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Question Number 107353 Answers: 2 Comments: 0
$${If}\:\mathrm{a}\:\mathrm{b}\:\mathrm{13}\:\mathrm{c}\:\mathrm{d}\:\mathrm{25}\:\mathrm{are}\:\mathrm{six}\:\mathrm{consecutive}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{A}{P}\:.{find}\:{tbe}\:{value}\:{of}\:{a}\:{b}\:{c}\:{and}\:{d}. \\ $$
Question Number 107350 Answers: 1 Comments: 0
Question Number 107342 Answers: 1 Comments: 0
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathscr{E}{valuate}: \\ $$$$\:\:\:\:\:\:\:\chi:=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} {x}^{\mathrm{2}} {tan}\left({x}\right){dx}=\:???\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\bigstar{prepared}\:{by}:\bigstar \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\clubsuit\clubsuit\clubsuit\:\:\:\mathscr{M}.\mathscr{N}\:\clubsuit\clubsuit\clubsuit \\ $$$$ \\ $$
Question Number 107385 Answers: 0 Comments: 7
$$\:\:\:\:\:\iddots\mathcal{B}{e}\mathcal{M}{ath}\iddots \\ $$$${Given}\:\mathrm{6}{x}^{\mathrm{2}} −\mathrm{6}{px}+\mathrm{14}{p}−\mathrm{2}=\mathrm{0} \\ $$$${has}\:{the}\:{roots}\:{are}\:\:{u}\:\&\:{v}\:{where}\:{u},{v}\:\notin\mathbb{Z} \\ $$$${If}\:{u},{v}\:\geqslant\:\mathrm{1}\:,\:{then}\:{the}\:{value}\:{of}\:\mid{u}−{v}\mid\:. \\ $$$$\left({a}\right)\mathrm{14}\:\:\:\:\:\left({b}\right)\mathrm{15}\:\:\:\:\:\left({c}\right)\mathrm{16}\:\:\:\:\:\left({d}\right)\mathrm{17}\:\:\:\left({e}\right)\:\mathrm{18} \\ $$
Question Number 107352 Answers: 5 Comments: 0
$$\:\:\:\:\:\:\:\:\circledcirc\mathcal{B}{e}\mathcal{M}{ath}\circledcirc \\ $$$$\left(\mathrm{1}\right)\mathrm{1}−\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:+\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}−\frac{\mathrm{1}}{\sqrt{\mathrm{4}}}+\frac{\mathrm{1}}{\sqrt{\mathrm{5}}}−\frac{\mathrm{1}}{\sqrt{\mathrm{6}}}+...=? \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{1}+\mathrm{sin}\:{x}\right)^{\frac{\mathrm{1}}{{x}}} \:? \\ $$
Question Number 107331 Answers: 1 Comments: 0
$$\mathrm{If}\:{A}\:\mathrm{is}\:\mathrm{an}\:\mathrm{invertible}\:\mathrm{matrix},\:\mathrm{then}\:\mathrm{det}\left({A}^{−\mathrm{1}} \right) \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$
Question Number 107330 Answers: 2 Comments: 0
$$\mathrm{If}\:\:\:{a}+{b}+{c}=\mathrm{0}\:\mathrm{one}\:\mathrm{root}\:\mathrm{of} \\ $$$$\begin{vmatrix}{{a}−{x}}&{\:\:\:\:{c}}&{\:\:\:{b}}\\{\:\:\:\:{c}}&{{b}−{x}}&{\:\:\:{a}}\\{\:\:\:\:{b}}&{\:\:\:{a}}&{{c}−{x}}\end{vmatrix}=\mathrm{0}\:\mathrm{is} \\ $$
Question Number 107328 Answers: 1 Comments: 6
Question Number 107327 Answers: 0 Comments: 0
$$\mathrm{1}\leqslant{p}\leqslant{k}\leqslant{n} \\ $$$${show}\:{that}\:\underset{{k}={p}} {\overset{{n}} {\sum}}\boldsymbol{{C}}_{{k}−\mathrm{1}} ^{{p}−\mathrm{1}} =\boldsymbol{{C}}_{{n}} ^{{p}} \\ $$$${please}\:{i}\:{need}\:{help} \\ $$
Question Number 107320 Answers: 1 Comments: 6
$${let}\:{x},{y},{z}\:{be}\:{a}\:{complex}\:{numbers} \\ $$$${as}\:\mid{x}\mid=\mid{y}\mid=\mid{z}\mid=\mathrm{1} \\ $$$$\begin{cases}{{x}+{y}+{z}=\mathrm{1}}\\{{xyz}=\mathrm{1}}\end{cases} \\ $$$${calcul}\:\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}+\frac{\mathrm{1}}{{z}}=? \\ $$$$\:\:{x}=?\:{y}=?\:{z}=? \\ $$$${please}\:{i}\:{need}\:{a}\:{help} \\ $$
Question Number 107314 Answers: 2 Comments: 0
$$\:\:\:\:\:\:\:\:\doublebarwedge{bemath}\doublebarwedge \\ $$$$\:\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}\:\mathrm{ln}\:\left(\mathrm{1}+\mathrm{sin}\:{x}\right)\:{dx}\:? \\ $$
Question Number 107313 Answers: 2 Comments: 0
$$\begin{cases}{{x}+{y}\sqrt{{x}}\:=\:\frac{\mathrm{95}}{\mathrm{8}}}\\{{y}+{x}\sqrt{{y}}\:=\:\frac{\mathrm{93}}{\mathrm{8}}}\end{cases}\:.\:\mathcal{F}{ind}\:\sqrt{{xy}} \\ $$
Question Number 107310 Answers: 3 Comments: 0
Question Number 107304 Answers: 2 Comments: 0
$$\:\:\:\curlyvee{bemath}\curlyvee \\ $$$$\left(\mathrm{1}\right){Find}\:{domain}\:{of}\:{function}\: \\ $$$${f}\left({x}\right)=\:\sqrt{\mathrm{log}\:_{\mathrm{0}.\mathrm{2}} \left(\frac{{x}+\mathrm{2}}{{x}−\mathrm{1}}\right)−\mathrm{1}}\: \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\mathrm{9}\pi}{\mathrm{8}}−\mathrm{2}{x}\right)−\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\mathrm{7}\pi}{\mathrm{8}}−\mathrm{2}{x}\right)}{\mathrm{sin}\:\left(\mathrm{2018}\pi+\mathrm{4}{x}\right)}=? \\ $$
Question Number 107299 Answers: 2 Comments: 0
$$\:\:\:\:\:\:\maltese\mathrm{bobhans}\maltese \\ $$$$\mathrm{find}\:\mathrm{without}\:\mathrm{L}'\mathrm{Hopital}\:\mathrm{and}\:\mathrm{series}\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}−\mathrm{sin}\:\mathrm{x}}{\mathrm{x}^{\mathrm{3}} }\:? \\ $$
Question Number 107292 Answers: 0 Comments: 0
$$\mathrm{let}\:\mathrm{U}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{sin}\left(\frac{\mathrm{k}}{\mathrm{n}}\right)\mathrm{sin}\left(\frac{\mathrm{k}}{\mathrm{n}^{\mathrm{2}} }\right) \\ $$$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{U}_{\mathrm{n}} \\ $$
Question Number 107291 Answers: 1 Comments: 0
$$\mathrm{let}\:\mathrm{x}\in\mathrm{R}−\left\{\mathrm{1},−\mathrm{1}\right\}\:\mathrm{explicit}\:\mathrm{the}\:\mathrm{function} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\mathrm{ln}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2xcos}\theta\:+\mathrm{1}\right)\mathrm{d}\theta \\ $$$$ \\ $$
Question Number 107290 Answers: 0 Comments: 0
$$\mathrm{find}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \frac{\mathrm{1}}{\mathrm{n}}\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}−\mathrm{1}} \:\frac{\mathrm{k}}{\sqrt{\mathrm{4n}^{\mathrm{2}} −\mathrm{k}^{\mathrm{2}} }} \\ $$
Question Number 107289 Answers: 1 Comments: 0
$$\mathrm{fnd}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\:\left(\frac{\left(\mathrm{2n}\right)!}{\mathrm{n}^{\mathrm{n}} \:\mathrm{n}!}\right)^{\frac{\mathrm{1}}{\mathrm{n}}} \\ $$
Question Number 107288 Answers: 1 Comments: 0
$$\mathrm{let}\:\mathrm{u}_{\mathrm{n}} =\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} }\prod_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \left(\mathrm{n}^{\mathrm{2}} \:+\mathrm{k}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{n}}} \\ $$$$\mathrm{determine}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \mathrm{u}_{\mathrm{n}} \\ $$
Question Number 107287 Answers: 1 Comments: 0
$$\mathrm{find}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \sum_{\mathrm{k}=\mathrm{n}} ^{\mathrm{2n}−\mathrm{1}} \:\frac{\mathrm{1}}{\mathrm{k}+\mathrm{n}} \\ $$
Question Number 107286 Answers: 1 Comments: 0
$$\mathrm{let}\:\mathrm{f}_{\mathrm{n}} \left(\mathrm{x}\right)\:=\mathrm{ne}^{−\mathrm{nx}} \:\:\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{f}_{\mathrm{n}} \left(\mathrm{x}\right)\mathrm{dx} \\ $$$$\mathrm{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \mathrm{f}_{\mathrm{n}} \left(\mathrm{x}\right)\mathrm{dx}\:\:\mathrm{is}\:\mathrm{the}\:\mathrm{convergence}\:\mathrm{uniform}\:\mathrm{on}\:\left[\mathrm{0},\mathrm{1}\right]? \\ $$
Question Number 107285 Answers: 0 Comments: 0
$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{arctan}\left(\mathrm{2x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$
Question Number 107284 Answers: 0 Comments: 0
$$\mathrm{find}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\mathrm{1}}{\sqrt{\left(\mathrm{k}+\mathrm{n}\right)\left(\mathrm{k}+\mathrm{n}+\mathrm{1}\right)}} \\ $$
Question Number 107283 Answers: 0 Comments: 0
$$\mathrm{f}\:\mathrm{integrable}\:\mathrm{continue}\:\mathrm{on}\:\left[\mathrm{a},\mathrm{b}\right]\:\mathrm{let}\:\mathrm{m}\:=\mathrm{inf}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{M}=\mathrm{sup}\:\mathrm{f}\left(\mathrm{x}\right) \\ $$$$\left(\mathrm{x}\:\in\left[\mathrm{a},\mathrm{b}\right]\:\mathrm{prove}\:\mathrm{that}\:\:\:\:\:\:\:\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} \leqslant\int_{\mathrm{a}} ^{\mathrm{b}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}×\int_{\mathrm{a}} ^{\mathrm{b}} \:\frac{\mathrm{dx}}{\mathrm{f}\left(\mathrm{x}\right)}\leqslant\frac{\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} }{\mathrm{4}}\frac{\left(\mathrm{m}+\mathrm{M}\right)^{\mathrm{2}} }{\mathrm{mM}}\right. \\ $$
Question Number 107279 Answers: 1 Comments: 1
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