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Question Number 63940 Answers: 1 Comments: 0
$$\mathrm{If}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\left(\mathrm{2}{r}+\mathrm{4}\right)\mathrm{th}\:\mathrm{and}\:\left({r}−\mathrm{2}\right)\mathrm{th} \\ $$$$\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{\mathrm{18}} \:\mathrm{are} \\ $$$$\mathrm{equal},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{r}\:\mathrm{is} \\ $$
Question Number 63939 Answers: 0 Comments: 0
$$\mathrm{If}\:\:\mathrm{the}\:\left({r}+\mathrm{1}\right)\mathrm{th}\:\mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\sqrt[{\mathrm{3}}]{\frac{{a}}{\sqrt{{b}}}}\:+\:\sqrt{\frac{{b}}{\sqrt[{\mathrm{3}}]{{a}}}}\right)^{\mathrm{21}} \:\mathrm{contains}\:{a}\:\mathrm{and}\:{b}\:\mathrm{to} \\ $$$$\mathrm{one}\:\mathrm{and}\:\mathrm{the}\:\mathrm{same}\:\mathrm{power},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:{r}\:\mathrm{is} \\ $$
Question Number 63938 Answers: 0 Comments: 0
$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{2}{x}+\mathrm{3}{y}−\mathrm{4}{z}\right)^{{n}} \:\mathrm{is} \\ $$
Question Number 63937 Answers: 1 Comments: 0
$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{r}} \:\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{1}−\mathrm{4}{x}\right)^{−\mathrm{1}/\mathrm{2}} \:\:\mathrm{is} \\ $$
Question Number 63936 Answers: 0 Comments: 0
$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{the}\:\mathrm{term}\:\mathrm{independent} \\ $$$$\mathrm{of}\:\:{x}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\: \\ $$$$\left(\mathrm{1}+{x}+\mathrm{2}{x}^{\mathrm{3}} \right)\left(\frac{\mathrm{3}}{\mathrm{2}}\:{x}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{3}{x}}\right)^{\mathrm{9}} \:\mathrm{is} \\ $$
Question Number 63935 Answers: 0 Comments: 0
$$\mathrm{If}\:\mathrm{the}\:\mathrm{9}^{\mathrm{th}} \mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left[\mathrm{3}^{\mathrm{log}_{\mathrm{3}} \:\sqrt{\mathrm{25}^{{x}−\mathrm{1}} +\mathrm{7}}} +\:\mathrm{3}^{−\:\frac{\mathrm{1}}{\mathrm{8}}\:\mathrm{log}_{\mathrm{3}} \left(\mathrm{5}^{{x}−\mathrm{1}} +\mathrm{1}\right)} \right]^{\mathrm{10}} \:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{180}\:\mathrm{and}\:\:{x}\:>\:\mathrm{1},\:\mathrm{then}\:{x}\:\mathrm{equals} \\ $$
Question Number 63934 Answers: 1 Comments: 1
$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:\:{x}^{\mathrm{5}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{2}−{x}+\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{6}} \:\:\mathrm{is} \\ $$
Question Number 63930 Answers: 0 Comments: 0
Question Number 63927 Answers: 0 Comments: 7
$$\int_{\mathrm{0}} ^{\pi} \frac{{dx}}{\left(\mathrm{3}+\mathrm{2}{cos}\:{x}\right)^{\mathrm{2}} } \\ $$
Question Number 63922 Answers: 0 Comments: 4
$$\mathrm{if}\:{f}\left({x}\right)=\begin{cases}{\frac{\mathrm{log}\:\left(\mathrm{1}+\mathrm{2}{ax}\right)−\mathrm{log}\:\left(\mathrm{1}−{bx}\right)}{{x}}}&{{x}\neq\mathrm{0}}\\{{k}}&{{x}=\mathrm{0}}\end{cases} \\ $$$$\mathrm{is}\:\mathrm{continuous}\:\mathrm{at}\:{x}=\mathrm{0}\:\mathrm{then}\:{k}=? \\ $$
Question Number 63921 Answers: 0 Comments: 2
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\left(\frac{{n}}{{n}+\mathrm{1}}\right)^{\alpha} +\mathrm{sin}\:\frac{\mathrm{1}}{{n}}\right)^{{n}} \mathrm{where}\:\alpha\in\mathbb{Q} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$
Question Number 63920 Answers: 0 Comments: 5
$$\mathrm{If}\:\alpha,\beta\:\mathrm{are}\:\mathrm{root}\:\mathrm{of}\:\mathrm{quadratic}\:\mathrm{equation} \\ $$$${ax}^{\mathrm{2}} +{bx}+{c}\:\mathrm{then} \\ $$$$\underset{{x}\rightarrow\alpha} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\:\left({ax}^{\mathrm{2}} +{bx}+{c}\right)}{\left({x}−\alpha\right)^{\mathrm{2}} }=? \\ $$
Question Number 63919 Answers: 0 Comments: 3
$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\prod}}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)=? \\ $$
Question Number 63918 Answers: 1 Comments: 0
$$\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\frac{\left[\mathrm{1}−\mathrm{tan}\:{x}/\mathrm{2}\right]\left[\mathrm{1}−\mathrm{sin}\:{x}\right]}{\left[\mathrm{1}+\mathrm{tan}\:{x}/\mathrm{2}\right]\left[\pi−\mathrm{2}{x}\right]^{\mathrm{3}} }=? \\ $$
Question Number 63917 Answers: 0 Comments: 2
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\sqrt{{x}+\sqrt{{x}+\sqrt{{x}}}}−\sqrt{{x}}\right) \\ $$
Question Number 63915 Answers: 0 Comments: 0
$$\mathrm{If}\:{p}\:\mathrm{is}\:\mathrm{nearly}\:\mathrm{equal}\:\:\mathrm{to}\:{q}\:\:\mathrm{and}\:\:{n}\:>\:\mathrm{1},\:\mathrm{such} \\ $$$$\mathrm{that}\:\frac{\left({n}+\mathrm{1}\right){p}+\left({n}−\mathrm{1}\right){q}}{\left({n}−\mathrm{1}\right){p}+\left({n}+\mathrm{1}\right){q}}\:=\:\left(\frac{{p}}{{q}}\right)^{{k}} ,\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{is} \\ $$
Question Number 63909 Answers: 1 Comments: 1
$$\mathrm{The}\:\mathrm{integral}\:\mathrm{part}\:\mathrm{of}\:\left(\mathrm{8}\:+\:\mathrm{3}\sqrt{\mathrm{7}}\:\right)^{\mathrm{20}} \:\mathrm{is}\:\mathrm{even}. \\ $$
Question Number 63908 Answers: 0 Comments: 1
$$\mathrm{If}\:{R}=\left(\mathrm{5}\sqrt{\mathrm{5}}\:+\mathrm{11}\right)^{\mathrm{2}{n}+\mathrm{1}} =\:\left[{R}\right]+\:{F},\:\mathrm{where} \\ $$$$\left[{R}\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{in}\:{R},\: \\ $$$$\mathrm{then}\:{RF}=\:\mathrm{2}^{\mathrm{2}{n}+\mathrm{1}} . \\ $$
Question Number 63907 Answers: 1 Comments: 1
$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{5}} \:\mathrm{in}\:\left(\mathrm{1}+\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} +...\right)^{−\mathrm{3}/\mathrm{2}} \\ $$$$\mathrm{is}\:\mathrm{21}. \\ $$
Question Number 63904 Answers: 0 Comments: 1
Question Number 63903 Answers: 0 Comments: 0
Question Number 63900 Answers: 0 Comments: 1
Question Number 63895 Answers: 0 Comments: 0
Question Number 63894 Answers: 0 Comments: 1
$${sove}\:{the}\:\left({de}\right)\:{x}^{\mathrm{2}} {y}^{'} \:−\left(\mathrm{2}{x}+\mathrm{3}\right){y}\:={sin}\left({x}^{\mathrm{2}} \right)\:\:{with}\:{y}\left(\mathrm{1}\right)=\mathrm{2}\:{and} \\ $$$${y}^{'} \left(\mathrm{1}\right)=\mathrm{1}\:. \\ $$
Question Number 63893 Answers: 0 Comments: 1
$$\left.\mathrm{1}\right)\:{simplify}\:{W}_{{n}} \left({z}\right)=\left(\mathrm{1}+{z}\right)\left(\mathrm{1}+{z}^{\mathrm{2}} \right)....\left(\mathrm{1}+{z}^{\mathrm{2}^{{n}} } \right)\:\left({z}\:{from}\:{C}\right) \\ $$$$\left.\mathrm{2}\right)\:{simplify}\:{P}_{{n}} \left(\theta\right)\:=\left(\mathrm{1}+{e}^{{i}\theta} \right)\left(\mathrm{1}+{e}^{\mathrm{2}{i}\theta} \right).....\left(\mathrm{1}+{e}^{{i}\mathrm{2}^{{n}} \theta} \right)\:{and}\:{sove} \\ $$$${P}_{{n}} \left(\theta\right)=\mathrm{0} \\ $$
Question Number 63892 Answers: 0 Comments: 3
$${calculate}\:{A}=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{2017}} }{\mathrm{1}+{x}^{\mathrm{2019}} }\:{dx}\:\:{and}\:{B}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2019}} }{\mathrm{1}+{x}^{\mathrm{2021}} }\:{dx} \\ $$$${calculate}\:{the}\:{fraction}\:\frac{{A}}{{B}} \\ $$
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