Question Number 118634 by ajfour last updated on 18/Oct/20 $${Prove}\:{that}\:{the}\:{equation}\:{of}\:{the}\:{circle} \\ $$$${passing}\:{through}\:{the}\:{points}\:{of} \\ $$$${intersection}\:{of}\:{these}\:{two}\:{curves}: \\ $$$$\:\:{y}=\mathrm{1}+\frac{{c}}{{x}}\:;\:\:{y}={x}^{\mathrm{2}} \:\:\:\:\:\left({c}\:<\:\frac{\mathrm{2}}{\mathrm{3}\sqrt{\mathrm{3}}}\:\right)\: \\ $$$${is}\:\:\:\left({x}−\frac{{c}}{\mathrm{2}}\right)^{\mathrm{2}} +\left({y}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{1}+\frac{{c}^{\mathrm{2}} }{\mathrm{4}}\:\:. \\ $$ Commented…
Question Number 184160 by cortano1 last updated on 03/Jan/23 $$\:\:{Given}\:\begin{cases}{{a}_{\mathrm{0}} =\mathrm{1}}\\{{a}_{{n}+\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{3}{a}_{{n}} +\sqrt{\mathrm{5}{a}_{{n}} ^{\mathrm{2}} −\mathrm{4}}\:\right)}\end{cases} \\ $$$$\:\forall{n}\geqslant\mathrm{0}\:,\:{n}\in{I}\: \\ $$$$\:\:{find}\:{a}_{{n}} . \\ $$ Commented by mr…
Question Number 53051 by Abror last updated on 16/Jan/19 $$\int\mathrm{sin}\:\left(\mathrm{2}{x}\right)\mathrm{cos}\:{xd}\left({x}\right)= \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 16/Jan/19 $$\frac{\mathrm{1}}{\mathrm{2}}\int\mathrm{2}{sin}\mathrm{2}{xcosxdx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\left({sin}\mathrm{3}{x}+{sinx}\right){dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{−{cos}\mathrm{3}{x}}{\mathrm{3}}+\frac{−{cosx}}{}\right]+{c} \\…
Question Number 184095 by Shrinava last updated on 02/Jan/23 Answered by mr W last updated on 03/Jan/23 Commented by mr W last updated on 03/Jan/23…
Question Number 184085 by Shrinava last updated on 02/Jan/23 $$\left(\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{xy}^{\mathrm{2}} \right)\mathrm{y}^{'} \:+\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{yx}^{\mathrm{2}} \:=\:\mathrm{0} \\ $$$$ \\ $$ Answered by mr W last updated…
Question Number 184084 by HeferH last updated on 02/Jan/23 Answered by Rasheed.Sindhi last updated on 03/Jan/23 $${w}^{\mathrm{3}} =\mathrm{1}\Rightarrow{w}=\mathrm{1},\omega,\omega^{\mathrm{2}} \\ $$$${Let}\:{w}_{\mathrm{1}} =\omega\:\&\:{w}_{\mathrm{2}} =\omega^{\mathrm{2}} \\ $$$${x}={a}+{b} \\…
Question Number 184083 by mnjuly1970 last updated on 02/Jan/23 $$ \\ $$$$\:\:\:\:{f}\left({x}\right)=\:{x}^{\:\mathrm{3}} \:+\mathrm{3}{x}^{\:\mathrm{2}} −{ax}\:\:\:{is}\:\: \\ $$$$\:\:\:\:\:{decreasing}\:{on}\:\:\left[\:−\mathrm{1}\:,\:\mathrm{2}\right] \\ $$$$\:\:\:\:\:\:{then}\:\:{which}\:\:{is}\:{correct}… \\ $$$$\:\:\:\:\:\mathrm{1}:\:\:\:\left[\:−\mathrm{3}\:,\mathrm{24}\right] \\ $$$$\:\:\:\:\:\mathrm{2}:\:\:\left[\:\mathrm{24}\:,\:+\infty\right) \\ $$$$\:\:\:\:\:\:\mathrm{3}:\:\left(−\infty\:,−\mathrm{3}\right] \\…
Question Number 184019 by HeferH last updated on 02/Jan/23 $$\begin{cases}{{y}^{{x}} =\:\mathrm{64}}\\{{y}^{\frac{{x}\:+\:\mathrm{1}}{{x}\:−\:\mathrm{1}}} \:=\:\mathrm{16}}\end{cases} \\ $$$$\:{find}\:“{x}'' \\ $$ Commented by a.lgnaoui last updated on 02/Jan/23 $${Hapy}\:{New}\:{year} \\…
Question Number 118482 by peter frank last updated on 17/Oct/20 $$\mathrm{If}\:\mathrm{the}\:\mathrm{tangents}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{end}\:\mathrm{of}\:\mathrm{a}\:\mathrm{focal}\:\:\mathrm{chord}\:\mathrm{of} \\ $$$$\mathrm{parabola}\:\mathrm{meet}\:\mathrm{the} \\ $$$$\mathrm{tangent}\:\mathrm{at}\:\mathrm{the}\:\:\mathrm{vertex} \\ $$$$\mathrm{in}\:\mathrm{C},\mathrm{D}.\mathrm{prove}\:\mathrm{that}\:\mathrm{CD} \\ $$$$\mathrm{substends}\:\mathrm{a}\:\mathrm{right}\:\mathrm{angle} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{focus} \\ $$…
Question Number 118452 by 2004 last updated on 17/Oct/20 $$\boldsymbol{{Question}}: \\ $$$$\mathrm{2}^{\boldsymbol{{x}}} +\mathrm{2}^{\mathrm{2}\boldsymbol{{x}}+\mathrm{1}} +\mathrm{1}=\boldsymbol{{y}}^{\mathrm{2}} \:\:\:\boldsymbol{{solve}}\:\boldsymbol{{this}}\:\boldsymbol{{equation}}\:\boldsymbol{{if}} \\ $$$$\boldsymbol{{x}},\boldsymbol{{y}\epsilon}\mathbb{Z} \\ $$ Commented by prakash jain last updated…