Question Number 6722 by Rasheed Soomro last updated on 16/Jul/16 $${If}\: \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{8789}=\mathrm{89} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{7690}=\mathrm{79} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{5478}=\mathrm{69} \\ $$$${then} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{5230}=? \\ $$ Terms of…
Question Number 6716 by Tawakalitu. last updated on 15/Jul/16 $${Prove}\:{that}\:{the}\:{locus}\:{of}\:{a}\:{point}\:{which}\:{moves}\:{its}\:{distance}\:{from}\: \\ $$$${the}\:{point}\:\left(−{b},\:\mathrm{0}\right)\:{is}\:{p}\:{times}\:{its}\:{distance}\:{from}\:{the}\:{point}\:\left({b},\:\mathrm{0}\right)\:{is} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({p}^{\mathrm{2}} \:−\:\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \right)\:−\:\mathrm{2}{b}\left({p}^{\mathrm{2}} \:+\:\mathrm{1}\right){x}\:=\:\mathrm{0} \\ $$$${Show}\:{that}\:{this}\:{locus}\:{is}\:{a}\:{circle}\:{and}\:{find}\:{its}\:{radius}. \\ $$ Answered by…
Question Number 72250 by aliesam last updated on 26/Oct/19 $${f}\left({x}\right)=\left({ax}+{b}\right){sin}\left({x}\right)+\left({cx}+{d}\right){cos}\left({x}\right) \\ $$$${and}\:\:\:\frac{{dy}}{{dx}}={xcos}\left({x}\right) \\ $$$${find}\:\:{a}\:,\:{b}\:,\:{c}\:,\:{d} \\ $$$$ \\ $$$$ \\ $$ Commented by kaivan.ahmadi last updated…
Question Number 6715 by Tawakalitu. last updated on 15/Jul/16 $${Solve}\:{the}\:{following}\:{equation}\:{for}\:\mathrm{0}\:<\:\Theta\:<\:\mathrm{360}^{{o}} \\ $$$${cosx}\:+\:{cos}\mathrm{3}{x}\:+\:{cos}\mathrm{5}{x}\:+\:{cos}\mathrm{7}{x}\:=\:\mathrm{0} \\ $$ Answered by Yozzii last updated on 16/Jul/16 $${We}\:{know}\:{the}\:{identity}\:{cosa}+{cosb}=\mathrm{2}{cos}\frac{{a}+{b}}{\mathrm{2}}{cos}\frac{{a}−{b}}{\mathrm{2}}\:{for}\:{a},{b}\in\mathbb{R}….\left(\mathrm{1}\right) \\ $$$${Let}\:{u}={cosx}+{cos}\mathrm{3}{x}+{cos}\mathrm{5}{x}+{cos}\mathrm{7}{x}. \\…
Question Number 137784 by mathocean1 last updated on 06/Apr/21 $$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{6}}} \frac{{dx}}{{sinx}}=? \\ $$$$ \\ $$ Answered by EnterUsername last updated on 06/Apr/21 $$\int\frac{{dx}}{{sinx}}={ln}\mid{cosecx}−{cotx}\mid+{C} \\…
Question Number 72251 by aliesam last updated on 26/Oct/19 Commented by mathmax by abdo last updated on 26/Oct/19 $${let}\:{f}\left({x}\right)=\frac{{ln}\left(\mathrm{1}+{ax}+{bx}^{\mathrm{2}} \right)−{ln}\left(\mathrm{1}−{ax}+{bx}^{\mathrm{2}} \right)}{\mathrm{1}−{cosx}} \\ $$$${we}\:{have}\:{ln}^{'} \left(\mathrm{1}+{u}\right)=\frac{\mathrm{1}}{\mathrm{1}+{u}}\:=\mathrm{1}−{u}\:+{o}\left({u}\right)\:\left({u}\rightarrow\mathrm{0}\right)\:\Rightarrow \\…
Question Number 137787 by physicstutes last updated on 06/Apr/21 $$\mathrm{A}\:\mathrm{string}\:{AB}\:\mathrm{of}\:\mathrm{lenght}\:\mathrm{2}{l}\:\mathrm{has}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{attached}\:\mathrm{to}\:\mathrm{its}\:\mathrm{midpoint}\:\mathrm{C}.\: \\ $$$$\mathrm{The}\:\mathrm{ends}\:{A}\:\mathrm{and}\:{B}\:\mathrm{of}\:\mathrm{the}\:\mathrm{string}\:\mathrm{are}\:\mathrm{fastened}\:\mathrm{to}\:\mathrm{two}\:\mathrm{fixed}\:\mathrm{points} \\ $$$$\mathrm{with}\:{A}\:\mathrm{distance}\:{l}\:\mathrm{vertically}\:\mathrm{above}\:\mathrm{B}.\mathrm{With}\:\mathrm{both}\:\mathrm{parts}\:\mathrm{of}\:\mathrm{the}\:\mathrm{string} \\ $$$$\mathrm{taunt},\mathrm{the}\:\mathrm{particle}\:\mathrm{describes}\:\mathrm{a}\:\mathrm{horizontal}\:\mathrm{circle}\:\mathrm{about}\:\mathrm{the}\:\mathrm{line}\:{AB}\:\mathrm{with} \\ $$$$\mathrm{constant}\:\mathrm{angular}\:\mathrm{speed}\:\omega.\:\mathrm{If}\:\mathrm{the}\:\mathrm{tension}\:\mathrm{in}\:{CA}\:\mathrm{is}\:\mathrm{three}\:\mathrm{times} \\ $$$$\mathrm{that}\:\mathrm{in}\:{CB},\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{angular}\:\mathrm{velocity}\:\mathrm{is}\:\mathrm{2}\sqrt{\frac{\mathrm{g}}{{l}}}\:. \\ $$ Answered by mr…
Question Number 72248 by Omer Alattas last updated on 26/Oct/19 Commented by Omer Alattas last updated on 26/Oct/19 $${solv}\:{to}\:{x}\: \\ $$ Answered by MJS last…
Question Number 72247 by TawaTawa last updated on 26/Oct/19 Answered by mind is power last updated on 26/Oct/19 $$\mathrm{s}_{\mathrm{1}} =\mathrm{s}_{\mathrm{2}} =\mathrm{s} \\ $$$$\frac{\mathrm{TS}}{\mathrm{PR}}=\frac{\mathrm{QT}}{\mathrm{QP}}\Rightarrow\mathrm{TS}=\frac{\mathrm{41}}{\mathrm{3}} \\ $$$$\mathrm{a}=\angle\mathrm{QST}\:,\mathrm{r}=\angle\mathrm{QST}=\angle\mathrm{PRQ}…
Question Number 6710 by Tawakalitu. last updated on 15/Jul/16 $$\int\:\frac{{dx}}{{sinx}\:+\:{sin}\mathrm{2}{x}}\:{dx} \\ $$ Answered by Yozzii last updated on 15/Jul/16 $${I}=\int\frac{{dx}}{{sinx}+{sin}\mathrm{2}{x}}=\int\frac{{sinx}}{{sin}^{\mathrm{2}} {x}\left(\mathrm{1}+\mathrm{2}{cosx}\right)}{dx} \\ $$$${I}=\int\frac{−{sinx}}{−\left(\mathrm{1}−{cos}^{\mathrm{2}} {x}\right)\left(\mathrm{1}+\mathrm{2}{cosx}\right)}{dx} \\…