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A robot is designed to move in a peculiar way and it can be set in motion by a microprocessor program. The program can be initiated by assigning a positive rational value to its variable n. The program directs the robot to move in the following way. As soon as the program is started, the robot starts from the point O, moves 2n metres northward and changes its direction by n° to the right. It then moves 2n metres forward and again changes its direction by n° to the right and continues in this manner till it reaches the starting point O, or till it covers a total distance of 1000 m, whichever happens first, and then it stops.

a. I assigned a value for n and started the program. If the robot finally came back to O and stopped, what is the total distance that it has covered?

1. 180 m

2. 360 m

3. 720 m

4. Cannot be determined.

b. For how many values of n in the intervals [1, 60] does the robot cover less than 1000 m, before it stops?

1. 19

2. 60

3. 355

4. Infinte

a)

$Ifnisfactorof360,thenaccordingtothepatternofmovementfollowedbytherobot,\phantom{\rule{0ex}{0ex}}itwillcoveraregularpolygonofanexternalangleofn\xb0andnumberofsides=\frac{360}{n}\phantom{\rule{0ex}{0ex}}Thelengthofeachsidewillbe2nmetres.HencetherobotwillcomebacktoOinthiscase.\phantom{\rule{0ex}{0ex}}However,ifnisnotafactorof360\xb0,thentherobotwillnotcomebacktoO,butwillcontinue\phantom{\rule{0ex}{0ex}}movingtillitcovers1000mandthenstop.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Note:TherobotmaycomebacktoOforothervaluesofn,whicharenotfactorsof360\xb0butare\phantom{\rule{0ex}{0ex}}factorsof720\xb0,1080\xb0....etc.However,insuchcasesthedis\mathrm{tan}cerequiredtobecoveredbefore\phantom{\rule{0ex}{0ex}}reachingOwillbegreaterthan1000m.\phantom{\rule{0ex}{0ex}}SincetherobotcamebacktoO,nmustbeafactorof360\xb0\phantom{\rule{0ex}{0ex}}andalsototaldis\mathrm{tan}cecovered=\left(no.ofsidesoftheregularpolygon\right)\times \left(lengthofeachside\right)\phantom{\rule{0ex}{0ex}}=\frac{360}{n}\times 2n=720m$

thus option (3) is correct.

b)

$Ifrobotcoverlessthan1000m,thenitmusthavecomebacktoO.Thefactors\phantom{\rule{0ex}{0ex}}of360intherange[1,60]are\frac{360}{1}=360sidesto\frac{360}{60}=6sides.\phantom{\rule{0ex}{0ex}}Allotherrationalvaluesofn,for359sides,358sidesandsoontill6sidesarepossible.\phantom{\rule{0ex}{0ex}}Henceatotalof\left(360-6\right)+1=355valuesarepossible.\phantom{\rule{0ex}{0ex}}Thusoption\left(3\right)iscorrect.$

Regards

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