Category: Relation and Functions

Question Number 186103 by MASANJAJJ last updated on 01/Feb/23 $$\mathrm{Let}\mathrm{R}\mathrm{denote}\mathrm{a}\mathrm{set}\mathrm{of}\mathrm{all}\mathrm{ordered}\mathrm{pairs}(\mathrm{x},\mathrm{y})$$$$\mathrm{of}\mathrm{integers}\mathrm{such}\mathrm{that}\mathrm{x}-\mathrm{y}\mathrm{is}\mathrm{an}\mathrm{integral}$$$$\mathrm{multiple}\mathrm{of}3.\mathrm{Which}\mathrm{of}\mathrm{the}\mathrm{followings}\mathrm{ordered}\mathrm{pairs}$$$$\mathrm{belong}\mathrm{to}\mathrm{R}(9,3)(3,9),(1,2)(1,5),(7,2)$$$$(0,4),(4,7).$$$$(\mathrm{note}:\mathrm{a}\mathrm{is}\mathrm{an}\mathrm{integral}\mathrm{multiple}\mathrm{of}\mathrm{b}\mathrm{if}\mathrm{a}=\mathrm{kb}$$$$\mathrm{whe}re\mathrm{k}\mathrm{is}\mathrm{an}\mathrm{integer})$$ Terms…

Question Number 54830 by maxmathsup by imad last updated on 12/Feb/19 $$let{V}_n={\int }_0^{\mathrm{\infty }}\frac{cos\left(nx\right)}{n+x^2}dxwithnintegrnsturalnot0.$$$$1)calculate{V}_n$$$$2)calculate{lim}_{n\to +\mathrm{\infty }}{nV}_n$$$$\left.\mathrm{3}\right)\:{calculate}\:{the}\:{sum}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{V}_{{n}}…

Question Number 120300 by bemath last updated on 30/Oct/20 $$Determineallfunctionf:R\to R$$$$whichsatisfyf(a+x)-f(a-x)=4ax$$$$forallrealaandx.$$ Commented by benjo_mathlover last updated on 30/Oct/20 $${let}\:{f}\left({x}\right)={px}^{\mathrm{2}} +{qx}+{r}…

Question Number 120272 by cantor last updated on 30/Oct/20 $$\phi \left(x\right)={\mathit{x}}^{\mathit{r}}$$$$\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathit{x}\in [0,+\mathrm{\infty }[\mathit{a}\mathit{n}\mathit{d}0<\mathit{r}⩽1$$$$shownthatforany(\mathit{x},\mathit{y})\in {\mathbb{R}}_{+}^2$$$$\mathit{\phi }(\mathit{x}+\mathit{y})⩽\mathit{\phi }\left(\mathit{x}\right)+\mathit{\phi }\left(\mathit{y}\right)$$$$\mathit{p}\mathit{l}\mathit{e}\mathit{a}\mathit{s}\mathit{e}$$ Terms of Service Privacy…

Question Number 54675 by maxmathsup by imad last updated on 09/Feb/19 $${simplify}\:\:{A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{2}} \:{C}_{{n}} ^{{k}} \:{cos}\left({k}\theta\right)\:{and}\:{B}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{2}} \:{C}_{{n}} ^{{k}} \:{sin}\left({k}\theta\right)\:. \…