Question Number 176303 by infinityaction last updated on 16/Sep/22 Answered by cortano1 last updated on 16/Sep/22 $$\:\mathrm{let}\:\begin{cases}{{a}+{b}={x}}\\{{c}+{d}={y}}\end{cases}\Rightarrow{x}+{y}=\mathrm{2}\:;\:\mathrm{y}=\mathrm{2}−\mathrm{x} \\ $$$$\:\mathrm{M}=\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{x}\left(\mathrm{2}−\mathrm{x}\right)=\mathrm{2x}−\mathrm{x}^{\mathrm{2}} \:;\:\mathrm{x}\geqslant\mathrm{0} \\ $$$$\Rightarrow\mathrm{0}\leqslant\mathrm{f}\left(\mathrm{x}\right)\leqslant\mathrm{1} \\ $$$$\Rightarrow\mathrm{0}\leqslant\:\mathrm{M}\leqslant\mathrm{1} \\…
Question Number 176300 by cortano1 last updated on 16/Sep/22 $$\:\mathrm{coeffisien}\:\mathrm{x}\:^{\mathrm{2}\:} \:\mathrm{from}\:\left(\mathrm{2}+\frac{\mathrm{1}}{\mathrm{4}}\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{5}} \left(\mathrm{1}−\mathrm{4x}\right)^{\mathrm{5}} \\ $$ Commented by mr W last updated on 16/Sep/22 $${C}_{\mathrm{1}} ^{\mathrm{5}}…
Question Number 45213 by MrW3 last updated on 10/Oct/18 $${Find}\:\int\sqrt{{a}^{\mathrm{2}} −\mathrm{tan}^{\mathrm{2}} \:{x}}\:{dx}\:\left({with}\:{a}>\mathrm{0}\right) \\ $$$$\left({related}\:{to}\:{Q}\mathrm{45187}\right) \\ $$ Answered by MrW3 last updated on 10/Oct/18 $${I}=\int\sqrt{{a}^{\mathrm{2}} −\mathrm{tan}^{\mathrm{2}}…
Question Number 176276 by cortano1 last updated on 15/Sep/22 $$\mathrm{coeff}\:\mathrm{of}\:\mathrm{x}^{\mathrm{2}} \:\mathrm{from}\: \\ $$$$\:\:\left[\mathrm{6}+\mathrm{8x}−\frac{\mathrm{3}}{\mathrm{2}}\mathrm{x}^{\mathrm{2}} \:\right]^{\mathrm{5}} \: \\ $$ Answered by mr W last updated on 15/Sep/22…
Question Number 176279 by CrispyXYZ last updated on 15/Sep/22 $$\mathrm{Proof}\:\mathrm{that}\:: \\ $$$$\nexists{n}\in\mathbb{Z},\:{n}^{\mathrm{2}} +\mathrm{1}\equiv\mathrm{0}\left(\mathrm{mod}\:\mathrm{4}\right) \\ $$ Answered by mahdipoor last updated on 15/Sep/22 $${I}>\:{n}=\mathrm{2}{k}+\mathrm{1}\: \\ $$$$\Rightarrow{n}^{\mathrm{2}}…
Question Number 45204 by Meritguide1234 last updated on 10/Oct/18 Commented by Tawa1 last updated on 10/Oct/18 $$. \\ $$ Commented by Meritguide1234 last updated on…
Question Number 110728 by Study last updated on 30/Aug/20 $${a}+{b}=\mathrm{9}\:\:,\:\:{ab}=\mathrm{20}\:\:\:\:\:{a}−{b}=? \\ $$ Answered by som(math1967) last updated on 30/Aug/20 $$\left(\mathrm{a}−\mathrm{b}\right)^{\mathrm{2}} =\left(\mathrm{a}+\mathrm{b}\right)^{\mathrm{2}} −\mathrm{4ab} \\ $$$$\left(\mathrm{a}−\mathrm{b}\right)^{\mathrm{2}} =\mathrm{81}−\mathrm{80}…
Question Number 110727 by Study last updated on 30/Aug/20 $${a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{10}\:\:\:,\:\:{ab}=\mathrm{13}\:\:\:,\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} =? \\ $$ Answered by mr W last updated on 30/Aug/20 $$\left({a}+{b}\right)^{\mathrm{2}}…
Question Number 110725 by Study last updated on 30/Aug/20 $${x}^{{x}^{{x}^{{x}^{…} } } } =\mathrm{2}\:\:\:\:\:\:\:{x}=? \\ $$ Answered by Her_Majesty last updated on 30/Aug/20 $${x}^{{x}^{{x}^{…} }…
Question Number 110715 by Aina Samuel Temidayo last updated on 30/Aug/20 $$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\mathrm{are}\:\mathrm{unit}\:\mathrm{digits}\:\mathrm{whose} \\ $$$$\mathrm{pairwise}\:\mathrm{sums}\:\mathrm{form}\:\mathrm{an}\:\mathrm{arithmetic} \\ $$$$\mathrm{progression}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\:\mathrm{is} \\ $$$$\mathrm{even},\:\mathrm{find}\:\mathrm{the}\:\mathrm{common}\:\mathrm{positive} \\ $$$$\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{arithmetic} \\ $$$$\mathrm{progression}. \\ $$ Commented…