Question Number 151082 by mathdanisur last updated on 18/Aug/21 Commented by Mokmokhi last updated on 18/Aug/21 $$\mathrm{This}\:\mathrm{can}\:\mathrm{be}\:\mathrm{proven}\:\mathrm{starting}\:\mathrm{from}\:\mathrm{left}. \\ $$$$\mathrm{By}\:\mathrm{substituting}\:{u}=\frac{\pi}{\mathrm{2}}−{x}. \\ $$$$\mathrm{After}\:\mathrm{evaluation}.\:\mathrm{By}\:\mathrm{dummy}\:\mathrm{variables}\:\mathrm{done}. \\ $$ Terms of…
Question Number 151080 by malwan last updated on 18/Aug/21 $${if}\:\left({f}\circ{f}\circ{f}\circ{f}\right)\left({x}\right)=\mathrm{16}{x}+\mathrm{15} \\ $$$${find}\:{f}\left({x}\right) \\ $$ Answered by Mokmokhi last updated on 18/Aug/21 $$\mathrm{It}\:\mathrm{is}\:\mathrm{reasonable}\:\mathrm{to}\:\mathrm{say}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{linear}\:\mathrm{by}\:\mathrm{rejecting}\:\mathrm{other}\:\mathrm{possibilities}. \\ $$$$\mathrm{Then}\:{f}\left({x}\right)={ax}+{b}\:\mathrm{for}\:\mathrm{some}\:\mathrm{unknowns}\:{a}\:\mathrm{and}\:{b}. \\…
Question Number 151078 by mathdanisur last updated on 18/Aug/21 $$\Omega_{\boldsymbol{\mathrm{k}}} =\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{n}}\:\centerdot\underset{\boldsymbol{\mathrm{p}}=\mathrm{0}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\frac{\begin{pmatrix}{\mathrm{n}}\\{\mathrm{p}}\end{pmatrix}}{\begin{pmatrix}{\mathrm{n}+\mathrm{k}}\\{\mathrm{n}+\mathrm{p}}\end{pmatrix}}\:\:;\:\:\mathrm{k}\in\mathbb{N}^{\ast} -\mathrm{fixed} \\ $$$$\mathrm{find}\:\:\Omega=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\Omega_{\boldsymbol{\mathrm{n}}-\mathrm{1}} }\:\centerdot\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\sqrt[{\boldsymbol{\mathrm{i}}^{\mathrm{2}} }]{\boldsymbol{\mathrm{i}}!}\: \\ $$ Terms…
Question Number 20001 by Tinkutara last updated on 20/Aug/17 $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{quadratic} \\ $$$$\mathrm{equation}\:\mathrm{8sec}^{\mathrm{2}} \theta\:−\:\mathrm{6sec}\theta\:+\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{is} \\ $$ Answered by mrW1 last updated on 20/Aug/17 $$\mathrm{8sec}^{\mathrm{2}} \theta\:−\:\mathrm{6sec}\theta\:+\:\mathrm{1}\:=\:\mathrm{0} \\…
Question Number 151073 by mathdanisur last updated on 18/Aug/21 $$\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\begin{pmatrix}{\mathrm{50}}\\{\mathrm{0}}\end{pmatrix}^{\mathrm{2}} +\begin{pmatrix}{\mathrm{50}}\\{\mathrm{1}}\end{pmatrix}^{\mathrm{2}} +\begin{pmatrix}{\mathrm{50}}\\{\mathrm{2}}\end{pmatrix}^{\mathrm{2}} +…+\begin{pmatrix}{\mathrm{50}}\\{\mathrm{49}}\end{pmatrix}^{\mathrm{2}} +\begin{pmatrix}{\mathrm{50}}\\{\mathrm{50}}\end{pmatrix}^{\mathrm{2}} \\ $$ Answered by ArielVyny last updated on 18/Aug/21…
Question Number 85532 by oustmuchiya@gmail.com last updated on 22/Mar/20 $${Find}\:{the}\:{term}\:{independent}\:{of}\:\boldsymbol{\mathrm{x}}\:{in}\:{the}\:{expression}\:{of}\:\left(\mathrm{2}{x}−\frac{\mathrm{1}}{\mathrm{2}{x}}\right)^{\mathrm{9}} \\ $$ Answered by mind is power last updated on 22/Mar/20 $$\left(\mathrm{2}{a}−\frac{\mathrm{1}}{\mathrm{2}{a}}\right)^{{k}} \\ $$$$=\underset{{i}=\mathrm{0}} {\overset{{k}}…
Question Number 151070 by mathdanisur last updated on 18/Aug/21 $$\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{30}\:\:\:\mathrm{and}\:\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{1}}{\mathrm{a}}\:+\:\frac{\mathrm{1}}{\mathrm{b}}\:+\:\frac{\mathrm{1}}{\mathrm{c}} \\ $$ Commented by john_santu last updated on 18/Aug/21 $$\frac{\mathrm{1}}{\mathrm{a}}+\frac{\mathrm{1}}{\mathrm{b}}+\frac{\mathrm{1}}{\mathrm{c}}\geqslant\frac{\left(\mathrm{1}+\mathrm{1}+\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{a}+\mathrm{b}+\mathrm{c}}=\frac{\mathrm{9}}{\mathrm{30}}=\frac{\mathrm{3}}{\mathrm{10}}…
Question Number 151059 by john_santu last updated on 18/Aug/21 $$\:\:\:\:\sqrt[{\mathrm{3}}]{\sqrt[{\mathrm{3}}]{\mathrm{x}−\mathrm{2}}\:+\mathrm{2}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{2}−\sqrt[{\mathrm{3}}]{\mathrm{x}+\mathrm{2}}}\:=\:\mathrm{2}\: \\ $$$$\:\:\:\:\mathrm{x}\:=? \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 151052 by mathdanisur last updated on 17/Aug/21 $$\mathrm{if}\:\:\:\mathrm{a};\mathrm{b};\mathrm{c}\:\:\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers}\:\:\mathrm{and} \\ $$$$\frac{\mathrm{a}}{\mathrm{1}+\mathrm{a}}\:+\:\frac{\mathrm{b}}{\mathrm{1}+\mathrm{b}}\:+\:\frac{\mathrm{c}}{\mathrm{1}+\mathrm{c}}\:=\:\mathrm{1}\:\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{abc}\:\leqslant\:\frac{\mathrm{1}}{\mathrm{8}} \\ $$ Answered by dumitrel last updated on 18/Aug/21 $$\mathrm{3}{r}+\mathrm{2}{q}+{p}=\mathrm{1}+{p}+{q}+{r}\Rightarrow\mathrm{2}{r}+{q}=\mathrm{1} \\…
Question Number 151045 by mathdanisur last updated on 17/Aug/21 Terms of Service Privacy Policy Contact: info@tinkutara.com