UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. B.Sc. Mathematics. (2011 Admn.) Semester Core Course VECTOR CALCULUS

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1 Shool of Dtne Eution UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B.S. Mthemt 2011 Amn. V Semester Core Course VECTOR CALCULUS Question Bnk & Answer Key 1. The Components of the vetor with initil point p : 621 n terminl point re The length of the vetor with initil point p: 325 n terminl point The ngle etween the vetors [12 3 ] n [0 21] os os 4. The norml vetor to the line x2y2 0 os os [12] [12] [12] [12] 5. The stright line through the point 13 in the x y plne n perpeniulr to the stright line 2y2 0 3 y2 y1 2 y5 2 y5 6. The unit vetor perpeniulr to the plne 4x2y4z 7 [ ] [ ] [ [ ] ] 7. If [11 0] n [300] in the righthne oorintes then x [003] [003] [300] [300] Vetor Clulus 1

2 Shool of Dtne Eution 8. The volume of the prllelopipe whose oterminl eges representing the vetors 3i4j 2i3j4k n 5k The volume of the tetrheron with oterminl eges representing the vetors i j ij n 2k 10.if the vetors 2ijk i2j3k n 3i j5k re oplnr then the vlue of The eqution of the plne etermine y the points n y13z y13z y13z y13z30 12.The prmetri equtions for the line through 323 n 114 re 14t y13t z47t 24t y23t 14t y13t z47t 34t y83t z57t z47t 13.The tne etween the point 115 n the line LX 1t y3t z2t The eqution of the plne through the point 307 perpeniulr to the vetor 5i2jk 5x2yz22 5x2yz22 5x2yz22 5x2yz22 15.The point of intersetion of the line x 2t y2t z1t n the plne 3x2y6z The spheril oorinte eqution for the one z ф 17.if rt sin ф /4 y3k then sin 3k os 6i2j6k 2 z Vetor Clulus os ф /2 none of these j3k j sin j then the elertion t 1 6i3k 19. If Ft t i 2 j 3k then 5i6j3k A prtile moves long the urve 3 20 t 8i3jk 6i6k i 6i2j3k j 3k i j3k 2

3 Shool of Dtne Eution 20. The length of one turn of the helix rt os tk The unit tngent vetor t point to the urve r os sin os os sin os sin sint sin os 22. The rius of urvture of 23. The vlue of 2 2 t 34 2r none of these 24. The omin of the funtion fxyz xylnz Entire Spe {xyz : xyz 0} hlf spe z>0 hlf spe z<0 25. The rnge of the funtion f xy os [01] [11] [0&] [11] 26. Whih of the following hols for the funtion fxy lim lim exts lim 0 none of these 27. Whih of the following hols for the funtion fxy oesn t exts f ontinuous everywhere f ontinuous nowhere f ontinuous on {xy R2: x y} f ontinuous on {x y R2:xy} 28. Let fxy x y n gzy gfxy e two ontinuous funtions. Then the omposition funtion ontinuous ontinuous Continuous t origin None of these. 29. If fxy 0 sin then the vlue of t If fxy Vetor Clulus then the vlue of ʄ 3

4 Shool of Dtne Eution 31. The plne x2 intersets the proloi z to the prol t then 33. If then 5 1 NOT 0 1 ƒ then If w. 1 1 t the point The unit norml to the surfe 7 os then Vetor Clulus 3 NOT /t t the point 20 in the iretion A t the point y t the point The eqution for the tngent plne to the surfe The eqution for the tngent to the ellipse 3 pv The erivtive of 3 D then the prtil erivtive W given y: 37. The lineriztion of Let ᴡ pv gpv NOT 35. If u log tn tn tn then sin 2 4 in prol. Then the slope of the tngent If fxy 34. If ztn t

5 Shool of Dtne Eution For the K z the vlue of / k 44. The funtion NOT K hs lol mximum lol minimum oth lol mximum & minimum no lol extreme vlues The Centre of urvture t the point 22 to the urve The solute mximum vlue of the gurnt oune y the lines on the tringulr plte in The minimum vlue tht the funtion The mximum vlue tht the funtion 3 4 tkes on the irle None of these Whih mong the following the vlue of 52. Wht the vlue of ʃ ʃ The vlue of the integrl 54. The re enlose etween in n ellipse. The points on the 010 n n 011 over the first gurnt of the irle n n 010 tkes on the ellipse 50. The plne 1 uts the yliner ellipse tht lies losest to the origin re Vetor Clulus losest to the origin on the plne The point n 1 n

6 Shool of Dtne Eution 55. The re enlose y the ellipse 2 4π3 1 None of these. 56. The volume of the soli enlose y the sphere of rius 3π3 1/3 1/6 ¼ The volume enlose y the oorinte plnes n the portion of the plne x y z 1 in the first otnt ½ 58. The vlue of the urve y 1 1 where R the semi irulr region oune y the x x n 59. Whih mong the following the vlue of / 26 log log log log The volume of the region D enlose y the surfes z z8 8π 2 4π 2 2π 2 π 2 3 & 61. Let V e the volume oune y the plnes x 0 y 0 z 0 n x y z 1. Then wht the vlue of log 2 log 2 log 2 log The entroi soli of the with ensity given y δ 1 enlose y the yliner x2 y2 4 oune ove y the proloi z x 2 y2 n elow y the xy plne lies insie the soli lies outsie the soli lies on the soli None of these 63. The volume of the upper region D ut form the soli sphere π/3 π/2 π/4 1 y the one π/6 π/3 π/8 π/4 64. A soli of onstnt ensity δ 1 oupies the upper region D ut from the soli the one π/3. The soli s moment of inerti out the z x given y. π/ The volue of 4/3 Vetor Clulus π/6 2 8/11 7/5 2/9 1 y 6

7 Shool of Dtne Eution 66.Consier the trnsformtion x u osv y u sin v the join for the trnsformtion v uv u v u 1 0 1/ Wht the vlue otine y integrting the funtion fxyz x3y2z over the line segment joining the origin n the point 111 None of these 68. A oil spring lies long the helix rt os4ti sin4tj k 0 t 2π. The spring s ensity onstnt δ 1. Then the rius of gyrtion of the spring out the zx A slener metl rh enser t the ottom thn top lies long the semi irle y2 z2 1 z 0 in the yzplne. If the ensity t the point x y z on the rh δx y z 2z then the entre of mss of the rh 70. The grient fiel of fx y z xyz yzi xzj xyk xzi yzj xyk xyi xzj yzk None of these 71. The unit norml to the surfe x2y 2xz 4 t the point If y y2 z2i x z 2xyj y 2xzk n then wht xz xy yz2 1 xz yz xz2 xy2 xz2 1 xy xy2 xz2 yz Whih mong the following the work one in moving prtile one roun irle C in the xyplne. Given the irle hs entre t the origin n rius 3 n the fore fiel given y F 2x y zi x y z2 j 3x 2y 4zk. 8π 80π 88π π 74. If F 3x2 6yj 14yzj 20xz2k then the vlue of 000 to 111 with prmetri from x t y t2 z t3. 18π where urve from A flui s veloity fiel F xi zj yk. Then the flow long the helix rt osti sin tj t k 0 t π/2 76. The irultion of the fiel F x yi xj roun the irle rt osti sin tj 0 t 2π π Vetor Clulus 2π 7

Shool of Dtne Eution 77. The flux of F x y i xj ross the irle x2 y2 1 in the xy plne π.

The work one y the onservtive fiel xyz long ny smooth urve joining the point 1 3 9 to 1 6 14 2 3 79.

The ifferentil form yx xy 4z ext 1 0 my e ext note of these not onseutive my not e onseutive not ext 81.

The url of the vetor fiel F xy x2 2yi xy y2 j y 2 x 2 xy 1 x y 84.

A prmeteriztion of the sphere x2 y2 z2 2 given y r θ os osθi sin sin θj sin θk 0 π 0 θ 2π r θ sin osθi sin sin θj os k 0 π 0 θ 2π

8 Shool of Dtne Eution 77. The flux of F x y i xj ross the irle x2 y2 1 in the xy plne π. The rest flow ross the urve outwr inwr no flow None of these 78. The work one y the onservtive fiel xyz long ny smooth urve joining the point to Let F 2x 3i xj os zk F lwys onseutive my e onseutive 80. The ifferentil form yx xy 4z ext 1 0 my e ext note of these not onseutive my not e onseutive not ext 81. The irultion ensity or url of vetor fiel F Mi Nj t the poing xy 82. The ivergene of Fxy x2 2yi xy y2j x 2y 3x y 2x 3y 3x 2y 83. The url of the vetor fiel F xy x2 2yi xy y2 j y 2 x 2 xy 1 x y 84. The re of the p ut from the hemphere x2 y2 z2 2 z 0 y the yliner x2 y2 1 2π 2 2 π 2 2 2π 2 2 2π π π 85. A prmeteriztion of the sphere x2 y2 z2 2 given y r θ os osθi sin sin θj sin θk 0 π 0 θ 2π r θ sin osθi sin sin θj os k 0 π 0 θ 2π r θ sin θ os i sin sin θj sin k 0 2π 0 θ 2π None of these 86. The surfe re of the one z π The irultion of the fiel F plne z2 meets the one z 88. url gr ɤ π π 5 4 k roun the urve C in whih the ounterlokwe s viewe from ove If r 0 z 1 n r r then gr r ɤ ɤ None of these 90. A vetor lle senoil if its ivergene zero url zero 91. The net outwr flux of th fiel F the region D: Vetor Clulus geeint zero zk None of these ross the ounry of 8

9 Shool of Dtne Eution If ^ 4 r zr F 0 x F 0 None of these x 0 0 o 0 the unit vetor in the iretion of r n r r then i v ^ 93. A vetor F irrottionl if F A vetor lle orthogonl to vetor if Vetor prout Commuttive ntiommuttive ssoitive not triutive wet vetor ition. 96. Slr triple prout of three oplnr vetors less thn 0 greter thn 0 equl to 0 None of these 97. The neessry n suffiient onition for the vetor funtion Ft to hve onstnt mgnitue 0 F. 0 F x 0 None of these 98. If FG re ifferentile vetor funtions n ifferentile slr funtion. Then url F x G gr x F url F iv F X G F url G g url F iv F x G G F G Fiv G G iv F url F x G F url G G url F 99. The unit vetor long the vetor u The inequlity. lle Prllelogrm ientity Shwrz inequlity Vetor Clulus Tringle ineuqlity None of these. 9

10 Shool of Dtne Eution ANSWER KEY Reserve Vetor Clulus 10

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion