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Question Number 193192 Answers: 1 Comments: 0
$$\mathrm{solve}\:\mathrm{and}\:\mathrm{solution} \\ $$$$\Omega=\int\sqrt{\mathrm{sin}^{−\mathrm{1}} \mathrm{x}}\mathrm{dx}=? \\ $$
Question Number 193203 Answers: 1 Comments: 0
Question Number 193183 Answers: 1 Comments: 0
$$ \\ $$$${There}\:{exists}\:{a}\:{unique}\:{positive}\:{integer}\:{a}\:{for} \\ $$$${which}\:{The}\:{sum}\:{u}\:\:=\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{2023}} {\sum}}\lfloor\frac{{n}^{\mathrm{2}} −{na}}{\mathrm{5}}\rfloor\:{is}\:{an}\:{integer} \\ $$$${strictly}\:{between}\:−\mathrm{1000}\:\&\:\mathrm{1000}\:{find}\:{a}+{u}. \\ $$
Question Number 193182 Answers: 1 Comments: 0
$$ \\ $$$$\frac{{x}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} −\mathrm{1}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} }+\frac{{z}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} −\mathrm{5}^{\mathrm{2}} }+\frac{{w}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} −\mathrm{7}^{\mathrm{2}} }=\mathrm{1} \\ $$$$\frac{{x}^{\mathrm{2}} }{\mathrm{4}^{\mathrm{2}} −\mathrm{1}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{\mathrm{4}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} }+\frac{{z}^{\mathrm{2}} }{\mathrm{4}^{\mathrm{2}} −\mathrm{5}^{\mathrm{2}} }+\frac{{w}^{\mathrm{2}} }{\mathrm{4}^{\mathrm{2}} −\mathrm{7}^{\mathrm{2}} }=\mathrm{1} \\ $$$$\frac{{x}^{\mathrm{2}} }{\mathrm{6}^{\mathrm{2}} −\mathrm{1}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{\mathrm{6}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} }+\frac{{z}^{\mathrm{2}} }{\mathrm{6}^{\mathrm{2}} −\mathrm{5}^{\mathrm{2}} }+\frac{{w}^{\mathrm{2}} }{\mathrm{6}^{\mathrm{2}} −\mathrm{7}^{\mathrm{2}} }=\mathrm{1} \\ $$$$\frac{{x}^{\mathrm{2}} }{\mathrm{8}^{\mathrm{2}} −\mathrm{1}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{\mathrm{8}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} }+\frac{{z}^{\mathrm{2}} }{\mathrm{8}^{\mathrm{2}} −\mathrm{5}^{\mathrm{2}} }+\frac{{w}^{\mathrm{2}} }{\mathrm{8}^{\mathrm{2}} −\mathrm{7}^{\mathrm{2}} }=\mathrm{1}\: \\ $$$$ \\ $$$${find}\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +{w}^{\mathrm{2}} \right). \\ $$
Question Number 193180 Answers: 3 Comments: 0
$$\mathrm{If}\:{a}\:+\:{b}\:=\:\mathrm{1001}\:\&\:\mathrm{HCF}\left({a},\:{b}\right)\:=\:\mathrm{13}\: \\ $$$$\mathrm{then}\:\mathrm{how}\:\mathrm{many}\:\mathrm{set}\:\mathrm{of}\:{a}\:\&\:{b}. \\ $$
Question Number 193179 Answers: 0 Comments: 0
$$\:\:{f}\left(\theta\right)\:=\:\frac{{v}^{\mathrm{2}} \mathrm{sin}\theta\mathrm{cos}\theta+{v}\mathrm{cos}\:\sqrt{{v}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \theta+{Hg}\:}\:}{{g}} \\ $$$$\:\:\:{f}\left(\theta\right)_{{max}} =? \\ $$
Question Number 193175 Answers: 3 Comments: 0
$${please}\:{solve}\:{for}\:{x}\:{if} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} =\mathrm{2}^{{x}} \\ $$
Question Number 193171 Answers: 0 Comments: 0
Question Number 193170 Answers: 1 Comments: 0
$$\mathrm{solve}\::\:\mathrm{7}^{\mathrm{x}} =−\mathrm{2} \\ $$
Question Number 193162 Answers: 1 Comments: 2
Question Number 193161 Answers: 1 Comments: 0
Question Number 193153 Answers: 0 Comments: 0
Question Number 193149 Answers: 1 Comments: 0
Question Number 193139 Answers: 3 Comments: 0
$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{inequalities}: \\ $$$$\mid\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}\mid>\mathrm{1} \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you} \\ $$
Question Number 193138 Answers: 2 Comments: 0
$$\mathrm{Show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all}\:\mathrm{a},\mathrm{b}\in\mathbb{R} \\ $$$$\left.\mathrm{i}\right)\:\mathrm{ab}\leqslant\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{ii}\right)\:\left(\frac{\mathrm{a}+\mathrm{b}}{\mathrm{2}}\right)^{\mathrm{2}} \leqslant\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{iii}\right)\:\sqrt{\mathrm{ab}}\leqslant\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{a}+\mathrm{b}\right),\:\mathrm{for}\:\mathrm{a},\mathrm{b}\geqslant\mathrm{0}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{have}\:\mathrm{square}\:\mathrm{roots}. \\ $$$$ \\ $$$$\mathrm{Help} \\ $$
Question Number 193137 Answers: 2 Comments: 0
$$\left.\mathrm{1}\right)\:\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mid\mathrm{a}+\mathrm{b}+\mathrm{c}\mid\geqslant\mid\mathrm{a}\mid−\mid\mathrm{b}\mid−\mid\mathrm{c}\mid \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\mathrm{Find}\:\mathrm{all}\:\mathrm{x}\in\mathbb{R}\:\mathrm{that}\:\mathrm{satify}\:\mathrm{the}\:\mathrm{follow}− \\ $$$$\mathrm{ing}\:\mathrm{inequalities}\: \\ $$$$\left.\mathrm{i}\right)\:\mid\mathrm{x}^{\mathrm{2}} −\mathrm{4}\mid<\mathrm{5} \\ $$$$\left.\mathrm{ii}\right)\:\mid\mathrm{x}\mid+\mid\mathrm{x}+\mathrm{2}\mid<\mathrm{5} \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$
Question Number 193117 Answers: 3 Comments: 0
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{cosx}\right)^{\frac{\mathrm{1}}{\mathrm{x}}} \\ $$
Question Number 193116 Answers: 2 Comments: 0
$$\mathrm{Show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\in\:\mathbb{R}\:\mathrm{with} \\ $$$$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\geqslant\:\mathrm{0}\: \\ $$$$\left.\mathrm{1}\right)\:\sqrt{\mathrm{ab}}\sqrt{\mathrm{cd}}\:\leqslant\:\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{2}\right)\:\left(\mathrm{abcd}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} \:\leqslant\:\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\right) \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$
Question Number 193111 Answers: 2 Comments: 0
$$\:\:\:\:\:\:\downharpoonleft\underline{\:} \\ $$
Question Number 193109 Answers: 1 Comments: 0
Question Number 193105 Answers: 0 Comments: 1
Question Number 193099 Answers: 1 Comments: 0
Question Number 193093 Answers: 1 Comments: 0
Question Number 193080 Answers: 1 Comments: 0
$$\mathrm{Show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{a},\mathrm{b}\in\mathbb{R}, \\ $$$$\mid\mathrm{a}−\mathrm{b}\mid\geqslant\mid\mathrm{a}\mid−\mid\mathrm{b}\mid \\ $$$$ \\ $$$$\left(\mathrm{Hint}:\:\mathrm{write}\:\mathrm{a}=\left(\mathrm{a}−\mathrm{b}\right)+\mathrm{b}\right. \\ $$
Question Number 193079 Answers: 0 Comments: 2
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{x}\in\mathbb{R}\:\mathrm{that}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{inequalities}: \\ $$$$\left.\mathrm{a}\right)\:\mid\mathrm{4x}−\mathrm{3}\mid\leqslant\mathrm{11} \\ $$$$\left.\mathrm{b}\right)\:\mid\mathrm{x}−\mathrm{2}\mid>\mid\mathrm{x}+\mathrm{1}\mid \\ $$$$\left.\mathrm{c}\right)\:\mid\mathrm{x}\mid\:+\:\mid\mathrm{x}+\mathrm{2}\mid\:+\:\mid\mathrm{2}−\mathrm{x}\mid\leqslant\mathrm{8} \\ $$
Question Number 193078 Answers: 1 Comments: 0
$$\left.\mathrm{1}\right)\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:\varepsilon>\mathrm{0}\:\mathrm{and}\:\mathrm{a},\mathrm{x}\in\mathbb{R},\:\mathrm{then} \\ $$$$\mid\mathrm{a}−\mathrm{x}\mid<\varepsilon\:\mathrm{iff}\:\mathrm{x}−\varepsilon<\mathrm{a}<\mathrm{x}+\varepsilon \\ $$$$ \\ $$$$\mathrm{help} \\ $$
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