Question Number 108453 by gospelkenny last updated on 17/Aug/20 $$\mathrm{If}\:\:{R}=\left(\mathrm{7}+\mathrm{4}\sqrt{\mathrm{3}}\right)^{\mathrm{2}{n}} \:=\:{I}+{f},\:\mathrm{where}\:{I}\:\in\:{N} \\ $$$$\mathrm{and}\:\:\:\mathrm{0}<{f}<\mathrm{1},\:\mathrm{then}\:{R}\left(\mathrm{1}−{f}\right)\:\mathrm{equals} \\ $$ Answered by 1549442205PVT last updated on 17/Aug/20 $$\mathrm{Set}\:\mathrm{Q}=\left(\mathrm{7}−\mathrm{4}\sqrt{\mathrm{3}}\right)^{\mathrm{2n}} \Rightarrow\sqrt{\mathrm{RQ}}=\left[\left(\mathrm{7}+\mathrm{4}\sqrt{\mathrm{3}}\right)^{\mathrm{n}} \left(\mathrm{7}−\mathrm{4}\sqrt{\mathrm{3}}\right)^{\mathrm{n}}…
Question Number 108407 by ZiYangLee last updated on 16/Aug/20 $$\mathrm{The}\:\mathrm{sides}\:\mathrm{AB},\:\mathrm{BC},\:\mathrm{CA}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangleABC} \\ $$$$\mathrm{have}\:\mathrm{3},\:\mathrm{4}\:\mathrm{and}\:\mathrm{5}\:\mathrm{interior}\:\mathrm{points}\:\mathrm{respectively} \\ $$$$\mathrm{on}\:\mathrm{them}.\:\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{triangles} \\ $$$$\mathrm{that}\:\mathrm{can}\:\mathrm{be}\:\mathrm{constructed}\:\mathrm{by}\:\mathrm{using}\:\mathrm{these} \\ $$$$\mathrm{points}\:\mathrm{as}\:\mathrm{vertices}\:\mathrm{is} \\ $$ Answered by mbertrand658 last updated…
Question Number 42728 by RANGAM kumar last updated on 01/Sep/18 $$\mathrm{If}\:{A}=\begin{bmatrix}{\:\:\:\mathrm{cos}\:{x}}&{\mathrm{sin}\:{x}}\\{−\mathrm{sin}\:{x}}&{\mathrm{cos}\:{x}}\end{bmatrix}\:\mathrm{and}\: \\ $$$${A}\:\mathrm{adj}\:{A}\:=\:{k}\begin{bmatrix}{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}\end{bmatrix},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{is} \\ $$ Answered by math1967 last updated on 01/Sep/18 $${adjA}=\begin{bmatrix}{{cosx}}&{{sinx}}\\{−{sinx}}&{{cosx}}\end{bmatrix}^{{t}} =\begin{bmatrix}{{cosx}}&{−{sinx}}\\{{sinx}}&{{cosx}}\end{bmatrix} \\…
Question Number 42684 by 1996ajaykmar@email.com last updated on 31/Aug/18 $$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{4}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\frac{{x}}{\mathrm{2}}\:−\:\frac{\mathrm{3}}{{x}^{\mathrm{2}} }\right)^{\mathrm{10}} \:\mathrm{is} \\ $$ Commented by maxmathsup by imad last updated on…
Question Number 42668 by bbnnftm833 last updated on 31/Aug/18 $$\mathrm{If} \\ $$$$\mid{a}\:\mathrm{sin}^{\mathrm{2}} \theta+{b}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta+{c}\:\mathrm{cos}^{\mathrm{2}} \theta−\frac{\mathrm{1}}{\mathrm{2}}\left({a}−{c}\right)\mid \\ $$$$\:\:\:\leqslant\:\frac{\mathrm{1}}{\mathrm{2}}\:{k},\:\mathrm{then}\:{k}^{\mathrm{2}} \:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 42410 by soufiane zarik last updated on 25/Aug/18 $$\underset{\:\mathrm{0}} {\overset{\mathrm{2}{a}} {\int}}\:\:\frac{{f}\left({x}\right)}{{f}\left({x}\right)+{f}\left(\mathrm{2}{a}−{x}\right)}\:{dx}\:= \\ $$ Commented by maxmathsup by imad last updated on 25/Aug/18 $${let}\:{A}\:=\:\int_{\mathrm{0}}…
Question Number 42345 by soufiane zarik last updated on 23/Aug/18 $$\mathrm{tan}\:\mathrm{15}°\:= \\ $$ Commented by MJS last updated on 24/Aug/18 $$\mathrm{tan}\:\mathrm{15}°\:=\mathrm{tan}\:\frac{\mathrm{30}°}{\mathrm{2}}\:=\frac{\mathrm{sin}\:\mathrm{30}°}{\mathrm{1}+\mathrm{cos}\:\mathrm{30}°}=\frac{\frac{\mathrm{1}}{\mathrm{2}}}{\mathrm{1}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}=\frac{\frac{\mathrm{1}}{\mathrm{2}}}{\frac{\mathrm{2}+\sqrt{\mathrm{3}}}{\mathrm{2}}}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}+\sqrt{\mathrm{3}}}=\frac{\mathrm{2}−\sqrt{\mathrm{3}}}{\left(\mathrm{2}+\sqrt{\mathrm{3}}\right)\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)}=\mathrm{2}−\sqrt{\mathrm{3}} \\ $$…
Question Number 42094 by amansingh@123 last updated on 17/Aug/18 $$\mathrm{If}\:\:\mathrm{7}\:\mathrm{points}\:\mathrm{out}\:\mathrm{of}\:\:\mathrm{12}\:\mathrm{are}\:\mathrm{in}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{straight}\:\mathrm{line},\:\mathrm{then}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\: \\ $$$$\mathrm{triangles}\:\mathrm{formed}\:\mathrm{is} \\ $$ Answered by $@ty@m last updated on 17/Aug/18 $$\:^{\mathrm{7}} {C}_{\mathrm{2}}…
Question Number 42093 by amansingh@123 last updated on 17/Aug/18 $$\mathrm{If}\:\mathrm{the}\:\mathrm{permutations}\:\mathrm{of}\:{a},\:{b},\:{c},\:{d},\:{e}\:\mathrm{taken} \\ $$$$\mathrm{all}\:\mathrm{together}\:\mathrm{be}\:\mathrm{written}\:\mathrm{down}\:\mathrm{in}\: \\ $$$$\mathrm{alphabetical}\:\mathrm{order}\:\mathrm{as}\:\mathrm{in}\:\mathrm{dictionary} \\ $$$$\mathrm{and}\:\mathrm{numbered},\:\mathrm{then}\:\mathrm{the}\:\mathrm{rank}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{permutation}\:\:{debac}\:\:\mathrm{is} \\ $$ Answered by $@ty@m last updated…
Question Number 41896 by Ugguvj3 last updated on 15/Aug/18 $$\underset{−\mathrm{1}/\mathrm{2}} {\overset{\mathrm{1}/\mathrm{2}} {\int}}\:\:\left[\:\left(\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right)^{\mathrm{2}} +\left(\frac{{x}−\mathrm{1}}{{x}+\mathrm{1}}\right)^{\mathrm{2}} −\mathrm{2}\right]^{\mathrm{1}/\mathrm{2}} {dx}\:= \\ $$ Commented by maxmathsup by imad last updated on…