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 7 1 2 re 133 133 133 133 2. The length of the vetor with initil point p: 325 n terminl point 513 3 4 5 6 3. 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
Shool of Dtne Eution 8. The volume of the prllelopipe whose oterminl eges representing the vetors 3i4j 2i3j4k n 5k 3 4 5 6 9.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 8 6 2 4 11.The eqution of the plne etermine y the points211 321 n 132 11 5y13z30 11 5y13z30 11 5y13z30 11 5y13z30 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 3 2 5 7 14.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 3x2y6z6 112 201 16. 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 013 18. A prtile moves long the urve 3 20 t 8i3jk 6i6k i 6i2j3k j 3k i j3k 2
Shool of Dtne Eution 20. The length of one turn of the helix rt os tk 4 2 2 2 21. The unit tngent vetor t point to the urve r os sin os os sin os sin 2 3 2 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 3 1 1 2 30. If fxy Vetor Clulus then the vlue of ʄ 3
Shool of Dtne Eution 31. The plne x2 intersets the proloi z to the prol t 524 6 then 33. If then 5 1 NOT 0 1 ƒ then 1 7 2 1 38 If w. 1 1 t the point 11 0 41. The unit norml to the surfe 7 os then Vetor Clulus 3 NOT /t 3 2 3 t the point 20 in the iretion A 3 4 1 2 2 t the point 32 8 3 25 8 y2 3 25 4 t the point 2 2 3 42. The eqution for the tngent plne to the surfe 2 2 2 2 1 8 25 8 25 3 0 2 2 7 40. The eqution for the tngent to the ellipse 3 pv 1 39. The erivtive of 3 D then the prtil erivtive W given y: 37. The lineriztion of 2 36. Let ᴡ pv gpv NOT 35. If u log tn tn tn then sin 2 4 in prol. Then the slope of the tngent 4 32. If fxy 34. If ztn 2 5 3 4 7 t 1 1 2 2 5 6 4
Shool of Dtne Eution 7 3 8 26 43. For the K 7 3 8 5 z the vlue of / k 44. The funtion NOT K hs lol mximum lol minimum oth lol mximum & minimum no lol extreme vlues 45. 4 The Centre of urvture t the point 22 to the urve 22 24 42 The solute mximum vlue of 2 2 2 the gurnt oune y the lines 0 9 46. 4 2 on the tringulr plte in 3 61 5 0. 48. The minimum vlue tht the funtion 2 2 49. The mximum vlue tht the funtion 3 4 tkes on the irle 5 4 3 None of these 1 51. Whih mong the following the vlue of 52. Wht the vlue of ʃ ʃ 0 1 5 6 53. The vlue of the integrl 54. The re enlose etween in n ellipse. The points on the 010 n 001 100 n 011 over the first gurnt of the irle 5 1 4 4 1 2 100 n 001 100 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 2 47. The point 44 10 n 1 n 25 5 36 5
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 1 1 57. 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 27 1 27 log 26 27 log 26 26 log 27 60. 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 2 62. 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/12 65. The volue of 4/3 Vetor Clulus π/6 2 8/11 7/5 2/9 1 y 6
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 1/2 1 2 3 000 00.570 00 0.57 0.57 0 0 67.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 4 69. 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 2 2. 3 72. If y y2 z2i x z 2xyj y 2xzk n 111 3 then wht xz xy yz2 1 xz yz xz2 xy2 xz2 1 xy xy2 xz2 yz 1 73. 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 13 7 5 3π 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 11 75. 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 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 1 3 9 to 1 6 14 2 3 79. 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 2 87. The irultion of the fiel F plne z2 meets the one z 88. url gr ɤ π 3 2 4 0 π 5 4 k roun the urve C in whih the ounterlokwe s viewe from ove 1 89. 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
Shool of Dtne Eution 2 92. 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 0 94. A vetor lle orthogonl to vetor if. 0 95. 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 u2 3 100. The inequlity. lle Prllelogrm ientity Shwrz inequlity Vetor Clulus Tringle ineuqlity None of these. 9
Shool of Dtne Eution ANSWER KEY 1 26 51 76 2 27 52 77 3 28 53 78 4 29 54 79 5 30 55 80 6 31 56 81 7 32 57 82 8 33 58 83 9 34 59 84 10 35 60 85 11 36 61 86 12 37 62 87 13 38 63 88 14 39 64 89 15 40 65 90 16 41 66 91 17 42 67 92 18 43 68 93 19 44 69 94 20 45 70 95 21 46 71 96 22 47 72 97 23 48 73 98 24 49 74 99 25 50 75 100 Reserve Vetor Clulus 10