Question Number 218345 by Spillover last updated on 07/Apr/25 Answered by som(math1967) last updated on 07/Apr/25 $${let}\:{AF}={AE}={x} \\ $$$$\:\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{3}×\left(\mathrm{6}+{x}\right)+\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{3}×\left(\mathrm{8}+{x}\right)+\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{3}×\mathrm{14}=\mathrm{63} \\ $$$$\Rightarrow\frac{\mathrm{3}}{\mathrm{2}}×\left(\mathrm{28}+\mathrm{2}{x}\right)=\mathrm{63} \\ $$$$\mathrm{28}+\mathrm{2}{x}=\mathrm{42} \\ $$$$\:\therefore{x}=\mathrm{7}…
Question Number 217084 by maths_plus last updated on 28/Feb/25 $$\boldsymbol{\mathrm{Geometr}}\mathrm{i}\boldsymbol{\mathrm{e}}\:\boldsymbol{\mathrm{dans}}\:\boldsymbol{\mathrm{le}}\:\boldsymbol{\mathrm{plan}}. \\ $$$$\overset{\rightarrow} {\mathrm{AB}}\:\mathrm{et}\:\overset{\rightarrow} {\mathrm{CD}}\:\mathrm{sont}\:\mathrm{deux}\:\mathrm{vecteurs}\:\mathrm{du}\:\mathrm{plan}. \\ $$$$\overset{\rightarrow} {\mathrm{AB}}\:\mathrm{n}'\mathrm{est}\:\mathrm{pas}\:\mathrm{nul}. \\ $$$$\mathrm{Demontre}\:\mathrm{que}\:\mathrm{si}\:\overset{\rightarrow} {\mathrm{AB}}\:\mathrm{et}\:\overset{\rightarrow} {\mathrm{CD}}\:\mathrm{sont}\:\mathrm{colineaires} \\ $$$$\mathrm{alors}\:\mathrm{il}\:\mathrm{existe}\:\mathrm{un}\:\mathrm{nombre}\:\mathrm{reel}\:\mathrm{k}\:\mathrm{tel}\:\mathrm{que} \\ $$$$\overset{\rightarrow}…
Question Number 217075 by maths_plus last updated on 28/Feb/25 $$ \\ $$Help me please… $\vv{AB}$ and $\vv{CD}$ are two vectors, and $\vv{AB}$ is not the…
Question Number 217088 by alcohol last updated on 28/Feb/25 $${show}\:{that}\:\int_{\:{n}} ^{\:{n}\:+\:\mathrm{1}} {ln}\left({t}\right)\:{dt}\:\leqslant\:{ln}\left({n}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$${Given}\:{u}_{{n}} \:=\:\frac{\left(\mathrm{4}{n}\right)^{{n}} {n}!{e}^{−{n}} }{\left(\mathrm{2}{n}\right)!},\:\forall{n}\:\geqslant\:\mathrm{1} \\ $$$${prove},\:{using}\:{the}\:{preceding}\:{question}\:{that} \\ $$$${u}_{{n}} \:{is}\:{decreasing}\:{and}\:{convergent} \\ $$ Answered…
Question Number 216859 by ajfour last updated on 23/Feb/25 Commented by ajfour last updated on 23/Feb/25 $${Radius}\:{of}\:{inner}\:{disc}\:{is}\:{R}.\:{As}\:{it}\:{rolls}\:{up} \\ $$$${the}\:{outer}\:{circular}\:{track}\:{of}\:{radius}\:\mathrm{2}{R},\:{find} \\ $$$${equation}\:{of}\:{trajectory}\:{of}\:{a}\:{point}\:\boldsymbol{{P}}\:{on}\:{the} \\ $$$${wheel}\:{until}\:{it}\:{comes}\:{into}\:{contact}\:{with} \\ $$$${the}\:{outer}\:{track}.…
Question Number 216750 by sniper237 last updated on 17/Feb/25 $${If}\:\:{a}−{b}=\sqrt{{ab}}\:\:\:{find}\:\:\:{the}\:{value}\:{of}\:\frac{{a}−{b}}{{a}+{b}} \\ $$ Answered by mr W last updated on 17/Feb/25 $$\left({a}+{b}\right)^{\mathrm{2}} =\left({a}−{b}\right)^{\mathrm{2}} +\mathrm{4}{ab}={ab}+\mathrm{4}{ab}=\mathrm{5}{ab} \\ $$$$\Rightarrow{a}+{b}=\pm\sqrt{\mathrm{5}{ab}}…
Question Number 216695 by sniper237 last updated on 16/Feb/25 $$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \frac{{dx}}{\mathrm{1}+{sinxcosx}}\overset{?} {=}\:\frac{\mathrm{4}\pi{ln}\mathrm{2}}{\:\sqrt{\mathrm{3}}}\:\: \\ $$ Answered by Ghisom last updated on 16/Feb/25 $$\underset{\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}\frac{{dx}}{\mathrm{1}+\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}}=\mathrm{8}\underset{\pi/\mathrm{4}}…
Question Number 216607 by Tawa11 last updated on 12/Feb/25 Answered by Rasheed.Sindhi last updated on 12/Feb/25 $$\mathrm{4}{x}^{\mathrm{2}} +{bx}−\mathrm{45}=\left({hx}+{k}\right)\left({x}+{j}\right);{h},{k},{j}\in\mathbb{Z} \\ $$$$\:\:\:\:\:\:\:\:={hx}^{\mathrm{2}} +\left({hj}+{k}\right){x}+{kj} \\ $$$${kj}=−\mathrm{45}\Rightarrow{j}=−\frac{\mathrm{45}}{{k}} \\ $$$$\frac{\mathrm{45}}{{k}}=−{j}\:\in\:\mathbb{Z}\:\rightarrow\left({D}\right)…
Question Number 216630 by Tawa11 last updated on 12/Feb/25 the circles x² + y² -4x -2y +3 =0 and x² +y² + 2x +4y -3…
Question Number 216387 by Tawa11 last updated on 06/Feb/25 Answered by mr W last updated on 06/Feb/25 $$\left(\mathrm{6},\:\mathrm{0}\right)\: \\ $$$$\left(\mathrm{6}−\mathrm{6}\right)^{\mathrm{2}} +\left(\mathrm{0}−\mathrm{3}+{a}\right)^{\mathrm{2}} =\mathrm{5}^{\mathrm{2}} \\ $$$$\Rightarrow−\mathrm{3}+{a}=\mathrm{5}\:\Rightarrow{a}=\mathrm{8} \\…