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Question Number 12760 by tawa last updated on 30/Apr/17

Answered by mrW1 last updated on 01/May/17

$$Q5$$$$totalnumberofwaysforSofiaandTesa$$$$eachtochooseoneintegerfrom1to10:$$$$10\times 10=100$$$$$$$$onlyfollowingchoiceshavethefeature$$$$thatoneofthechoosenintegeris$$$$thesquareoftheother:$$$$SofiaTesa$$$$11$$$$24$$$$39$$$$42$$$$93$$$$--------$$$$therearetotally5ways.$$$$$$$$inanotherwordintheother95ways$$$$neitherofthechoosenintegeris$$$$thesquareoftheother,thecorresponding$$$$probabilityis95/100=0.95.$$$$$$$$\Rightarrow AnswerE$$

Commented by tawa last updated on 01/May/17

$$\mathrm{wow},\mathrm{i}\mathrm{really}\mathrm{appreciate}\mathrm{sir}.\mathrm{God}\mathrm{bless}\mathrm{you}$$

Answered by mrW1 last updated on 01/May/17

$$Q6$$$$a=2^{\frac{1}{2}}$$$$b=3^{\frac{1}{3}}$$$$c=6^{\frac{1}{6}}$$$$$$$$\ln a=\frac{1}{2}\ln 2$$$$$$$$\ln b=\frac{1}{3}\ln 3=\frac{1}{3}\ln 9^{\frac{1}{2}}>\frac{1}{3}\ln 8^{\frac{1}{2}}=\frac{1}{3}\ln 2^{\frac{3}{2}}=\frac{1}{2}\ln 2=\ln a$$$$\Rightarrow \ln b>\ln a$$$$\Rightarrow 3^{\frac{1}{3}}>2^{\frac{1}{2}}$$$$$$$$\ln c=\frac{1}{6}\ln 6=\frac{1}{2}\ln 6^{\frac{1}{3}}<\frac{1}{2}\ln 8^{\frac{1}{3}}=\frac{1}{2}\ln 2=\ln a$$$$\Rightarrow \ln c<\ln a$$$$\Rightarrow 6^{\frac{1}{6}}<2^{\frac{1}{2}}$$$$$$$$i.e.6^{\frac{1}{6}}<2^{\frac{1}{2}}<3^{\frac{1}{3}}$$$$$$$$\Rightarrow \mathit{A}nswerC$$

Commented by tawa last updated on 01/May/17

$$\mathrm{Nice}\mathrm{one},\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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