Question Number 23928 by Tinkutara last updated on 10/Nov/17 $$\mathrm{Find}\:^{{n}} {C}_{\mathrm{1}} \:−\:\frac{\mathrm{1}}{\mathrm{2}}\:^{{n}} {C}_{\mathrm{2}} \:+\:\frac{\mathrm{1}}{\mathrm{3}}\:^{{n}} {C}_{\mathrm{3}} \:−\:….\:+ \\ $$$$\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \frac{\mathrm{1}}{{n}}\:\centerdot\:^{{n}} {C}_{{n}} . \\ $$ Commented by…
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Question Number 154994 by mathdanisur last updated on 24/Sep/21 $$\mathrm{Find} \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{x}^{\mathrm{2}} \:\mathrm{tan}^{-\mathrm{1}} \left(\mathrm{2x}\right)\mathrm{ln}^{\mathrm{2}} \left(\mathrm{3x}\right)\mathrm{dx}=? \\ $$ Answered by phanphuoc last updated on…
Question Number 89454 by student work last updated on 17/Apr/20 Commented by Tony Lin last updated on 17/Apr/20 $$\mathrm{32}{mod}\mathrm{7}=\mathrm{4} \\ $$$$\mathrm{32}^{\mathrm{2}} {mod}\mathrm{7}=\mathrm{2} \\ $$$$\mathrm{32}^{\mathrm{3}} {mod}\mathrm{7}=\mathrm{1}…
Question Number 154986 by mathdanisur last updated on 23/Sep/21 $$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}\:\mathrm{in}\:\mathbb{R} \\ $$$$\mathrm{2x}\:=\:\frac{\mathrm{y}^{\mathrm{2}} \:-\:\mathrm{4y}\:+\:\mathrm{1}}{\mathrm{y}^{\mathrm{2}} \:-\:\mathrm{y}\:+\:\mathrm{1}} \\ $$$$\mathrm{y}\:=\:\frac{-\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{6x}\:-\:\mathrm{1}}{\mathrm{3x}^{\mathrm{2}} \:-\:\mathrm{2x}\:+\:\mathrm{3}} \\ $$ Answered by Rasheed.Sindhi last updated…
Question Number 154973 by mathdanisur last updated on 23/Sep/21 $$\mathrm{If}\:\:\mathrm{x}\:+\:\frac{\mathrm{7}}{\:\sqrt{\mathrm{x}}}\:=\:\mathrm{50} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression} \\ $$$$\sqrt{\mathrm{x}}\:-\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{x}}}\:=\:? \\ $$ Answered by mr W last updated on 23/Sep/21 $${let}\:{t}=\sqrt{{x}}>\mathrm{0}…
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Question Number 154980 by mathdanisur last updated on 23/Sep/21 $$\mathrm{if}\:\:\boldsymbol{\mathrm{a}}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{b}}\:\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{determine}\:\mathrm{a}\:\mathrm{necessary}\:\mathrm{and}\:\mathrm{sufficient} \\ $$$$\mathrm{condition}\:\mathrm{so}\:\mathrm{as}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{x}^{\mathrm{2}} \:+\:\frac{\mathrm{a}\sqrt{\mathrm{b}}}{\mathrm{x}}\:=\:\mathrm{a}\:+\:\mathrm{b} \\ $$$$\mathrm{to}\:\mathrm{have}\:\mathrm{three}\:\mathrm{real}\:\mathrm{distinct}\:\mathrm{roots}. \\ $$ Answered by mr W…
Question Number 154959 by mathdanisur last updated on 23/Sep/21 $$\mathrm{Consider}\:\mathrm{the}\:\mathrm{polynomial} \\ $$$$\mathrm{P}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{10}} -\mathrm{6x}^{\mathrm{9}} -\mathrm{11x}^{\mathrm{8}} +\mathrm{3x}^{\mathrm{2}} -\mathrm{18x}-\mathrm{7} \\ $$$$\mathrm{Calculate}:\:\:\mathrm{P}\left(\mathrm{3}+\mathrm{2}\sqrt{\mathrm{5}}\right) \\ $$ Commented by prakash jain last…
Question Number 23884 by Tinkutara last updated on 09/Nov/17 $$\mathrm{If}\:{C}_{{r}} \:\mathrm{stands}\:\mathrm{for}\:^{{n}} {C}_{{r}} \:=\:\frac{{n}!}{{r}!\:{n}\:−\:{r}!}\:\mathrm{and} \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}.{C}_{{r}} ^{\mathrm{2}} \:=\:\lambda\:\mathrm{for}\:{n}\:\geqslant\:\mathrm{2},\:\mathrm{then}\:\lambda\:\mathrm{is}\:\mathrm{divisible} \\ $$$$\mathrm{by} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{3}\:\left({n}\:−\:\mathrm{1}\right) \\ $$$$\left(\mathrm{2}\right)\:{n}\:+\:\mathrm{1}…