Question Number 2253 by 123456 last updated on 11/Nov/15 $${f}_{{n}} :\left[\mathrm{0},\mathrm{1}\right]\rightarrow\left[\mathrm{0},\mathrm{1}\right],{g}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\left[\mathrm{0},\mathrm{1}\right] \\ $$$${f}_{{n}+\mathrm{1}} \left({x}\right)={g}\left[{f}_{{n}} \left({x}\right)\right]+{f}_{{n}} \left[{g}\left({x}\right)\right] \\ $$$${f}_{\mathrm{0}} \left({x}\right)={x} \\ $$$${f}_{\mathrm{4}} \left({x}\right)=? \\ $$$${g}\left({x}\right)={x}^{\mathrm{2}} ,{f}_{\mathrm{2}}…
Question Number 133321 by 777316 last updated on 21/Feb/21 $${Find}\:{x}\:: \\ $$$${sin}\left(\mathrm{3}{x}\right)−{sin}\left(\mathrm{2}{x}\right)−\mathrm{2}{sin}\left({x}\right)\:=\:\sqrt{\mathrm{3}}{cos}\left({x}\right) \\ $$ Commented by bramlexs22 last updated on 21/Feb/21 $$\mathrm{x}=\frac{\mathrm{2}\pi}{\mathrm{3}} \\ $$$$\mathrm{sin}\:\left(\mathrm{3}×\frac{\mathrm{2}\pi}{\mathrm{3}}\right)−\mathrm{sin}\:\left(\mathrm{2}×\frac{\mathrm{2}\pi}{\mathrm{3}}\right)−\mathrm{2sin}\:\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right)= \\…
Question Number 133320 by bramlexs22 last updated on 21/Feb/21 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{n}\:\mathrm{for}\:\mathrm{which}\:\mathrm{n}^{\mathrm{2}} +\mathrm{2n}+\mathrm{4}\: \\ $$$$\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{7}\: \\ $$ Answered by EDWIN88 last updated on 21/Feb/21 $$\mathrm{let}\:\mathrm{n}\:=\:\mathrm{7k}+\mathrm{r}\:\mathrm{then}\:\mathrm{n}^{\mathrm{2}} +\mathrm{2n}+\mathrm{4}\:=\:\left(\mathrm{7k}+\mathrm{r}\right)^{\mathrm{2}} +\mathrm{2}\left(\mathrm{7k}+\mathrm{r}\right)+\mathrm{4}…
Question Number 2249 by Filup last updated on 11/Nov/15 $$\mathrm{With}\:\mathrm{linear}\:\mathrm{functions}\:{f}\left({x}\right)\:\mathrm{and}\:{g}\left({x}\right), \\ $$$$\mathrm{if}\:{f}\left({x}\right)\bot{g}\left({x}\right),\:\mathrm{then}: \\ $$$${m}_{{f}} {m}_{{g}} =−\mathrm{1}\:\:\:\:\mathrm{where}\:{m}_{{i}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{gradient} \\ $$$$\mathrm{of}\:\mathrm{function}\:{i}\left({x}\right). \\ $$$$ \\ $$$$\mathrm{Does}\:\mathrm{that}\:\mathrm{therefore}\:\mathrm{mean}\:\mathrm{that},\:\mathrm{if}\:\mathrm{given} \\ $$$$\mathrm{function}\:\left(\mathrm{including}\:\mathrm{non}−\mathrm{linear}\right)\:{f}\left({x}\right),…
Question Number 133316 by bramlexs22 last updated on 21/Feb/21 $$\mathrm{How}\:\mathrm{many}\:\mathrm{6}−\mathrm{letter}\:\mathrm{words}\:\mathrm{in}\: \\ $$$$\mathrm{which}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{letter}\:\mathrm{appears} \\ $$$$\mathrm{more}\:\mathrm{than}\:\mathrm{once}\:,\mathrm{can}\:\mathrm{be}\:\mathrm{made}\:\mathrm{from} \\ $$$$\mathrm{the}\:\mathrm{letters}\:\mathrm{in}\:\mathrm{the}\:\mathrm{word}\:\mathrm{FLIGHT} \\ $$$$ \\ $$ Commented by mr W last…
Question Number 133318 by bramlexs22 last updated on 21/Feb/21 $$\left(\mathrm{a}\right)\:\mathrm{Are}\:\mathrm{there}\:\mathrm{any}\:\mathrm{graphs}\:\mathrm{with}\:\mathrm{5} \\ $$$$\mathrm{vertices}\:\mathrm{which}\:\mathrm{have}\:\mathrm{vertices}\:\mathrm{of}\: \\ $$$$\mathrm{degrees}\:\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}\:\mathrm{and}\:\mathrm{5}?\: \\ $$ Answered by EDWIN88 last updated on 21/Feb/21 $$\mathrm{No};\:\mathrm{if}\:\mathrm{a}\:\mathrm{graph}\:\mathrm{has}\:\mathrm{n}\:\mathrm{vertices}\:\mathrm{it}\:\mathrm{can}\:\mathrm{have}\:\mathrm{no}\: \\…
Question Number 2241 by Yozzi last updated on 10/Nov/15 $${Evaluate} \\ $$$$\:\:\:\:\:\:\:\:\frac{\mathrm{9}}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \left(\frac{{sin}\theta{cos}\theta}{{cos}^{\mathrm{3}} \theta+{sin}^{\mathrm{3}} \theta}\right)^{\mathrm{2}} {d}\theta. \\ $$ Answered by Filup last updated on…
Question Number 133314 by bramlexs22 last updated on 21/Feb/21 $$\mathrm{How}\:\mathrm{many}\:\mathrm{rearrangements}\:\mathrm{are}\: \\ $$$$\mathrm{there}\:\mathrm{of}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{in}\:\mathrm{the}\:\mathrm{world} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{ENGINEERING} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{MATHEMATICAL}\: \\ $$ Answered by EDWIN88 last updated on 21/Feb/21…
Question Number 2240 by Rasheed Soomro last updated on 10/Nov/15 $$\mathcal{GENERALIZE}: \\ $$$$\left({a}+{b}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{ab}\right)={a}^{\mathrm{3}} +{b}^{\mathrm{3}} \\ $$$$\left({a}+{b}+{c}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}}…
Question Number 67774 by TawaTawa last updated on 31/Aug/19 Answered by MJS last updated on 31/Aug/19 $${DE}={AB}=\mathrm{2}{r} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{semicircle}\:=\frac{\pi}{\mathrm{2}}{r}^{\mathrm{2}} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{triangle}\:=\frac{\mathrm{1}}{\mathrm{2}}\mid{AB}\mid{h}={rh} \\ $$$$\:\:\:\:\:\mathrm{with}\:{h}={r}\mathrm{tan}\:{x} \\ $$$$\:\:\:\:\:={r}^{\mathrm{2}}…