DEEPAK SERIES DEEPAK SERIES DEEPAK SERIES FREE BOOKLET CSIR-UGC/NET MATHEMATICAL SCIENCES

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1 DEEPAK SERIES DEEPAK SERIES DEEPAK SERIES FREE BOOKLET DEEPAK SERIES CSIR-UGC/NET MATHEMATICAL SCIENCES SOLVED PAPER DEC- DEEPAK SERIES DEEPAK SERIES DEEPAK SERIES

2 Note : This material is issued as complimetary for educatioal, learig ad promotioal activity as well as to showcase the patter of the so called book / material CSIR-UGC/NET MATHEMATICAL SCIENCES by DEEPAK SERIES for TEST PREPARATION ad therefore does t claim to reder ay professioal services. However, the iformatio cotaied has bee obtaied by the author from sources believed to be reliable ad are correct to the best of her kowledge. The book is desiged i a friedly maer ad the syllabus is strictly accordig to CSIR-UGC/NET MATHEMATICAL SCIENCES which will certaily help the aspirats to clear CSIR-NET with high score. + Objective Type Questios based o recet patter ad tred with iformative epalatios. Uitwise Blueprit Aalysis of previous year papers with their solutios. CSIR-UGC/NET MATHEMATICAL SCIENCES by DEEPAK SERIES MRP : ` 785/- ISBN NO :

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4 . Let f : [,] [, ] be give by f ( ) ( ) for all o-egative itegers. Let f( ) lim f ( ) for. The which of the followig is true? (a) f is a cotiuous fuctio o [, ] (b) f coverges uiformly to f o [, ] as (c) lim f ( ) d f ( ) d (d) for ay a (, ) we have lim f ( a) f ( a) Ep: Give f :[,,] [,] be give by f ( ) ( ) V f( ) lim f ( ) If ; lim ( ) f ( ),, f ( ) d f ( ) ( ) d ( ) Lim f ( ) d As: (c). For a fied positive iteger, let A be the matri defied as A I J, where I is the idetity matri ad J is the matri with all etries equal to. Which of the followig statemets is NOT true? (a) A k A for every positive iteger k. (b) Trace ( A) (c) Rak (A) + Rak ( A). (d) A is ivertible. 4

5 Ep: A I J Suppose take matri 4 det( A ) = A is ot ivertible. As:(d) A. Let A be a 5 4 matri with real etries such that A if ad oly if where is a 4 vector ad is a ull vector. The, the rak of A is (a) 4 (b) 5 (c) (d) Ep: A be a 5 4 matri A iff. If there are o free variables rark( A) colum. The oly solutio is =. rak( A) 4 As:(a) si if 4. Let f( ). The f is if (a) discotiuous (b) cotiuous but ot differetiable (c) differetiable oly oce (d) differetiable more tha oce si Ep: f ( ) if =, = The Taylor series for si is si ( ) f ( ) ( )! si ( ) ( )! Puttig i the above series we get which agrees with the defiitio of f. Let f ( ) ( ) ( )! The easily we ca show that f coverges uiformly to f o [, ] ad f also coverges uiformly o [, ] ad f also coverges uiformly o [, ] ad so o. Hece f is ifiitely times differetiable. As:(d) 5

6 5. Cosider the followig row vectors : a,,,,,), a (,,,,, ), 6 ( a (,,,,,) a4 (,,,,,), a5 (,,,,,), a6 (,,,,,) The dimesio of the vector space spaed by these row vectors is (a) 6 (b) 5 (c) 4 (d) Ep:The vectors,,, 4 are liearly idepedet Also 4 (,,,,, ) Satisfies 5 & 6 Thus vector 4 vector (,,, ) (,,,,, ) & (,,, 4 ) is a basis. The space has dimesio 4. As:(c) 6. Let A (( aij )),, where a b b, i, j,,..., ij ( i j ) for some distict real umbers b, b... b. The b, b... b. The det(a) is (a) i j( b i bj ) (b) i j( b i bj ) (c) (d) Ep: A ( aij ) Suppose = a b b a b b a b b a b b a b b a b b a b b a b b a b b b b b b A b b b b b b b b As:(c) det( A) ( b b )( ( b b ))( b b ) ( b b )( b b )( b b ) ( b b ) ( b b ) ( b b ) ( b b ) ( b b ) ( b b ) =

7 7. Let { a },{ b } be sequeces of real umbers satisfyig a b for all. The (a) a coverges wheever b coverges (b) a coverges absolutely wheever b coverges absolutely (c) b coverges wheever a coverges. (d) b coverges absolutely wheever a coverges absolutely As:(b) 8. If true? 7 a is absolutely coverget, the which of the followig is NOT (a) a m as (c) m a e (b) a si is coverget is diverget (d) a is diverget Ep: Give a coverges absolutely a is ot diverget. a, we have lima Hece k N a V k a a V k k a k a a k a is diverget. As: (d) 9. Let A be a matri with real etries. Which of the followig is correct? (a) If A, the A is diagoalisable overcomple umbers (b) If A I, the A is diagoalisable over realumbers (c) If A A, the A is diagoalisable oly overcomple umbers (d) The oly matri of size satisfyig thecharacteristic polyomial of A is A Ep: (a) It is ot true i geeral For eample A

8 (b) True, if a matri is aihilated by a polyomial with sigle roots, the it is diagoalizable ad its eigevalues are roots of the polyomial. (c) They are diagoalizable as the polyomial has oly sigle roots. But ot ecessarily over comple umbers (ie.) A (d) Not true A & B satisfy p( ) which is their characteristic polyomials. As: (b). Let f :[,] [, ] be ay twice differetiable fuctio satisfyig f( a ( a) y) af( ) ( a) f( y) for all, y [, ] ad ay a [, ]. The for all (, ) (a) f ( ) (b) f ( ) (c) f ( ) (d) f( ) Ep:Give f :[,] [,] be twice differetiable. As:(b) f ( a ( a) y) a f ( ) ( a) f ( y) V, y [,] V a[,] f is a cove fuctio, we kow that f is cove iff f ( ).. Let A be a 4 4 ivertible real matri. Which of the followig is NOT ecessarily true? (a) The rows of A form a basis of R 4 (b) Null space of A cotais oly the vector (c) A has 4 distict eigevalues (d) Image of the liear trasformatio A o 4 R is 4 R Ep: Give A to be a 4 4 ivertible matri the (c) is ot correct as if we take A to be. The eige values of A are,,, As: (c) 8

9 u u. The partial differetial equatio u t ca be trasformed to v v t. For (a) t v e u (b) t v e u (c) v tu (d) v tu Ep: V e t u u e t v v e t u u e u e e t t t t u u u t As: (a) u 9 v t e u e t v t u e u u u t. The itegral equatio ( ) f( ) K(, y) ( y) dy For K(, y) y has a solutio (a) ( ) f( ) (b) ( ) K(, ) (c) 4 ( ) (d) ( ) f( ) f( ) d Ep: ( ) f ( ) y ( y) dy = f ( ) y ( y) dy ( ) f ( ) c where c y ( y) dy y f ( y) dy c c ( ) 4 y f y dy y cdy 4 ( ) f ( ) ( ) y f y dy As:(d) t u t c y ( f ( y) y c) dy c 4 c c y f ( y) dy 4 y f ( y) dy

10 4. For ay itegers a, b let N a, b deote the umber of positive itegers satisfyig a(mod 7) ad b(mod 7). The, (a) there eist a, b such that N. (b) for all a, b,, N a b (c) for all a, b, N a, b a, b (d) there eist a, b such that N, adthere eist a, b such that N a, b a, b Ep: Give N a, b is the umber of positive itegers ; satisfyig a (mod 7) b (mod 7) gcd (7,7) a(mod 7) By Chiese - remaider theorem. There eist a iteger b(mod 7) ad all the solutios are give by k k where k Z. Hece N a, b = As:(b) 5. Let be the product (stadard) topology o R geerated by the base. B {( s, t) ( u, v) : s t, u v where s, t, u, v R} ( B is the collectio of product of ope itervals) Give r, R R with r R ad a ( a, a) R, R Let C( a, r, R) {(, ) } r ( a ) ( a ) R } R Let B { C( a, r, R); a R, r, R R, r } Let be the topology geerated by the base B. The (a), (b), (c) (d), Ep: is geerated by ope rectagles ad is geerated by ope auluses. Cosider ay ope rectagle say

11 Take ay poit iside it The we ca get a ope aulus iside it. easily, Cetre of aulus The we ca get a ope rectagle arrow iside it Cetre of aulus The we ca get a ope rectagle arrow iside the ope rectagle As:(c) Cetre of aulus 6. The umber of group homomorphisms from the symmetric group S to the additive group Z / 6Z is (a) (b) (c) (d) Ep: Let : S Z6 be a homomorphism. The ker is a ormal subgroup of S. The possible ormal subgroups of S are e, S,{ e,(,,),(,,)}. If kerel ( e ), the by first isomorphism theorem, S ad Z 6 will be isomorphic which is ot true. If ker S the ( S ). It is a homomorphism. If ker { e,(,,),(,, )} the O( (,,)) O((,)). Hece O( (,)) = or. If it is, the (,) =. Sice S is geerated by (, ) & (,,). Hece O( (, )) (, ). Similarly ((,)) ((,)) We ca easily check that this is a homomorphism As:(b) 7. If f :[,] (, ) is a cotiuous mappig the which of the followig is NOT true? (a) F [, ] is a closed set implies f (F ) is closed i R. (b) If f( ) f() the f([, ]) must be equal to [ f (), f()]. (c) There must eist (, ) such that f( ). (d) f : ([,] (, ).

12 Ep: If f () f () the f ([,]) may ot be [ f (), f ()] cosider f give by, f (), f () ½ As:(b) 8. How may ormal subgroups does a o-abelia group G of order have other tha the idetity subgroup {e} ad G? (a) (b) (c) (d) 7 Ep: G.7 No. of sylow subgroups + k dividig 7 is or 7 No. of sylow 7 subgroups + 7k dividig is So we have a uique sylow 7 subgroup ad so it is ormal. Sylow subgroups is ot ormal.g has oly oe o trivial ormal subgroup. As:(b) 9. Suppose X, X..., X are idepedet ad idetically distributed radom variables each havig a epoetial distributio with parameter. Let X () X ()... X ( ) be the correspodig order statistics. The the probability distributio of ( X ( ) X ( ) ) / X () is (a) Chi-square with degree of freedom (b) Beta with parameters ad (c) F with parameters ad (d) F with parameters ad. As:(c) 4. A populatio cotais three uits u, u ad u. For i,, lety i be the value of a study variable for u i. A simple radom sample of size two is draw from the populatio without replacemet. Let T deote the usual sample mea ad let T adt be two other estimators, defied as follows : ( Y Y ) if u, u are i the sample T ( Y Y ) if u, u are i the sample Y Y if u, u are i the sample

13 ( Y Y ) if u, u are i the sample T Y Y if u, u are i the sample Y Y if u, u are i the sample IfY is the populatio mea, the which of the followig statemets is true? (a) All the three estimators T, T, T areubiased for Y. (b) T adt are biased estimator for Y but T is ot. (c) T adt are ubiased for Y but T is ot. (d) T adt are ubiased for Y but T is ot. As: (c) 4. Let X, X,..., X be a radom sample from N (, ) where is kow. Suppose has the Cauchy prior with desity., with ad kow. The with referece to the posterior distributio of (a) the posterior mea does ot eist ad theposterior variace is (b) the posterior mea eists but the posteriorvariace is (c) the posterior mea eists ad the posteriorvariace is fiite (d) the posterior variace is fiite but theposterior mea does ot eist As:(c) 4. I a cotigecy table if we multiply a particular colum by a iteger k ( ), the the odds ratio (a) will icrease (b) will decrease (c) remais same (d) will icrease if k ad will decrease if k. As:(c) 4. A popular car comes i both a petrol ad diesel versio. Each of these is further available i models, L, V ad Z. Amog all owers of the petrol

14 versio of this car, 5% have model V ad % have model Z. Amog diesel car customers, 5% have model L ad % model V, 6% of all customers have bought diesel cars. If a radomly chose customer has model V, what is the probability that the car is a diesel car? (a) /8 (b) /5 (c) /5 (d) / As:(a) 44. Let X, X,..., X be a radom sample from uiform,. Cosider the problem of testig H : agaist H :. Defie X( ) mi{ X, X,..., X }. Cosider the followig test : Reject H if X (), accept otherwise. Which of the followig is true? (a) power of the test =, size of the test =. (b) power of the test =, size of the test =. (c) power of the test =, size of the test =. (d) power of the test =, size of the test =. As:(c) 45. Suppose the cumulative distributio fuctio of failure time T of a compoet is ep( ct ), t,, c The the hazard rate of (t) is (a) costat (b) o-costat mootoically icreasig i t. (c) o-costat mootoically decreasig i t. (d) ot a mootoe fuctio i t. As: (b) 46. Let X,... be a Markov chai with state space {,,, 4}. Let the, X / / / 4 / 4 / 4 / 4 P trasitio probability matri P be give be / / Which of the followig is a statioary distributio for the Markov chai? (a) / 4 / 4 / 4 / 4 (b) / / (c) As: (d) / 4 / / 4 (d) / / / 4

15 47. A factorial eperimet ivolvig 4 factors, F, F, F ad F 4 each of levels, ad, is plaed i 4 blocks each of size 4. Oe of these blocks has the F F F F4 followiog cotets : 5 The cofouded factorial effect are (a) F F, F F, F F (b) F F, F F F4, F F F4 (c) F F4, F F F, F F F 4 (d) F F4, F F, F F F F4 As: (b) 48. Let X,,, ad 4 be idepedet ad idetically distributed ormal radom variables with mea ad variace. Defie X i i i X where, for i,,,. Let. deote the partial correlatio betwee X i give X j give X k. The p 4. = (a) As: (d) (b) (c) (d) 49. Cosider the followig probability mass fuctio P ( ), where the parameters, ) take values i the parameter space (,,,,,,, ; ij k (, ),,,, / / 7 / 8 / 9 / / 4 / 6 / 9 8/ 5/ 7 / 4 / 4 / / 4 / 7 / 9 Let X be a radom observatio from the distributio. If the observed value of X is, the (a) MLE of /, MLE of (b) MLE of /, MLE of (c) MLE of, MLE of / (d) MLE of, MLE of /. As: (c)

16 DEEPAK SERIES DEEPAK SERIES DEEPAK SERIES Dear Aspirats, You have goe through a sample of the patter followed i our book of CSIR- UGC/NET Mathematical Scieces by DEEPAK SERIES where you will come accross + questios alogwith study material based o latest syllabi. THE BOOK IS AVAILABLE AT NEAREST BOOKSELLER AND FOR MORE DEATILS WRITE US AT ifo.dstp@gmail.com DEEPAK SERIES DEEPAK SERIES DEEPAK SERIES 6

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Mathematical Statistics - MS Paper Specific Istructios. The examiatio is of hours duratio. There are a total of 60 questios carryig 00 marks. The etire paper is divided ito three sectios, A, B ad C. All sectios are compulsory. Questios