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K-2614 ±Üâ.£.®æãà./P.T.O. 1 13. There is no negative marks for incorrect answers. 13. ÓÜÄ AÆÉ¨Ü EñܤÃÜWÜÚWæ Má| AíPÜ CÃÜáÊÜâ©ÆÉ . 12. PÝÂÆáRÇàæ oà A¥ÜÊÝ ÇÝW pæàÇ CñÝ©¿á E±ÜÁãàWÜÊ® Ü áÜ ° ¯ÐæàÓÜÇÝX¨æ. 12. Use of any calculator or log table etc., is prohibited. 11. Use only Blue/Black Ball point pen. 11. ¯àÈ/PܱÜâ³ ¸ÝÇ ±ÝÀáíp ±æ® ÊÜÞñÜÅÊæà E±ÜÁãàXÔÄ. OMR Answer Sheet soon after the examination. ¯Êæã¾í©Wæ ñæWæ¨ÜáPæãívÜá ÖæãàWÜÖÜá¨Üá. 10. You can take away question booklet and carbon copy of 10. ±ÜÄàPæÒ¿á ®ÜíñÜÃÜ, ±ÜÄàPÝÒ ±ÜÅÍæ° ±Ü£ÅPæ¿á®Üá° ÊÜáñÜᤠ®ÜPÜÆá OMR EñܤÃÜ ÖÝÙæ¿á®Üá° carry it with you outside the Examination Hall. Pæãívæã¿á PÜãvܨÜá. at the end of the examination compulsorily and must NOT ¯àÊÜâ ×í£ÃÜáXÓܸàæ PÜá ÊÜáñÜᤠ±ÜÄàPÝÒ PæãsÜw¿á ÖæãÃÜWæ OMR ®Üá° ¯Êæã¾í©Wæ 9. 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EñܤÃÜWÜÚÊæ. ¯àÊÜâ ±ÜÅÍæ°¿á G¨ÜáÃÜá ÓÜÄ¿Þ¨Ü EñܤÃÜ¨Ü ÊæáàÇæ, PæÙÜWæ PÝ~Ô¨Üíñæ 4. ±ÜÅ£Áãí¨Üá ±ÜÅÍW °æ ã Ü (A), (B), (C) ÊÜáñÜᤠ(D) Gí¨Üá WÜáÃÜá£Ô¨Ü ®ÝÆáR ±Ü¿Þì¿á ¨ÜÇÝÀáÓÜÇÝWÜáÊÜâ©ÆÉ , ¿ÞÊÜâ¨æà ÖæaáÜ c ÓÜÊáÜ ¿áÊÜ®ã Ü ° PæãvÜÇÝWÜáÊÜâ©ÆÉ . CÃÜáÊÜ ±ÜâÔ¤PæWæ ¨ÜÇÝÀáÔPæãÙÜÛ¸æàPÜá. B ÚPÜ ±ÜÅÍæ° ±Ü£ÅPæ¿á®Üá° ¨æãàÐܱäÜ ÄñÜ ±ÜâÔ¤P¿ æ á®Üá° PÜãvÜÇæ 5 ¯ËáÐÜ¨Ü AÊÜ JÙÜW,æ ÓÜíËàPÜÒPÄÜ í¨Ü ÓÜÄ A¥ÜÊÝ ©Ì±ÜÅ£ A¥ÜÊÝ A®ÜáPÜÅÊÜáÊÝXÆÉ¨Ü A¥ÜÊÝ CñÜÃÜ ¿ÞÊÜâ¨æà ÊÜÂñÝÂÓÜ¨Ü ÊÜáá©Åst ÊÜÞ×£Áãí©Wæ ñÝÙæ ®æãàwÄ. ±ÜâoWÜÙáÜ /±ÜÅÍW°æ ÙÜ áÜ PÝOæ¿Þ¨Ü, (ii) ±ÜâÔ¤Pæ¿áÈÉ®Ü ±ÜÅÍæ°WÜÙÜ ÓÜíTæ ÊÜáñÜᤠ±ÜâoWÜÙÜ ÓÜíTæ¿á®Üá° ÊÜááS±Üâo¨Ü ÊæáàÇæ ±ÜâÔ¤P¿ æ á®Üá° ÔÌàPÜÄÓܸàæ w. ±æà±Üà ÔàÆ®Üá° ÖÜÄÀáÄ. ÔrPRÜ Ã ÔàÇ CÆÉ¨Ü ±ÜÅͱ °æ âÜ Ô¤Pæ ÔÌàPÜÄÓܸàæ w. ñæè æ Ü (i) ±ÜÅÍ°æ ±ÜâÔ¤PW æ æ ±ÜÅÊàæ ÍÝÊÜPÝÍÜ ±Üv¿ æ áÆá, D Öæã©Pæ ±Üâo¨Ü Aíb®Ü ÊæáàÈÃÜáÊÜ ¯àÊÜâ ±ÜâÔ¤P¿ æ á®Üá° ñæÿ æ áÆá ÊÜáñÜᤠPæÙXÜ ®Üíñæ PÜvÝx¿áÊÝX ±ÜÄàQÒÓÆ Ü á PæãàÃÜÇÝX¨æ. 3. ±ÜÄàPæÒ¿á ±ÝÅÃíÜ »Ü¨È Ü É , ±ÜÅÍ° æ ±ÜâÔ¤P¿ æ á®Üá° ¯ÊÜáWæ ¯àvÜÇÝWÜáÊÜâ¨Üá. Êæã¨ÜÆ 5 ¯ËáÐÜWÙÜ È Ü É 2. D ±Ü£ÅPæ¿áá ÖÜá BÁáR Ë«Ü¨Ü G±Ü³ñæô¨Üá ±ÜÅÍæ°WÜÙÜ®Üá° JÙÜWæãíw¨æ. 1. D ±Üâo¨Ü ÊæáàÆᤩ¿áÈÉ J¨ÜXst Óܧ٨ Ü È Ü É ¯ÊÜá¾ ÃæãàÇ ®ÜíÃÜ®áÜ ° ÃæÀáÄ. A»Ü¦ìWÜÚWæ ÓÜãaÜ®æWÜÙÜá entries, which may disclose your identity, you will render yourself Answer Sheet, except for the space allotted for the relevant 8. If you write your name or put any mark on any part of the OMR 7. Rough Work is to be done in the end of this booklet. 6. Read the instructions given in OMR carefully. evaluated. place other than in the ovals in OMR Answer Sheet, it will not be in the OMR Sheet kept inside the Booklet. If you mark at any 5. Your responses to the question of Paper III are to be indicated where (C) is the correct response. Example : 4. 3. 2. 1. B C D Instructions for the Candidates Number of Pages in this Booklet : 16 Time : 2 Hours 30 Minutes A correct response against each item. and (D). You have to darken the oval as indicated below on the Each item has four alternative responses marked (A), (B), (C) Booklet will be replaced nor any extra time will be given. period of 5 minutes. Afterwards, neither the Question by a correct booklet from the invigilator within the other discrepancy should be got replaced immediately missing or duplicate or not in serial order or any cover page. Faulty booklets due to pages/questions in the booklet with the information printed on the (ii) Tally the number of pages and number of questions without sticker-seal and do not accept an open booklet. seal on the edge of this cover page. Do not accept a booklet (i) To have access to the Question Booklet, tear off the paper open the booklet and compulsorily examine it as below : be given to you. In the first 5 minutes, you are requested to At the commencement of examination, the question booklet will This paper consists of seventy five multiple-choice type of questions. Write your roll number in the space provided on the top of this page. Number of Questions in this Booklet : 75 Maximum Marks : 150 Subject : MATHEMATICAL SCIENCE Paper : III Name : ____________________________________ Signature: ____________________________________ Name : ___________________________________ Signature : ___________________________________ Name & Signature of Invigilator/s (Figures as per admission card) Test Subject Code : K-2614 Test Subject : MATHEMATICAL SCIENCE Test Paper : III Roll No. OMR Sheet No. : _________________________________ Test Booklet Serial No. : _______________________ Paper III K-2614 2 e locally compact space and A is open in X, then A is (D) u (x, t) = (D) If X is locally compact Hausdorff compact space is regular (B) Every locally compact Hausdorff (C) u(x, t) = metrizable space is compact false ? 3. Which one of the following statements is (B) u(x, t) = (D) π log a e a (C) π log a π log a π log a ∫ ∞ x2 + a2 log x (C) 1 (C) ∞ (B) 1 (A) 0 n→∞ n 1. lim 1 + 2 + 3 3 + .... + n n is equal to 1 Note : (D) dx is equal to 2 ( ∞ n 1 2 + π 3 12 n =1 ∑ ∞ 4 n e 1 − 3n2 π 2 t 2 π + 3 12 n =1 ∑ ∞ n (−1)n+1 2 π + 3 12 n =1 n ∞ ( −1)n + 1 ∑ ux (0, t) = ux (2, t) = 0, u (x, 0) = 3x, is ut = 3uxx, 0 < x < 2, t > 0, 5. The solution to the heat equation 2a 0 (D) ∑ 4 2 cos − 3n2 π 2 t nπx (A) u(x, t ) = 2 2. If a > 0 then n =1 4 4 e cos − 3n2 π2 t nπx (A) Every sequentially compact (A) 2 4 + 3 12 2 sin nπx (C) Every limit point compact space is (B) 4 2 − 3n2 π 2 t cos nπx ) (B) (A) dx 2 d2 y λ = n2 , n = 1, 2, 3, ..... has non-trivial solutions if λ = 2n, n = 1, 2, 3, .... has no non-trivial solutions if λ = n, n = 1, 2, 3, .... has a non-trivial solution if has a non-trivial solution if λ ≤ 0 + λy = 0 , y (0) = 0, y(π) = 0 4. The sturm-Liouville problem carries two (2) marks. All questions are compulsory. This paper contains seventy-five (75) objective type questions. Each question PAPER – III MATHEMATICAL SCIENCE *K2614* Total Number of Pages : 16 K-2614 Paper III 3 (C) 1.2428 (D) 1.2424 (D) (A × B) ∩ (C × D) = (A ∪ C) × (B ∪ D) (A) 1.2426 (B) 1.2425 (C) (A × B) ∪ (C × D) ⊂ (A ∪ C) × (B ∪ D) increment h = 0.2 is dx = x + y, y(0) = 1 with given that dy Runge-Kutta method of fourth order, 8. The value of y (0.2) obtained by (D) φ(x ) = f(x) + (C) φ(x ) = f(x ) + λ2 − 1 λ2 0 ex ∫ 1 e − y f(y) dy λ −1 0 ∫ λ e x e − y φ(y) dy 1 λ − 1 0 (B) φ(x ) = f(x) − ∫ λ + 1 e x e − y f(y ) dy 1 0 λ −1 e x ∫ e − y f(y ) dy (A) φ(x ) = f(x ) − λ 1 solution of (1) ? then which one of the following is the given real function f(x) (0 ≤ x ≤ 1) . If λ ≠ 1, (B) (A × B) ∪ (C × D) ⊃ (A ∪ C) × (B ∪ D) (A) (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D) true ? which one of the following statements is 10. If A, B, C, D are nonempty sets, then (D) φ (t) ≠ 0 for all t ∈ [a, b] with t ≠ t 0 of t0 ≠ t ∈ [a, b] (C) φ (t) ≠ 0 and φ′(t ) = 0 for all values (B) φ (t) = 0 ∀ t ∈ [a, b] [a, b] (A) φ (t) ≠ 0 for atleast one value of t in system such that φ( t 0 ) = 0 . Then φn ⋅ ⋅ φ = be a solution of the above ⋅ φ2 1 φ Let to be any point of [a, b] and let 0 φ(x) = f(x ) + λ ∫ 1 e x − y φ(y ) dy .... (1) for a 7. Consider the Fredholm integral equation (D) (C) (B) (A) y (x) = e y (x) = 0 y (x) = c, where c is any real constant ≠ 0 y (x) = x a ∫ b (x + y 2 + 3y′) dx is given by 6. The extremal for the functional *K2614* dt = an1(t)x1 + ... + ann (t)xn. dx n • • • • • • • • • • • • dt = a 21(t ) x1 + ... + a 2n (t )x n dx 2 dt = a11(t )x1 + ... + a1n (t )x n dx1 system 9. Consider the homogeneous linear Total Number of Pages : 16 Paper III connected. (D) Every connected space is path connected. under a continuous map is (C) The image of a connected space common is connected. connected sets that have a point in (B) The union of a collection of have Bd A ≠ φ . nonempty proper subset A of X, we (A) If X is connected, then for every not true ? 14. Which one of the following statements is K-2614 4 subfield. (D) No Q-vector subspace of E is a E which are subfields is infinite. (C) Number of Q-vector subspaces of subspaces of E are subfields of E. (B) Only finitely many Q-vector subfield of E. (A) Every Q - vector subspace of E is a statements is true ? [E : Q] > 2. Which one of the following spaces is compact. (D) The product of finitely many compact under a continuous map is compact. (C) The image of a compact space topological space is closed. (B) Every compact subspace of any compact space is compact. (A) Every closed subspace of a false ? 13. Which one of the following statements is is normal. (D) Every subspace of a normal space not be normal. (C) Product of two normal spaces need (B) Every regular space is normal. (A) Every Hausdorff space is regular. true ? 12. Which one of the following statements is (D) closed (C) path connected (B) bounded (A) compact usual topology, then the set IR 2 – A is 11. If A is a countable subset of IR 2 with *K2614* field of rational numbers Q. Assume 17. Let E be a finite Galois extension of the (C) pq – p – q (D) pq – 1 (A) pq – p – q + 1 (B) pq – p – q – 1 which are coprime to pq is Then the number of integers a, 1 < a < pq, 16. Let p and q be two distinct prime numbers. other. IR are not homeomorphic to each 2 (D) Under the usual topologies, IR and homeomorphism. by f(t) = (cos2 π t, sin2 π t) is a (C) The mapping f : [0, 1) S′ defined to each other. (B) (–1, 1) and IR are homeomorphic IR are homeomorphic to each other. (A) The subspaces [a, b] and [0, 1] of not true ? 15. Which one of the following statements is Total Number of Pages : 16 K-2614 Paper III 5 25 elements (D) irreducible over any finite field with elements 5 (C) irreducible over the field IF of five numbers IR (B) reducible over the field of real (A) irreducible over ring of integers 9 (D) does not exist (C) exists, but not 3 (B) is 3 (A) is 3 A = {p ∈ Q + | p 2 < 3} in Q 24. The supremum of the set 20. The polynomial f(x) = x4 + x3 + x2 + x + 1 is a=b (D) For a, b, c ∈ R − {0} , if ac = bc then has a solution (C) The equation x = a, a ∈ R always 2 group under multiplication (B) Non-zero elements can never be a always a C ∈ R such that a ⋅ c = b (A) Given any a, b ∈ R − {0} there is following holds ? 19. In an integral domain R, which one of the zero ideal of 9. 9[i], the intersection 9 p is a non(D) For any non-zero prime ideal p of prime ideal. ideal generated by ‘p’ in 9[i] is a (C) For any prime number ‘p ’ in 9, the extension of its prime filed. 9[i]/ p is always a degree 2 (B) If p is a prime ideal of 9[i], then 9[i]/p is a field. (A) If p is a prime ideal of 9[i], then statements is true ? integers. Which one of the following 18. Let 9 [ i ] denote the ring of Gaussian *K2614* relation (D) both transitive and symmetric (C) a symmetric relation (B) a transitive relation (A) an equivalence relation “a divides b” is 23. In the set of integers, the relation (D) G cannot be a cyclic group (C) G is a cyclic group (B) G cannot be an abelian group (A) G is an abelian group center. Assume G Z is cyclic. Then 22. Let G be a finite group and Z ⊂ G be its (D) an associative ring identity (C) a non commutative ring without (B) a commutative ring with identity (A) a non-associative ring 21. The ring M2 (IR) of all 2 × 2 real matrices is Total Number of Pages : 16 Paper III 2 1 (D) is (C) is 1 (B) is 0 (A) does not exist f(x) → 0 as x ∞ . The xlim∞ f ′(x) (0, + ∞ ), f′′ is bounded on (0, + ∞ ) and 27. Suppose f is twice differentiable on Stielties integrable functions (D) f is a product of two Riemann- U(p, f, α) − L(p, f, α ) < ∈ partition p such that (C) for every ∈ > 0, there exists a (B) f is monotonic on [a, b] (A) f is continuous on [a, b] respect to α if and only if Riemann-stielties in tegrable with [a, b]. Then the function f : [a, b] IR is 26. Let α be monotonically increasing on 0 ∫ (D) 2 1 0 (C) ∫ ∞ 3 log x dx 1 0 ∫ 1 1 + 2x 2 7e − x − 1 4 x log x (D) nlim∞ xn = 1 n ∞ (C) lim n xn = 2 2 (B) nlim∞ nxn = 1 (A) nlim∞ nxn = 0 monotonically. Then which the terms xn decrease n =1 30. Let ∑ xn be a convergent series in ∞ (C) 2 1 (D) 1 (A) 0 (B) n =1 ∑ 4 1 n4 + n2 + 1 n ∞ dx dx integrals diverges ? 25. Which one of the following improper *K2614* (D) is either 1 or –1 (C) is always 1 (B) may be any non-zero integer (A) may not be an integer Then det (A) Assume that A–1 also has integer entries. 31. Let A be a 3 × 3 matrix with integer entries. 29. The sum of the series 0 (B) ∫ e dx −x 2 ∞ (A) K-2614 6 (D) Diverges for a > is 3 1 (C) Diverges for all real values of a (B) Converges for a > 3 1 (A) Converges for all real values of a n=1 28. The series ∑ ∞ n − sin 1n 1 ( ) a Total Number of Pages : 16 K-2614 T is diagonal with respect to which the matrix of (D) There may not be any basis of V digonal respect to which the matrix of T is the field of complex numbers with (C) There is always a basis of V over diagonal respect to which the matrix of T is the field of real numbers with (B) There is always a basis of V over (A) The matrix of T is always diagonal following is true ? of rational numbers. Which one of the dimensional vector space V over the field 34. Let T be an endomorphism of a two values cannot be distinct (D) If A is invertible then the eigen values must be distinct (C) If A is invertible then the eigen a non-zero real eigen value for A. (B) If A is invertible then there is always complex eigen value (A) A is invertible if A has a non real, of the following is true ? 33. Let A be a 2 × 2 real matrix. Then which composition of endomorphisms EndF(V) always form a group under (D) Non-zero endomorphisms of F (C) End (V) is always a F-vector space endomorphisms addition and composition of commutative ring under usual F (B) End (V) can never be a structure F (A) End (V) has no F-vector space following statements is true ? F-endomorphisms of V. Which one of the 32. EndF (V) denotes the set of all *K2614* Paper III 7 0 0 1 (D) 0 1 0 1 1 0 0 0 0 (C) 0 − 1 0 1 0 0 0 2 0 (B) 1 − 1 0 1 − 1 0 0 2 0 (A) 1 0 0 0 0 0 respect to the basis {1 + x, x, x2} is dx . Then the matrix of T with T( f ) = f ′ = df field F. Define T : F 2[x] F 2 [x] by polynomials of degree atmost 2 over a 36. Let F 2 [x] be the vector space of all 0 (D) 1 0 1 1 and 1 2 1 0 (C) 1 0 1 1 and 1 0 1 1 0 0 0 0 0 (B) 0 0 1 and 0 0 1 0 1 0 0 1 0 1 (A) 1 0 0 1 and 3 0 1 over IR are similar ? 35. Which of the following pairs of matrices Total Number of Pages : 16 Paper III K-2614 8 y d z c (D) and a x b w (D) 2πi e4 y d x d (C) and a x a y (B) 2πi (e2 − e4 ) z c y a and (B) b w d x y b z d and (A) a x c w matrices can never be similar ? numbers, which pair of the following and w, x, y and z represent any real 39. If a < b < c < d are fixed real numbers (D) 4 (C) 2πi e2 (A) 2πi (e4 − e2 ) |z| = 5, positively oriented, is C ∫ (z − 1) (z − 2) dz where C is the circle e2z 42. The value of the integral (C) (B) (C) 3 (B) 2 (A) 1 The dimension of V over IR is and d = a + b} ( ) 2 2 cos (α ) 2 2 sin (α ) 2 2 sin (α ) (D) 2 cos α 2 (A) then | z| is 41. Suppose α is real and z = 1 – cosα + i sin α , 38. Let V = {(a, b, c, d) | a, b, c, d ∈ IR , a = c (D) e + z (D) have absolute value 1 z (C) have multiplicity 2 (C) zeiz (B) are real (B) e – iz (A) are purely imaginary (A) e + iz values of A i − 3 − 4i 5 37. If A = 0 2 − 3 + 4i then the eigen 1 0 −i *K2614* z z f(z) can be for all z ∈ " and u(x, y) = y + ex cosy. Then 40. Given that f(z) = u(x, y) + i ν (x, y) is analytic Total Number of Pages : 16 K-2614 Paper III 9 (D) Standard exponential (C) Degenerate at 1 (B) Degenerate at 0 (A) Uniform over [0, 1] Y = f(U), then what is the distribution of Y ? uniform random variable over [0, 1], U and 46. If f is the probability density function of (D) an essential singularity at 1 (C) a pole at 1 of order 2 (B) a pole at 0 of order 2 Then 2 . f(z) has 1 45. Consider the complex valued function 2 (D) z = nπ + (−1)n π , n ∈ 9 2 (C) z = nπ + π , n ∈ 9 ( ) 2 z = nπ + ( −1)n (π − 4i), n ∈ 9 (B) z = (−1) ⋅ nπ + π 2 − 4i , n ∈ 9 n (A) X+Y+Z is Cauchy X distribution (C) X +Y +Z has chi-square 2 2 2 (B) ex + ey + ez is exponential (A) X given Y and Z is normal correct. distribution, which one of the following is 49. If (X, Y, Z) has tri-variate normal (D) (A) a removable singularity at 1 f(z) = (z – 1)2 ⋅ e ( z −1) normal random variables (D) X + Y and X – Y are independent distribution (C) (X + Y) (X – Y) has chi-square distributed (B) X + Y and X – Y are identically random variables (A) X + Y and X – Y are dependent following is true ? random variables, which one of the 50. If X and Y are independent normal z ∈ " are 44. The roots of the equation sinz = cosh4, (D) Beta distribution (C) Uniform [0, 1] distribution (B) Degenerate distribution at 1 (A) Degenerate distribution at 0 distribution over [0, 1] ? converge to, U having uniform 48. What does the distribution function of U (D) 0 (C) 4πi 2 (D) Binomial l, 1 (B) 1 (A) 2πi 4 (C) Binomial l, 1 2 (B) Geometric 1 C ∫ z −1 dz is f (z ) anticlockwise. Then the value of radius 2 with center as (1, 0), oriented 43. Let f(z) = z3 – 1 and C denote the circle of *K2614* n 1 (A) Poisson (2) of X given X + Y = l ? random variables, what is the distribution 47. If X and Y are independent Poisson (2) Total Number of Pages : 16 Paper III K-2614 10 moments finite (D) Student’s t-distribution has all (D) 5 1 distribution generalization of Cauchy (C) Student’s t-distribution is a (C) 5 2 (B) 5 3 (B) Student’s t-distribution is symmetric sampling distribution (A) Student’s t-distribution is a 53. Which of the following is not true ? (A) 1 →∞ 11 what is nlim p( n) ? states 1, 2, given that p12 = (D) MLE of θ does not exist (C) Sample range is the MLE of θ 56. With reference to a Markov chain with (D) i is non-null (B) Sample mean is the MLE of θ (C) i is recurrent null (A) Sample median is the MLE of θ (B) i is aperiodic the following is correct ? f(x; θ) = 2 e , . Which of 1 −|x −θ| x ∈ IR , θ ∈ IR the probability density function 52. Let {X1, ..., Xn} be a random sample from 6 3 , p 21 = , 1 2 (A) i is ergodic sufficient condition for existence of ai ? in n-steps. Which of the following is a probability of going to state i from state i n→∞ ii ai = lim p(nii) where p(n) is the transition 55. Let i be a state of a Markov chain and 3 (D) 2F + G 4 (C) F+G (B) FG 2 (A) F+G function ? of the following is not a distribution 51. If F and G are distribution functions, which *K2614* (D) The errors have zero variances (C) The errors have zero expectations (B) The error variances are different (A) The error variances are same ‘heteroscedastic’ mean ? 54. In a linear model, what does Total Number of Pages : 16 K-2614 (D) 6 1 (D) e , t ∈ IR t2 (C) e , t ∈ IR 3 (C) 2 (B) –t 2 (B) e–|t|, t ∈ IR (A) e , t ∈ IR 5 2 –t n X1 + ... + X n 5 (A) 1 is idle ? steady state probability that the system service rate 2 and no waiting, what is the 59. In an M/M/1 queue with arrival rate 3, 4 (D) B , t ≥ 0 t (C) {| B(t) |, t ≥ 0} (B) {eB( t ) , t ≥ 0} (A) {B(t + 2) − B(t ), t ≥ 0} is a Brownian motion ? Brownian motion, which of the following 58. If {B(t ), t ≥ 0} denotes a standard (D) n =1 ∑ ∞ (C) n =1 ∑ ∞ converges to 0 p(kk n) is divergent p(kk n) n→∞ (B) lim p kk = 0 (n) n=1 (A) Paper III 11 ∑ ∞ kk p(n) converges following ? Markov chain is which one of the 57. A criterion for state k to be recurrent in a *K2614* as n → ∞ ? of the characteristic function of 1 − | t | if | t | ≤ 1; what is the limit φ(t ) = 0 if | t | > 1, characteristic function 62. If X1, X2, ... , Xn are independent with (D) Tn is never unbiased for θ regularity conditions (C) Tn is consistent for θ under Cramer’s (B) Tn is never consistent for θ (A) Tn is always unbiased for θ is true ? distribution F(⋅ ; θ) , which of the following 61. If Tn is the MLE of a parameter θ in a →∞ (D) tlim t µ = a.s. N(t ) 1 t→∞ t = µ a.s N(t ) t→∞ t = 0 a.s. N(t ) t→∞ t = 1 a.s. N(t ) (C) lim (B) lim (A) lim µ . Which of the following is true ? mean finite and mean inter-arrival time 60. Let {N(t ), t ≥ 0} be a renewal process with Total Number of Pages : 16 Paper III (D) K-2614 12 ( ) EY X − Cov. , Y Y ( ) EX (C) X − Cov. , Y Y (D) 64e2 (C) 64e2 (B) 64e2 (A) 32e2 ( ) EY (B) X − Cov. , X Y 27 81 11 27 P (X + Y = 1) if X and Y are independent ? defined for appropriate values of t, what is ( ) EX X (A) − Cov. , X Y Mx (t ) 4 = 3 + et in a simple random sampling ? 65. What is the bias of the ratio estimator x y (D) θ1 = θ2 − θ3 (C) θ3 = θ1 + θ2 (B) θ2 = θ1 + θ3 (A) θ1 = θ2 + θ3 a′θ = a1θ1 + a 2θ2 + a 3θ3 estimable when is the linear parametric function 0 1 1 , A = 1 0 1 1 1 2 64. Given the linear model (Y, Aθ, σ 2 I 3 ) with (C) 0.6 (D) – 0.3 (A) – 0.4 (B) 0.3 between 1 + X and 1 – 2Y ? Y is 0.3, what is the correlation coefficient 63. If correlation coefficient between X and *K2614* and My(t) = e2(e –1) t 3 68. Given the moment generating functions (D) (C) (B) (A) 5 10 0 –5 x ≥ 0, y ≥ 0 minimize x + y subject to x – y = – 5, at an optimal solution of the LPP. 67. Find the value of the objective function contrast (D) a′α is an elementary treatment of the design matrix (C) Vector a belongs to the row space space of the C-matrix of the design (B) Vector a belongs to the column space of the design matrix (A) Vector a belongs to the column effects alone, in a general block design ? parametric function a′α of treatment condition for estimability of a linear 66. What is a necessary and sufficient Total Number of Pages : 16 K-2614 Paper III 13 (D) P [| X | ≤ ∈] ≥ free n (D) None of D , Dn+ , Dn− are distribution not D (C) P [| X | ≤ ∈] ≤ n (B) P [| X | ≥ ∈] ≤ (C) Dn+ and Dn− are distribution free but n (B) D is distribution free but not Dn+ and Dn− n (A) D , Dn+ and Dn− are distribution free following is true ? x D −n = sup (F(x ) − Fn (x )) , which of the D +n = x sup ( Fn ( x ) − F( x ) ), x sup | Fn (x ) − F(x) |, = distribution function. Let Dn Fn be the corresponding empirical a continuous distribution function F and 70. Let {X1, ..., Xn} be a random sample from (D) 0 (C) 3 1 (B) 2 1 (A) 1 often) equal to ? n = 1, 2, ... , what is P({Xn = 0} infinitely 2n = 1 − P( Xn = 1) , 69. Given that P( Xn = 0) = 1 *K2614* (A) P [| X | ≥ ∈] ≤ ∈k E | X |k ∈k E | X |k E | X |k ∈k ∈k E | X |k Then the Markov inequality states that 73. Let X be a random variable with E |X|k < ∞ . components (D) Median survival time of its components (C) Mean survival time of its components (B) Maximum of the survival times of its components (A) Minimum of the survival times of its system survival time is 72. In a parallel system of k components, the (D) θ σ 2n (C) σ 2n ⋅θ (B) σ 2n ⋅θ2 (A) θ2 σ 2n variance σ2n then log Tn is CAN for log θ with 71. If Tn is a CAN estimator of θ with variance Total Number of Pages : 16 Paper III K-2614 14 ___________ (D) there is multicollinearity in the model (D) s p2 = n −1 p2q (C) s p = 2 n −1 npq (B) sp = 2 n pq (A) s p = 2 n −1 pq are linearly independent (C) the columns of the regression matrix vector is singular (B) the dispersion matrix of the error (A) the errors are autocorrelated estimator is proposed when linear regression model, a ridge 74. While estimating the parameters of a *K2614* variance of proportion p is proportion q of category II then the has proportion p of items of category I and dichotomous population. If the sample 75. A sample of size n is drawn from a Total Number of Pages : 16 K-2614 15 Paper III Space for Rough Work bñÜᤠÃÜÖÜPÝRX ÓܧÙÜ *K2614* Total Number of Pages : 16 Paper III 16 K-2614 Space for Rough Work bñÜᤠÃÜÖÜPÝRX ÓܧÙÜ *K2614* Total Number of Pages : 16