Question Number 16363 by Tinkutara last updated on 21/Jun/17 $$\mathrm{A}\:\mathrm{car}\:\mathrm{drives}\:\mathrm{due}\:\mathrm{north}\:\mathrm{at}\:\mathrm{50}\:\mathrm{km}/\mathrm{hr}. \\ $$$$\mathrm{Wind}\:\mathrm{blows}\:\mathrm{due}\:\mathrm{North}-\mathrm{West}\:\mathrm{at}\:\mathrm{50}\sqrt{\mathrm{2}} \\ $$$$\mathrm{km}/\mathrm{hr}.\:\mathrm{In}\:\mathrm{what}\:\mathrm{direction},\:\mathrm{a}\:\mathrm{flag} \\ $$$$\mathrm{hoisted}\:\mathrm{on}\:\mathrm{the}\:\mathrm{roof}\:\mathrm{of}\:\mathrm{the}\:\mathrm{car}\:\mathrm{points}? \\ $$ Commented by ajfour last updated on 22/Jun/17…
Question Number 147432 by mathdanisur last updated on 20/Jul/21 $${if}\:\:\begin{cases}{\mathrm{2}{x}\:+\:{a}\:\:;\:\:{x}\:<\:−\mathrm{3}}\\{{x}^{\mathrm{2}} \:-\:\mathrm{4}\:\:;\:\:−\mathrm{3}\:\leqslant\:{x}\:<\:\mathrm{2}}\\{{x}^{\mathrm{2}} \:+\:{ax}\:+\:{b}\:\:;\:\:{x}\:\geqslant\:\mathrm{2}}\end{cases} \\ $$$${find}\:\:\:\mathrm{3}{a}\:-\:{b}\:=\:? \\ $$ Answered by liberty last updated on 21/Jul/21 $$\left(\bullet\right)\:\underset{{x}\rightarrow−\mathrm{3}} {\mathrm{lim}}\:{f}\left({x}\right)=\underset{{x}\rightarrow−\mathrm{3}}…
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Question Number 16360 by ajfour last updated on 21/Jun/17 Answered by ajfour last updated on 21/Jun/17 $$\:{To}\:{find}\:{x}={CD},\:\:{y}={BD},\:{z}={AD} \\ $$$${BE}\:{is}\:{drawn}\:\bot\:{to}\:{AD},\:{AF}\:{is} \\ $$$${drawn}\:\bot\:{to}\:{AD}\:{produced}. \\ $$$$\:\bigtriangleup{BDE}\:\sim\:\bigtriangleup{ADF}\:\:\left({right}\:\angle\:{and}\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:{vertically}\:{opposite}\:{angles}\right)…
Question Number 16359 by Tinkutara last updated on 21/Jun/17 $$\mathrm{In}\:\mathrm{a}\:\Delta{ABC}\:\mathrm{if}\:\frac{{s}\:−\:{a}}{{a}\:−\:{b}}\:=\:\frac{{s}\:−\:{c}}{{b}\:−\:{c}}\:,\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:{r}_{\mathrm{1}} ,\:{r}_{\mathrm{2}} ,\:{r}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{in}\:\mathrm{A}.\mathrm{P}. \\ $$$$\mathrm{Here}\:{r}_{\mathrm{1}} ,\:{r}_{\mathrm{2}} \:\mathrm{and}\:{r}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{exradii} \\ $$$$\mathrm{opposite}\:\mathrm{to}\:\mathrm{angles}\:{A},\:{B}\:\mathrm{and}\:{C}\:\mathrm{respectively}. \\ $$ Answered…
Question Number 16358 by Tinkutara last updated on 21/Jun/17 $$\mathrm{In}\:\mathrm{any}\:\mathrm{triangle}\:{ABC},\:{a}\:\mathrm{cot}\:{A}\:+\:{b}\:\mathrm{cot}\:{B} \\ $$$$+\:{c}\:\mathrm{cot}\:{C}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{1}\right)\:{r}\:+\:{R} \\ $$$$\left(\mathrm{2}\right)\:{r}\:−\:{R} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{2}\left({r}\:+\:{R}\right) \\ $$$$\left(\mathrm{4}\right)\:\mathrm{2}\left({r}\:−\:{R}\right) \\ $$ Commented by b.e.h.i.8.3.4.1.7@gmail.com…
Question Number 147431 by vvvv last updated on 20/Jul/21 Answered by SEKRET last updated on 21/Jul/21 $$\boldsymbol{\mathrm{a}}=\boldsymbol{\mathrm{b}}=\boldsymbol{\mathrm{c}}=\boldsymbol{\mathrm{d}}=\boldsymbol{\mathrm{x}} \\ $$$$ \\ $$$$\:\:\boldsymbol{\mathrm{x}}_{\mathrm{1}} \centerdot\boldsymbol{\mathrm{x}}_{\mathrm{2}} \centerdot\boldsymbol{\mathrm{x}}_{\mathrm{3}} \centerdot\boldsymbol{\mathrm{x}}_{\mathrm{4}} =\:\boldsymbol{\mathrm{w}}=\:\boldsymbol{\mathrm{abcd}}…
Question Number 81892 by Power last updated on 16/Feb/20 Answered by MJS last updated on 16/Feb/20 $$\mathrm{we}\:\mathrm{can}\:\mathrm{find}\:\mathrm{a}\:\mathrm{formula}\:\mathrm{for}\:\mathrm{any}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{next}\:\mathrm{number}\:\Rightarrow\:\mathrm{the}\:\mathrm{answer}\:\mathrm{is}\:\mathrm{not}\:\mathrm{unique} \\ $$ Commented by mr W…
Question Number 16354 by Tinkutara last updated on 21/Jun/17 Answered by myintkhaing last updated on 21/Jun/17 $$\frac{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{2}{bc}}+\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{c}^{\mathrm{2}} }{\mathrm{2}{ab}}=\mathrm{2}−\frac{{c}^{\mathrm{2}} +{a}^{\mathrm{2}} −{b}^{\mathrm{2}}…
Question Number 81888 by rajesh4661kumar@gmail.com last updated on 16/Feb/20 Commented by mathmax by abdo last updated on 16/Feb/20 $${let}\:{A}\:=\int\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}\:\:{we}\:{do}\:{the}\:{changement}\:{x}\:={cht}\:\Rightarrow \\ $$$${A}\:=\int\:\:\:\frac{{sht}\:{dt}}{\left({ch}^{\mathrm{2}} {t}\:+\mathrm{1}\right){sht}}\:=\int\:\:\frac{{dt}}{\mathrm{1}+\frac{\mathrm{1}+{ch}\left(\mathrm{2}{t}\right)}{\mathrm{2}}}\:=\int\:\:\frac{\mathrm{2}{dt}}{\mathrm{3}+{ch}\left(\mathrm{2}{t}\right)} \\…