Question Number 133053 by pete last updated on 18/Feb/21 $$\mathrm{When}\:\mathrm{the}\:\mathrm{polynomial}\:\mathrm{f}\left({x}\right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\left(\mathrm{x}−\mathrm{2}\right)\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{4}\:\mathrm{and}\:\mathrm{when}\:\mathrm{it}\:\mathrm{is}\:\mathrm{divided} \\ $$$$\left(\mathrm{x}−\mathrm{3}\right)\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{7}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{f}\left({x}\right) \\ $$$$\mathrm{may}\:\mathrm{be}\:\mathrm{written}\:\mathrm{in}\:\mathrm{the}\:\mathrm{formf}\left({x}\right)=\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)\mathrm{Q}\left(\mathrm{x}\right)+\mathrm{ax}+\mathrm{b}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right).\:\mathrm{If}\:\mathrm{also}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{cubic}\:\mathrm{function} \\ $$$$\mathrm{in}\:\mathrm{which}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} \:\mathrm{is}\:\mathrm{unity}\:\mathrm{and} \\ $$$$\mathrm{f}\left(\mathrm{1}\right)=\mathrm{1},\:\mathrm{determine}\:\mathrm{Q}\left(\mathrm{x}\right).…
Question Number 67501 by TawaTawa last updated on 28/Aug/19 $$\mathrm{Show}\:\mathrm{that}\:\:\mathrm{1n}^{\mathrm{3}} \:+\:\mathrm{2n}\:+\:\mathrm{3n}^{\mathrm{2}} \:\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{2}\:\mathrm{and}\:\mathrm{3}\:\mathrm{for}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{n}. \\ $$ Commented by Prithwish sen last updated on 28/Aug/19 $$\mathrm{Another}\:\mathrm{approch} \\ $$$$\mathrm{We}\:\mathrm{know}\:\mathrm{that}…
Question Number 67492 by MJS last updated on 28/Aug/19 $$\mathrm{please}\:\mathrm{check}\:\mathrm{my}\:\mathrm{comment}\:\mathrm{to}\:\mathrm{qu}.\:\mathrm{67471} \\ $$$$\mathrm{I}'\mathrm{ve}\:\mathrm{been}\:\mathrm{confusing}\:\mathrm{myself}… \\ $$ Commented by MJS last updated on 28/Aug/19 �� Terms of Service…
Question Number 1952 by prakash jain last updated on 25/Oct/15 $$\mathrm{Inequality}\:\mathrm{relation}\:\mathrm{starting}\:\mathrm{a}\:\mathrm{new}\:\mathrm{thread} \\ $$$$\frac{{x}^{{p}} }{{p}\left({p}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{p}}\geqslant\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{q}} \\ $$$${p}=\mathrm{2},\:{q}=\mathrm{1},\:{x}=\mathrm{1} \\ $$$$\frac{{x}^{{p}} }{{p}\left({p}+\mathrm{1}\right)}=\frac{\mathrm{1}}{\mathrm{6}} \\ $$$$\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\:\frac{{x}^{{p}}…
Question Number 1937 by Rasheed Soomro last updated on 25/Oct/15 $$\bullet{Is}\:\:\:'\Leftrightarrow'\:\:{necessary}\:{and}\:{suficient}\:{for}\:{two} \\ $$$${inequalities}\:{to}\:{be}\:{equivalent}? \\ $$$$\bullet{If}\:\:\boldsymbol{\mathrm{a}}>\boldsymbol{\mathrm{b}}\:: \\ $$$${Are}\:\:\boldsymbol{\mathrm{A}}>\boldsymbol{\mathrm{B}}\:{and}\:\boldsymbol{\mathrm{A}}+\boldsymbol{\mathrm{a}}\:>\:\boldsymbol{\mathrm{B}}+\boldsymbol{\mathrm{b}}\:{equivalent}? \\ $$ Answered by 123456 last updated on…
Question Number 133009 by shaker last updated on 18/Feb/21 Commented by liberty last updated on 18/Feb/21 $$?\:=\:\mathrm{1} \\ $$ Answered by Rasheed.Sindhi last updated on…
Question Number 1930 by Rasheed Soomro last updated on 24/Oct/15 $${f}\:'\left({x}\right)−{g}\left({x}\right)=\mathrm{0} \\ $$$${f}\left({x}\right)+{g}'\left({x}\right)=\mathrm{0} \\ $$$${f}\left({x}\right)=? \\ $$$${g}\left({x}\right)=? \\ $$ Answered by prakash jain last updated…
Question Number 1902 by Rasheed Soomro last updated on 23/Oct/15 $${f}^{\:\mathrm{2}} \left({x}\right)−{f}\left({x}^{\mathrm{2}} \right)=\mathrm{2}\:,\:{f}^{\:\mathrm{2}} \left({x}\right)\:{stands}\:{for}\:\left[{f}\left({x}\right)\right]^{\mathrm{2}} \\ $$$${f}\left({x}\right)=? \\ $$$$\left({If}\:{possible}\:{solve}\:{stepwise}\right) \\ $$ Commented by 123456 last updated…
Question Number 1899 by Yozzy last updated on 22/Oct/15 $${Consider}\:{the}\:{system}\:{of}\:{equations} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}{yz}+{zx}−\mathrm{5}{xy}=\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{yz}−{zx}+\mathrm{2}{xy}=\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{yz}−\mathrm{2}{zx}+\mathrm{6}{xy}=\mathrm{3}. \\ $$$${Show}\:{that}\:{xyz}=\pm\mathrm{6}\: \\ $$$${and}\:{find}\:{the}\:{possible}\:{values} \\ $$$${of}\:{x},{y}\:{and}\:{z}. \\ $$ Commented…
Question Number 132961 by danielasebhofoh last updated on 17/Feb/21 Answered by Ar Brandon last updated on 17/Feb/21 $$\mathrm{2n}−\mathrm{3}+\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{3}} {\sum}}\mathrm{k}=\mathrm{2n}−\mathrm{3}+\frac{\left(\mathrm{n}−\mathrm{3}\right)\left(\mathrm{n}−\mathrm{2}\right)}{\mathrm{2}} \\ $$$$\frac{\mathrm{4n}−\mathrm{6}+\mathrm{n}^{\mathrm{2}} −\mathrm{5n}+\mathrm{6}}{\mathrm{2}}=\frac{\mathrm{n}\left(\mathrm{n}−\mathrm{1}\right)}{\mathrm{2}} \\ $$$$\overset{\mathrm{n}}…