Question Number 116603 by ZiYangLee last updated on 05/Oct/20 $$\mathrm{When}\:{f}\left({x}\right)\:\mathrm{divided}\:\mathrm{by}\:\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right), \\ $$$$\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\left({x}+\mathrm{3}\right) \\ $$$$\mathrm{When}\:{f}\left({x}\right)\:\mathrm{divided}\:\mathrm{by}\:\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}\right), \\ $$$$\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\left(\mathrm{2}{x}+\mathrm{1}\right) \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{if}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{divided} \\ $$$$\mathrm{by}\left({x}−\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}\right) \\ $$ Answered…
Question Number 116601 by ZiYangLee last updated on 05/Oct/20 $$\mathrm{If}\:\left({x},{y}\right)\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations} \\ $$$$\begin{cases}{\mid{x}\mid−{x}−{y}+\mathrm{2}=\mathrm{0}}\\{\mid{y}\mid+{y}+\mathrm{5}{x}=\mathrm{1}}\end{cases}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}+{y}.\: \\ $$ Answered by mr W last updated on 05/Oct/20 $$\begin{cases}{{x}−{x}−{y}+\mathrm{2}=\mathrm{0}}\\{{y}+{y}+\mathrm{5}{x}=\mathrm{1}}\end{cases}\:…
Question Number 116595 by ZiYangLee last updated on 05/Oct/20 $$\mathrm{If}\:\left(\mathrm{1}−\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{10}} ={a}_{\mathrm{0}} +{a}_{\mathrm{1}} {x}+{a}_{\mathrm{2}} {x}^{\mathrm{2}} +…+{a}_{\mathrm{20}} {x}^{\mathrm{20}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +…+{a}_{\mathrm{20}} \\ $$ Answered by…
Question Number 182082 by SEKRET last updated on 04/Dec/22 $$\:\:\:\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \underset{\boldsymbol{\mathrm{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\underset{\boldsymbol{\mathrm{r}}=\mathrm{0}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\left(\mathrm{1}\right)^{\boldsymbol{\mathrm{r}}} \:\centerdot\frac{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{r}}} \boldsymbol{\mathrm{y}}^{\mathrm{2022}\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right)} }{\left(\boldsymbol{\mathrm{n}}−\boldsymbol{\mathrm{r}}\right)!\left(\boldsymbol{\mathrm{r}}!\right)^{\mathrm{2}} \left(\mathrm{2022}\boldsymbol{\mathrm{y}}^{\mathrm{2022}} +\mathrm{2023}\right)^{\mathrm{2}} }\boldsymbol{\mathrm{dxdy}} \\…
Question Number 116538 by Khalmohmmad last updated on 04/Oct/20 $$\mathrm{x}+\sqrt{\mathrm{y}}=\mathrm{11} \\ $$$$\sqrt{\mathrm{x}}−\mathrm{y}=\mathrm{24} \\ $$$$\mathrm{x},\mathrm{y}=? \\ $$ Commented by mr W last updated on 04/Oct/20 $${no}\:{real}\:{solution}!…
Question Number 50991 by ggururajguru0219@gmail.com last updated on 23/Dec/18 $${gururaja}\:{chitra} \\ $$ Commented by peter frank last updated on 23/Dec/18 $$???? \\ $$ Terms of…
Question Number 50987 by naka3546 last updated on 23/Dec/18 $$\sqrt{{x}\:+\:{y}\:+\:\mathrm{9}}\:\:+\:\:\sqrt{{x}\:−\:{y}\:+\:\mathrm{8}}\:\:=\:\:\mathrm{33}^{\mathrm{2}} \\ $$$${x}\:>\:{y} \\ $$$${x},\:{y}\:\:\in\:\:\mathbb{Z}^{+} \\ $$$$\left({x}^{{y}} \:+\:{y}^{{x}} \right)\:\:{mod}\:\:\left(\mathrm{1000}\right)\:\:=\:\:? \\ $$ Commented by Rasheed.Sindhi last updated…
Question Number 116523 by zakirullah last updated on 04/Oct/20 Answered by Olaf last updated on 04/Oct/20 $$\left(\mathrm{1}+{a}\right)\begin{vmatrix}{\mathrm{1}+{b}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{1}+{c}}\end{vmatrix}−\begin{vmatrix}{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{1}+{c}}\end{vmatrix}+\begin{vmatrix}{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{1}+{b}}&{\mathrm{1}}\end{vmatrix} \\ $$$$ \\ $$$$=\:\left(\mathrm{1}+{a}\right)\left[\left(\mathrm{1}+{b}\right)\left(\mathrm{1}+{c}\right)−\mathrm{1}\right] \\ $$$$−\left(\mathrm{1}+{c}−\mathrm{1}\right)+\left(\mathrm{1}−\mathrm{1}−{b}\right) \\ $$$$…
Question Number 116516 by sandy_delta last updated on 04/Oct/20 Answered by som(math1967) last updated on 04/Oct/20 $$\left(\frac{\mathrm{1}}{\mathrm{a}+\mathrm{b}}+\frac{\mathrm{1}}{\mathrm{b}+\mathrm{c}}+\frac{\mathrm{1}}{\mathrm{c}+\mathrm{a}}\right)\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)=\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)×\frac{\mathrm{7}}{\mathrm{10}} \\ $$$$\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{a}+\mathrm{b}}+\frac{\mathrm{a}+\mathrm{b}+{c}}{\mathrm{b}+\mathrm{c}}+\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{c}+\mathrm{a}}=\frac{\mathrm{49}}{\mathrm{10}} \\ $$$$\left[\because\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{7}\right] \\ $$$$\mathrm{1}+\frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}}+\mathrm{1}+\frac{\mathrm{a}}{\mathrm{b}+\mathrm{c}}+\mathrm{1}+\frac{\mathrm{b}}{\mathrm{a}+\mathrm{c}}=\frac{\mathrm{49}}{\mathrm{10}} \\ $$$$\therefore\frac{\mathrm{a}}{\mathrm{b}+\mathrm{c}}+\frac{\mathrm{b}}{\mathrm{a}+\mathrm{c}}+\frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}}=\frac{\mathrm{49}}{\mathrm{10}}−\mathrm{3}=\frac{\mathrm{19}}{\mathrm{10}}\mathrm{ans}…
Question Number 116512 by bounhome last updated on 04/Oct/20 $${give}\:{x}\in{R}\:{and}\:{prove}\:{that}\:{if}\:\:{x}=\sqrt{\mathrm{2}{x}+\mathrm{3}}\:{than}\:{x}=\mathrm{3} \\ $$ Commented by bemath last updated on 04/Oct/20 $$\mathrm{x}^{\mathrm{2}} −\mathrm{2x}−\mathrm{3}=\mathrm{0}\:;\:\mathrm{x}\:\geqslant\mathrm{0} \\ $$$$\left(\mathrm{x}−\mathrm{3}\right)\left(\mathrm{x}+\mathrm{1}\right)\:=\:\mathrm{0}\Rightarrow\:\mathrm{x}\:=\:\mathrm{3} \\ $$…