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Question Number 142379 by Rexzie last updated on 30/May/21 $${Show}\:{that}\:\mathrm{1}+\mathrm{3}{n}<{n}^{\mathrm{2}} \:{for}\:{every}\:{positive}\:{integer}\:{n}\geqslant\mathrm{4} \\ $$ Commented by mr W last updated on 30/May/21 $${n}\geqslant\mathrm{4} \\ $$$${n}^{\mathrm{2}} \geqslant\mathrm{4}{n}=\mathrm{3}{n}+{n}\geqslant\mathrm{3}{n}+\mathrm{4}>\mathrm{3}{n}+\mathrm{1}…
Question Number 11302 by uni last updated on 19/Mar/17 $$\frac{\mathrm{sin10x}−\mathrm{sin6x}−\mathrm{sin2x}}{\mathrm{sin9x}−\mathrm{sin7x}−\mathrm{sinx}}=? \\ $$ Answered by sandy_suhendra last updated on 20/Mar/17 $$=\frac{\left(\mathrm{sin10x}−\mathrm{sin6x}\right)−\mathrm{sin2x}}{\left(\mathrm{sin9x}−\mathrm{sin7x}\right)−\mathrm{sinx}} \\ $$$$=\frac{\mathrm{2cos8xsin2x}−\mathrm{sin2x}}{\mathrm{2cos8xsinx}−\mathrm{sinx}} \\ $$$$=\frac{\mathrm{sin2x}\left(\mathrm{2cos8x}−\mathrm{1}\right)}{\mathrm{sinx}\left(\mathrm{2cos8x}−\mathrm{1}\right)} \\…
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Question Number 11285 by uni last updated on 19/Mar/17 $$\frac{{sin}\mathrm{20}}{{cos}\mathrm{80}−{tan}\mathrm{30}×{sin}\mathrm{80}}=? \\ $$ Answered by mrW1 last updated on 19/Mar/17 $$=\frac{{sin}\mathrm{20}}{{cos}\mathrm{80}−\frac{\mathrm{sin}\:\mathrm{30}}{\mathrm{cos}\:\mathrm{30}}×{sin}\mathrm{80}} \\ $$$$=\frac{{sin}\mathrm{20}×\mathrm{cos}\:\mathrm{30}}{\mathrm{cos}\:\mathrm{30}×{cos}\mathrm{80}−\mathrm{sin}\:\mathrm{30}×{sin}\mathrm{80}} \\ $$$$=\frac{{sin}\mathrm{20}×\mathrm{cos}\:\mathrm{30}}{\mathrm{cos}\:\mathrm{110}} \\…
Question Number 142348 by mathdanisur last updated on 30/May/21 $$\frac{\sqrt[{\mathrm{6}}]{\mathrm{4}−\sqrt{\mathrm{15}}}}{\:\sqrt{\mathrm{4}−\sqrt{\mathrm{15}}}\:\centerdot\:\sqrt[{\mathrm{3}}]{\mathrm{4}+\sqrt{\mathrm{15}}}}\:=\:? \\ $$ Answered by bramlexs22 last updated on 30/May/21 $$\:\frac{\sqrt[{\mathrm{6}\:}]{\mathrm{4}−\sqrt{\mathrm{15}}}}{\:\sqrt[{\mathrm{6}\:}]{\left(\mathrm{4}−\sqrt{\mathrm{15}}\right)^{\mathrm{3}} }.\sqrt[{\mathrm{3}\:}]{\mathrm{4}+\sqrt{\mathrm{15}}}}\:= \\ $$$$\:\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{6}}]{\left(\mathrm{4}−\sqrt{\mathrm{15}}\right)^{\mathrm{2}} }.\sqrt[{\mathrm{3}\:}]{\mathrm{4}+\sqrt{\mathrm{15}}}}\:= \\…
Question Number 11274 by uni last updated on 18/Mar/17 $$\frac{{sin}\mathrm{20}+\sqrt{\mathrm{3}}×{cos}\mathrm{20}}{{cos}\mathrm{10}}=? \\ $$ Answered by bahmanfeshki1 last updated on 18/Mar/17 $${sin}\:\mathrm{20}+\sqrt{\mathrm{3}}×{cos}\mathrm{20}=\mathrm{2}×{cos}\left(\mathrm{30}−\mathrm{20}\right)=\mathrm{2}×{cos}\mathrm{10} \\ $$$$\Rightarrow\frac{\mathrm{2}{cos}\mathrm{10}}{{cos}\mathrm{10}}=\mathrm{2} \\ $$ Answered…
Question Number 11268 by tawa last updated on 18/Mar/17 $$\mathrm{Solve}\:\mathrm{x},\:\mathrm{y}\:\mathrm{and}\:\mathrm{z}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{p},\:\mathrm{q}\:\mathrm{and}\:\mathrm{r} \\ $$$$\mathrm{yz}\:=\:\mathrm{py}\:+\:\mathrm{qz}\:\:\:\:\:……..\:\left(\mathrm{i}\right) \\ $$$$\mathrm{zx}\:=\:\mathrm{qz}\:+\:\mathrm{rx}\:\:\:\:\:\:…….\:\left(\mathrm{ii}\right) \\ $$$$\mathrm{xy}\:=\:\mathrm{rx}\:+\:\mathrm{py}\:\:\:\:\:…….\:\left(\mathrm{iii}\right) \\ $$ Answered by mrW1 last updated on 18/Mar/17…
Question Number 11263 by tawa last updated on 18/Mar/17 $$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}\:\mathrm{in}\:\mathrm{the}\:\mathrm{equation}\:. \\ $$$$\mathrm{625}^{\mathrm{x}\:−\:\mathrm{5}} \:=\:\mathrm{200}\sqrt{\mathrm{x}^{\mathrm{3}} } \\ $$ Answered by ajfour last updated on 18/Mar/17 $$\mathrm{5}^{\mathrm{4}\left(\mathrm{x}−\mathrm{5}\right)} \:=\:\mathrm{200x}^{\mathrm{3}/\mathrm{2}}…