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Question Number 156186 Answers: 1 Comments: 2
Question Number 156184 Answers: 2 Comments: 1
Question Number 155810 Answers: 1 Comments: 0
$$\mathrm{Verify}\:\mathrm{the}\:\mathrm{identity}\:\mathrm{in}\:\mathrm{Excercise}\:\mathrm{below} \\ $$$$\left.\mathrm{1}\right).\:\mathrm{cos}\:\theta\mathrm{sec}\:\theta=\mathrm{1} \\ $$$$\left.\mathrm{2}\right).\:\left(\mathrm{1}+\mathrm{cos}\:\beta\right)\left(\mathrm{1}−\mathrm{cos}\:\beta\right)=\mathrm{sin}\:^{\mathrm{2}} \beta \\ $$$$\left.\mathrm{3}\right).\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\left(\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}−\mathrm{1}\right)=\mathrm{sin}\:^{\mathrm{2}} \mathrm{x} \\ $$$$\left.\mathrm{4}\right).\:\frac{\mathrm{sin}\:\mathrm{t}}{\mathrm{cosec}\:\mathrm{t}}+\frac{\mathrm{cos}\:\mathrm{t}}{\mathrm{sec}\:\mathrm{t}}=\mathrm{1} \\ $$$$\left.\mathrm{5}\right).\:\frac{\mathrm{cosec}\:^{\mathrm{2}} \theta}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{2}} \theta}\:=\:\mathrm{cot}\:^{\mathrm{2}} \theta \\ $$
Question Number 155809 Answers: 1 Comments: 0
$$\:\:\mathrm{Given}\:\mathrm{that}\:{f}\circ{g}\:=\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}{x}^{\mathrm{2}} \:−\:{x}\:+\:\mathrm{4}}\:\:\mathrm{and} \\ $$$$\:\:{g}\left({x}\right)\:=\:\frac{{x}}{{x}\:−\:\mathrm{2}},\:\mathrm{find}\:{f}\left({x}\right)\:? \\ $$$$ \\ $$
Question Number 155806 Answers: 1 Comments: 0
Question Number 155803 Answers: 2 Comments: 0
$$\mathrm{proof}\:\mathrm{that}\: \\ $$$$\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\mathrm{n}×\mathrm{n}!\:=\:\mathrm{11}!−\mathrm{1} \\ $$
Question Number 155802 Answers: 1 Comments: 0
$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\:\mathrm{2}×\mathrm{1}!+\mathrm{4}×\mathrm{2}!+\mathrm{6}×\mathrm{3}!+...+\mathrm{200}×\mathrm{100}! \\ $$
Question Number 155801 Answers: 3 Comments: 0
$$\:\begin{cases}{\mathrm{m}^{\mathrm{2}} =\mathrm{n}+\mathrm{2}}\\{\mathrm{n}^{\mathrm{2}} =\mathrm{m}+\mathrm{2}}\end{cases}\:\Rightarrow\mathrm{m}\neq\mathrm{n} \\ $$$$\:\mathrm{4mn}−\mathrm{m}^{\mathrm{3}} −\mathrm{n}^{\mathrm{3}} =?\: \\ $$
Question Number 155797 Answers: 0 Comments: 6
$${a},{b},{c},{d},{e}\:\:\left({kids}\right)\:{are}\:{in}\:{ascending} \\ $$$${order}\:{of}\:{heights}.\:{If}\:{they}\:{are} \\ $$$${to}\:{stand}\:{in}\:{a}\:{circle}\:{in}\:{a}\:{way}\:{so} \\ $$$${that}\:{the}\:{sum}\:{of}\:\mid{difference}\:{in} \\ $$$${heights}\:{of}\:{adjacent}\:{pairs}\mid \\ $$$$\:{is}\:{a}\:{minimum},\:{find}\:{this} \\ $$$$\:{minimum}. \\ $$
Question Number 155775 Answers: 0 Comments: 0
Question Number 155774 Answers: 0 Comments: 0
$$\mathrm{if}\:\:\mathrm{x};\mathrm{y}>\mathrm{0}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{xy}\right)^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{4}} \left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} +\left(\mathrm{1}+\mathrm{y}\right)^{\mathrm{2}} }{\mathrm{1}\:+\:\mathrm{xy}\:+\:\mathrm{x}^{\mathrm{2}} \mathrm{y}\:+\:\mathrm{xy}^{\mathrm{2}} }\:\geqslant\:\mathrm{3xy} \\ $$
Question Number 155772 Answers: 1 Comments: 0
Question Number 155770 Answers: 1 Comments: 1
Question Number 155759 Answers: 0 Comments: 1
$$\: \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{ln}\left(\left(\mathrm{sin}\left({x}\right)+\mathrm{cos}\left({x}\right)\right)^{\mathrm{2}} +\mathrm{1}\right)\:{dx} \\ $$$$\: \\ $$
Question Number 155757 Answers: 0 Comments: 3
$$\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} \:\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}\:−\:\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} \:=\:\left(\boldsymbol{{x}}−\mathrm{1}\right)\:\boldsymbol{{y}}^{\mathrm{2}} \:\:;\:\boldsymbol{{y}}\left(\mathrm{1}\right)\:=\:\mathrm{2}\: \\ $$$$ \\ $$$$\square\:\boldsymbol{{M}}\: \\ $$
Question Number 155745 Answers: 2 Comments: 0
$$\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left(\frac{\mathrm{k}}{\left(\mathrm{4k}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{2k}+\mathrm{3}\right)}\right)=? \\ $$
Question Number 155736 Answers: 0 Comments: 0
Question Number 155735 Answers: 1 Comments: 0
Question Number 155729 Answers: 1 Comments: 0
$$\mathrm{A}=\left\{\left({a},{b}\right)\in\mathrm{IR}^{\mathrm{2}} \:/\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} \leqslant\mathrm{1}\right\} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{A}\:\mathrm{can}'\mathrm{t}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{the}\:\mathrm{cartesian} \\ $$$$\mathrm{product}\:\mathrm{of}\:\mathrm{two}\:\mathrm{parts}\:\mathrm{of}\:\mathrm{IR}. \\ $$
Question Number 155724 Answers: 1 Comments: 0
$$\mathrm{Find}: \\ $$$$\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{6}^{°} }\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{42}°}\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{66}°}\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{78}°}\:=\:? \\ $$
Question Number 155720 Answers: 3 Comments: 1
Question Number 155776 Answers: 0 Comments: 0
$$\mathrm{if}\:\:\:\mathrm{x}\in\left(\mathrm{0};\frac{\pi}{\mathrm{2}}\right)\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{2}\:+\:\left(\mathrm{1}+\mathrm{cot}\boldsymbol{\mathrm{x}}\right)\left(\mathrm{tan}^{\mathrm{3}} \boldsymbol{\mathrm{x}}+\mathrm{cot}^{\mathrm{3}} \boldsymbol{\mathrm{x}}\right)}{\left(\mathrm{1}+\mathrm{tan}\boldsymbol{\mathrm{x}}\right)\left(\mathrm{1}+\mathrm{cot}\boldsymbol{\mathrm{x}}\right)}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$
Question Number 155710 Answers: 3 Comments: 1
$$\mathrm{li}\underset{{x}−{oo}} {\mathrm{m}}\:\:\:\frac{\mathrm{1}}{{n}\sqrt{{n}}}\:\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{E}\left(\sqrt{\left.{k}\right)}\right. \\ $$$$ \\ $$
Question Number 155701 Answers: 0 Comments: 3
Question Number 155692 Answers: 1 Comments: 5
Question Number 155686 Answers: 2 Comments: 0
$$\:\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{dx}}{\mathrm{1}+\mathrm{tan}\:{x}}\:=? \\ $$
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