Question Number 208468 by Tawa11 last updated on 16/Jun/24 Answered by A5T last updated on 17/Jun/24 $$\frac{{OD}×{OC}}{{AO}×{OB}}=\frac{\mathrm{1}}{{k}^{\mathrm{2}} }=\frac{\mathrm{9}}{\mathrm{25}}\Rightarrow{k}=\frac{\mathrm{5}}{\mathrm{3}} \\ $$$$\frac{{AB}}{{DC}}=\frac{\mathrm{5}}{\mathrm{3}};\:{AB}=\mathrm{5}{x},{DC}=\mathrm{3}{x} \\ $$$$\left[{ABCD}\right]=\frac{\left(\mathrm{8}{x}\right){h}}{\mathrm{2}}=\mathrm{4}{xh} \\ $$$$\left[{OBC}\right]=\left[{OAD}\right]=\frac{\mathrm{5}{xh}}{\mathrm{2}}−\mathrm{25}=\frac{\mathrm{3}{xh}}{\mathrm{2}}−\mathrm{9}\Rightarrow{xh}=\mathrm{16} \\…
Question Number 208469 by hardmath last updated on 16/Jun/24 $$\mathrm{Find}:\:\:\:\:\:\frac{\mathrm{61}^{\mathrm{3}} \:\:+\:\:\mathrm{24}^{\mathrm{3}} }{\mathrm{61}^{\mathrm{3}} \:\:+\:\:\mathrm{37}^{\mathrm{3}} }\:\:=\:\:? \\ $$ Answered by Frix last updated on 16/Jun/24 $${a}=\mathrm{61};\:{b}=\mathrm{24};\:{a}−{b}=\mathrm{37} \\…
Question Number 208437 by efronzo1 last updated on 16/Jun/24 Answered by som(math1967) last updated on 16/Jun/24 $$\:{a}={sin}\mathrm{84}+{cos}\mathrm{66} \\ $$$$\Rightarrow{a}={sin}\mathrm{84}+{sin}\mathrm{24} \\ $$$$\Rightarrow{a}=\mathrm{2}{sin}\mathrm{54}{cos}\mathrm{30} \\ $$$$\:\therefore{a}=\mathrm{2}×\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{4}}×\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}=\frac{\sqrt{\mathrm{3}}\left(\sqrt{\mathrm{5}}+\mathrm{1}\right)}{\mathrm{4}} \\ $$$${b}={sin}\mathrm{48}−{sin}\mathrm{12}…
Question Number 208465 by Tawa11 last updated on 16/Jun/24 Answered by A5T last updated on 17/Jun/24 $$\frac{\mathrm{3}\sqrt{\mathrm{2}}}{{s}}=\frac{\mathrm{3}\sqrt{\mathrm{5}}}{\mathrm{10}}\Rightarrow{s}=\frac{\mathrm{10}\sqrt{\mathrm{10}}}{\mathrm{5}}=\mathrm{2}\sqrt{\mathrm{10}}\Rightarrow{s}^{\mathrm{2}} =\mathrm{40} \\ $$ Commented by Tawa11 last updated…
Question Number 208466 by Tawa11 last updated on 16/Jun/24 Answered by A5T last updated on 17/Jun/24 Commented by A5T last updated on 17/Jun/24 $$\mathrm{4}{x}+\mathrm{3}{x}+\mathrm{2}{x}=\mathrm{180}\Rightarrow{x}=\mathrm{20}° \\…
Question Number 208467 by Tawa11 last updated on 16/Jun/24 Answered by A5T last updated on 17/Jun/24 Commented by A5T last updated on 17/Jun/24 $${sin}\theta=\frac{\mathrm{3}\sqrt{\mathrm{10}}}{\:\mathrm{10}}\Rightarrow{cos}\theta=\frac{\sqrt{\mathrm{10}}}{\mathrm{10}} \\…
Question Number 208463 by hardmath last updated on 16/Jun/24 $$\mathrm{Compare}\:\mathrm{it}: \\ $$$$\mathrm{a}\:=\:\mathrm{log}_{\mathrm{3}} \:\mathrm{4} \\ $$$$\mathrm{b}\:=\:\mathrm{log}_{\mathrm{5}} \:\mathrm{6} \\ $$$$\mathrm{c}\:=\:\mathrm{log}_{\mathrm{6}} \:\mathrm{2} \\ $$ Answered by Frix last…
Question Number 208453 by hardmath last updated on 16/Jun/24 $$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\left(\mathrm{2a}\:+\:\mathrm{1}\right)\centerdot\mathrm{x}\:+\:\mathrm{1}}{\mathrm{x}\:−\:\mathrm{a}}\:\:\:\mathrm{and}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right) \\ $$$$\mathrm{Find}:\:\:\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{3}\:=\:? \\ $$ Answered by efronzo1 last updated on 16/Jun/24 $$\:\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right)=\frac{\mathrm{ax}+\mathrm{1}}{\mathrm{x}−\left(\mathrm{2a}+\mathrm{1}\right)}\:=\:\frac{\left(\mathrm{2a}+\mathrm{1}\right)\mathrm{x}+\mathrm{1}}{\mathrm{x}−\mathrm{a}}…
Question Number 208455 by efronzo1 last updated on 16/Jun/24 Commented by mr W last updated on 16/Jun/24 $${the}\:{green}\:{one}\:{even}\:{doesn}'{t}\:{need}\:{to}\:{be}\: \\ $$$${a}\:{semicircle}. \\ $$ Commented by mr…
Question Number 208412 by efronzo1 last updated on 15/Jun/24 Answered by A5T last updated on 15/Jun/24 $$\mathrm{8}=\mathrm{24}^{\frac{\mathrm{1}}{{a}}} ,\mathrm{27}=\mathrm{24}^{\frac{\mathrm{1}}{{b}}} ,\mathrm{64}=\mathrm{24}^{\frac{\mathrm{1}}{{c}}} \\ $$$$\Rightarrow\mathrm{24}^{\mathrm{3}} =\mathrm{8}×\mathrm{27}×\mathrm{64}=\mathrm{24}^{\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}=\frac{{ab}+{bc}+{ca}}{{abc}}\right)} \\ $$$$\Rightarrow\frac{{ab}+{bc}+{ca}}{{abc}}=\mathrm{3}\Rightarrow\frac{\mathrm{2022}{abc}}{{ab}+{bc}+{ca}}=\mathrm{2022}×\frac{\mathrm{1}}{\mathrm{3}}=\mathrm{674} \\…