Question Number 202198 by MATHEMATICSAM last updated on 22/Dec/23 $$\mathrm{If}\:\alpha,\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{x}^{\mathrm{2}} \:+\:{ax}\:−\:{b}\:=\:\mathrm{0}\: \\ $$$$\mathrm{and}\:\gamma,\:\delta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{x}^{\mathrm{2}} \:+\:{ax}\:+\:{c}\:=\:\mathrm{0}\: \\ $$$$\mathrm{then}\:\mathrm{show}\:\mathrm{that}\:\frac{\alpha\:−\:\gamma}{\beta\:−\:\gamma}\:=\:\frac{\beta\:−\:\delta}{\alpha\:−\:\delta}\:\:. \\ $$ Answered by MM42 last updated on 22/Dec/23…
Question Number 202167 by tri26112004 last updated on 22/Dec/23 $$\underset{\mathrm{0}} {\int}^{\mathrm{1}} \underset{{x}} {\int}^{\mathrm{1}} {sin}\left({y}^{\mathrm{2}} \right){dydx}\:=\:¿ \\ $$ Answered by mnjuly1970 last updated on 22/Dec/23 $$\:{answer}:=\:{sin}^{\mathrm{2}}…
Question Number 202160 by Calculusboy last updated on 22/Dec/23 Answered by MathematicalUser2357 last updated on 22/Dec/23 $$\frac{\mathrm{1}}{\mathrm{7}!}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{d}^{\mathrm{7}} }{{dx}^{\mathrm{7}} }\:\left(\mathrm{sin}\left(\mathrm{tan}\:{x}\right)−\mathrm{tan}\left(\mathrm{sin}\:{x}\right)\right) \\ $$ Terms of Service…
Question Number 202193 by MATHEMATICSAM last updated on 22/Dec/23 $$\mathrm{If}\:\alpha,\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{x}^{\mathrm{2}} \:+\:{ax}\:+\:{b}\:=\:\:\mathrm{0}\: \\ $$$$\mathrm{and}\:\alpha\:+\:\delta,\:\beta\:+\:\delta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$${x}^{\mathrm{2}} \:+\:{px}\:+\:{q}\:=\:\mathrm{0}\:\mathrm{then}\:\mathrm{show}\:\mathrm{that}\: \\ $$$${a}^{\mathrm{2}} \:−\:{p}^{\mathrm{2}} \:=\:\mathrm{4}\left({b}\:−\:{q}\right). \\ $$ Answered by Rasheed.Sindhi…
Question Number 202161 by Calculusboy last updated on 22/Dec/23 Answered by som(math1967) last updated on 22/Dec/23 $$\frac{\left({m}+\mathrm{1}\right)!\left(\mathrm{1}+\mathrm{3}+\mathrm{5}+…+\mathrm{2}{m}+\mathrm{3}\right.}{\mathrm{2}{m}\left({m}+\mathrm{2}\right)\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+…+{m}+\mathrm{1}\right)} \\ $$$$=\frac{\left({m}+\mathrm{1}\right)!\frac{{m}+\mathrm{2}}{\mathrm{2}}×\mathrm{2}\left\{\mathrm{1}+\left({m}+\mathrm{2}−\mathrm{1}\right)\right\}}{\mathrm{2}{m}\left({m}+\mathrm{2}\right)\frac{\left({m}+\mathrm{1}\right)\left({m}+\mathrm{2}\right)}{\mathrm{2}}} \\ $$$$=\frac{\left({m}+\mathrm{1}\right)!\left({m}+\mathrm{2}\right)^{\mathrm{2}} }{{m}\left({m}+\mathrm{2}\right)^{\mathrm{2}} \left({m}+\mathrm{1}\right)} \\ $$$$=\frac{{m}\left({m}+\mathrm{1}\right)×\left({m}−\mathrm{1}\right)!}{{m}\left({m}+\mathrm{1}\right)}…
Question Number 202184 by mr W last updated on 22/Dec/23 $${with}\:{f}\left({x}\right)={x}^{\mathrm{2}} +\mathrm{12}{x}+\mathrm{30}\:{and}\:{x}\in{R} \\ $$$${solve}\:{f}\left({f}\left({f}\left({f}\left({f}\left({x}\right)\right)\right)\right)\right)=\mathrm{0} \\ $$ Answered by cortano12 last updated on 22/Dec/23 $$\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{x}+\mathrm{6}\right)^{\mathrm{2}} −\mathrm{6}\:\Rightarrow\mathrm{x}=\sqrt{\mathrm{f}\left(\mathrm{x}\right)+\mathrm{6}}−\mathrm{6}…
Question Number 202153 by sonukgindia last updated on 22/Dec/23 Answered by witcher3 last updated on 22/Dec/23 $$\mathrm{ln}\left(\mathrm{x}\right)\mathrm{y}'+\frac{\mathrm{y}}{\mathrm{x}}=\left(\mathrm{ln}\left(\mathrm{x}\right)\right)\mathrm{y}'+\left(\mathrm{ln}\left(\mathrm{x}\right)\right)'\mathrm{y} \\ $$$$=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ln}\left(\mathrm{x}\right)\mathrm{y}\right) \\ $$$$\Leftrightarrow\begin{cases}{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{y}.\left(\mathrm{ln}\left(\mathrm{x}\right)\right)=\mathrm{x}^{\mathrm{2}} \mathrm{sin}\left(\mathrm{x}\right)\right.}\\{\mathrm{y}\left(\pi\right)=\frac{\pi^{\mathrm{2}} }{\mathrm{ln}\left(\pi\right)}}\end{cases} \\ $$$$\Rightarrow\int_{\pi}…
Question Number 202154 by cortano12 last updated on 22/Dec/23 $$\:\:\:\downharpoonleft\underline{\:} \\ $$ Commented by mr W last updated on 22/Dec/23 $$\mathrm{f}\left(\mathrm{xy}+\mathrm{1}\right)=\:\:\boldsymbol{\nu} \\ $$ Answered by…
Question Number 202187 by sonukgindia last updated on 22/Dec/23 Commented by mr W last updated on 22/Dec/23 $${is}\:{AB}={AC}\:? \\ $$ Commented by AST last updated…
Question Number 202212 by Calculusboy last updated on 22/Dec/23 Commented by BOYQOBILOV last updated on 23/Dec/23 $$ \\ $$ Answered by shunmisaki007 last updated on…