Question Number 109307 by john santu last updated on 22/Aug/20 Answered by mr W last updated on 22/Aug/20 $${x}^{\mathrm{2}} +\mathrm{2}{yz}−\mathrm{2}\left({y}+{z}\right)=\mathrm{82} \\ $$$${y}^{\mathrm{2}} +\mathrm{2}{zx}−\mathrm{2}\left({z}+{x}\right)=\mathrm{83} \\ $$$${z}^{\mathrm{2}}…
Question Number 174837 by mnjuly1970 last updated on 12/Aug/22 $$ \\ $$$$\:\:\:\:\:{sin}\left({A}\right)+\:{sin}\left({B}\:\right)+\:{sin}\left({C}\right)\leqslant\:\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\succ\:\:\:{Solution}\:\prec\:\:\:\:\:\:\left(\:{A}+{B}\:+{C}\:=\pi\:\right) \\ $$$$\:\:\:\:\:\:{l}.{h}.{s}\:=\:\mathrm{2}{sin}\left(\frac{{A}+{B}}{\mathrm{2}}\:\right){cos}\left(\frac{{A}−{B}}{\mathrm{2}}\right)+\mathrm{2}{sin}\left(\frac{{C}}{\mathrm{2}}\right){cos}\left(\frac{{C}}{\mathrm{2}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2}{cos}\:\left(\frac{{C}}{\mathrm{2}}\right)\left\{\mathrm{2}{cos}\frac{{A}}{\mathrm{2}}\:.{cos}\left(\frac{{B}}{\mathrm{2}}\:\right)\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{4}{cos}\left(\frac{{A}}{\mathrm{2}}\right).{cos}\left(\frac{{B}}{\mathrm{2}}\right).{cos}\left(\frac{{C}}{\mathrm{2}}\right)\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:{l}.{h}.{s}\:\underset{{post}} {\overset{{previoue}} {\leqslant}}\:\mathrm{4}\:\left(\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{8}}\:\right)=\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{2}} \\…
Question Number 43759 by ajfour last updated on 15/Sep/18 Answered by MrW3 last updated on 15/Sep/18 $${eqn}.\:{of}\:{line}: \\ $$$${x}=\frac{{y}}{{m}}+{a} \\ $$$${b}^{\mathrm{2}} {y}={x}\left({x}^{\mathrm{2}} −{a}^{\mathrm{2}} \right) \\…
Question Number 43756 by ajfour last updated on 15/Sep/18 $$\:\:\boldsymbol{{x}}^{\mathrm{3}} +\boldsymbol{{px}}+\boldsymbol{{q}}\:=\:\mathrm{0} \\ $$$$\boldsymbol{{If}}\:\boldsymbol{{equation}}\:\boldsymbol{{has}}\:\boldsymbol{{all}}\:\boldsymbol{{its}}\:\boldsymbol{{roots}} \\ $$$$\boldsymbol{{real}},\:\boldsymbol{{find}}\:\boldsymbol{{them}}. \\ $$ Answered by ajfour last updated on 15/Sep/18 $$\:\:{A}\:{poor}\:{self}\:{attempt}\::…
Question Number 43757 by Rauny last updated on 15/Sep/18 $$\mathrm{Probably}\:\mathrm{if}\:{x}^{{n}} ={Am}\:\left(\frac{{a}}{{b}}\pi\right),\:{x}={e}^{\frac{\mathrm{2}{k}+{a}}{{bn}}{i}\pi} \\ $$$$\mathrm{about}\:\mathrm{0}<\left({k}\in\mathbb{N}\cup\left\{\mathrm{0}\right\}\right)<\left({n}\in\mathbb{N}\right)\:\mathrm{and}\:{b}\neq\mathrm{0}. \\ $$$$\mathrm{p}.\mathrm{s}.\:{Am}\:\left(\mathrm{0}°\right)=\mathrm{1},\:{Am}\:\left(\mathrm{90}°\right)={i}\:\mathrm{etc}., \\ $$$$\mathrm{and}\:{s}°=\frac{\pi}{\mathrm{180}}{s}\:\mathrm{rad}\left(\mathrm{ians}\right)=\frac{\pi}{\mathrm{180}}{s}. \\ $$ Terms of Service Privacy Policy Contact:…
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Question Number 174806 by cortano1 last updated on 11/Aug/22 $${Given}\:{f}\left({x}\right)={x}^{\mathrm{4}} +{ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d} \\ $$$${where}\:{a},{b},{c}\:{and}\:{d}\:{are}\:{real}\:{number} \\ $$$${suppose}\:{the}\:{graph}\:{f}\left({x}\right)\:{intersects} \\ $$$${the}\:{graph}\:{of}\:{y}=\mathrm{2}{x}−\mathrm{1}\:{at}\:{x}=\mathrm{1},\mathrm{2},\mathrm{3}. \\ $$$$\:{Find}\:{the}\:{value}\:{of}\:\:{f}\left(\mathrm{0}\right)+{f}\left(\mathrm{4}\right). \\ $$ Commented by…
Question Number 174800 by AgniMath last updated on 11/Aug/22 $$\mathrm{For}\:\mathrm{which}\:\mathrm{value}\:\mathrm{of}\:{x}\:\mathrm{this}\:\mathrm{cubic}\:\mathrm{equation}\:\mathrm{will} \\ $$$$\mathrm{be}\:\mathrm{0}\:?\:{a}^{\mathrm{3}} \:−\:\mathrm{16}{a}\:−\:\mathrm{3} \\ $$ Answered by MJS_new last updated on 11/Aug/22 $${a}_{\mathrm{1}} =−\frac{\mathrm{8}\sqrt{\mathrm{3}}}{\mathrm{3}}\mathrm{cos}\:\left(\frac{\pi}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{3}}\mathrm{arcsin}\:\frac{\mathrm{9}\sqrt{\mathrm{3}}}{\mathrm{128}}\right) \\…
Question Number 174799 by Shrinava last updated on 11/Aug/22 Answered by behi834171 last updated on 12/Aug/22 $${cot}\frac{{A}}{\mathrm{2}}=\sqrt{\frac{{p}\left({p}−{a}\right)}{\left({p}−{b}\right)\left({p}−{c}\right)}}\:\:=\frac{\boldsymbol{{S}}}{\left(\boldsymbol{{p}}−\boldsymbol{{b}}\right)\left(\boldsymbol{{p}}−\boldsymbol{{c}}\right)}= \\ $$$$=\frac{\boldsymbol{{p}}\left(\boldsymbol{{p}}−\boldsymbol{{a}}\right)\:}{\boldsymbol{{S}}}=\frac{\boldsymbol{{p}}−\boldsymbol{{a}}}{\boldsymbol{{r}}}\:\:\:\:\:{and}\:{so}\:{on}… \\ $$$$\Pi\boldsymbol{{cot}}\frac{\boldsymbol{{A}}}{\mathrm{2}}=\frac{\boldsymbol{{S}}^{\mathrm{2}} }{\boldsymbol{{p}}.\boldsymbol{{r}}^{\mathrm{3}} }=\frac{\boldsymbol{{S}}}{\boldsymbol{{r}}^{\mathrm{2}} } \\…
Question Number 43716 by behi83417@gmail.com last updated on 14/Sep/18 Commented by tanmay.chaudhury50@gmail.com last updated on 15/Sep/18 Commented by tanmay.chaudhury50@gmail.com last updated on 15/Sep/18 Commented by…