Question Number 98443 by Ar Brandon last updated on 14/Jun/20 $$\mathcal{G}\mathrm{iven}\:\mathrm{the}\:\mathrm{sequence}\:\left(\mathrm{U}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \:\mathrm{defined}\:\mathrm{by}\:\mathrm{U}_{\mathrm{0}} =\mathrm{1}\:\mathrm{and} \\ $$$$\mathrm{U}_{\mathrm{n}+\mathrm{1}} =\mathrm{f}\left(\mathrm{U}_{\mathrm{n}} \right)\:\mathrm{where}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{x}}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\: \\ $$$$\mathcal{S}\mathrm{how}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that}\:\forall\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\mathrm{0}<\mathrm{U}_{\mathrm{n}} \leqslant\frac{\mathrm{1}}{\mathrm{n}} \\…
Question Number 32869 by Rio Mike last updated on 04/Apr/18 $${A}=\begin{pmatrix}{\mathrm{2}\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{1}\:\:\:\:\:\:\:\mathrm{2}}\end{pmatrix}\:;{h}\:\mathrm{and}\:{k}\:\mathrm{are}\:\mathrm{numbers}\:\mathrm{so} \\ $$$$\mathrm{that}\:\mathrm{A}^{\mathrm{2}} ={hA}\:+\:{kI},\mathrm{where}\:\mathrm{I}=\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix}. \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{h}\:\mathrm{and}\:{k}. \\ $$ Commented by abdo mathsup last updated on…
Question Number 32850 by Rio Mike last updated on 03/Apr/18 $${evaluate} \\ $$$$\int_{\mathrm{0}} ^{\pi} \mathrm{42}\left(\mathrm{2}\right){dx} \\ $$ Commented by MJS last updated on 03/Apr/18 $$?…
Question Number 32844 by Rio Mike last updated on 03/Apr/18 $${sketch}\:{on}\:{the}\:{x}−{y}\:{plain}\:{the}\:{locus} \\ $$$${of}\:{a}\:{point},{P}\:{which}\:{moves}\:{such}\:{that} \\ $$$${its}\:{x}\:{and}\:\:{y}\:{coordinates}\:{are}\:{same}. \\ $$$${state}\:{the}\:{locus}. \\ $$ Commented by Tinkutara last updated on…
Question Number 32845 by Rio Mike last updated on 03/Apr/18 $${how}\:{many}\:{moles}\:{are}\:{there}\:{in}\:\mathrm{3}{g}\:{of} \\ $$$${Na}_{\mathrm{2}_{} } {CO}_{\mathrm{3}} \\ $$ Answered by Joel578 last updated on 04/Apr/18 $${Mr}\:=\:\mathrm{2}\left(\mathrm{23}\right)\:+\:\mathrm{12}\:+\:\mathrm{3}\left(\mathrm{16}\right)\:=\:\mathrm{106}…
Question Number 32838 by Rio Mike last updated on 03/Apr/18 $${the}\:{points}\:{M}\left(\mathrm{1},\mathrm{1}\right),{N}\left(\mathrm{2},\mathrm{4}\right)\:{andR}\:\left(\mathrm{3}\:,{q}\right) \\ $$$${lie}\:{on}\:{theresame}\:{straight}\:{line}. \\ $$$${find}\:{the}\:{coordinates}\:{of}\:{R} \\ $$ Commented by Rasheed.Sindhi last updated on 03/Apr/18 $${M}\left(\mathrm{2},\mathrm{4}\right)={N}\left(\mathrm{2},\mathrm{4}\right)\:\mathrm{is}\:\mathrm{one}\:\mathrm{point}.\mathrm{So}\:\mathrm{there}\:\mathrm{are}…
Question Number 32837 by Rio Mike last updated on 03/Apr/18 $${the}\:{length}\:{of}\:{the}\:{line}\:{segment}\:{joining} \\ $$$${A}\:{and}\:{B}\:{is}\:\sqrt{\mathrm{10}}\:.{Given}\:{that}\:{its}\:{double} \\ $$$${the}\:{line}\:{segment}\:{joining}\:{the}\:{points} \\ $$$$\left(\mathrm{7},{n}\right)\:{and}\:\left(\mathrm{6},\mathrm{2}\right)\:{find}\:{the}\:{possible}\:{values}\:{of}\:{n}. \\ $$$$ \\ $$ Commented by MJS last…
Question Number 32834 by Rio Mike last updated on 03/Apr/18 $${find}\:\:{the}\:{value}\:{of}\:{k}\:{for}\:{which}\:{the}\: \\ $$$${length}\:{of}\:{the}\:{line}\:{segment}\:{joining} \\ $$$$\left(\bar {{k}},\mathrm{2}\right)\:{and}\:\left(−\mathrm{1},\mathrm{4}\right)\:{is}\:\mathrm{2}\sqrt{\mathrm{2}}\:{units}. \\ $$$$\:\:\:\:{show}\:{full}\:{working}… \\ $$ Answered by MJS last updated…
Question Number 32793 by Rio Mike last updated on 02/Apr/18 $${Given}\:{that}\:{the}\:{point}\:\left(−\bar {\mathrm{3}},\mathrm{2}\right)\:{lies}\:{on}\:{the} \\ $$$${line}\:\:\:\:{y}=\mathrm{2}{x}\:+\:\mathrm{c}.{find}\:{the}\:{coordinates}\: \\ $$$${of}\:{the}\:{point}\:{of}\:{intersection}\:{of}\:{this}\:{line}\:{and}\:{the}\:{y}−{axis} \\ $$ Answered by Joel578 last updated on 02/Apr/18…
Question Number 32794 by Rio Mike last updated on 02/Apr/18 $${given}\:{that}\:{the}\:{point}\:\left(\bar {{t}},\mathrm{0}\right)\:{lies}\:{on}\:{the} \\ $$$${curve}\:{y}=\mathrm{2}{x}^{\mathrm{2}} −{x}.\:{find}\:{the}\:{value}\:{of}\:{t}. \\ $$ Answered by Joel578 last updated on 02/Apr/18 $${y}\:=\:\mathrm{2}{x}^{\mathrm{2}}…