UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. (2014 Admn. onwards) III Semester. B.Sc. Mathematics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY

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1 UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION (0 Adm. owrds) III Semester B.Sc. Mthemtics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY Questio Bk & Aswer Key. l l () = b) 0 c). l d =... c (l ) b) c. t d =... l c) c l c l cos c b) l sec c c) sec +c l si c d. (l k) =... d k b) k d. ( ) =... d c) k l b) c) log 6. d =... l b) l l c) log 7. d =... b) l c) l l

2 School of Distce Eductio 8. Rge of si h is... [, ] b) (, ) c) (, 0] (0, ) 9. Rge of cos h is... [, ] b) (, ) c) (, ] (, ) 0. si h =. The cos h is... b) c). d (sec h)=... d sec h b) sec h t h c) sec h t h t h. cot h =... th d. cosh.. 6. d sih l b) th c) coth =... b) =... c) b) + c) 0 si 0 =... b) 0 c) 0 b l b =... b b) l 7. Rge of t h is... c) l (b) th [, ] b) (, ) c) (,) (0, ) 8. cos h is... fuctio. Odd b) Eve c) Fluctutig Neither eve or odd CALCULUS AND ANALYTICAL GEOMETRY Pge

3 School of Distce Eductio 9. t h is... fuctio. 0. Odd b) Eve c) Neither eve or odd Noe of these cot = b) c). Which of the followig is ot idetermite form b) 0 c) 0. lim l =... 0 b) c) Noe of these. Emple of costt sequece is... {() =, N} b) {() =, N} c) {() = +, N} {() =, N}. =, + = +. The =... b) c) 6 7. Third term of the sequece () + si, b) 0 c) Noe of these 6. The sequece {() + } is Coverges b) Diverges 7. = c) Hs coverget subsequece Noe. The =... b) 8. Wht is wrog bout the sequece {() } c) Diverges b) Coverges c) Icresig Absolutely coverget 9. The sequece{ } is Diverges b) Icresig c) Decresig Noe of these 0. The sequece {() } is Coverges b) Bouded bove c) Bouded below Not bouded CALCULUS AND ANALYTICAL GEOMETRY Pge

4 School of Distce Eductio. Which is true? A) Every bouded sequece is coverged B) Every coverged sequece is bouded A d B re true b) A is true b is flse c) A is flse B is true Both re flse =, b =, the ( + b ) =... 6 b) 0 c) 8 6 =... 0 b) c) =... 0 b) =... b) c) c) l =... 0 b) c) Noe of these =... 0 b) c) Noe of these ( > 0) =... 0 b) c) Noe of these 9. If r >, r =... 0 b) c) Noe of these 0. If r <,. r =... 0 b) c) Noe of these (+ ) =... b) 0 c) e e. Fourth term of the sequece ( ) is b) c) CALCULUS AND ANALYTICAL GEOMETRY Pge

5 School of Distce Eductio. Which of the followig series diverges? 0 b) (). The series coverges, the = c, costt b) c) =. The series c) = 0 does ot eist Coverges to b) Coverges to c) Diverges Noe of these 6. The series p Coverges for P b) Diverges for P c) Coverges for P > Diverges for P > 7. Let { } of {b } is such tht b. The coverges if b coverges b) b coverges if coverges c) coverges if b diverges b diverges if diverges 8. Let > 0 d b > 0 d b = l. The d b coverges if l = 0 b) l > 0 c) l = Noe of these 9. Let > 0 d b > 0 d b = 0. The coverges if b coverges b) b coverges if coverges c) Both d b coverges b coverges coverges 0. is Coverges to b) Coverges to 0 c) Diverges Noe of these. be series such tht = l. The coverges if l > b) diverges if l > 0 c) coverges if l < diverges if l = 0. be series such with 0 ( ) / = l. The coverges if l < b) diverges if l < c) coverges if l = diverges if l = CALCULUS AND ANALYTICAL GEOMETRY Pge

6 School of Distce Eductio. Let { } be sequece of positive terms such tht + d coverges b) () coverges c) coverges but () diverges Noe of these. If the series (c) coverges for = k. The it coverges i = 0, the r c < k b) r c k c) r c > k r c k. If series coverges bsolutely for ll. The its rdius of coverges if Fiite b) Ifiite c)cot be determied Noe of these 6. The rdius of covergece of the power series ( ) is... 0 b) c) 7. The rdius of covergece of the series 0 is... b) c) 8. The series 0 is Coverges bsolutely for < b) Coverges for > c) Hs rdius of coverges ½ Noe of these 9. Coefficiet of ( ) i the Tylor series epsio of f() = t = is b) ½ c) ½ Coefficiet of i the Mcluri s Series epsio of f() = l ( +) is b) ½ c) ½ 6. Coefficiet of i the Mcluri s series epsio of f() = cos is b) 0 c) 6. Focus of the prbol y = 8 is (0, ) b) (0, ) c) (, 0) (, 0) 6. Equtio of the directri of the prbol y = 8 is = b) y = c) = y = 6. Equtio of the directri of the prbol = 6y is = b) y = c) = y = 6. The verte of the prbol whose focus is (, ) d whose directri psses through (, ) is (, ) b) (, ) c) (, ) (, ) CALCULUS AND ANALYTICAL GEOMETRY Pge 6

7 School of Distce Eductio y 66. Focus of the ellipse + = is (, 0) b) (0, ) c) (, 0) (0, ) 67. Electricity of the ellipse 6 + 9y = is... y 68. Equtio of the symptotes to the hyperbol 6 9 b) 7 c) = is 7 y = b) = y c) y = = y y 69. Equtio of the symptote to the hyperbol 9 6 = is y = b) = y c) y = = y 70. Equtio of the hyperbol with foci (0, ) d symptote y = y = b) y = c) y = 7. Equtio of hyperbol with foci (, 0) d eccetricity y = 9y 9 9y 6 0y = b) = c) = Noe 7. Equtio of hyperbol with oe focus t (, 0) d correspodig directri = is y = 8 b) y = 8 c) 7. Eccetricity of the hyperbol 9 6y = is 7 b) c) 6 y y = = y 7. Foci of the hyperbol 8 = is ( 0, 0) b) (0, 0 ) c) (,0) (0, ) 7. The gle to be rotted so tht the y term i the equtio 6y + y = 8 c remove is 6 b) c) 76. The equtio A + Cy + D + Ey +F = 0 represet ellipse if (A > 0 d C > 0) b) (A > 0 d C < 0) c) (A < 0 d C > 0) (A 0, C 0) CALCULUS AND ANALYTICAL GEOMETRY Pge 7

8 School of Distce Eductio 77. The equtio A + Cy + D + Ey +F = 0 represet circle if (A = 0, C 0) b) (A 0, C 0) c) (A 0, C = 0) Noe 78. A + Cy + D + Ey + F = 0 represet Prbol if AC 0 b) Ellipse if AC > 0 c) Ellipse if AC < 0 Hyperbol if AC > y + y = 0 represet Prbol b) Ellipse c) Hyperbol Circle 80. +y + y + y + = 0 represet Prbol b) Ellipse c) Hyperbol Circle 8. Which of the followig represet prmetric equtio of circle ( = cos t, y = b si t) b) ( = cos t, y = si t) c) ( = sec t, y = b t t) Noe 8. Tget to the curve, = si t, y = cos t t t = is b) c) 8. Legth of the circle = cos t, y = si t i the first qudrt is. b).7 c) 6. oe d y 8. Fid if = cos t, y = si t d cosec t b) cosec t c) cosec t cosec t 8. The polr co-ordite equl to (, ) is (, ) b) (, 9 ) c) (, 9 ) (, ) 86. Crtesi equtio of r = is si cos = y + b) y = + c) y = + = The equtio r = si is symmetric bout is b) y is c) Origi The lie = 88. Which of the followig represet equtio of circle? r = b) = c) r = r si = 89. Crtesi equtio of r si ( + 6 ) = is + y = b) + y = c) y + = y + = CALCULUS AND ANALYTICAL GEOMETRY Pge 8

9 School of Distce Eductio 90. Which of the followig poit lie o the curve r = si (, ) b) (, ) c) (, ) Noe 9. Oe of the poit of itersectio of the Crdiods r = + cos, t r =- cos, is (0, ) b) (, ) c) (, ) (, 0) 9. Polr equtio of circle pssig through origi, rdius d cetre o +ve -is is r = si b) r = cos c) r = cos r = cos 9. Polr equtio of the circle with cetre t (, ) d pssig through origi is r = si b) r = si c) r = cos r = cos 9. Polr equtio for the hyperbol with eccetricity t directio t = 6 is r = 6 cos 9. The polr equtio r = b) r = si 6 si represet c) r = 0 si r = 0 si Prbol b) Ellipse c) Hyperbol Noe of these t d =... 0 l 97. l =... b) l c) l l b) e c) e 98. l (e ) =... e b) c) e 99. si e cos d =... 0 e b) e c) e ½ 00. log ( ) =... b) c) 0. si + cos =... b) c) CALCULUS AND ANALYTICAL GEOMETRY Pge 9

10 School of Distce Eductio t e 0. 0 d 0. 7 d =... e b) e c) e e t = () is... b) t 7 c) 7 t Coverges b) Diverges c) Absolutely coverget Noe dy 0. Let y = si h cos h the =... d Noe cos h b) si h c) cos h + cos h si h 06. l [ + ] =... cos h b) si h c)t h sec h 07. y = sec h + t h dy the =... d 08. sec h t h b) 0 c) sec h t h =... 0 b) c) Noe y 09. Legth of Ltus Rectum of the ellipse + b) c) 0. Which of the followig fuctio is odd? f() =. If f() = ( )., if, if = is b) si + cos c) log( + f() =... Noe 0 b) 6 c) Does ot eist is equl to... 0 b) c) Does ot eist CALCULUS AND ANALYTICAL GEOMETRY Pge 0

11 School of Distce Eductio dy. y = si (si ). The cos (si ) cos =... d 0 b) cos c) si. For differetibility of fuctio cotiuity is... Sufficiet b) Necessry c) Sufficiet d ecessry Noe of these. The fuctio f() = is... Cotiuous t = 0 b) Discotiuous t = 0 c) Differetible t = 0 Not differetible t =0 6. The series is... Coverget b) Absolutely coverget c) Diverget All of the bove 7. I series of +ve terms U if U 0 the the series Coverget b) Diverget c) Not coverged Oscilltory 8. The series m is coverged if... > d = whe m < b) > d = whe m > c) < d = whe m < < d = whe m > 9. The series is Coverget but ot bsolutely b) Oscilltory c) Diverget Absolutely coverget 0. = sect y=btt is the prmetric represettio of Prbol b) Ellipse c)hyperbol Circle. CALCULUS AND ANALYTICAL GEOMETRY Pge

12 School of Distce Eductio Aswer Key.b,.b,.b,.c,.c,6., 7.b, 8.b, 9.c, 0.b,.c,,b,.d,.,.c, 6., 7.b, 8.b, 9., 0.b,.b,.,.b,,,.c,6.b,7.c,8.c,9.c,0.d,.c,.b,.c,,c,.d, 6.,7.b,8,b,9.c,0.,.c,.b,.c,.c,.c,6.c,7,,8.,9.,0.c,.c,.,.b.b,.b,6.b,7.,8.,9.b,60.b,6.b,6.c,6.d,6.b,6.b,66.c,67.b,68.,69.c,70.c, 7.,7.,7.b,7.,7.b,76.,77.b,78.b,79.c,80.b,8.b,8.,8.b,8.d,8.c,86.b,87.b,88.,89.c,90.,9.b,9.b,9.,9.c,9.,96b,97.b,98.b,99.b,00.d,0.b,0.,0.,0.c, 0.d,06.,07.d,08.b,09.d,0.d,.b,.d,.,.b,.d,6.b,7.b,8.c, 9.c,0.c Prepred by: Scrutiised by : Aboobcker P Assistt Professor, Deprtmet of Mthemtics, WMO Arts & Sciece College, Muttil Wyd 67 Dr.D.Jyprsd, Pricipl, Sreekrish College, Guruvyur Chirm, Bord of Studies i Mthemtics (UG) CALCULUS AND ANALYTICAL GEOMETRY Pge

is continuous at x 2 and g(x) 2. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a

is continuous at x 2 and g(x) 2. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a . Cosider two fuctios f () d g () defied o itervl I cotiig. f () is cotiuous t d g() is discotiuous t. Which of the followig is true bout fuctios f g d f g, the sum d the product of f d g, respectively?