Question Number 198249 by liuxinnan last updated on 15/Oct/23 $${if}\:\:{sin}\left({x}+\varphi\right)+{cos}\mathrm{2}{x}\leqslant\sqrt{\mathrm{3}\:}\: \\ $$$$\varphi=? \\ $$ Commented by Frix last updated on 16/Oct/23 $$\mathrm{I}\:\mathrm{get}\:\mathrm{one}\:\mathrm{solution} \\ $$$$\varphi=\frac{\pi}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}^{−\mathrm{1}} \:\frac{\left(\mathrm{7}\sqrt{\mathrm{7}}−\mathrm{19}\right)\sqrt{\mathrm{3}}}{\mathrm{9}}…
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Question Number 198246 by liuxinnan last updated on 15/Oct/23 $${prove}\:{that} \\ $$$$\:\underset{{i}=\mathrm{1}} {\overset{\mathrm{2}{n}−\mathrm{1}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{i}−\mathrm{1}} }{{i}}>{ln}\mathrm{2}+\frac{\mathrm{1}}{\mathrm{4}{n}} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 198241 by sonukgindia last updated on 15/Oct/23 Answered by Frix last updated on 15/Oct/23 $$\frac{{x}^{\mathrm{2}} }{\mathrm{9}}−\frac{{y}^{\mathrm{2}} }{\mathrm{4}}=\mathrm{1}\:\mathrm{Hyperbola},\:−\mathrm{3}\geqslant{x}\vee{x}\geqslant\mathrm{3} \\ $$$$\left({x}−\mathrm{1}\right)^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{16}\:\mathrm{Circle},\:−\mathrm{3}\leqslant{x}\leqslant\mathrm{5} \\ $$$$\Rightarrow\:\mathrm{3}\:\mathrm{intersections}…
Question Number 198242 by Tawa11 last updated on 15/Oct/23 How many numbers with a maximum of 5 digits, greater than 4000, can be formed with…
Question Number 198243 by mr W last updated on 15/Oct/23 $${find}\:{the}\:{sum}\:{of}\:{the}\:{first}\:{n}\:{terms}\:{from} \\ $$$$\mathrm{1},\:\mathrm{2}+\mathrm{3},\:\mathrm{4}+\mathrm{5}+\mathrm{6},\:\mathrm{7}+\mathrm{8}+\mathrm{9}+\mathrm{10},\:… \\ $$ Answered by som(math1967) last updated on 15/Oct/23 $$\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+\mathrm{5}+\mathrm{6}+\mathrm{7}+\mathrm{8}+\mathrm{9}+\mathrm{10}+..{n} \\ $$$${no}\:{of}\:{term}\:=\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}…
Question Number 198222 by cortano12 last updated on 14/Oct/23 $$\:\:\:\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{5}^{\mathrm{x}} −\mathrm{3}^{\mathrm{x}} \right)\:=\:\mathrm{log}\:_{\mathrm{5}} \left(\mathrm{4}^{\mathrm{x}} +\mathrm{3}^{\mathrm{x}\:} \right) \\ $$ Commented by Rasheed.Sindhi last updated on 14/Oct/23…
Question Number 198232 by SANOGO last updated on 16/Oct/23 $${find}\: \\ $$$$\underset{{k}={o}} {\overset{{n}} {\sum}}{sin}\left({k}\right) \\ $$ Answered by witcher3 last updated on 14/Oct/23 $$\mathrm{sin}\left(\mathrm{k}\right)\in\mathbb{R} \\…
Question Number 198228 by sulaymonnorboyev140 last updated on 14/Oct/23 $$\left(\mathrm{4}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}\right)^{{x}^{\mathrm{2}} −{x}} >\mathrm{1} \\ $$ Answered by MM42 last updated on 14/Oct/23 $$\begin{cases}{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}>\mathrm{1}}\\{{x}^{\mathrm{2}} −{x}>\mathrm{0}}\end{cases}\Rightarrow\left(−\infty,−\frac{\mathrm{1}}{\mathrm{2}}\right)\cup\left(\mathrm{0},+\infty\right)={A}\:\:\&\:\left(−\infty,\mathrm{0}\right)\cup\left(\mathrm{1},+\infty\right)={B}…
Question Number 198231 by pticantor last updated on 14/Oct/23 Answered by witcher3 last updated on 14/Oct/23 $$\mathrm{C}_{\mathrm{n}} ^{\mathrm{p}} =\frac{\mathrm{n}!}{\mathrm{p}!.\left(\mathrm{n}−\mathrm{p}\right)!} \\ $$$$\mathrm{pC}_{\mathrm{n}} ^{\mathrm{p}} =\frac{\mathrm{n}!}{\left(\mathrm{p}−\mathrm{1}\right)!.\left(\mathrm{n}−\mathrm{p}\right)!}=\mathrm{n}.\frac{\left(\mathrm{n}−\mathrm{1}\right)!}{\left(\mathrm{p}−\mathrm{1}\right)!.\left(\mathrm{n}−\mathrm{1}−\left(\mathrm{p}−\mathrm{1}\right)\right)!} \\ $$$$=\mathrm{nC}_{\mathrm{n}−\mathrm{1}}…