A MATHEMATICAL SCIENCES II

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1 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBJECT CODE SUBJECT PAPER CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC A-5-0 MATHEMATICAL SCIENCES II DURATION MAXIMUM MARKS NUMBER OF PAGES NUMBER OF QUESTIONS HOUR 5 MINUTES This is to certify that, the entries made in the above portion are correctly written and verified. Candidate s Signature HALL TICKET NUMBER OMR SHEET NUMBER Instructions for the Candidates. Write your Hall Ticket Number in the space provided on the top of this page.. This paper consists of fifty multiple-choice type of questions. 3. At the commencement of eamination, the question booklet will be given to you. In the first 5 minutes, you are requested to open the booklet and compulsorily eamine it as below : (i) To have access to the Question Booklet, tear off the paper seal on the edge of this cover page. Do not accept a booklet without sticker-seal and do not accept an open booklet. (ii) Tally the number of pages and number of questions in the booklet with the information printed on the cover page. Faulty booklets due to pages/questions missing or duplicate or not in serial order or any other discrepancy should be got replaced immediately by a correct booklet from the invigilator within the period of 5 minutes. Afterwards, neither the Question Booklet will be replaced nor any etra time will be given. (iii) After this verification is over, the Test Booklet Number should be entered in the OMR Sheet and the OMR Sheet Number should be entered on this Test Booklet. 4. Each item has four alternative responses marked,, and. You have to darken the circle as indicated below on the correct response against each item. QUESTION BOOKLET NUMBER Name and Signature of Invigilator A æýåææÿ$ ËMæü$ çü*^èþ èþë$. D ç³#r ò³o êvæü ÌZ CÐèþÓºyìþ èþ çü Ë ÌZ Ò$ çàìœý sìýmðüsœý èþ ºÆæÿ$ Æ>Äæý$ yìþ.. D {ç³ôèý² ç³{ èþðèþ$$ Äæý* ñýo ºçßý$âñýO_eMæü {ç³ôèý²ë èþ$ MæüÍW E. 3. ç³è æü { ëææÿ æýðèþ$$ èþ D {ç³ôé²ç³{ èþðèþ$$ Ò$Mæü$ CÐèþÓºyæþ$ èþ$. Ððþ$$ æþsìý I æþ$ Ñ$çÙÐèþ$$ËÌZ D {ç³ôé²ç³{ èþðèþ$$ èþ$ ðþç_ Mìü æþ ðþíí³ èþ A ÔéË èþ$ èþç³µ çüçv> çüç^èþ*çü$mø yìþ. (i) D {ç³ôèý² ç³{ èþðèþ$$ èþ$ ^èþ*yæþyé Mìü MæüÐèþÆŠÿõ³h A ^èþ$ èþ E èþ² M>W èþç³# ïüë$ èþ$ _ ^èþ yìþ. íütmæüpæšÿ ïüë$ìôý Ðèþ$ÇÄæý$$ C ÐèþÆæÿMóü ðþç_ E èþ² {ç³ôé²ç³{ èþðèþ$$ èþ$ Ò$Ææÿ$ A XMæüÇ ^èþðèþ æþ$ª. (ii) MæüÐèþÆæÿ$ õ³h ò³o Ðèþ$${ _ èþ çüðèþ*^éææÿ {ç³m>ææÿ D {ç³ôèý²ç³{ èþðèþ$$ìz õ³ië çü QÅ èþ$ Ðèþ$ÇÄæý$$ {ç³ôèý²ë çü QÅ èþ$ çüç^èþ*çü$mø yìþ. õ³ië çü QÅMæü$ çü º «_ V>± Ìôý é çü*_ _ èþ çü QÅÌZ {ç³ôèý²ë$ ÌôýMæü ùðèþ#r Ìôý é f{ç³ M>Mæü ùðèþ#r Ìôý é {ç³ôèý²ë$ {MæüÐèþ$ç³ æþ ÌZ ÌôýMæü ùðèþ#r Ìôý é HÐðþŌ é óþyéë$ yæþ$r Ðèþ sìý øçùç³nç èþððþ$ō èþ {ç³ôèý² ç³{ é ² Ððþ r óþ Ððþ$$ æþsìý I æþ$ Ñ$ÚëÌZÏ ç³è > ç³ææÿåðóþ æümæü$ Mìü ÇW C_aÐóþíÜ é Mìü º æþ$ë$v> çüçv>y E èþ² {ç³ôèý²ç³{ é ² çü$mø yìþ. èþ æþ èþ èþææÿ {ç³ôèý²ç³{ èþðèþ$$ Ðèþ*Ææÿaºyæþ æþ$ A æþ èþç³# çüðèþ$äæý$ CÐèþÓºyæþ æþ$. (iii) ò³o Ñ«æþ V> çüç^èþ*çü$mö èþ² èþæ>ó èþ {ç³ôé²ç³{ èþ çü QÅ èþ$ OMR ç³{ èþðèþ$$ ò³o A óþñ«æþ V> OMR ç³{ èþðèþ$$ çü QÅ èþ$ D {ç³ôé²ç³{ èþðèþ$$ ò³o Çª çùt t çü Ë ÌZ Æ>Äæý$ÐèþÌñý èþ$. 4. {ç³ {ç³ôèý²mæü$ éë$væü$ {ç³ éåðèþ*²äæý$ {ç³ çüµ æþ èþë$,, Ðèþ$ÇÄæý$$ Ë$V> CÐèþÓºyézÆÿ$$. {ç³ {ç³ôèý²mæü$ çüæðÿō èþ {ç³ çüµ æþ èþ èþ$ G èþ$²mö Mìü æþ ðþíí³ èþ Ñ«æþ V> OMR ç³{ èþðèþ$$ìz {ç³ {ç³ôé² çü QÅMæü$ CÐèþÓºyìþ èþ éë$væü$ Ðèþ é ÌZÏ çüæðÿo èþ {ç³ çüµ æþ èþ èþ$ çü*_ ^óþ Ðèþ é ²»êÌŒý ëæÿ$$ sœý ò³ Œþ ø Mìü æþ ðþíí³ èþ Ñ«æþ V> ç³nç ^éí. E éçßýææÿ ý : A B C D çüæðÿō èþ {ç³ çüµ æþ èþ AÆÿ$$ óþ 5. {ç³ôèý²ëmæü$ {ç³ çüµ æþ èþë èþ$ D {ç³ôèý²ç³{ èþðèþ$$ ø CÐèþÓºyìþ èþ OMR ç³{ èþðèþ$$ ò³o èþ CÐèþÓºyìþ èþ Ðèþ é ÌZÏ óþ ç³nç _ Væü$Ç ^éí. AÌêM>Mæü çüðèþ*«é èþ ç³{ èþ ò³o ÐóþÆöMæü ^ør Væü$Ç õü Ò$ {ç³ çüµ æþ èþ Ðèþ$*ÌêÅ Mæü èþ ^óþäæý$ºyæþ æþ$. 6. {ç³ôèý² ç³{ èþðèþ$$ ÌZç³Ë C_a èþ çü*^èþ èþë èþ$ gê{væü èþ V> ^èþ æþðèþ yìþ. Eample : A B C D where is the correct response. 5. Your responses to the items are to be indicated in the OMR Answer Sheet given to you. If you mark at any place other than in the circle in the Answer Sheet, it will not be evaluated. 6. Read instructions given inside carefully. 7. Rough Work is to be done in the end of this booklet. 8. If you write your name or put any mark on any part of the OMR Answer Sheet, ecept for the space allotted for the relevant 7. _ èþ$ ç³ {ç³ôèý²ç³{ èþðèþ$$ _ÐèþÆæÿ C_a èþ RêäçÜ ËÐèþ$$ÌZ ^óþäæý*í. entries, which may disclose your identity, you will render yourself 8. OMR ç³{ èþðèþ$$ ò³o È~ èþ çü Ë ÌZ çü*_ ^èþðèþëíü èþ ÑÐèþÆ>Ë$ èþí³µ _ C èþææÿ çü Ë ÌZ liable to disqualification. Ò$ Væü$Ç ç³# èþ$ ðþíõ³ Ñ«æþ V> Ò$ õ³ææÿ$ Æ>Äæý$yæþ V>± Ìôý é C èþææÿ _à²ë èþ$ 9. The candidate must handover the OMR Answer Sheet to ò³rtyæþ V>± ^óþíü èþrïæÿ$$ óþ Ò$ A èþææÿá èþmæü$ Ò$Æóÿ»ê«æþ$ÅËÐèþ# éææÿ$. the invigilators at the end of the eamination compulsorily 9. ç³è æü ç³nææÿ Æÿ$$ èþ èþæ>ó èþ Ò$ OMR ç³{ é ² èþç³µ çüçv> ç³è æü ç³ææÿåðóþ æümæü$yìþmìü CÐéÓÍ. and must not carry it with you outside the Eamination Hall. The candidate is allowed to take away the carbon copy of OMR Ðésìý ç³è æü Væü ºÄæý$rMæü$ çü$mæü$ððþâæýïmæü*yæþ æþ$. ç³è æü ç³nææÿ Æÿ$$ èþ èþææÿ$ðé èþ A æýåææÿ$ Ë$ Sheet and used Question paper booklet at the end of the {ç³ôèý² ç³{ é ², OMR ç³{ èþ Äñý$$MæüP M>Ææÿ¾ Œþ M>ï³ çü$mæü$ððþâæýïðèþ^èþ$a. eamination. 0. ±Í/ èþëï Ææÿ Væü$»êÌŒý ëæÿ$$ sœý ò³ Œþ Ðèþ*{ èþðóþ$ Eç³Äñý*W ^éí. 0. Use only Blue/Black Ball point pen.. ÌêVæüÇ æþðœþ$ sôýº$ìœýþ, M>ÅÍMæü$ÅÌôýrÆŠÿË$, GË[M>t MŠü ç³çmæüæ>ë$ Ððþ$$ æþëvæü$ èþñ ç³è æüvæü ÌZ. Use of any calculator or log table etc., is prohibited. Eç³Äñý*W ^èþyæþ õù«æþ.. There is no negative marks for incorrect answers.. èþç³š çüðèþ*«é éëmæü$ Ðèþ*Ææÿ$PË èþwy ç³# Ìôý æþ$. II æ A-5-0 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

2 DO NOT WRITE HERE II æ A-5-0

3 MATHEMATICAL SCIENCE Paper II. Let ( ) S n = ; n =,, 3,... ; n and and be the ifimum and supremum of S respectively. Then the ordered pair (, ) = (0, ) (0, ) (, ). If { n } = n such that (, ) is a sequence of real numbers a > 0and n+ ( n ) n, where a > 0, then = + for { n } is monotonic increasing and lim n = + n { n } is monotonic decreasing and lim n = n { n } is monotonic increasing and n 3. If f: is defined by for rational f ( ) = 0 for irrational, then set of points of continuity for f is, the set of all rational numbers I, the set of all irrational numbers, the set of all real numbers φ, the empty set 4. Suppose f: is a monotonic decreasing continuous function with f (0) = a. ( ) Then the image, ( ) f 0,, of the interval ( 0, ), under f is a sub set of (, a) lim n = n a (, a) { n } is monotonic decreasing and ( a, + ) lim = n n a ( a, + ) II æ 3 A-5-0

4 5. If f :a, [ b ] is such that for some δ> 0, ( ) ( ) for all, y [ a, b] f f y < y +δ then f( ) = e for all [ a, b] f ( ) = cos for all [ a, b] f () = k, a constant, for all [ a, b] f () = log for all [ a, b] 6. In the vector space 3 over, if (,, 5) (,, ) (,, 3) =λ +λ + 3 (,,) λ then (,, ) ( 6,, 3) λ λ λ = 3 7. If B = {,,..., } 8. If 9. If α α α is a basis for a vector n space V and S = {,,..., m} is a linearly independent set in V then B S m n S B n < m 3 3 T: is defined by ( ) ( z, + z) T, y, z = + y, y for (, y, z) 3 then the rank of T is 3 0 T : 3 3 is given by T(, y, z ) = ( + y + z, y + z, z ) for (, y, z) 3 then ( ) T, y, z = ( 6, 3, ) ( y, y z, z ) ( 6,, 3) (, y z, z ) ( y, y z, z) ( 6,, 3) ( + y, y + z, z) II æ 4 A-5-0

5 0. Consider the vector space p ( ) of all. Let C be the circle = positively z polynomials of degree over. If (a, b, c) oriented. Then zz ( ) dz = C represents the co-ordinates of the i polynomial 3t + 4t + 5 with respect to the i basis {, + t, + t } then (a, b, c) = i (, 4, 3) i (, 4, 3) 3. Suppose f(z) = u + iv is an entire function ( 3, 4, 3) with u v z. If f (0) = then f (04) = (4, 4, 3) 0. Let C be the circle z 3 = positively 04 oriented. Then 0 z cosec z dz = C 4. The number of zeroes of the polynomial z 8 + 0z that lie in the domain i < z < is πi 5 6 πi 7 8 II æ 5 A-5-0

6 5. If R is a Boolean ring, then the 8. A field among the following quotient rings is characteristic of R is 0 6. The number of normal sub groups of order 3 in a group of order 43 is [ ] ++ [ ] [ ] 3 [ ] The Galois group of the polynomial 0 9. Let X = {a, b, c, d, e}. Consider the { T = φ, X, a, a, b, {a, b, e}, topology { } { } {(a, c, d}, {a, b, c, d}} on X. If A = {a, c, d, e} over is isomorphic to the group ({, 3, 7, 9 }, X mod 0) ( 0,,, 3, 4, 5, 6, 7, 8, 9, + mod0) { } ({,, 3, 4 }, X mod 5) ( 0, + ) then the interior of A is {a, b, e} {a, c, d} {a} {a, b} II æ 6 A-5-0

7 0. Let S be an infinite subset of a discrete 3. If () y = and if the solution of topological space X. Then S is Compact Not Compact Not a T space Not a T space. The general solution of the differential equation ( ) y = c y d + y dy = 0 is dy dy is d d ( )( ) y = y + then k = y = ky, + y = c y = cy 4. If the solution of the partial differential + y = cy. If the solution of the differential equation dy d ( ) 3 + ycot = cos + y sin 4 is y f ( ) tan π ( cos ) = 4 4 when ( π ) = then ( ) y, tan f = π π 4 equation z 3 z + 4 z = e 3 y y + y ( α ) ( β ) ( γ ) z = f y + + f y + + f y y +δ e, then α+β+γ = δ is II æ 7 A-5-0

8 5. The solution of the Volterra integral equation, y t dt + is + ( ) () y = 0 ( + ) 3 ( ) 3 ( + ) 8. The etremals of the functional 0 ( y ) +y sin d under boundary conditions π ( ) ( ) y 0 = 0, y = is y = 4 + cos π y = 4 sin π y = cos π ( ) y = + sin π 6. Using Newton-Raphson method, a root of the equation 3 5= 0 is When [, 3] is divided in to 4 equal sub 3 intervals, by Simpson s rule, d = If y( 0 ) =, y ( 0 ) = 0, and y ( ) = ( ) then the integral equation corresponding to y +y+y=0 is ( ) ( ) ( ) + 3 u u du + = 0 0 ( ) ( ) ( ) + 3 u u du = 0 0 ( ) ( ) ( ) ψ + u ψ u du+ = 0 0 ψ + u ψ u du = 0 ( ) ( ) ( ) 0 II æ 8 A-5-0

9 30. The number of degrees of freedom of rigid body moving freely in space is 33. The joint probability mass function (pmf) of (, y) is given below: For any two events A and B which of the following is true? PAB ( ) PA ( ) PA ( B) 5 6 y The marginal pmf of is given by PAB ( ) PA ( ) PA ( B) PA ( ) + PB ( ) PA ( B) PA ( ) PB ( ) PA ( B) 3. Let A and B be two events such that PA=0.3, ( ) ( ) PA B= 0.8and P = P. Then for what values of P, A and B are independent P() P() P() P() 5 5 II æ 9 A-5-0

10 34. The characteristic function of standard normal distribution is given by sint t t e +t t e 35. Which of the following matri is not a transition probability matri? 36. If X and Y are two i.i.d. random variables such that X+Y and X Y are independently distributed, then the distribution of X is Eponential Normal Cauchy Laplace 37. State which of the following is correct? Unbiased estimator is always the best estimator Unbiased estimator is also a consistent estimator Unbiased estimator is unique Unbiased estimator need not be consistent II æ 0 A-5-0

11 38. Consider the following statements on the maimum likelihood estimator (MLE) of θ involved in the uniform U ( θ, θ+ ) distribution: P : MLE of θ is unique; Q : MLE of θ is always unbiased Both P and Q are true P is true but not Q Both P and Q are not true Q is true but not P 39. For a fied confidence coefficient ( α ), the most preferred confidence interval for the parameter θ is the one with Shortest length Largest length Average length None of these 40. The nonparametric alternative to the two sample t test is Sign test Mann-Whitney U test Wilcoon signed rank test Runs test for randomness 4. The Gauss Markov theorem establishes that the OLS estimator of ( ), = X X X Y is BLUE Error free estimator uve only Not a BLUE 4. A quadratic form in p-variables,,..., p is a homogeneous function consisting of all possible second order terms namely p a ij ij i j p aiji ij p aij j ij p a jiij ij II æ A-5-0

12 43. The standard procedure for maimising a 45. Let r.3 be the partial correlation coefficient function of several variables, subject to one or more constraints, is the method of Lagrange Multipliers First Principal Component Principal of Least Squares Data Reduction Method 44. In Cluster Analysis the dissimilarity measure equation dab Min( dij ) = is i A jb known as Dendrogram Complete linkage clustering Single linkage clustering Both and between X and X after linear effect of X 3 has been eliminated, then the relationship between total correlation, partial correlation and multiple correlation can be epressed as R = ( r )( r ).3 3. R.3 = ( r )( r ) 3. ( ) R.3 = r ( r 3. ) R.3 = ( r )( r ) 46. Let f (, y) 3., 0< < y; 0< y< = 0, Otherwise be the joint probability density function of and y, then and y are uncorrelated are correlated with coefficient of correlation equal to are correlated with coefficient of correlation equal to 0.5 are correlated with coefficient of correlation equal to 0. II æ A-5-0

13 47. The estimator of reliability function for eponential failure distribution with parameter '' is given by ep( λt) λ ep( λt) 49. Any solution to a general linear programming problem which also satisfies the non negative restrictions is called a Optimum solution Non basic solution λ Feasible solution 48. The structure function φ( ) of K out of n Basic solution system is 50. Single item inventory models occur when n i = n i = n i = n i = i i i i K K = K K an item is ordered only once to satisfy the Supply for the period Demand for the period EOQ problem with shortages Supply and Demand for the period II æ 3 A-5-0

14 Space for Rough Work II æ 4 A-5-0

15 Space for Rough Work II æ 5 A-5-0

16 Space for Rough Work II æ 6 A-5-0

A PHYSICAL SCIENCES II

A PHYSICAL SCIENCES II CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBJECT CODE SUBJECT PAPER CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC