Q. 1 Q. 5 carry one mark each.

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1 Geeral Aptitude (GA) Set-8 Q Q 5 carry oe mark each Q The fisherme, the flood victims owed their lives, were rewarded by the govermet (A) whom (B) to which (C) to whom (D) that Q Some studets were ot ivolved i the strike If the above statemet is true, which of the followig coclusios is/are logically ecessary? Some who were ivolved i the strike were studets No studet was ivolved i the strike 3 At least oe studet was ivolved i the strike 4 Some who were ot ivolved i the strike were studets (A) ad (B) 3 (C) 4 (D) ad 3 Q3 The radius as well as the height of a circular coe icreases by 0% The percetage icrease i its volume is (A) 7 (B) 0 (C) 33 (D) 78 Q4 Five umbers 0, 7, 5, 4 ad are to be arraged i a sequece from left to right followig the directios give below: No two odd or eve umbers are et to each other The secod umber from the left is eactly half of the left-most umber 3 The middle umber is eactly twice the right-most umber Which is the secod umber from the right? (A) (B) 4 (C) 7 (D) 0 Q5 Util Ira came alog, Idia had ever bee i kabaddi (A) defeated (B) defeatig (C) defeat (D) defeatist GA /3

2 Geeral Aptitude (GA) Set-8 Q 6 Q 0 carry two marks each Q6 Sice the last oe year, after a 5 basis poit reductio i repo rate by the Reserve Bak of Idia, bakig istitutios have bee makig a demad to reduce iterest rates o small savig schemes Fially, the govermet aouced yesterday a reductio i iterest rates o small savig schemes to brig them o par with fied deposit iterest rates Which oe of the followig statemets ca be iferred from the give passage? (A) Wheever the Reserve Bak of Idia reduces the repo rate, the iterest rates o small savig schemes are also reduced (B) Iterest rates o small savig schemes are always maitaied o par with fied deposit iterest rates (C) The govermet sometimes takes ito cosideratio the demads of bakig istitutios before reducig the iterest rates o small savig schemes (D) A reductio i iterest rates o small savig schemes follow oly after a reductio i repo rate by the Reserve Bak of Idia Q7 I a coutry of 400 millio populatio, 70% ow mobile phoes Amog the mobile phoe owers, oly 94 millio access the Iteret Amog these Iteret users, oly half buy goods from e-commerce portals What is the percetage of these buyers i the coutry? (A) 050 (B) 470 (C) 500 (D) 5000 Q8 The omeclature of Hidustai music has chaged over the ceturies Sice the medieval period dhrupad styles were idetified as baais Terms like gayaki ad baaj were used to refer to vocal ad istrumetal styles, respectively With the istitutioalizatio of music educatio the term gharaa became acceptable Gharaa origially referred to hereditary musicias from a particular lieage, icludig disciples ad grad disciples Which oe of the followig pairigs is NOT correct? (A) dhrupad, baai (B) gayaki, vocal (C) baaj, istitutio (D) gharaa, lieage Q9 Two trais started at 7AM from the same poit The first trai travelled orth at a speed of 80km/h ad the secod trai travelled south at a speed of 00 km/h The time at which they were 540 km apart is AM (A) 9 (B) 0 (C) (D) 30 GA /3

3 Geeral Aptitude (GA) Set-8 Q0 I read somewhere that i aciet times the prestige of a kigdom depeded upo the umber of taes that it was able to levy o its people It was very much like the prestige of a head-huter i his ow commuity Based o the paragraph above, the prestige of a head-huter depeded upo (A) the prestige of the kigdom (B) the prestige of the heads (C) the umber of taes he could levy (D) the umber of heads he could gather END OF THE QUESTION PAPER GA 3/3

4 Q Q 5 carry oe mark each Q For a balaced trasportatio problem with three sources ad three destiatios where costs, availabilities ad demads are all fiite ad positive, which oe of the followig statemets is FALSE? (A) The trasportatio problem does ot have ubouded solutio (B) The umber of o-basic variables of the trasportatio problem is 4 (C) The dual variables of the trasportatio problem are urestricted i sig (D) The trasportatio problem has at most 5 basic feasible solutios Q Let f :[ a, b] (the set of all real umbers) be ay fuctio which is twice differetiable i ( ab, ) with oly oe root i ( ab, ) Let f( ) ad f( ) deote the first ad secod order derivatives of f( ) with respect to If is a simple root ad is computed by the Newto-Raphso method, the the method coverges if (A) (C) (B) ( ) ( ) ( ), for all (, ) f ( ) f ( ) f ( ), for all ( a, b) f f f a b (D) ( ) ( ) ( ), for all (, ) f ( ) f ( ) f ( ), for all ( a, b) f f f a b Q3 Let f : (the set of all comple umbers) be defied by Let f () z deote the derivative of f i y y i y y i 3 3 ( ) 3 3, f with respect to The which oe of the followig statemets is TRUE? z (A) f( i) eists ad f( i) 3 5 (B) f is aalytic at the origi (C) f is ot differetiable at i (D) f is differetiable at Q4 The partial differetial equatio is u u u 0 y y y y (A) parabolic i the regio y (B) hyperbolic i the regio y (C) elliptic i the regio 0 y (D) hyperbolic i the regio 0 y MA /7

5 Q5 If t,,,3,, u e dt the which oe of the followig statemets is TRUE? (A) Both the sequece u ad the series u are coverget (B) Both the sequece u ad the series u are diverget (C) The sequece u is coverget but the series u is diverget (D) limu e Q6 Let (, y, z) 3 :, y, z ad : be a fuctio whose all secod order partial derivatives eist ad are cotiuous If satisfies the Laplace equatio 0 for all (, y, z), the which oe of the followig statemets is TRUE i? 3 ( is the set of all real umbers, ad (, y, z) :, y, z (A) (B) (C) (D) is soleoidal but ot irrotatioal is irrotatioal but ot soleoidal is both soleoidal ad irrotatioal is either soleoidal or irrotatioal ) MA /7

6 Q7 Let X (,,) : ad oly fiitely may 's are o-zero ad d : X be a metric o X defied by i d(, y) sup i y i for (,,), y ( y, y,) i X i ( is the set of all real umbers ad is the set of all atural umbers) Cosider the followig statemets: P : ( X, d) is a complete metric space i X Q :The set { : (0, ) } X d is compact, where 0 is the zero elemet of X Which of the above statemets is/are TRUE? (A) Both P ad Q (B) P oly (C) Q oly (D) Neither P or Q Q8 Cosider the followig statemets: I The set is ucoutable II The set { f : f is a fuctio from to {0, }} is ucoutable III The set { p: p is a prime umber} is ucoutable IV For ay ifiite set, there eists a bijectio from the set to oe of its proper subsets ( is the set of all ratioal umbers, is the set of all itegers ad is the set of all atural umbers) Which of the above statemets are TRUE? (A) I ad IV oly (B) II ad IV oly (C) II ad III oly (D) I, II ad IV oly Q9 Let f be defied by : f y y y y 6 4 (, ) ( is the set of all real umbers ad (, y) :, y Which oe of the followig statemets is TRUE? ) (A) f has a local maimum at origi (B) f has a local miimum at origi (C) f has a saddle poit at origi (D) The origi is ot a critical poit of f MA 3/7

7 Q0 Let a 0 be ay sequece of real umbers such that a If the radius of 0 covergece of TRUE? 0 a is r, the which oe of the followig statemets is ecessarily (A) (B) r or r r is ifiite (C) (D) r 0 0 a r a Q Let topology o T be the co-coutable topology o (the set of real umbers) ad T be the co-fiite Cosider the followig statemets: I I (, T ), the sequece II I (, T ), the sequece coverges to 0 coverges to 0 III I (, T ), there is o sequece of ratioal umbers which coverges to IV I (, T ), there is o sequece of ratioal umbers which coverges to 3 3 Which of the above statemets are TRUE? (A) I ad II oly (C) III ad IV oly (B) II ad III oly (D) I ad IV oly Q Let X ad Y be ormed liear spaces, ad let T : X closed graph The which oe of the followig statemets is TRUE? (A) The graph of T is equal to X (C) The graph of Y (B) T Y be ay bijective liear map with is cotiuous T is closed (D) T is cotiuous MA 4/7

8 Q3 Let g : be a fuctio defied by (, ) ( cos, si ) ( is the set of all real umbers ad g y e y e y ad ( a, b) g, 3 (, y) :, y ) Which oe of the followig statemets is TRUE? (A) g is ijective (B) If h is the cotiuous iverse of that h( a, b), 3, the the Jacobia of h at (, ) is (C) If h is the cotiuous iverse of g, ab,, such defied i some eighbourhood of ab that h( a, b), 3, the the Jacobia of h at (, ) is (D) g is surjective e ab,, such g, defied i some eighbourhood of ab e Q4 Let The limu u!, 35( ) is equal to (the set of all atural umbers) Q5 If the differetial equatio dy y, y() d is solved usig the Euler s method with step-size h 0, the y () is equal to (roud off to places of decimal) Q6 Let f be ay polyomial fuctio of degree at most over (the set of all real umbers) If the costats a ad b are such that df a f ( ) f ( ) b f ( ), for all, d the 4a + 3b is equal to (roud off to places of decimal) MA 5/7

9 Q7 Let L deote the value of the lie itegral (3 4 ) (4 ) C y d y y dy, where C, a circle of radius with cetre at origi of the y-plae, is traversed oce i the ati-clockwise directio The L is equal to Q8 The temperature 3 T : \{(0,0,0)} at ay poit P(, y, z) is iversely proportioal to the square of the distace of P from the origi If the value of the temperature T at the poit R(0,0,) is 3, the the rate of chage of T at the poit Q (,,) i the directio of equal to (roud off to places of decimal) QR is ad 3 ( is the set of all real umbers, (, y, z) :, y, z ecludig the origi) 3 \{(0,0,0)} deotes 3 Q9 Let f be a cotiuous fuctio defied o [0,] such that f( ) 0 for all [0,] If the area bouded by y f ( ), 0, y 0 ad b is 3 b 3, where b (0,], the f () is equal to (roud off to place of decimal) Q0 If the characteristic polyomial ad miimal polyomial of a square matri A are ( )( ) ( ) 4 5 ad ( )( )( ), respectively, the the rak of the matri A I is, where I is the idetity matri of appropriate order Q Let be a primitive comple cube root of uity ad i The the degree of the field etesio i, 3, over (the field of ratioal umbers) is MA 6/7

10 Q Let i z e dz, C : cost isi t, 0 t, i z 5z C The the greatest iteger less tha or equal to is Q3 Cosider the system: 3 a, 3 4 3, 3 4,,, If,, 3 0, 4 b c is a basic feasible solutio of the above system (where a, b ad c are real costats), the abc is equal to Q4 Let f f : be a fuctio defied by i z : z is ( is the set of all comple umbers) f z z z The the umber of zeros of 6 4 ( ) 5 0 Q5 Let be a ormed liear space with the orm i i i { (,,) :, } i i Let g : be the bouded liear fuctioal defied by g ( ) 3 for all,, The sup ( ) : g is equal to (roud off to 3 places of decimal) ( is the set of all comple umbers) MA 7/7

11 Q 6 Q 55 carry two marks each Q6 For the liear programmig problem (LPP): Maimize Z 4 ( is the set of all real umbers) cosider the followig statemets: subject to 4, 3 6,, 0,, I The LPP always has a fiite optimal value for ay 0 II The dual of the LPP may be ifeasible for some 0 III If for some, feasible solutio the poit (,) is feasible to the dual of the LPP, the, of the LPP Z 6, for ay IV If for some, ad are the basic variables i the optimal table of the LPP with, the the optimal value of dual of the LPP is 0 The which of the above statemets are TRUE? (A) I ad III oly (C) III ad IV oly (B) I, III ad IV oly (D) II ad IV oly Q7 Let f be defied by : ( y )si, if (, y) (0,0) f (, y) y 0, if ( y, ) =(0,0) Cosider the followig statemets: f f I The partial derivatives, eist at (0, 0) but are ubouded i ay eighbourhood of y (0, 0) II f is cotiuous but ot differetiable at (0, 0) III is ot cotiuous at (0, 0) IV f is differetiable at (0, 0) f ) ( is the set of all real umbers ad (, y) :, y Which of the above statemets is/are TRUE? (A) I ad II oly (B) I ad IV oly (C) IV oly (D) III oly MA 8/7

12 Q8 Let K k i, j be a ifiite matri over (the set of all comple umbers) such that i, j (i) for each i (the set of all atural umbers), the row ki,, k i,, of K is i ad,,, (ii) for every where y k, i i j j j j k i, j j th i is summable for all i, ad ( y, y,), Let the set of all rows of K be deoted by E Cosider the followig statemets: P: E is a bouded set i Q: E is a dese set i {(,,) : i, i } i {(,,) : i, sup i } i Which of the above statemets is/are TRUE? (A) Both P ad Q (B) P oly (C) Q oly (D) Neither P or Q Q9 Cosider the followig heat coductio problem for a fiite rod u u t, 0, 0, e t t t with the boudary coditios u t t u t e t t ad the iitial t (0, ), (, ), 0 coditio 3 u(,0) si si, 0 If v(, t) u(, t) e t t, the which oe of the followig is CORRECT? 4 t (, ) 7 si 9 t v t e e si3 4 t (, ) si 3 t v t e e si3 4 t (, ) 3 si 3 t v t e e si3 4 t (A) (, ) si 9 t v t e e si3 (B) (C) (D) MA 9/7

13 Q30 Let f : be o-zero ad aalytic at all poits i If F( z) f ( z)cot( z) for z \, the the residue of F at ( is the set of all comple umbers, is the set of all itegers ad all comple umbers ecludig itegers) is \ deotes the set of (A) f( ) (B) f( ) (C) f( ) (D) df dz z Q3 Let the geeral itegral of the partial differetial equatio be give by F( u, v) 0 z (y ) z ( yz), where z y F is a cotiuously differetiable fuctio ( is the set of all real umbers ad (, y) :, y The which oe of the followig is TRUE? : ) (A) (C) u y z, v z y (B) u y z, v y z (D) u y z, v z y u y z, v y z Q3 Cosider the followig statemets: I If deotes the additive group of ratioal umbers ad f : is a o-trivial homomorphism, the f is a isomorphism II Ay quotiet group of a cyclic group is cyclic III If every subgroup of a group G is a ormal subgroup, the G IV Every group of order 33 is cyclic Which of the above statemets are TRUE? (A) II ad IV oly (B) II ad III oly (C) I, II ad IV oly (D) I, III ad IV oly is abelia MA 0/7

14 Q33 A solutio of the Dirichlet problem u( r, ) 0, 0 r,, u(, ),, is give by (A) u( r, ) r cos( ) (B) u( r, ) r cos( ) (C) u( r, ) r cos( ) (D) u( r, ) r cos( ) Q34 Cosider the subspace Y (, ) : If is a bouded liear fuctioal o followig sets is equal to of the ormed liear space Y, defied by,,, the which oe of the (,0) : is a orm preservig etesio of to, ( is the set of all comple umbers, (, y) :, y, sup, ) (A) (B) 3, ad (C), (D) 0, Q35 Cosider the followig statemets: I The rig [ ] is a uique factorizatio domai II The rig [ 5] is a pricipal ideal domai III I the polyomial rig [ ], the ideal geerated by 3 is a maimal ideal IV I the polyomial rig [ ], the ideal geerated by 6 3 is a prime ideal ( deotes the set of all itegers, positive iteger ) deotes the set of all itegers modulo, for ay Which of the above statemets are TRUE? (A) I, II ad III oly (C) I, II ad IV oly (B) I ad III oly (D) II ad III oly MA /7

15 Q36 Let M be a 3 3 real symmetric matri with eigevalues 0, eigevectors (4,, ) T T T u b c, v (,,0) ad w (,,) Cosider the followig statemets: I a b c 0 3 II The vector 0,, satisfies M v w d spa u, v, w, M d has a solutio III For ay IV The trace of the matri ( T y T M M is 8 deotes the traspose of the vector y ) ad a with the respective Which of the above statemets are TRUE? (A) I, II ad III oly (C) II ad IV oly (B) I ad II oly (D) III ad IV oly Q37 Cosider the regio i y :, y, i 3 3 i y i the comple plae The trasformatio i y e maps the regio oto the regio S (the set of all comple umbers) The the area of the regio S is equal to 3 e e 3 e e 4 e e 6 e e 4 4 (A) (B) 4 4 (C) (D) Q38 Cosider the sequece is the derivative of g( ) with respect to g of fuctios, where g ( ),, ( is the set of all real umbers, is the set of all atural umbers) The which oe of the followig statemets is TRUE? ad g ( ) (A) g does NOT coverge uiformly o (B)g coverges uiformly o ay closed iterval which does NOT cotai (C)g coverges poit-wise to a cotiuous fuctio o (D) g coverges uiformly o ay closed iterval which does NOT cotai 0 MA /7

16 Q39 Cosider the boudary value problem (BVP) d y y ( ) 0, (the set of all real umbers), d with the boudary coditios y(0) 0, y( ) k ( k is a o-zero real umber) The which oe of the followig statemets is TRUE? (A) For (B) For, the BVP has ifiitely may solutios, the BVP has a uique solutio (C) For, k 0, the BVP has a solutio y ( ) such that y ( ) 0 for all (0, ) (D) For, k 0, the BVP has a solutio y ( ) such that y ( ) 0 for all (0, ) Q40 Cosider the ordered square the geeral elemet of subset is (A) (C) I 0 I 0, the set [0,] [0,] with the dictioary order topology Let be deoted by y, 3 S : 0 a b 4 i (, ] 0 [, ) (B) (, ) 0 (, ) (D) S ( a, b] 0 S a b a b S a b a b where y, [0,] The the closure of the I 0 [, ) 0 (, ] S a b a b Q4 Let P be the vector space of all polyomials of degree at most over umbers) Let a liear trasformatio T : P P be defied by (the set of real Cosider the followig statemets: T a b c a b b c a c ( ) ( ) ( ) I The ull space of T : II The rage space of T is spaed by the set, is III T( T( )) IV If M is the matri represetatio of T with respect to the stadard basis,, of P, the the trace of the matri M is 3 Which of the above statemets are TRUE? (A) I ad II oly (C) I, II ad IV oly (B) I, III ad IV oly (D) II ad IV oly MA 3/7

17 Q4 Let T ad T be two topologies defied o (the set of all atural umbers), where T is the topology geerated by Ɓ {{, }: } ad T is the discrete topology o Cosider the followig statemets : I I (, T ), every ifiite subset has a limit poit II The fuctio f : (, T) (, T) defied by, if is eve f( ), if isodd is a cotiuous fuctio Which of the above statemets is/are TRUE? (A) Both I ad II (C) II oly (B) I oly (D) Neither I or II Q43 Let p q Cosider the followig statemets: I p q II p q L [0,] L[0,], p p where {(,,) : i, i } ad i p p L [0,] f :[0,] : f is -measurable, f d, where is the Lebesgue measure [0,] ( is the set of all real umbers) Which of the above statemets is/are TRUE? (A) Both I ad II (B) I oly (C) II oly (D) Neither I or II MA 4/7

18 Q44 Cosider the differetial equatio d y dy dy dt dt dt t 0 t t y 0, t 0, y(0 ), 0 If Ys () is the Laplace trasform of places of decimal) yt (), the the value of Y() (Here, the iverse trigoometric fuctios assume pricipal values oly) is (roud off to Q45 Let y, y 4, y ad y 5 R be the regio i the y-plae bouded by the curves The the value of the itegral R y dy d is equal to Q46 Let V be the vector space of all 3 3 matrices with comple etries over the real field If T : ad W A V : trace of A 0 W A V A A the the dimesio of W W is equal to ( deotes the cojugate traspose of A) T A, Q47 The umber of elemets of order 5 i the additive group is ( deotes the group of itegers modulo, uder the operatio of additio modulo, for ay positive iteger ) Q48 Cosider the followig cost matri of assigig four jobs to four persos: Jobs Persos J J J3 J4 P P P P The the miimum cost of the assigmet problem subject to the costrait that job J4 is assiged to perso P, is MA 5/7

19 Q49 Let y :[, ] with y() satisfy the Legedre differetial equatio The the value of ( ) d y dy 6 y 0 d d for y d is equal to (roud off to places of decimal) Q50 Let 5 be the rig of itegers modulo 5 uder the operatios of additio modulo 5 ad multiplicatio modulo 5 If m is the umber of maimal ideals of umber of o-uits of, the m 5 is equal to 5 ad is the Q5 The maimum value of the error term of the composite Trapezoidal rule whe it is used to evaluate the defiite itegral 4 0 si log e d with sub-itervals of equal legth, is equal to (roud off to 3 places of decimal) Q5 By the Simple method, the optimal table of the liear programmig problem: Maimize Z 3 subject to 8, 3, 4,,, 0, 3 4 where,, are real costats, is c j Basic variable 3 4 Solutio z c j j The the value of is MA 6/7

20 Q53 Cosider the ier product space P of all polyomials of degree at most over the field of real umbers with the ier product Let f0, f, f be a orthogoal set i f, g f ( t) g( t) dt for P 0, where f, gp f, f t c, f t c f c ad 0 3 c, c, c3 are real costats The the value of c c 3c3 is equal to Q54 Cosider the system of liear differetial equatios d 5, dt d 4, dt with the iitial coditios (0) 0, (0) The log () () e is equal to Q55 Cosider the differetial equatio ( d y ) dy y d d The sum of the roots of the idicial equatio of the Frobeius series solutio for the above differetial equatio i a eighborhood of = 0 is equal to END OF THE QUESTION PAPER MA 7/7

Q. 1 Q. 5 carry one mark each.

Q. 1 Q. 5 carry one mark each. Geeral Aptitude (GA) Set-8 Q. Q. 5 carry oe mark each. Q. The fisherme, the flood victims owed their lives, were rewarded by the govermet. (A) whom (B) to which (C) to whom (D) that Q.2 Some studets were