Question Number 67026 by TawaTawa last updated on 21/Aug/19 Commented by Tony Lin last updated on 22/Aug/19 $$\because{slope}\:{of}\:{L}_{{BC}} ={tan}\mathrm{150}°=−\frac{\sqrt{\mathrm{3}}}{\mathrm{3}} \\ $$$$\therefore{L}_{{BC}} :\:{y}=−\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\left({x}+\mathrm{4}\right) \\ $$$$\because{L}_{{BC}} \bot{L}_{{AB}}…
Question Number 1395 by Rasheed Soomro last updated on 28/Jul/15 $$\:\:\:\:\:\:\:\:\:\:\mathrm{C}{onsider}\:\boldsymbol{\mathrm{quadrilateral}}\:\boldsymbol{\mathrm{ABCD}}\:\:{same}\:{as}\:{in}\:{Q}\:\mathrm{1378}\:{with}\: \\ $$$${same}\:{conditions}/{restrictions}\:\left({Pl}\:\:{refer}\:\:{the}\:{Question}\:{again}\right). \\ $$$$\:\:\:\:\:\:\:\:\:\:\bullet\:{What}\:{could}\:{be}\:{possible}\:\boldsymbol{\mathrm{minimum}}\:{and}\:\boldsymbol{\mathrm{maximum}}\:\boldsymbol{\mathrm{area}} \\ $$$${of}\:{the}\:{quadrilateral}? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\bullet{When}\:{qusdrilateral}\:{has}\:\boldsymbol{\mathrm{minimum}}\:\boldsymbol{\mathrm{area}}\:{what}\:{is}\:{the}\:{value}/{s} \\ $$$${of}\:\boldsymbol{{m}}\angle\boldsymbol{\mathrm{A}}?\:{Similarly}\:{what}\:{is}\:{value}/{s}\:\:{of}\:\boldsymbol{{m}}\angle\boldsymbol{\mathrm{A}}\:{in}\:{case}\:{of} \\ $$$$\boldsymbol{\mathrm{maximum}}\:\boldsymbol{\mathrm{area}}? \\ $$$$…
Question Number 1387 by navajyoti.tamuli.tamuli@gmail. last updated on 27/Jul/15 $${Q}.\:{is}\:{there}\:{any}\:{angle}\:{in}\:{a}\:{circle}? \\ $$ Commented by Rasheed Soomro last updated on 28/Jul/15 $${Two}\:{answers}\:{can}\:{be}\:{given}.{Each}\:{has}\:\:{its}\:\:{own}\:{reasoning}. \\ $$$$\left({i}\right)\:{There}\:{are}\:{infinity}\:{number}\:{of}\:{angles}. \\ $$$${Consider}\:{a}\:{polygon}\:{of}\:{n}\:{angles}\:{inscribed}\:{in}\:{a}\:{circle}:…
Question Number 1378 by Rasheed Soomro last updated on 28/Jul/15 $$\:\:\:\:\:{Four}\:{sides}\:{m}\overline {\boldsymbol{\mathrm{AB}}}\:,\:{m}\overline {\boldsymbol{\mathrm{BC}}}\:,\:{m}\overline {\boldsymbol{\mathrm{CD}}}\:\:{and}\:\:{m}\overline {\boldsymbol{\mathrm{DA}}}\:{of}\:{a}\:\boldsymbol{\mathrm{quadrilateral}}\: \\ $$$$\boldsymbol{\mathrm{ABCD}}\:\:{have}\:{measurement}\:\boldsymbol{{a}}\:,\:\boldsymbol{{b}}\:,\:\boldsymbol{{c}}\:\:{and}\:\boldsymbol{{d}}\:{units}\:{respectively}. \\ $$$$\:\:\:\:\:{Let}\:{the}\:{sum}\:{of}\:{any}\:{adjacent}\:{sides}\:{is}\:{not}\:{equal}\:{to}\:{the}\:{sum}\:{of} \\ $$$${remaining}\:{adjacent}\:{sides}\:\:{and}\:{measurement}\:{of}\:{all}\:{the}\:{sides}\: \\ $$$${is}\:{positive}\:{and}\:{real}. \\ $$$$\:\:\:\:\:\:{What}\:{could}\:{be}\:{the}\:{possible}\:{minimum}\:{and}\:{maximum}\:{values}…
Question Number 1343 by Rasheed Soomro last updated on 24/Jul/15 $$\mathrm{3}^{{log}\:\mathrm{3}{x}+\mathrm{4}} =\mathrm{4}^{{log}\:\mathrm{4}{x}+\mathrm{3}} \\ $$ Answered by 112358 last updated on 24/Jul/15 $${Taking}\:{logs}\:{to}\:{base}\:{e}\:{on}\:{both}\:{sides} \\ $$$$\Rightarrow\left({log}\mathrm{3}{x}+\mathrm{4}\right){ln}\mathrm{3}=\left({log}\mathrm{4}{x}+\mathrm{3}\right){ln}\mathrm{4} \\…
Question Number 132373 by Ari last updated on 13/Feb/21 Answered by Ari last updated on 13/Feb/21 Answered by mr W last updated on 14/Feb/21 Commented…
Question Number 1304 by 314159 last updated on 20/Jul/15 $${If}\:\mathrm{3}^{{log}\:\mathrm{3}{x}} =\mathrm{4}^{{log}\:\mathrm{4}{x}} ,\:{find}\:{x}. \\ $$$$ \\ $$ Answered by Rasheed Soomro last updated on 20/Jul/15 $$\mathrm{3}^{{log}\:\mathrm{3}{x}}…
Question Number 1268 by 314159 last updated on 18/Jul/15 $$\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{AM}}\:>\:\boldsymbol{\mathrm{HM}}. \\ $$ Answered by prakash jain last updated on 18/Jul/15 $$\mathrm{AM}=\frac{{a}+{b}}{\mathrm{2}},\:\mathrm{HM}=\frac{\mathrm{2}{ab}}{{a}+{b}} \\ $$$$\mathrm{AM}−\mathrm{HM}=\frac{{a}+{b}}{\mathrm{2}}−\frac{\mathrm{2}{ab}}{{a}+{b}}=\:\frac{\left({a}−{b}\right)^{\mathrm{2}} }{\mathrm{2}\left({a}+{b}\right)} \\…
Question Number 132327 by liberty last updated on 13/Feb/21 $$\:\mathrm{If}\:\mathrm{the}\:\mathrm{line}\:\begin{cases}{\frac{\mathrm{x}−\mathrm{1}}{\mathrm{2}}=\frac{\mathrm{y}+\mathrm{1}}{\mathrm{3}}=\frac{\mathrm{z}−\mathrm{1}}{\mathrm{4}}}\\{\frac{\mathrm{x}−\mathrm{3}}{\mathrm{1}}=\frac{\mathrm{y}−\mathrm{k}}{\mathrm{2}}=\frac{\mathrm{z}}{\mathrm{1}}}\end{cases} \\ $$$$\:\mathrm{intersect}\:.\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{is}\: \\ $$ Answered by Ar Brandon last updated on 13/Feb/21 $$\mathrm{L}_{\mathrm{1}} :\:\mathrm{i}−\mathrm{j}+\mathrm{k}+\lambda\left(\mathrm{2i}+\mathrm{3j}+\mathrm{4k}\right)=\left(\mathrm{1}+\mathrm{2}\lambda\right)\mathrm{i}+\left(\mathrm{3}\lambda−\mathrm{1}\right)\mathrm{j}+\left(\mathrm{4}\lambda+\mathrm{1}\right)\mathrm{k} \\…
Question Number 1157 by navajyoti.tamuli.tamuli@gmail. last updated on 06/Jul/15 $${show}\:{that} \\ $$$${tan}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }−\mathrm{1}}{{x}}\right)=\frac{\mathrm{1}}{\mathrm{2}}{tan}^{−\mathrm{1}} {x} \\ $$$$ \\ $$ Answered by prakash jain last updated…