MATHEMATICAL SCIENCES

Transcription

1 MATHEMATICAL SCIENCES This Test Booklet will cotai 0 (0 Part `A +40 Part `B+60 Part C ) Multiple Choice Questios (MCQs) Both i Hidi ad Eglish. Cadidates are required to aswer 5 i part A, 5 i Part B ad 0 questios i Part C respectively (No. of questios to attempt may vary from exam to exam). I case ay cadidate aswers more tha 5, 5 ad 0 questios i Part A, B ad C respectively oly first 5, 5 ad 0 questios i Parts A, B ad C respectively will be evaluated. Each questios i Parts `A carries two marks, Part B three marks ad Part C 4.75 marks respectively. There will be egative marks i Part A ad 0.75 i part B for each wrog aswers. Below each questio i Part A ad Part B, four alteratives or resposes are give. Oly oe of these alteratives is the CORRECT aswer to the questio. Part C shall have oe or more correct optios. Credit i a questio shall be give oly o idetificatio of ALL the correct optios i Part C. No credit shall be allowed i a questio if ay icorrect optio is marked as correct aswer. No partial credit is allowed. MODEL QUESTION PAPER PART A May be viewed uder headig Geeral Sciece PART B. The sequece a =... ( ) ( ) coverges to 0 coverges to / 3 coverges to /4 4 does ot coverge.. Let x = / ad y = (!) /, be two sequeces of real umbers. The (x ) coverges, but (y ) does ot coverge (y ) coverges, but (x ) does ot coverge

2 3 both (x ) ad (y ) coverge 4 Neither (x ) or (y ) coverges 3. The set { x : x si x, x cos x } is a bouded closed set a bouded ope set 3 a ubouded closed set. 4 a ubouded ope set. 4. Let f:[0,] be cotiuous such that f(t) 0 for all t i [0, ]. Defie x g(x) = f () t dt the 0 g is mootoe ad bouded g is mootoe, but ot bouded 3 g is bouded, but ot mootoe 4 g is either mootoe or bouded 5. Let f be a cotiuous fuctio o [0, ] with f(0) =. Let G(a) = a a 0 f ( x) dx 3 lim Ga ( ) a0 lim Ga ( ) a0 lim Ga ( ) 0 a0 4 The limit lim Ga ( ) a0 dose ot exist 6. Let = si ( ), =,,. The coverges lim sup lim if 3 lim 4 diverges

3 7. If, for x, φ(x) deotes the iteger closest to x (if there are two such itegers take the larger oe), the ( x ) dx equals 0 8. Let P be a polyomial of degree k > 0 with a o-zero costat term. Let f (x) = P( x ) x (0, ) lim f ( x) x (0, ) x (0, ) such that lim f ( x) > P(0) 3 lim f ( x) 4 lim f ( x) =0 x (0, ) = P(0) x (0, ) 9. Let C [0, ] deote the space of all cotiuous fuctios with supremum orm. ì æ ö ü The, K = í f Î [0,]: lim f ç = 0ý î è ø þ is a. vector space but ot closed i C[0,].. closed but does ot form a vector space. 3. a closed vector space but ot a algebra. 4. a closed algebra. 30. Let u, v, w be three poits i 3 ot lyig i ay plae cotaiig the origi. The u + v + 3 w = 0 => = = 3 = 0 u, v, w are mutually orthogoal 3 oe of u, v, w has to be zero 4 u, v, w caot be pairwise orthogoal

4 3. Let x, y be liearly idepedet vectors i suppose T: is a liear trasformatio such that Ty = x ad Tx =0 The with respect to some basis i, T is of the form a 0, a > 0 0 a a 0, 0 b a, b > 0; a b Suppose A is a x real symmetric matrix with eigevalues,,..., the 3 i i i det( A) i det( A) i det( A) i 4 if det( A) the for j =,. 33. Let f be aalytic o D = { z : z < } ad f(0) =0. Defie f() z ; z 0 gz () z f(0); z 0 The g is discotiuous at z = 0 for all f g is cotiuous, but ot aalytic at z = 0 for all f 3 g is aalytic at z = 0 for all f 4 g is aalytic at z = 0 oly if f ' (0) = 0 j

5 34. Let be a domai ad let f(z) be a aalytic fuctio o such that f(z) = si z for all z the f(z) = si z for all z f(z) = si z for all z. 3 there is a costat c with c = such that f(z) = c si z for all z 4 such a fuctio f(z) does ot exist 35. The radius of covergece of the power series (4 3) z is Let be a fiite field such that for every a the equatio x =a has a solutio i. The the characteristic of must be must have a square umber of elemets 3 the order of is a power of 3 4 must be a field with prime umber of elemets 37. Let be a field with 5 elemets. What is the total umber of proper subfields of?

6 38. Let K be a extesio of the field Q of ratioal umbers If K is a fiite extesio the it is a algebraic extesio If K is a algebraic extesio the it must be a fiite extesio 3 If K is a algebraic extesio the it must be a ifiite extesio 4 If K is a fiite extesio the it eed ot be a algebraic extesio 39. Cosider the group S 9 of all the permutatios o a set with 9 elemets. What is the largest order of a permutatio i S 9? Suppose V is a real vector space of dimesio 3. The the umber of pairs of liearly idepedet vectors i V is oe ifiity 3 e Cosider the differetial equatio dy y,( x, y) dx. The,. all solutios of the differetial equatio are defied o (,).. o solutio of the differetial equatio is defied o (,). 3. the solutio of the differetial equatio satisfyig the iitial coditio y(x 0 ) = y 0, y 0 > 0, is defied o, x0. y0 4. the solutio of the differetial equatio satisfyig the iitial coditio æ ö y(x 0 )=y 0, y 0 >0, is defied o ç x 0 -, è y ø. 4. The secod order partial differetial equatio 0 xy xy u u u 0 x xy y is. hyperbolic i the secod ad the fourth quadrats. elliptic i the first ad the third quadrats

7 3. hyperbolic i the secod ad elliptic i the fourth quadrat 4. hyperbolic i the first ad the third quadrats 43. A geeral solutio of the equatio x. u( x, y) e f ( y) x. u( x, y) e f ( y) xe x 3. u( x, y) e f ( y) xe x 4. u( x, y) e f ( y) xe x x x u( x, y) + u( x, y) = e x - x is 44. Cosider the applicatio of Trapezoidal ad Simpso s rules to the followig itegral 4 3 (x 3x 5x ) dx 0. Both Trapezoidal ad Simpso s rules will give results with same accuracy.. The Simpso s rule will give more accuracy tha the Trapezoidal rule but less accurate tha the exact result. 3. The Simpso s rule will give the exact result. 4. Both Trapezoidal rule ad Simpso s rule will give the exact results. 45. The itegral equatio 46. Let g(x)y(x)=f(x)+λ k(x,t)y(t)dt β α with f(x), g(x) ad k(x,t) as kow fuctios, α ad β as kow costats, ad λ as a kow parameter, is a. liear itegral equatio of Volterra type. liear itegral equatio of Fredholm type 3. oliear itegral equatio of Volterra type 4. oliear itegral equatio of Fredholm type b y(x) = f(x)+λ k(x,t)y(t)dt, where f(x) ad k(x,t) are kow fuctios, a a ad b are kow costats ad λ is a kow parameter. If λ i be the eigevalues of the correspodig homogeeous equatio, the the above itegral equatio has i geeral,. may solutios for λ λ i. o solutio for λ λ i 3. a uique solutio for λ=λ i 4. either may solutios or o solutio at all for λ=λ i, depedig o the form of f(x) 47. The equatio of motio of a particle i the x-z plae is give by dv v kˆ dt

8 with v kˆ, where = (t) ad ˆk is the uit vector alog the z-directio. If iitially (i.e., t = 0) =, the the magitude of velocity at t = is. /e. (+e)/3 3. (e )/e Cosider the fuctioal / du dv F( u, v) u( x) v( x) dx dx dx 0 with u(0), v(0) ad u 0, v 0. The, the extremals satisfy. u( ), v( ). u( ) v( ) 0, u( ) v( ) 3. u( p) = -, v( p) = 4. u( ) v( ), u( ) v( ) The pairs of observatios o two radom variables X ad Y are X : Y : The the correlatio coefficiet betwee X ad Y is 0 /5 3 / Let X, X, X 3 be idepedet radom variables with P(X i = +) = P(X i = -) = /. Let Y = X X 3, Y = X X 3 ad Y 3 = X X. The which of the followig is NOT true?. Y i ad X i have same distributio for i =,, 3. (Y, Y, Y 3 ) are mutually idepedet 3. X ad (Y, Y 3 ) are idepedet 4. (X, X ) ad (Y, Y ) have the same distributio

9 5. Let X be a expoetial radom variable with parameter. Let Y = [X] where [x] deotes the largest iteger smaller tha x. The. Y has a Geometric distributio with parameter.. Y has a Geometric distributio with parameter - e l. 3. Y has a Poisso distributio with parameter 4. Y has mea [/ ] 5. Cosider a fiite state space Markov chai with trasitio probability matrix P=((p ij )). Suppose p ii =0 for all states i. The the Markov chai is. always irreducible with period.. may be reducible ad may have period >. 3. may be reducible but period is always. 4. always irreducible but may have period >. 53. Let X, X,. X be i.i.d. Normal radom variables with mea ad variace. ad let Z = (X +X +. +X )/ The. Z coverges i probability to. Z coverges i probability to 3. Z coverges i distributio to stadard ormal distributio 4. Z coverges i probability to Chi-square distributio. 54. Let X, X,. X be a radom sample of size ( 4) from uiform (0,) distributio. Which of the followig is NOT a acillary statistic?. X X ( ) (). X X X X X X X( ) X()

10 55. Suppose X, X, X are i.i.d, Uiform (0, q ), {,...}. The the MLE of is. X (). X 3. [X () ] where [a] is the iteger part of a. 4. [X () +] where [a] is the iteger part of a. 56. Let X, X,., X be idepedet ad idetically distributed radom variables with commo cotiuous distributio fuctio F(x). Let R i = Rak(X i ), i=,,,. The P ( R R ³ -) is. 0. ( ) 3. ( ) A simple radom sample of size is draw without replacemet from a populatio of size N (> ). If i (i=,, N) ad p ij (i j. i, j =,, N) deote respectively, the first ad secod order iclusio probabilities, the which of the followig statemets is NOT true? N å i= p i = N å p = ( - ) p j¹ i i j 3 p ip j p i j 4 p i j p i i for each pair ( i, j) < for each pair ( i, j). 58. Cosider a balaced icomplete block desig with usual parameters v, b, r, k ( ),. Let t i be the effect of the i th treatmet (i =,,,v) ad deote the variace of a observatio. The the variace of the best liear

11 ubiased estimator of v pt i i where i v pi 0 ad i v pi, uder the i itra-block model, is v k r k v 3 k v A aircraft has four egies two o the left side ad two o the right side. The aircraft fuctios oly if at least oe egie o each side fuctios. If the failures of egies are idepedet, ad the probability of ay egie failig i equal to p, the the reliability of the aircraft is equal to. p ( p ) C p ( - p ) 4 ( - p ) ( p ) 60. A compay maitais EOQ model for oe of its critical compoets. The setup cost is k, uit productio cost is c, demad is a uits per uit time, ad h is the cost of holdig oe uit per uit time. I view of the criticality of the compoet the compay maitais a safety stock of s uits at all times. The ecoomic order quatity for this problem is give by.. ak s h +. s + ak h 3. ak h 4. ak + s h

12 PART C 6. Suppose {a }, {b } are coverget sequeces of real umbers such that a > 0 ad b > 0 for all. Suppose lima a ad limb b. Let c = a /b. The. {c } coverges if b > 0. {c } coverges oly if a = 0 3. {c } coverges oly if b > 0 4. lim sup c = if b = Cosider the power series 0 ax where a 0 = 0 ad a = si(!)/! for. Let R be the radius of covergece of the power series. The. R. R π 3. R 4 4. R. 63. Suppose f is a icreasig real-valued fuctio o [0, ) with f(x)> 0 x ad let x g( x) f ( u) du; 0 x. x 0 The which of the followig are true:. g(x) f(x) for all x(0,). xg(x) f(x) for all x(0,) 3. xg(x) f(0) for all x(0,) 4. yg(y) xg(x) (y-x)f(y) for all x < y. 64. Let f: [0, ] be defied by xcos( /( x)) if x0, f( x) 0 if x0. The. f is cotiuous o [0, ]. f is of bouded variatio o [0, ] 3. f is differetiable o the ope iterval (0, ) ad its derivative f is bouded o (0,) 4. f is Riema itegrable o [0, ]. 65. For ay positive iteger, let f : [0, ] be defied by

13 x f( x) x for x[0,]. The. the sequece {f } coverges uiformly o [0, ]. the sequece { f } of derivatives of {f } coverges uiformly o [0, ] 3. the sequece 4. the sequece 0 0 ( ) f x dxis coverget ( ) f x dxis coverget. 66. Let f: [0, ) ad g : [0, ) be cotiuous fuctios satisfyig f ( x ) ( ) 3 x x t dt x ( x) ad g( t) dt x for all x[0, ). 0 0 The f() + g() is equal to Cosider f: defied by f(0, 0) = 0 ad x y f(x, y) = for (x, y) (0, 0). 4 x y The which of the followig statemets is correct?. Both the partial derivatives of f at (0, 0) exist. The directioal derivative D u f(0, 0) of f at (0, 0) exists for every uit vector u 3. f is cotiuous at (0, 0) 4. f is differetiable at (0, 0). 68. Let f: ad g: be defied by f(x, y) = x + y ad g(x, y) = xy. The. f is differetiable at (0, 0), but g is ot differetiable at (0, 0). g is differetiable at (0, 0), but f is ot differetiable at (0,0) 3. Both f ad g are differetiable at (0, 0) 4. Both f ad g are cotiuous at (0, 0). 69. Decide for which of the fuctios F : 3 3 give below, there exists a fuctio f : 3 such that ( f )( x) F( x).

14 . (4xyz z 3y, x z 6xy+, x y xz ). (x, xy, xyz) 3. (,,) 4. (xyz, yz, z). 70. Let f: be the fuctio defied by the rule f(x) = x.b, where b ad x.b deotes the usual ier product. The. [ f (x)] (b) = bb. xx. [ f (x)] (x) =, x 3. [ f (0)](e ) = b.e, where e = (, 0,,0). 4. [ f (e )](e j ) = 0, j, where e j = (0,,, 0) with i the j th slot. 7. Cosider the subsets A ad B of defied by Ax, xsi : x(0,] ad B A{(0,0)}. x The. A is compact. A is coected 3. B is compact 4. B is coected. 7. Lef f = be a cotiuous fuctio.which of the followig is always true?. f - (U) is ope for all ope sets U. f - (C) is closed for all closed sets C 3. f - (K) is compact for all compact sets K 4. f - (G) is coected for all coected sets G. 73. Let A be a matrix,, with characteristic polyomial x - (x ). The. A = A. Rak of A is 3. Rak of A is at least 4. There exist ozero vectors x ad y such that A(x + y) = x y. 74. Let A, B ad C be real matrices such that AB +B = C. Suppose C is osigular. Which of the followig is always true?. A is osigular. B is osigular 3. A ad B are both osigular 4. A + B is osigular.

15 75. Let V be a real vector space ad let {x, x, x 3 } be a basis for V. The. {x + x, x, x 3 } is a basis for V. The dimesio of V is 3 3. x, x, x 3 are pairwise orthogoal 4. {x x, x x 3, x x 3 } is a basis for V. 76. Cosider the system of m liear equatios i ukows give by Ax = b, where A= (a ij ) is a real m matrix, x ad b are colum vectors. The. There is at least oe solutio. There is at least oe solutio if b is the zero vector 3. If m = ad if the rak of A is, the there is a uique solutio 4. If m < ad if the rak of the augmeted matrix [A: b] equals the rak of A, the there are ifiitely may solutios. 77. Let V be the set of all real matrices A = (a ij ) with the property that a ij = a ji for all i, j =,,,. The. V is a vector space of dimesio. For every A i V, a ii = 0 for all i =,,, 3. V cosists of oly diagoal matrices 4. V is a vector space of dimesio. 78. Let W be the set of all 3 3 real matrices A = (a ij ) with the property that a ij = 0 if i > j ad a ii = for all i. Let B = (b ij ) be a 3 3 real matrix that satisfies AB = BA for all A i W. The. Every A i W has a iverse which is i W.. b = 0 3. b 3 = 0 4. b 3 = Let f(z) be a etire fuctio with Re(f(z) ) 0 for all z. The. Im (f(z)) 0 for all z. Im (f(z)) = a costat 3. f is a costat fuctio 4. Re(f(z)) = z for all z. 80. Let f be a aalytic fuctio defied o D = {z z < } such that f(z) for all z D. The. there exists z 0 D such that f(z 0 ) =. the image of f is a ope set 3. f(0) = 0

16 4. f is ecessarily a costat fuctio. si z cos z 8. Let f() z z z. The. f has a pole of order at z = 0. f has a simple pole at z = 0 3. f ( z) dz 0, where the itegral is take ati-clockwise z 4. the residue of f at z = 0 is -i. 8. Let f be a aalytic fuctio defied o D = { z: z <}.The g : D is aalytic if. g(z) = f( z ) for all zd. g(z) = f( z ) for all zd 3. g(z) = f() z for all zd 4. g(z) = i f ( z) for all zd. 83. Which of the followig statemets ivolvig Euler's fuctio is/are true?. () is eve as may times as it is odd. () is odd for oly two values of 3. () is eve whe > 4. () is odd whe = or is odd. 84. Let p be a prime umber ad d ( p ). The which of the followig statemets about the cogruece x d (mod p) is/are true?. It does ot have ay solutio. It has at most d icogruet solutios 3. It has exactly d icogruet solutios 4. It has at least d icogruet solutios. 85. Let K be a field, L a fiite extesio of K ad M a fiite extesio of L. The. [M:K] = [M:L] + [L:K]. [M:K] = [M:L] [L:K] 3. [M:L] divides [M:K] 4. [L:K] divides [M:K]. 86. Let R be a commutative rig ad R[x] be the polyomial rig i oe variable over R.. If R is a U.F.D., the R [x] is a U.F.D.. If R is a P.I.D., the R [x] is a P.I.D. 3. If R is a Euclidea domai, the R[x] is a Euclidea domai 4. If R is a field, the R[x] is a Euclidea domai.

17 87. Let G be a group of order 56. The. All 7-sylow subgroups of G are ormal. All -Sylow Subgroups of G are ormal 3. Either a 7-Sylow subgroup or a -Sylow subgroup of G is ormal 4. There is a proper ormal subgroup of G. 88. Which of the followig statemets is/are true?. 50! eds with a eve umber of zeros. 50! eds with a prime umber of zeros 3. 50! eds with0 zeros 4. 50! eds with zeros. 89. Let X = {(x, y) x + y = } Y = {(x, y) x + y = }, ad Z = {(x, y) x y = }. The. X is ot homeomorphic to Y. Y is ot homeomorphic to Z 3. X is ot homeomorphic to Z 4. No two of X, Y or Z are homeomorphic. 90. Let, ad 3 be topologies o a set X such that is a compact Hausdorff space. The. = if (X, ) is a Hausdorff space. = if (X, ) is a compact space 3. = 3 if (X, 3 ) is a Hausdorff space 4. = 3 if (X, 3 ) is a compact space. x t 3 x, x(0) 0 ; 9. The iitial value problem /3 i a iterval aroud t = 0, has. o solutio. a uique solutio 3. fiitely may liearly idepedet solutios 4. ifiitely may liearly idepedet solutios. 9. For the system of ordiary differetial equatios: ad (X, 3 )

18 d dt x( t) 0 x( t) x ( t) 0 x ( t),. every solutio is bouded. every solutio is periodic 3. there exists a bouded solutio 4. there exists a o periodic solutio. 93. The kerel p x, y y y is a solutio of x. the heat equatio. the wave equatio 3. the Laplace equatio 4. the Lagrage equatio. 94. The solutio of the Laplace equatio o the upper half plae, which takes the x value x e o the real lie is. the real part of a aalytic fuctio. the imagiary part of a aalytic fuctio 3. the absolute value of a aalytic fuctio 4. a ifiitely differetiable fuctio. 95. Which of the followig polyomials iterpolate the data x / 3 y (x ) 53 5 (x ) (x ). 3 (x ) (x ) 0(x ) (x 3) + 0 (x 3) (x ) 3. 3(x ) (x 3) 8 (x ) (x 3)+ 5 (x )(x ) 4. (x 3) (x +0) + (x +0) (x ) +3 (x ) (x 3).

19 96. The evaluatio of the quatity x ear x = 0 is achieved with miimum loss of sigificat digits if we use the expressio. x x x x x x x x. 97. If x(t) is a extremal of the fuctioal b. ( ) m x cx dt, where a, b,c are a arbitrary costats ad satisfies ẋ = dx/dt, the the fuctio x(t). mx cx 0. mx cx m( x) cx k with k as a arbitrary costat c x( t) k si( t k ) m 4. with k ad k as arbitrary costats. 98. If u(x) ad v(x) satisfyig u(0) =, v(0) =, u (/) = 0 ad v((/)=0 are du dv the extremals of the fuctioal u( ) v( ) uv dx, dx dx the u( ) v( ) u( ) v( ) u( ) v( ) Cosider the itegral equatio y( x) x xty( t) dt,

20 where is a real parameter. The the Neuma series for the itegral equatio coverges for all values of. except for =3. lyig i the iterval -3 < < 0 3. lyig i the iterval -3 < < 3 4. lyig i the iterval 0 < < 3. 5x 00. The solutio of the itegral equatio ( x) xt( t) dt 6 satisfies. (0) + ()=. + 3 = = =. 0. A particle of uit mass is costraied to move o the plae curve xy= uder gravity g. The 0. the kietic eergy of the system is ( ) x y. the potetial eergy of the system is g x 3. the Lagragia of the particle is x x 4 g x 4. the Lagragia of the particle is x x 4 g. x 0. Suppose a mechaical system has the sigle coordiate q ad Lagragia q L q. The 4 9. the Hamiltoia is p q +( ) 9

21 . Hamilto s equatios are q p, p (/9) q 3. q satisfies q (4/ 9) q 0 4. the path i the Hamiltoia phase-space, i.e. q pplae is a ellipse. 03. Let X,., X be i.i.d. observatios from a distributio with variace. Which of the followig is/are ubiased estimator(s) of?. i Xi i X i X. X 3. Xi X j i j 4. Xi X. i 04. Let X,X, be i.i.d. N(0,) ad let S = Which of the followig is/are true? Xi be the partial sums. i S. 0 almost surely S. E 0 S 3. Var 0 4. Var S Let (X, Y) be a pair of idepedet radom variables with X havig expoetial distributio with mea ad Y havig uiform distributio o {,,,m}. Defie Z=X+Y. The m. E(Z X) = X +

22 m. E(Z Y) = + 3. Var (Z X) = m 4. Var (Z Y) =. 06. A simple symmetric radom walk o the iteger lie is a Markov chai which is. recurret. ull recurret 3. irreducible 4. positive recurret. 07. Suppose X ad Y are radom variables with E(X) = E(Y) =0, V(X)=V(Y)= ad Cov (X,Y) = 0.5. The which of the followig is/are always true?. P { X+Y 4} 4 6. P { X+Y 4} P { X+Y 4} P { X+Y 4} Let X,., X be a radom sample from uiform, distributio. Which of the followig is/are maximum likelihood estimator(s) of?. X (). X () 3. X () - 4. X X ( ) () Let X (X,,X ) be a radom sample from uiform

23 (0, ). Which of the followig is/are uiformly most powerful size 0 test(s) for testig H 0 : = o agaist H : > o?. ( X ), if X () > 0 or X () < o / = 0, otherwise. ( X ), if X () > 0 =, if X () ( X ), if X () > 0 / = 0, if X () 0 / 4. 4 ( X ), if X () < 0 ( / ) = 0, otherwise / / or X () > ( / ) T X A X is chi- 0. Suppose X p has a N p ( OI, p ) distributio. The distributio of square with r degrees of freedom oly if. A is idempotet with rak r. Trace (A) = Rak (A) = r 3. A is positive defiite 4. A is o-egative defiite with rak r. 0. Let X, X,, X m be iid radom variables with commo cotiuous cdf F(x). Also let Y, Y,..,Y be iid radom variables with commo cotiuous cdf G(x) ad X s & Y s are idepedetly distributed. For testig H o : F(x) = G(x) for all x agaist H : F(x) G(x) for at least oe x, which of the followig test is/are used?. Wilcoxo siged rak test. Kolmogorov-Smirov test 3. Wald-Wolfowitz ru test 4. Sig test.. Radom variables X ad Y are such that E(X)=E(Y) = 0, V(X)=V(Y)=, correlatio (X,Y) = 0.5. The the. coditioal distributio Y give X = x is ormal with mea 0.5x ad variace 0.75

24 . least-squares liear regressio of Y o X is y=0.5x ad of X o Y is x=y 3. least-squares liear regressio of X o Y is x = 0.5y ad of Y o X is y = x. 4. least-squares liear regressio of Y o X is y=0.5x ad of X o Y is x = 0.5y. 3. X has a biomial (5,p) distributio o which a observatio x=4 has bee made. I a Bayesia approach to the estimatio of p, a beta (,3) prior distributio (with desity proportioal to p(-p) ) has bee formulated. The the posterior. distributio of p is uiform o (0.). mea of p is distributio of p is beta (6,4) 4. distributio of p is biomial (0,0.5). 4. I a study of voter prefereces i a electio, the followig data were obtaied Geder Party votig for Total C B Male Female Total The the. chi-square statistic for testig o associatio betwee party ad geder is 0.. expected frequecy uder the hypothesis of o associatio is 50 i each cell. 3. log-liear model for cell frequecy m ij, log(m ij ) = costat, i,j=,, fits perfectly to the data. 4. chi-square test of o geder-party associatio with degree of freedom has a p-value of.

25 5. Let X,Y ad N be idepedet radom variables with P(X=0) = ½=-P(X=) ad Y followig Poisso with parameter 0 ad N followig ormal with mea 0 ad variace. Defie Z Y if X 0 N if X The, the characteristic fuctio of Z is give by it e t it e e e e e it e t e / it e t e / / it e t / it e e e. 6. A simple radom sample of size is draw from a fiite populatio of N uits, with replacemet. The probability that the i th ( i N) uit is icluded i the sample is. /N N N N ( ) N( N ). 7. Uder a balaced icomplete block desig with usual parameters v, b, r, k,, which of the followig is/are true?. All treatmet cotrasts are estimable if k. The variace of the best liear ubiased estimator of ay ormalized treatmet cotrast is a costat depedig oly o the desig parameters ad the per observatio variace 3. The covariace betwee the best liear ubiased estimators of two mutually orthogoal treatmet cotrasts is strictly positive 4. The variace of the best liear ubiased estimator of a elemetary treatmet cotrast is strictly smaller tha that uder a radomized block desig with replicatio r.

26 8. Cosider a radomized (complete) block desig with v (>) treatmets ad r () replicates. Which of the followig statemets is/are true?. The desig is coected. The variace of the best liear ubiased estimator (BLUE) of every ormalized treatmet cotrast is the same 3. The BLUE of ay treatmet cotrast is ucorrelated with the BLUE of ay cotrast amog replicate effects 4. The variace of the BLUE of ay elemetary treatmet cotrast is /r, where is the variace of a observatio. 9. The startig ad optimal tableaus of a miimizatio problem are give below. The variables are x, x ad x 3. The slack variables are S ad S. Startig Tableau Z x x x 3 S S RHS Z a S 0 b 0 6 S Optimal Tableau Z x x x 3 S S RHS Z 0 -/3 -/3 -/3 0-4 x 0 c /3 /3 /3 0 e S 0 d 8/3 -/3 /3 3 Which of the followig are the correct values of the ukows a, b, c, d ad e. a =, b = 3, c =, d = 0, e =. a =, b = -3, c =, d = 0, e = - 3. a = -, b = 3, c =, d = 0, e = 4. a = -, b = 3, c = -, d = 0, e = 0. Cosider the followig liear programmig problem. Miimize Z = x + x subject to sx + tx x 0 x urestricted. The ecessary ad sufficiet coditio to make the LP. feasible is s 0 ad t = 0. ubouded is s > t or t < 0 3. have a uique solutio is s = t ad t > 0 4. have a fiite optimal solutio is x 0.