MATHEMATICAL SCIENCES

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1 SET7-Math.Sc.-II-D Roll No. 57 (Write Roll Number from left side exactly as i the Admit Card) Subject Code : 5 PAPER II Sigature of Ivigilators.. Questio Booklet Series Questio Booklet No. (Idetical with OMR Aswer Sheet Number) X MATHEMATICAL SCIENCES Time : Hour 5 Miutes Maximum Marks: 00 Istructios for the Cadidates. Write your Roll Number i the space provided o the top of this page as well as o the OMR Sheet provided.. At the commecemet of the examiatio, the questio booklet will be give to you. I the first 5 miutes, you are requested to ope the booklet ad verify it: (i) To have access to the Questio Booklet, tear off the paper seal o the edge of this cover page. (ii) Faulty booklet, if detected, should be get replaced immediately by a correct booklet from the ivigilator withi the period of 5 miutes. Afterwards, either the Questio Booklet will be replaced or ay extra time will be give. (iii) Verify whether the Questio Booklet No. is idetical with OMR Aswer Sheet No.; if ot, the full set to be replaced. (iv) After this verificatio is over, the Questio Booklet Series ad Questio Booklet Number should be etered o the OMR Sheet. 3. This paper cosists of fifty (50) multiple-choice type questios. All the questios are compulsory. Each questio carries two marks. 4. Each Questio has four alterative resposes marked: A B C D. You have to darke the circle as idicated below o the correct respose agaist each questio. Example: A B C D, where C is the correct respose. 5. Your resposes to the questios are to be idicated correctly i the OMR Sheet. If you mark your respose at ay place other tha i the circle i the OMR Sheet, it will ot be evaluated. 6. Rough work is to be doe at the ed of this booklet. 7. If you write your Name, Roll Number, Phoe Number or put ay mark o ay part of the OMR Sheet, except the space allotted for the relevat etries, which may disclose your idetity, or use abusive laguage or employ ay other ufair meas, such as chage of respose by scratchig or usig white fluid, you will reder yourself liable to disqualificatio. 8. Do ot tamper or fold the OMR Sheet i ay way. If you do so, your OMR Sheet will ot be evaluated. 9. You have to retur the Origial OMR Sheet to the ivigilator at the ed of the examiatio compulsorily ad must ot carry it with you outside the Examiatio Hall. You are, however, allowed to carry questio booklet ad duplicate copy of OMR Sheet after completio of examiatio. 0. Use oly Black Ball poit pe.. Use of ay calculator or mobile phoe etc. is strictly prohibited.. There are o egative marks for icorrect aswers. [Please Tur Over]

3 X 3 57 II MATHEMATICAL SCIENCES PAPER II. If f is a real valued fuctio which is cotiuous ad satisfies x + y f ( x) + f ( y) f =, x, y, f(0) = ad f (0) =, the f(5) is equal to 9 0 (D) 6. { } lim (+ )( + )... ( + ) is equal to 4e 7 7e 4 7 4e (D) 4 7e 3. Let p( z) = a0 + az az ad qz ( ) = bz+ bz bz be complex polyomials. If a0, b are o-zero complex p( z) umbers, the the residue of q( z) at 0 is equal to a0 b b a0 a b (D) a0 a 4. Let A be a real 3 4 matrix of rak. The the rak of A t A, where A t deotes the traspose of A, is exactly exactly 3 exactly 4 (D) at most but ot ecessarily 5. If A is a 5 5 real matrix with trace 5 ad if ad 3 are eigevalues of A, each with algebraic multiplicity, the the determiat of A is (D) Let V be the vector space of twice differetiable fuctios f o satisfyig f f + f = 0. Defie T : V by T( f) = ( f (0), f(0) ). The T is oe oe ad oto oe oe but ot oto oto but ot oe oe (D) either oe oe or oto 7. The probability that a teacher will give a surprize test durig ay class meetig is 3/5. If a studet is abset o two days, the the probability that he will miss at least oe test is /5 4/5 /5

4 57 II X 4 8. Suppose that the variables x 0 ad x 0 satisfy the costraits x+ x 3 ad x+ x 4 ad let Z = 5x+ 7 x. Which of the followig is true? The maximum value of Z is ad it does ot have ay fiite miimum. The miimum value of Z is 7 ad it does ot have ay fiite maximum. The maximum value of Z is ad its miimum value is 7. (D) Neither has a fiite maximum or a fiite miimum. is 9. The period of the fuctio π 0π π (D) 5π 0. Cosider the power series covergece of this series is si 0 (D) a real umber greater tha x + si x 5 3 5! z. The radius of =. Which of the followig is true? ad log diverges but log ad log (D) coverges but log. The radius of covergece of the series 3. Let z is! e e (D) 0 4. The co-efficiet of z si z is z 0 (D) i the Lauret series of 5. The umber of subgroups of 48 is (D) 0 f ( Z) = Z, Z. If γ : Z i =, = 0 f( Z) dz the 0 is equal to ( Z i) γ π i(+ 0 i) 30πi πi (D) 0

5 X 5 57 II 6. Let G be a group ad a, b G such that order of a is 5 ad aba = b. The the order of b is (D) Suppose I is the group of itegers ad H = {3 x: x I} is a ormal subgroup of I. The the elemets of I H are { H, H +, H + } { H +, H +, H + 3} { H, H +, H + 3} (D) { H, H +, H + 4} 8. I the set of itegers I, defie a b= a+ b+, a b= a+ b+ ab with a, b I. The ( I,, ) is a rig such that it is commutative a itegral domai a field 9. Let S : 3 4 ad T : 4 3 be liear trasformatios such that ToS is the idetity map of 3. The SoT is idetity map of 4 SoT is oe-oe but ot oto SoT is oto but ot oe-oe (D) SoT is either oe-oe or oto 0. Let M 3 ( ) be the vector space of all 3 3 real 0 matrices. Let A = 0 0. Which oe of the followig is ot a subspace of M 3 ( )? {X M 3 ( ) : XA = AX} {X M 3 ( ) : X + A = A + X} {X M 3 ( ) : trace (AX) = 0} (D) {X M 3 ( ) : det (AX) = 0}. Let A ad B be two matrices such that BA + B = I BA 3, where I is the idetity matrix. Which oe of the followig is always true? A is o-sigular B is o-sigular A + B is o-sigular (D) BA is o-sigular. Which of the followig is a subspace of 3? {(x,y,z) 3 : xyz = 0} {(x,y,z) 3 : x y = 3} {(x,y,z) 3 : x + y + z = } (D) {(x,y,z) 3 : x + y = 0} 3. Let W be the Wroskia of two liearly idepedet solutios of the ODE: d y dy + + t y = 0, t. dt dt The for all t, a costat c such that W(t) is t ce t ce (D) t ce t ce

6 57 II X 6 u u 4. The ature of = x y elliptic parabolic hyperbolic dy 3 5. The iitial value problem 3 y, y(0) 0 dx = = i a iterval aroud x = 0 has o solutio uique solutio more tha oe solutio (D) ifiite umber of liearly idepedet solutios is 8. If X ad Y are idepedet gamma variates with parameters μ ad ν respectively, the the ratio X/(X + Y) follows Beta distributio of type I with μ ad ν parameters Beta distributio of type II with μ ad ν parameters Gamma distributio with parameters μ ad ν 9. If X ~ N (0, ), the the pdf of Y = X is y g () y = e, y 0 Y y g ( y) = / π e, y 0 Y y g ( y) = / π e, y 0 Y 6. It is ecessary to fid cumulative frequecies i order to draw a/a Histogram Frequecy polygo Ogive curve (D) Colum chart 7. If oe regressio coefficiet of the two regressio lies is greater tha uity, the other will be greater tha less tha (D) If X ad Y are idepedet Poisso variates, the the coditioal distributio of X give X + Y, is ormal biomial hypergeometric (D) Poisso 3. Let E ad F be two idepedet evets with PE ( / F) + PF ( / E) =, PE ( F) = 9 ad PF ( ) < PE ( ). The PE ( ) equals 3 3 (D) 3 4

7 X 7 57 II 3. Let the radom variables X ad Y have the joit pmf p(x, y), the its distributio fuctio is 35. The test statistic for testig the sigificace of a observed partial correlatio coefficiet is i : x j : y i x j y F ( xy, ) px (, y) = i= j= i j r r (D) i : x j : y i > x j y F ( xy, ) px (, y) = i= j= i : x j : y i x j y F ( xy, ) px (, y) = i= j= i : x j : y i x j y F ( xy, ) px (, y) = i= j= i i i j j j r ( k+ ) ( r ( k+ ) ) r ( k+ ) ( r ( k+ ) ) r 34...( k+ ) (D) ( r ( k+ ) ) k k k 33. The stadard chi-squared test for a by cotigecy table is valid oly if all the expected frequecies are greater tha five both variables are cotiuous at least oe variable is from a ormal distributio 34. Let X be a radom variable such that E X <. The E X c is miimized if we choose c is equal to the mea of the distributio. the media of the distributio. the quartile deviatio of the distributio. (D) the stadard deviatio of the distributio. 36. A experimet is coducted uder the followig circumstaces: (a) whe there are pairs of observatios o two thigs beig compared. (b) for ay pair, each of the two observatios is made uder similar extraeous coditios. (c) differet pairs are observed uder differet coditios. I such a situatio which test ca be used? Paired t-test Sig test Media test (D) Idepedet t-test 37. Homogeeity of several variaces ca be tested by Bartlett s test Fisher s exact test F-test (D) t-test

8 57 II X Let X,..., X be a radom sample of size draw from R(0, θ). Defie T = X( ), T = X() + X( ). The which oe is correct? T is cosistet but ot T. T is cosistet but ot T. Both are cosistet. 39. Asymptotic distributio of U statistic is m m( m + ) N, (D) m m( m + + N, m m( m ) N, m m( m + ) N, 40. The statistic t for testig the hypothesis ρ = 0 based o a sample of size from a bivariate populatio has degrees of freedom (D) 3 4. If i a liear programmig problem the umber of variables i primal is, ad the umber of costraits i its dual is m, the m = m m (D) m Game is a situatio where players have same objectives. players have coflictig objectives. players have o objectives. 43. The demad for a item is determiistic ad costat over the time ad it is equal to 400 uit per year. The per uit cost of the item is ` 50 while the cost of placig a order is ` 5. The ivetory carryig cost is 0% of the cost of ivetory per aum ad the cost of shortage is ` per uit per year. Whe the stock outs are permitted the optimal orderig quatity is 33 uits 66 uits 8 uits (D) 88 uits 44. Whe the equipmet starts deterioratig with respect to time, its maiteace cost gradually starts/remais decreasig costat icreasig (D) zero 45. Uder the proportioal allocatio, the size of the sample from each stratum depeds o total sample size size of the stratum populatio size (D) All of the above 46. A populatio is perfectly homogeeous i respect of a characteristic. What size of sample would you prefer? A large sample A small sample A sigle sample (D) No item

9 X 9 57 II 47. Sowball samplig is a Radom samplig Cluster samplig No-radom samplig 48. I the ANOVA, treatmet refers to experimetal uits differet levels of a factor a factor 49. The Stadard Error (S.E.) of ay treatmet mea is give by for RBD S.E. ( ti) = se, i=,,..., t t = s r i= t S.E. ( i) E /,,,..., S.E. ( ) 3/ t = s / r, i =,,..., t i E (D) S.E. ( ) = /, =,,..., ti se r i t where s E stads for sum of squares due to error ad r the o. of replicatio 50. For a factorial experimet with three factors N, P ad K, each at two levels, the key block of a replicate is give below: () pk k p The cofouded effect is pk p k (D) pk

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Mathematical Statistics - MS

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