PART A Questions 1 10 : Choose the most grammatically correct and meaningful option from those provided at the end of each sentence: 1.

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Drusilla Watson
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1 RP-QP -4- *RPQP* PART A Questios : Choose the most grammatically correct ad meaigful optio from those provided at the ed of each setece:. To create ad perform somethig o the spur of the momet is to A) reflect B) improvise C) itellect D) explore. A hoorary job is oe that is A) praiseworthy B) urecogised C) uremuerated D) excellet. Everyoe i the departmet is reuired to with the official rules ad regulatios. A) compliace B) complied C) complyig D) comply 4. Nobody wated of the lavish desserts after the very satisfyig dier. A) ay B) may C) othig D) so much 5. Last witer was oe of the I have experieced i my whole life. A) severer B) more severest C) most severer D) severest 6. If I you, I would first verify the details before takig ay step. A) am B) was C) were D) had bee 7. No oe was i a mood to leave early,? A) was t it B) is t it C) were they D) was he 8. We thik someoe him o the head to make him ucoscious before robbig him. A) striked B) struck C) stricke D) stroke 9. The worst aspect of malutritio is that it ca cause permaet to the brai. A) destructio B) deformatio C) declamatio D) damage. of you would like to accompay me to the party? A) Which B) What C) Two D) Whichever. Choose the appropriate aswer for the followig : Tuberculosis : Lugs : : Cataract :? A) Ear B) Throat C) Ski D) Eye. Which umber will come i the blak space? 6,,, 6, 56, A) 4 B) 5 C) 8 D) 9. A girl itroduced a boy as the so of the daughter of the father of her ucle. The boy is girl s A) Brother B) So C) Ucle D) So-I-law

2 *RPQP* -5- RP-QP 4. I a examiatio, a studet scores 4 marks for every correct aswer ad loses mark for every wrog aswer. If he attempts all 75 uestios ad secures 5 marks, the umber of uestios he attempts correctly is A) 5 B) 4 C) 4 D) How may triagles are there i the followig figure? A)9 B) C) D) 6. Select the most suitable syoym for HECTIC A) orgaised B) clumsy C) fratic D) thrillig 7. Select the most suitable atoym for TRUNCATE A) legthe B) split C) digress D) eervate 8. Idetify the meaig of idiom A bird s eye view A) without care B) withi the walls C) a overall view D) out of place 9. Select the pair which shows the same relatioship as the capitalized pair of words SUN : SUNLIGHT A) Moo : moolight B) Bee-hive : hoey C) Tap : water D) Flower : fragrace. Choose the correct optio ( + 66 )/ 8 =? A) B) C) 5 D) Noe of these. The average of first five multiples of is A) B) 9 C) D) 5..5 ca be expressed i terms of percetage with respect to as A).5% B).5% C) 5% D) 5%. A shopkeeper sold a article for Rs.,9.4. Approximately, what will be the percetage profit if he sold that article for Rs.,6.58? A) 5% B) % C) 5% D) % 4. If A : B = : 4 ad B : C = 8 : 9, the A : C is A) : B) : C) : D) : 5. A ma ca do a job i 5 days. His father takes days ad his so fiishes it i 5 days. How log will they take to complete the job if they all work together? A) Less tha 6 days B) Exactly 6 days C) Approximately 6.4 days D) More tha days

3 RP-QP -6- *RPQP* PART B 6. If A, B ad a A, b B, a b, if A ad B have both supremum ad ifimum, the A) Sup A If B B) Sup A If B C) If A = Sup B D) Sup A = Sup B + 7. For what value of x the ifiite series x coverge? A) x < e B) x > e C) x = e D) x = 8. The x + x + x x +... is A) coverget, if x ad diverget, if x < B) coverget, if x ad diverget, if x > C) coverget, if x > ad diverget, if x D) coverget, if x < ad diverget, if x x cos if x 9. If f (x) = x the uder what coditios is f (x) differetiable at x = if x = A) > B) C) D) < [x]. The fuctio f (x) = is x [x] A) Cotiuous at x = ad x = B) Cotiuous at x = ad discotiuous at x = C) Discotiuous at x = ad cotiuous at x = D) Discotiuous at x = ad x =

4 *RPQP* -7- RP-QP. If the taget to the curve f (x) = x at ay poit (c, f (c)) is parallel to the lie joiig (a, f (a)) ad (b, f (b)) o the curve, the which oe of the followig is true? A) a, c, b i A.P. B) a, c, b i H.P. C) a, c, b i G.P. D) a, c, b do ot follow defiite seuece. The seuece f (x) = x, x +,, x x x A) coverges uiformly B) coverges poitwise to f (x) = x C) lim f (x)dx D) lim f = (x)dx f f (x)dx (x) dx. Give the fuctio f (x, y) = x xy + y + x y + x 5. The it A) has miimum value at origi B) has maximum value at origi C) has maximum value but o miimum value at origi D) has either maximum or miimum value at origi

5 RP-QP -8- *RPQP* 4. Let us cosider the pair of ordered pair of seueces (s, t ) ad (a, b ) such that lim (s, t ) = (s, t) ad A) lim f (s, t ) = B) lim f (s, t ) C) lim f (s, t ) D) lim f (s, t ) lim (a, b ) = (s, t) the (x,y) (s,t) lim f (a, b ) lim f (a, b ) lim f (a, b ) lim f (a, b ) lim does ot exists, if 5. Cosider the real vector space the subspace {(x, y, z) : y = x} of is geerated by which oe of the followig? A) {(,, ), (,, )} B) {(,, ), (,, )} C) {(,, ), (,, )} D) {(,, ), (,, )} 6. Miimal polyomial m (x) of A ( ) each of whose elemet is, is A) x B) x + x C) x x D) x + x 7. If S = { (,, ), (,, )} the which oe of the followig vectors of is ot i the spa [S]? A) (,, ) B) (,, ) C) (,, ) D) (4/,, ) 8. I the matrix euatio AX = B, which of the followig is a ecessary coditio for the existece of atleast oe solutio for the ukow vector X? A) Augmeted matrix [A : B] must have the same rak as matrix A B) Vector B must have oly o-zero elemets C) Matrix A must be sigular D) Matrix A must be suare 9. S = { (x, x, x,..., x 5 ). s.t.x = x =... = x 5 =, x 6 + x x 5 = }. The the dimesio of S is A) 5 B) 5 C) 4 D)

6 *RPQP* -9- RP-QP 4. Let [ ] = { a + b :a, b A) Q [ ] is ifiite dimesioal vector space over Q the which of the followig is correct? B) Q [ ] is two-dimesioal vector space over C) Q [ ] is two dimesioal vector space over D) is fiite dimesioal vector space over Q [ ] 4. Let {v, v, v, v 4 } be a basis of 4 ad v = c v + c v + c v + c 4 v 4, where c i, i =,,, 4. The {v v, v v, v v, v 4 v} is a basis of 4 if ad oly if A) c = c = c = c 4 B) c c c c 4 = C) c + c + c + c 4 D) c + c + c + c 4 4. Let V = {f (x) [x] degf(x) } where is a field of real umbers. Defie, e :V by e [ f (x)] = f (x) dx e whose dual basis is {, } A) { + x, x} e e is ad e [ f (x)] = f (x) dx. The, the basis of V B) C) D) + x x, ( x), + x (+ x), x 4. Let T : be a map defied by T (a, b) = (a + b, a b). Which of the followig statemet is correct? A) T is liear ad its kerel has ifiite umber of elemet of B) T is ot liear C) The kerel of T cosist of oly two elemet of D) Nullity of T is zero

7 RP-QP -- *RPQP* 44. Let F be ay field ad let T be a liear operator o F defied by T (s, t) = (s + t, s), the T (s, t) is eual to A) (s, t) B) (t, s t) C) (s t, t) D) (s, s + t) 45. If T be a liear operator o a vector space V such that T T + I =, the A) o such T exists B) T is oe to oe but may ot be oto C) T is ivertible D) T is oto but may ot be oe to oe 46. The caoical form of the uadratic form (x, y, z) = 6x + 6y + 5z 4xy xz yz is A) 8x + 6y z B) 8x + 6y z C) 8x 6y z D) 8x + 6y + z 47. If V (F) be a vector space of polyomials i x with ier product give by (p, ) = p (x) p(x) (x) dx. Let p (x) = x +, the p is A) B) 9 C) 7 D) The radius of covergece of = + z is A) e B) C) e D) e e 49. The C log z dz where C is uit circle, is A) πi B) C) πi D) π

8 *RPQP* -- RP-QP 5. Let f be a etire fuctio ad satisfied the followig two fuctios f ( z + ) = f (z), f (z + i) = f (z) for every z, the A) f (z) B) f = costat C) f (z) = f (z) D) f is a o-costat polyomial 5. If C is a regio ad f is a aalytic fuctio with zeros a, a,..., a. Also, if γ is a closed rectifiable curve i C which does ot pass through ay poit a k ad γ =, the f (z) (z) A) dz = B) dz f (z) γ does ot exist f (z) (z) C) dz = πi f (z) f γ D) dz = πi = ( γ, ak) γ f f (z) f (z) 5. The image of the uit circle z = uder the trasformatio w = z + z is A) circle B) straight lie C) parabola D) cardioid 5. The biliear trasformatio which maps the poits z =, i, oto the poits w = i,, i is A) z = i B) z = w k C) w z = i D) + w + w z = i w 54. If p is a prime iteger ad if a, b are ay itegers such that p ab, A) p a ad p b B) either p a or p b C) p a ad p b D) either p a or p b 55. If itegers a, b > the the set of all iteger of the form xa + yb (x, y Z) icludes A) both their gcd ad lcm B) their gcd but ot lcm C) their lcm but ot gcd D) either their lcm or gcd

9 RP-QP -- *RPQP* 56. If ac bc (modm) ad d = (m, c) m m A) a b mod B) a mod d d b C) a m (mod b) D) a m mod a 57. Let G be a group of order 5. The, the umber of Sylow subgroup of G of order is A) B) C)5 D) 58. Let S be the group of all permutatios o three symbols, with the idetity elemet e, the the umber of elemets i S that satisfy the euatio x = e is A) B) C) D) 59. Defie a map f : C R such that f (x + iy) = x. The, Ker f is A) {iy : y R} B) {x + iy : x ad y are ozero} C) {} D) {x : x } 6. Which of the followig statemet is true? A) The order of a elemet of a group is the same as that of its iverse B) The order of a elemet of a group is greater tha as of its ow iverse C) The order of a elemet of a group is less tha as of its ow iverse D) The order of a elemet of a group is ot same as of its ow iverse The sum of the series euals!!! e e A) e B) C) D) e + 6. Let V be the space of twice differetiable fuctios o satisfyig f f + f =. Defie T : V by T (f) = ( f (), f ()) the T is A) oe-to-oe ad oto B) oe-to-oe but ot oto C) oto but ot oe-to-oe D) either oe-to-oe or oto

10 *RPQP* -- RP-QP 6. Which of the followig subsets of is compact (with respect to the usual topology of )? A) {(x, x,..., x ) : <, i } x i B) {(x, x,..., x ) : = x = } C) {(x, x,..., x ) : x i >, i } D) {(x, x,..., x ) : x, i } 64. Which of the followig is ot a field? i i i i A) B) C) 4 D) Which of the followig is correct? A) The rig of itegers is a Euclidea rig B) The rig of polyomials over a field is a Euclidea rig C) The rig of Gaussia itegers is a Euclidea rig D) All the above 66. If K is a extesio of field F ad a K is algebraic over F, if A) F (a) is a extesio of F B) F (a) is a subfield of f C) F (a) is fiite extesio of F D) F (a) is ot a fiite extesio of F 67. Which of the followig is a topology o X = {,,, 4}? A) T = {φ, {}, {}, {, }} B) T = {φ, X, {}, {}} C) T = {X, {}, {, }} D) T = {φ, X, {}}

11 RP-QP -4- *RPQP* 68. Let X ad Y be topological spaces ad f : X Y. If f is cotiuous, the A) for every closed set B Y, set f (B) is closed i X B) for every ope set B Y, set f (B) is closed i X C) for every closed set B Y, set f (B) is ope i X D) all the above 69. Determie a iterval i which the solutio of the followig iitial value problem is certai to exist y + (ta θ) y = si θ, y( π ) = A) π < θ< B) π< θ< π C) < θ< π D) π < θ< π dy x 7. The solutio of = represets dx y A) a family of circle cetre at (, ) B) a family of circle cetre at (, ) C) a family of circle cetre at (, ) D) a family of straight lie with slope 7. Let y ad y be solutios of Bessel s euatio t y + ty + (t )y = o the iterval < t <, with y () =, y () =, y () = ad y () =. The W [y, y ] (t) o A) is o-zero B) is idetically zero C) is zero oly at fiite umber of poits D) 7. The solutio of the differetial euatio y (x + y + ) dy + [x (x + y ) ]dx = A) x + y + log (x + y ) = C B) x y + log (x + y ) = C C) x + y log (x + y ) = C D) x y log (x + y ) = C 7. The PDE yu xx + xu yy = is hyperbolic i A) the II ad IV uadrats B) the I ad II uadrats C) the II ad III uadrats D) the I ad III uadrats

12 *RPQP* -5- RP-QP 74. The iitial value problem u t + u x =, u (x, x) = si x, x, has A) a uiue solutio B) o solutio C) two solutios D) ifiitely may solutios 75. The variable ξ ad η which reduce the differetial euatio u xx x u yy = to the caoical form, are A) x x ξ = y +, η= y B) x x ξ = y +, η= y C) ξ = y + x, η= y x D) ξ = y + x, η= y x 76. Which of the followig cocer is true for the Neuma problem for Laplace s euatio o a smooth bouded domai? A) Solutio is ot uiue B) Solutio is uiue up to a additive costat C) Solutio is uiue up to a multiplicative costat D) No coclusio ca be made about uiueess 77. Let Ω be a smooth plae domai of uit area. Let u (x, y) = x + y. If u deotes u its outer ormal derivative o Ω, the boudary of Ω, the the value of ds Ω A) B) C) 9 D) 8 x θ follows : Y = k, if k X < k +, k =,,,... The the distributio of Y is 78. Suppose X has desity f ( x θ) = e θ,x > where θ > is ukow. Defie Y as A) geometric B) biomial C) poisso D) ormal 79. The umber of subfields of a field of cardiality is A) B) 4 C) D) 9

13 RP-QP -6- *RPQP* 8. The fuctio a + b log t, G (x, t) = c+ d log x, < x t t x is a Gree s fuctio for x y + y = subject to y beig bouded as x ad y () = y (), if A) a =, b =, c =, d = B) a =, b =, c =, d = C) a =, b =, c =, d = D) a =, b =, c =, d = 8. The eigevalues of the matrix A = (a ij ) ( ) where a ij = for all. A) λ = with multiplicity B) λ = with multiplicity C) λ = with multiplicity ad λ = with multiplicity D) λ = with multiplicity ad λ = with multiplicity 8. Let A ( ) be a matrix which is ot a diagoal matrix. The a) If tr (A) = ad det (A) =, the A = I b) If A = I, the tr (A) = ad det (A) = I c) If A = I, the A is diagoalizable over A) All the above are true B) oly (a) ad (b) true C) oly (a) ad (c) true D) oly (c) ad (b) true 8. If oe root of the euatio f (x) = is ear to x, the the first approximatio of this root as calculated by Newto-Raphso method is the abscissa of the poit, where the followig straight lie itersects the x axis A) the straight lie passig through the poit (x, f (x )) B) ormal to the curve y = f (x) at the poit (x, f (x )) C) taget to the curve y = f (x) at the poit (x, f (x )) D) the straight lie through the poit (x, f (x )) havig the gradiet f (x )

14 *RPQP* -7- RP-QP 84. Fid the ecessary coditio o a, so that the Gauss-Jacobi method coverges for the followig system of euatio AX = b where a A = 7, b is arbitrary a A) a B) a = C) a > D) a < 85. O what curves ca the fuctio I [y (x) ] = ( y ) [ + xy]dxwith y () = ad y () = be extremized? A) y = x B) y = x C) y = x D) y = x 86. The itegral euatio g(s) λ ( st ts ) g(t) dt = has A) real eigevalues B) real eigevector C) imagiary eigevector D) both real eigevalue ad eigevector 87. Extremals of the fuctioal I [y (x) ] = ( xy+ (y ) ) dx lies o the A) oe parameter family of curves B) three parameter family of curves C) six parameter family of curves D) five parameter family of curves 88. The itegral euatio y (x) = λ (x )ty (t) dt with λ as a parameter, has A) oly oe characteristic umber B) two characteristic umbers C) more tha two characteristic umbers D) o characteristic umber x x

15 RP-QP -8- *RPQP* 89. Let s + t k (s, t) = t s else The the itegral euatio g (s) = + λ k(s, t) g(t) dt has A) a uiue solutio for every value of λ B) o solutio for ay value of λ C) a uiue solutio for ifiitely may values of oly λ D) ifiite may solutios for ifiite may values of λ 9. O a frictioless iclied plae, makig a agle θ with a horizotal surface, the acceleratio of a slidig block is A) gcos θ B) g si θ C) g ta θ D) ot a costat 9. The system is said to be i euilibrium, if the geeralized forces actig o the system A) are ifiite B) are o-zero C) are eual to zero D) oe of the above 9. Lagrage s euatios for a coservative holoomic dyamical systems are T L A), k =,,..., = k k T L B), k =,,..., = k k d T L C), k =,,..., dt = k k T L D), k =,,..., = k k 9. Let X ad Y be two idepedet radom variables, each represetig the umber of failures precedig the first success i a seuece of Beroulli trials with p as probability of success i a sigle trial ad as probability of failure, the p(x=y) is A) + B) p + p C) + p D) p +

16 *RPQP* -9- RP-QP 94. I a competitio A, B ad C are participatig. The probability that A wis is twice that of B; the probability that B wis is twice that of C. The probability that A loses is A) 7 B) 7 C) 7 D) ( x ), 95. Let X has the probability desity fuctio f (x) =, What is probability P x? A).6 B).65 C).68 D).75 x [, ]. else 96. If X ad Y are idepedet N (, ) variables, the the characteristic fuctio of XY is t A) e B) ( + t ) / C) ( + t) / D) ( + t ) x e, x 97. Give the probability desity fuctio f (x) = the the cumulative, x < probability fuctio F() is A) + e B) e C) e D) e 98. If the joit desity of x, x ad x is give by f (x, x x (x + x ) e for < x <, < x <, x >, x ) =, elsewhere The, the regressio euatio of x o x ad x is A) x + x x B) x + x C) x x + D) x + x The maximum value of z = x + y of subject to the costraits x + y, y, y, x y ad x is A) 76 B) 7 C) D) 6. The multiplicative iverse of 4x+ i the field [x]/(x + ) is A) 6x + B) 6x + 5 C) 7x + 8 D) does t exist

17 RP-QP -- *RPQP* SPACE FOR ROUGH WORK

NBHM QUESTION 2007 Section 1 : Algebra Q1. Let G be a group of order n. Which of the following conditions imply that G is abelian?

NBHM QUESTION 2007 Section 1 : Algebra Q1. Let G be a group of order n. Which of the following conditions imply that G is abelian? NBHM QUESTION 7 NBHM QUESTION 7 NBHM QUESTION 7 Sectio : Algebra Q Let G be a group of order Which of the followig coditios imply that G is abelia? 5 36 Q Which of the followig subgroups are ecesarily