Question Number 12169 by indreshpatelindresh@435gmail.i last updated on 15/Apr/17 $$\mathrm{If}\:\:\mathrm{tan}\:\alpha\:\mathrm{equals}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{solution}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{inequality}\:\:\mathrm{4}{x}^{\mathrm{2}} −\mathrm{16}{x}+\mathrm{15}<\mathrm{0}\:\mathrm{and}\: \\ $$$$\mathrm{cos}\:\beta\:\:\mathrm{equals}\:\mathrm{to}\:\mathrm{the}\:\mathrm{slope}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bisector} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{quadrant},\:\mathrm{then}\: \\ $$$$\mathrm{sin}\:\left(\alpha+\beta\right)\:\mathrm{sin}\:\left(\alpha−\beta\right)\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$ Answered by mrW1 last…
Question Number 12170 by indreshpatelindresh@435gmail.i last updated on 15/Apr/17 $$\mathrm{In}\:\mathrm{any}\:\bigtriangleup{ABC}, \\ $$$$\:\mathrm{2}\left({bc}\:\mathrm{cos}\:{A}+{ca}\:\mathrm{cos}\:{B}+{ab}\:\mathrm{cos}\:{C}\right)\:= \\ $$ Answered by mrW1 last updated on 15/Apr/17 $${a}^{\mathrm{2}} ={b}^{\mathrm{2}} +{c}^{\mathrm{2}} −\mathrm{2}{bc}\mathrm{cos}\:{A}…
Question Number 12160 by indreshpatelindresh@435gmail.i last updated on 15/Apr/17 $$\mathrm{Two}\:\mathrm{angles}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{are}\:\mathrm{cot}^{−\mathrm{1}} \mathrm{2} \\ $$$$\mathrm{and}\:\mathrm{cot}^{−\mathrm{1}} \mathrm{3}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{third}\:\mathrm{angle}\:\mathrm{is} \\ $$ Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 15/Apr/17 $${tg}\left({cot}^{−\mathrm{1}} \mathrm{2}+{cot}^{−\mathrm{1}}…
Question Number 11979 by 786786AM last updated on 08/Apr/17 $$\mathrm{If}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}\:{p}\:\mathrm{terms},\:\mathrm{first}\:\:{q}\:\mathrm{terms}\:\mathrm{and} \\ $$$$\mathrm{first}\:{r}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}\:\mathrm{be}\:\:{x},\:{y}\:\:\mathrm{and}\:{z}\: \\ $$$$\mathrm{respectively}.\:\mathrm{Then} \\ $$$$\frac{{x}}{{p}}\left({q}−{r}\right)\:+\:\frac{{y}}{{q}}\left({r}−{p}\right)\:+\:\frac{{z}}{{r}}\left({p}−{q}\right)\:\:\mathrm{is} \\ $$ Answered by ajfour last updated on 09/Apr/17…
Question Number 11977 by 786786AM last updated on 08/Apr/17 $$\mathrm{If}\:\:{S}_{\mathrm{1}} ,\:{S}_{\mathrm{2}} ,\:{S}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:{n},\:\mathrm{2}{n},\:\mathrm{3}{n}\:\mathrm{terms} \\ $$$$\mathrm{respectively}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP},\:\mathrm{then} \\ $$ Answered by ajfour last updated on 08/Apr/17 $${S}_{\mathrm{1}}…
Question Number 11943 by 010418 last updated on 06/Apr/17 $$\mathrm{The}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$${x}^{\mathrm{2}} +\:{x}\:+\:\mathrm{1}\:=\:\mathrm{1}\:\:\:\mathrm{is} \\ $$ Answered by ajfour last updated on 06/Apr/17 $$\Rightarrow\:{x}\left({x}+\mathrm{1}\right)=\mathrm{0} \\ $$$${this}\:{happens}\:{when}\:…
Question Number 11889 by murtaza8 last updated on 04/Apr/17 $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{in}\:\mathrm{which}\:\mathrm{8}\: \\ $$$$\mathrm{different}\:\mathrm{flowers}\:\mathrm{can}\:\mathrm{be}\:\mathrm{strung}\:\mathrm{to} \\ $$$$\mathrm{form}\:\mathrm{a}\:\mathrm{garland}\:\mathrm{so}\:\mathrm{that}\:\mathrm{4}\:\mathrm{particular} \\ $$$$\mathrm{flowers}\:\mathrm{are}\:\mathrm{never}\:\mathrm{separated}\:\mathrm{is} \\ $$ Answered by ajfour last updated on 04/Apr/17…
Question Number 11744 by mnbvcxz89 last updated on 30/Mar/17 $$\mathrm{If}\:\:\:\mathrm{5}^{\mathrm{n}+\mathrm{2}} =\:\mathrm{625}\:,\:\mathrm{find}\:\left[\mathrm{5}\left(\mathrm{n}+\mathrm{3}\right)\right]^{\mathrm{1}/{n}} . \\ $$ Answered by mrW1 last updated on 30/Mar/17 $$\mathrm{5}^{{n}+\mathrm{2}} =\mathrm{625}=\mathrm{5}^{\mathrm{4}} \\ $$$$\Rightarrow{n}+\mathrm{2}=\mathrm{4}…
Question Number 11737 by Tadagbé last updated on 30/Mar/17 $$\int\:\frac{\mathrm{1}}{{x}\left({x}^{{n}} +\mathrm{1}\right)}\:{dx}\:= \\ $$ Answered by ajfour last updated on 30/Mar/17 $${x}^{{n}} ={t}\:\:\:\Rightarrow\:{n}\mathrm{ln}\:{x}=\mathrm{ln}\:{t} \\ $$$$\frac{{ndx}}{{x}}=\frac{{dt}}{{t}} \\…
Question Number 11419 by fadim727 last updated on 25/Mar/17 $$\mathrm{If}\:\mathrm{A}\:=\:\left\{\mathrm{whole}\:\mathrm{numbers}\right\}\:\mathrm{and}\: \\ $$$$\mathrm{B}=\left\{\mathrm{natural}\:\mathrm{numbers}\right\},\:\mathrm{then}\:\mathrm{A}\bigtriangleup\mathrm{B}=\_\_\_. \\ $$ Commented by FilupS last updated on 26/Mar/17 $$\mathrm{whole}\:\mathrm{numbers}\:=\:\mathbb{Z}\:=\:\left\{…,\:−\mathrm{2},\:−\mathrm{1},\:\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:…\right\} \\ $$$$\mathrm{natural}\:\mathrm{numbers}\:=\:\mathbb{N}\:=\:\left\{\mathrm{1},\:\mathrm{2},\:…\right\} \\…