathrm{0}} \:\mathrm{to}\:\mathrm{the}\:\mathrm{straight} \\ $$$$\mathrm{line}\:\bar {\alpha}{z}\:+\:\alpha\bar {{z}}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{is} \\ $$$${p}\:=\:\mid\frac{\bar {\alpha}{z}_{\mathrm{0}} \:+\:\alpha\bar {{z}}_{\mathrm{0}} \:+\:{c}}{\mathrm{2}\:\mid\alpha\mid}\mid. \\ $$…
Question Number 19505 by Tinkutara last updated on 12/Aug/17 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{two}\:\mathrm{straight}\:\mathrm{lines}\:\mathrm{with} \\ $$$$\mathrm{complex}\:\mathrm{slopes}\:\mu_{\mathrm{1}} \:\mathrm{and}\:\mu_{\mathrm{2}} \:\mathrm{are}\:\mathrm{parallel} \\ $$$$\mathrm{and}\:\mathrm{perpendicular}\:\mathrm{according}\:\mathrm{as}\:\mu_{\mathrm{1}} \:=\:\mu_{\mathrm{2}} \\ $$$$\mathrm{and}\:\mu_{\mathrm{1}} \:+\:\mu_{\mathrm{2}} \:=\:\mathrm{0}.\:\mathrm{Hence}\:\mathrm{if}\:\mathrm{the}\:\mathrm{straight} \\ $$$$\mathrm{lines}\:\bar {\alpha}{z}\:+\:\alpha\bar {{z}}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{and}\:\bar…
Question Number 19506 by Tinkutara last updated on 12/Aug/17 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line} \\ $$$$\mathrm{joining}\:\mathrm{the}\:\mathrm{points}\:{z}_{\mathrm{1}} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{can}\:\mathrm{be}\:\mathrm{put} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:{z}\:=\:{tz}_{\mathrm{1}} \:+\:\left(\mathrm{1}\:−\:{t}\right){z}_{\mathrm{2}} ,\:\mathrm{where} \\ $$$${t}\:\mathrm{is}\:\mathrm{a}\:\mathrm{parameter}. \\ $$ Answered by ajfour…
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 $$\overline {\mathrm{abcd}}\in\boldsymbol{\mathrm{A}};\:\:\frac{\mathrm{a}}{\mathrm{d}\:+\:\mathrm{1}}\:=\:\frac{\mathrm{c}\:-\:\mathrm{b}}{\mathrm{c}}\:=\:\frac{\mathrm{a}\:+\:\mathrm{b}}{\mathrm{b}\:+\:\mathrm{c}};\mathrm{A}=? \\…
Question Number 150559 by mathdanisur last updated on 13/Aug/21 $$\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 $$\mathrm{if}\:{x}=\mathrm{0}:\:{y}^{\mathrm{4}} =\mathrm{2022}\:\Rightarrow\:{y}\approx\mathrm{6}.\mathrm{7}…
Question Number 150558 by mathdanisur last updated on 13/Aug/21 $$\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 $$\mathrm{2021}=\mathrm{43}×\mathrm{47} \\…
Question Number 150549 by mathdanisur last updated on 13/Aug/21 $$\mathrm{Prove}\:\mathrm{that}:\:\:\forall\mathrm{n}\in\mathbb{N} \\ $$$$\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\prod}}\mathrm{k}!\:\centerdot\:\mathrm{k}^{\boldsymbol{\mathrm{n}}−\boldsymbol{\mathrm{k}}+\mathrm{1}} \:\leqslant\:\left(\frac{\mathrm{n}+\mathrm{2}}{\mathrm{3}}\right)^{\boldsymbol{\mathrm{n}}\centerdot\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com