++++++++++ Calculus Previous ear Questios from 6 to 99 Ramaasri 7 S H O P NO- 4, S T F L O O R, N E A R R A P I D F L O U R M I L L S, O L D R A J E N D E R N A G A R, N E W D E L H I. W E B S I T E : M A T H E M A T I C S O P T I O N A L. C O M C O N T A C T : 875766/66/6464
. Evaluate: I log d 6 ( marks) z. Fid the matri ad miimum values of z subject to the coditios ad 4 5 5 z ( marks) 4 5 5,, ) (,). Let f(, ) ( ) fid a,, ) (,) wheever such that (, ) (,). f f (5 marks) 4. Fid the surface area of the plae z cut off b, ad 6 (5 marks), if 5. Evaluate f(, ) dd, over the rectagle R [,;,] where f(, ) R, elsewhere (5 marks) 5 6. Evaluate the followig limit lim a a / ta a ( Marks) si 7. Evaluate the followig itegral: d ( Marks) /6 si cos 8. A coical tet is of give capacit. For the least amout of Cavas required, for it, fid the ratio of its height to the radius of its base. ( Marks) 9. Which poit of the sphere z is at the maimum distace from the poit(,,) ( Marks). Evaluate the itegral ( ) cos ( ) dd where R is the rhombus with successive vertices as (,),(, ),(, ),(, ) R ( Marks). Evaluate dd where R [,;,] ( Marks) R, (, ) (,). For the fuctio f(, ). Eamie the cotiuit ad differetiabilit., (, ) (,) ( Marks) 4. Prove that betwee two real roots e cos, a real root of e si lies. ( Marks) Reputed Istitute for IAS, IFoS Eams Page
4. Evaluate: loge d. ( Marks) 5. B usig the trasformatio u, uv evaluate the itegral dd take over the area eclosed b the straight lies, ad. (5 Marks) 6. Fid the height of the clider of maimum volume that ca be iscribed i a sphere of radius a. (5 Marks) 7. Fid the maimum or miimum values of z subject to the coditio a b cz ad l m z iterpret result geometricall 8. Evaluate si cos d ( Marks) 9. Usig Lagrage s multiplier method fid the shortest distace betwee the lie ad the ellipse 4 9 f, f,. Compute ad for the fuctio f, Also distace the cotiuit of. Evaluate dawhere D D. Defie a fuctio f f ad f,,,,,, at,. (5 Marks) is the regio bouded b the lie ad the parabola 6. of two real variables i the plae b f, Check the cotiuit ad differetiabilit of f (5 Marks) cos cos for,, otherewis at,. ( Marks). Let p ad q be positive real umbers such that p q a b show that for real umbers ab ab. p q p q ( Marks) 4. Fid the poit of local etrema ad saddle poits of the fuctio f for two variable defied b f, 6 5. Defied a sequece s of real umbers b s the value of this limit ad justif our aswer i log i log does lim s eist? If so compute Reputed Istitute for IAS, IFoS Eams Page
6. Fid all the real values of lim 7. Fid,, p if it eists 8. Let f be a fuctio defied o ca 9. Evaluate: (i) (ii) f possibl be? lim f d. ad q so that the itegral such that 4, Where f,. A twice differetiable fuctio there be is at least oe poit,. Dose the itegral f p log f ad Reputed Istitute for IAS, IFoS Eams Page 4 q d coverges f ' 5 for all values of i is such that f a f b ad a b for which eist if so fid its value f '' f c for ( Marks) How large ( Marks) a c b prove that ( Marks) ( Marks). Show that a bo (rectagular parallelepiped) of maimum volume V with prescribed surface area is a cube. (Marks). Let D be the regio determie b the iequalities,, z 8 ad z compute dddz. D 4. If, f is a homogeeous fuctio of degree i ad, ad has cotiuous first ad secod order partial derivatives the show that f f i f f f f ii f 9 5. Suppose the f '' is cotiuous o, ad that f has three zeroes i the iterval, show that f '' has least oe zero i the iterval,. ( Marks) 6. If is the derivative of same fuctio defied o, ab, such f b a that 7. If. abprove that there eists a umber f t dt f b a ( Marks) ad 4.with approimatel what accurac ca ou calculate the polar coordiate r ad of the poit P, Epress ou estimates as percetage chages of the value that r ad have at the poit,4
8. A space probe i the shape of the ellipsoid 4 4z 6 eters the earth atmosphere ad its,, z surface beigs to heat. After oe hour, the temperature at the poit give b o the probe surface is T,, z 8 4z 6z 6 Fid the hottest poit o the probe surface. s 9. Evaluate I ddz dzd z dd where S is the outer side of the part of the sphere z i the first octat. lim 4. Fid the value of 4. Evaluate 8 cot. ( Marks) d. ( Marks) 4. Determie the maimum ad miimum distaces of the origi from the curve give b the equatio 4 6 4. 4. Evaluate the double itegral a dd b chagig the order of itegratio 44. Obtai the volume bouded b the elliptic paraboloid give b the equatios z 9 & z 8 9 7 45. Letl, be defied b f si is f cotiuous o, differetiable o,? 46. A figure bouded b oe arch of a ccloid si, cos,, if it is cotiuous the is it ( Marks) a t t a t t ad the -ais is revolved about the -ais. Fid the volume of the solid of revolutio ( Marks) 47. Fi a rectagular parallelepiped of greatest volume for a give total surface area S usig Lagrage s method of multipliers z z 48. Prove that if z a a the a for a twice differetiable ad a is a costat. (5 Marks) 49. Show that e is bouded o, for all positive itegral values of usig this result show that e d ais s. 6 (5 Marks) 5. Fid a ad b so that '() m 5. Epress p f eists where f, if a b if d i terms of Gamma fuctio ad hece evaluate the itegral ( Marks) Reputed Istitute for IAS, IFoS Eams Page 5
6 d ( Marks) 5. lim a si b log cos Fid the values of a ad b such that. 4 (5 Marks) 5. If z f g show that z z. (5 Marks) 54. e Chage the order of itegratio i dd ad hece evaluate it. (5 Marks) z 55. Fid the volume of the uiform ellipsoid a b c 5 56. Show that the fuctio give below is ot cotiuous at the origi f, 57. Let, R R be defied as but f is ot differetiable at 58. If u z, uv z ad 59. Evaluate (5 Marks) if ( Marks) if f,,,, f, prove that f ad, uvw z. ( Marks),, z the fid m d i terms of Beta fuctio. m u, v, w (5 Marks) (5 Marks) f eist at 6. Evaluate zdv where V the volume is bouded below b the coe z ad above b the v sphere z lig o the positive side of the -ais. (5 Marks) 6. Fid the -coordiate of the ceter of gravit of the solid lig iside the clider a betwee the plae z ad the paraboloid az. (5 Marks) 6. Prove that the fuctio f defied o 4 ad that f d 6. 6. Shaw that 4 log.,4 f greatest iteger,,4 64. Let the roots of the equatio i u, v, w z z,, z u vv ww u is itegrable o,4 z be u, v, w provig that ( Marks) ( Marks). (5 Marks) 65. Prove that a equatio of the form where e N ad is a real umber has a positive root (5 Marks) Reputed Istitute for IAS, IFoS Eams Page 6
ab whe the itegral is take roud the ellipse p 4 66. Prove that d 4 a b a b ad is three legth of three perpedicular from the ceter to the taget. (5 Marks) a b,,, the show that possesses both the,,, partial derivative at but it is ot cotiuous thereat. (5 Marks) 67. If the fuctio f is defied b f, p f, 68. Let f be a real fuctio defied as follow: Show that f is discotiuous at ever odd f, R iteger, f ( ) ( Marks) e asi, 69. For all real umbers is give as f( ) Fid values of a ad b for which b( ), is differetiable at. ( Marks) 7. A rectagular bo ope at the top is to have a volume of 4 Usig Lagrage s method of multipliers fid the dimesio of the bo so that the material of a give tpe required to costruct it ma be least. (5 Marks) 7. Test the coverget of the itegrals(i) 7. Evaluate the itegral a ( ) a dd a a d (ii) si d (5 Marks) 7. Fid the volume geerated b revolvig b the real bouded b the curves 4a 8 a,v ad about the -ais. (5 Marks) 74. Show that 75. Show that 76. Let b a b a si b si a for a b e dd 4 si, p f, Obtai coditio o p such that(i)f is cotiuous at a b. ( Marks) ( Marks) ad (ii) f is differetiable at (5 Marks) 77. Cosider the set of triagle havig a give base ad a give verte agle show that the triagle havig the maimum area will be isosceles (5 Marks) 78. If the roots of the equatio,, z u vv ww u u, v, w z z u v w i are,, z show that. (5 Marks) Reputed Istitute for IAS, IFoS Eams Page 7
79. Fid the ceter of gravit of the regio bouded b the curve a b quadrat the desit beig k where k is costat. ad both aes I the first (5 Marks) f 8. Let be defied o b settig cotiuous at 8. Test the covergece of f if is ratioal ad but is discotiuous at ever other poit. if is irratioal show that is (Marks) si d. ( Marks) 8. Fid the equatio of the cubic curve which has the same asmptotes as which touches the ais at the origi ad passes though the poit, ad. (5 Marks) 8. Fid the maimum ad miimum radii vectors of the sectio of the surface z a b c z b the plae l m z (5 Marks) 84. Evaluate z dddz over the regio defied b,, z, z (5 Marks) 85. Fid the volume of the solid geerated b revolvig the cardioid cos 86. Use the mea value theorem to prove that log.4. 7 5 87. Show that t m lm m dd r 4 l m r a about the iitial lie for all positive values of ad laig the circle (5 Marks) ( Marks) r. ( Marks) z 88. Fid the ceter of gravit of the positive octat of the ellipsoid a b c z 89. Let f show that if is ot Riema itegrable o[ ab, ], is irratioal, is ratioa d log! log... d 9. Show that 9. Fid costat a ad b for which, log if the desit varies as (5 Marks) (5 Marks (5 Marks) F a b a b d is a miimum (5 Marks) Reputed Istitute for IAS, IFoS Eams Page 8
eed to tpe) 999( questios 9. If f is Riema itegral over ever iterval of fiite legth ad f f f for ever pair of real umbers ad show that c f f c where si 9. Show that the area bouded b cissoids asi t, a ad its asmptote is cost m m 94. Show that m ab over the positive quadrat of the ellipse is a b 4 m 998 ( Mistake Real Aalsis questios) 95. Test the covergece of the itegral 96. Test the series g 97. Let f ad e a 4 si a d. a. for uiform covergece does f dg eist? If it eists the fid its value 997 9 d determie f 98. Suppose f( ) 7 4 6 9 d 99. Prove that the volume of the greatest parallelepiped that ca be iscribe i the ellipsoid 8 z abc a b c. Show that the asmptotes of the cut the curve if it eists. ( )( 4 ) 6 5 z agai i eight poits which lie o a circle of radius.. A area bouded b a quadrat of a circle of radius a ad the taget at its etremities revolve about oe of the taget Fid the volume so geerated. e si. Show how the chages of order i the itegral si dd leads to the evaluatio of d hece evaluate it.. Show that i where ad deote gamma fuctio. Reputed Istitute for IAS, IFoS Eams Page 9
996 4 4 4. Fid the asmptotes of all curves4( )7 4 (4 ) ( ) ad show that the pass thought the poit of itersectio of the curve with the ellipse 4 4. 5. Show that a cotiuous fuctio defied for all real ad satisfig the equatio f( ) f( ) for all must be a costat fuctio. 6. Show that the maimum ad miimum of the radii vectors of the sectio of the surface z z b the plae vz are give b the equatio a b c a b a. I a r I b r I c r z u u u 7. If u f,, a b c prove that z z 8. Evaluate r dd. 9. The area cut off from the parabola 4a b chord joiig the verte to a ed of the latus rectum is rotated though four right agle about the chord. Fid the volume of the solid so formed. f 995. If g is the iverse of ad f'( ) prove that g( ) [ g( )] ( I ). Takig the th derivative of( ) i tow differet was show that... to I I I,, ( )! ( ) term (!). Let f(, ) which possesses cotiuous partial derivatives of secod order be a homogeeous fuctio of ad off degree prove that f f f ( ) f.. Fid the area bouded b the curve. 4 9 4 9 4. Let f( ), be such that the area bouded b the curve f( ) ad the lies, b is equal to b for allb does f attai its miimum? If so what is its values? ( ) 5. Show that.... 994 Reputed Istitute for IAS, IFoS Eams Page
6. f( ) but ( ) of f b a a a Is defied as follows: f( ) b of a b 6 f'( ) is discotiuous. b a 7. If ad lie betwee the least ad greatest values of of b a, b, c prove that prove that f( ) ad f'( ) are cotiuous f( a) f( b) f( c) f( a) f '( ) f( ) ( a) ( b) ( c) K ( a) '( ) ( ) where K ( b c )( c a )( a b ) ( ) ( b) ( c) ( ) '( ) ( ) 8. Prove that all rectagular parallelepipeds of same volume, the cube has the least surface 9. Show that meas of beta fuctio that z d ( f ( z ) ( t) si.. Prove that the value of dddz ( z ) take over the volume bouded b the co-ordiate plaes ad the plae z is log 5 8.. The sphere z a is pierced b the clider( ) a ( ) prove b the clider 8a 5 4 ( ) a ( ) is 4 99. Prove that f( ) si, ad f( ) for is cotiuous ad differetiable at but its derivative is ot cotiuous there.. If f( ), ( ), ( ) have derivative whea b show that there is a values c of lig betwee a ad b such that f( a) ( a) ( a) f( b) ( b) ( b).. f f( c) ( c) ( c) 4. Fid the triagle of maimum area which ca be iscribed i a circle a 5. Prove that e d ( a ) deduce that [..5...( )] e d a 6. Defied Gamma fuctio ad prove that 7. Show that volume commo to the sphere z a a ad the clider a is 9 99 ( 4). Reputed Istitute for IAS, IFoS Eams Page
a 8. If e cosb prove that a ( a b ) ad hece epad e cosb i powers of Deduce e a the epasio of ad cosb. 9. If r si cos, r si si, z r cos the prove that d d dz dr r d r si d.. z Fid the dimesio of the rectagular parallelepiped iscribed i the ellipsoid a b c that has greatest volume. Prove that the volume eclosed b the cliders a, z ais 8 a /5. Fid the cetre of gravit of the volume formed b revolvig the area bouded b the parabolas 4b about the -ais 4a ad. Evaluate the followig itegral i terms of Gamma fuctio ( ) ( ) d,[ p, q ] ad prove that p q Reputed Istitute for IAS, IFoS Eams Page