State Eligibility Test Mathematical Sciences. (Model Answer Key) Part-A

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1 State Eligibility Test 2017 Mathematical Sciences (Model nswer Key) Part- Q1 mong the following, choose the odd one न न ल खत म स ज स म लत नह ह, क च नय The axiom of choice एग जयम ऑफ च इस Well - ordering principle व ल - आड र ग स त Pigeon - hole principle पजन-ह ल स त Zorn's lemma ज र स ल म ( म यक ) nswer Key Q2 If denotes the null set, then among the following the number of true statements is य द र त स म य क द श त करत ह त न न ल खत म स स य कथन क स य ह

2 one एक Two द Three त न Four च र nswer Key Q3 f(x) is an non-negative function in [0, [ such that f'(x) 2f(x) and f(0) = 0. Then f(x) पर [0, [ एक ऋण तर फलन इस क र ह क f'(x) 2f(x) तथ f(0) = 0 तब f(x) is always a constant function f(x) सद व एक अचर फलन ह f(x) is increasing function f(x) एक वध म न फलन ह f(x) is decreasing function f(x) घटत ह आ फलन ह f'(x) changes sign f'(x) क च ह बदलत ह nswer Key Q4 The value of क म न

3 is ह can be ह सकत ह does not exist क अ त व नह ह both (is ) and (does not exist) are true ( ह ) तथ (क अ त व नह ह ) द न सह ह nswer Key Q5 Which of the following, subsets of 2 is ompact 2 क न न उपसम चय म स क न स स हत ह {(x, y) x 1, y 2} {(x, y) x 1, y 2} {(x, y) x2 y 2 + 5} {(x, y) x 2 y 2 + 5} {(x, y) x = y} {(x, y) x = y} {(x, y) x 1, y 2 3} {(x, y) x 1, y 2 3} nswer Key Q6 For each j = 1,2,3,..., let j be a finite set containing at least two distinct elements. Then which of the following is true? य क j = 1,2,3,..., क लय म नय क j एक प र मत सम चय ह जसम कम स कम द भनन अवयव ह तब न न ल खत म स क न स स य ह? is uncountable अगणन य ह is uncountable is uncountable अगणन य ह

4 अगणन य ह ll of the these सभ वक प सह ह nswer Key Q7 The value of is /2 /2 nswer Key क म न ह Q8 "If f[a,b] is a continuous function and f(a). f(b) < 0, then there is x, a < x < b such that f(x) = 0". This result of the consequence of ''य द f[a,b] यह कथन एक नत ज ह एक सतत फलन ह तथ f(a). f(b) < 0 त एक ऐस x, a < x < b ह त ह क f(x) = 0 '' oundedness of [a,b] [a,b] क प रब त क ompleteness of [a,b] [a,b] क प ण त क ompactness of [a,b] [a,b] क स स हतत क onnectness of [a,b] [a,b] क स ब त क nswer Key

5 Q9 Which of the following statements are not true? न न ल खत म स क न स कथन स य नह ह? Every convergent sequence is bounded य क अ भस र अन म प रब ह त ह Every bounded sequence is convergent य क प रब अन म अ भस र ह त ह Every auchy sequence of real numbers is convergent य क व त वक स य ओ क क श अन म अ भस र ह त ह Every convergent sequence is a auchy sequence य क अ भस र अन म एक क श अन म ह त ह nswer Key Q Let f(x) and g(x) be differentiable functions for 0 x 2 such that f(0) = 4, f(2) = 8, g(0) = 0 and f'(x) = g'(x) for all x in [0,2] then the value of g(2) must be म न ल जय क f(x)और g(x), 0 x 2 क लय ऐस अवकलन य फलन ह क f(0) = 4, f(2) = 8, g(0) = 0 तथ f'(x) = g'(x),[0,2] म x क सभ म न क लय, तब g(2)क म न ह ग nswer Key Q11 Let be a function such that and exist. Then at (0,0) म नय क क ई फलन इस क र ह क तथ क अ त व ह तब (0,0)पर f must be continuous but not differentiable f अवशय ह सतत ह ग ल कन अवकलन य न ह ग f must be differentiable

6 f अवशय ह अवकलन य ह ग f need not to be continuous f क सतत ह न ज र नह ह f'(x) is continuous f'(x) सतत ह ग nswer Key Q12 The integral dx, p > 0 converges absolutely if सम कल dx, p > 0 नर प य अ भस र ह य द p > 1 p > 1 p = 1 p = 1 p < 1 p < 1 p > 0 p > 0 nswer Key Q13 Which of the following statements is correct? न न ल खत म स क न स कथन स य ह? f is measurable f is measurable f म य ह f म य ह f is measurable f is measurable f म य ह f म य ह f is measurable f + f is measurable f म य ह f + f म य ह None of these are correct

7 क ई वकलप सह नह nswer Key Q Let be a 5 X 4 matrix with real entries such that X = 0 if and only if X = 0, where X is a 4 X 1 matrix and 0 is null matrix. Then the rank of is य द एक व त वक व टय क 5 X 4 क ऐस आ य ह ह क X = 0 य द और क वल य द X = 0 जह X एक 4 X 1 क आ य ह ह तथ 0 श य आ य ह ह तब क ज त ह nswer Key Q15 Let be a real 3 X 4 matrix of rank 2. Then rank of *, where * denotes the transpose of, is म नय क ज त 2 क एक व त वक 3 X 4 आ य ह ह तब * क ज त, जह *, प रवत क द श त करत ह, ह ग exactly 2 ठ क 2 exactly 3 ठ क 3 exactly 4 ठ क 4 at most 2 but not necessarily 2 अ धक स अ धक 2 ल कन ज र नह क 2 ह nswer Key Q16 If is a 5 X 5 real matrix with trace 14 and if 2 and 3 are eigen values of, each with algebraic multiplicity 2, then the determinant of is equal to य द एक 5 X 5 क व त वक आ य ह ह जसक अन र ख 14 ह तथ य द 2 और 3, क अ भल णक म न,

8 य क ब ज य बह कत 2 क स थ ह त क स र णक क म न ह nswer Key Q17 If, then 50 is य द तब 50 ह

9 nswer Key Q18 Let V 1 and V 2 be subspaces of a vector space V. onsider the following statements P & Q P V 1 V 2 is a subspace of V Q V 1 + V 2 = {x+y x V 1, y V 2 } is not subspace of V. Then य द V 1 तथ V 2 एक स दश सम ट V क उपसम टय ह त न न ख लत कथन P तथ Q पर वच र कर P V 1 V 2 एक उपसम ट ह V क Q V 1 + V 2 = {x+y x V 1, y V 2 }, V क एक उपसम ट नह ह P is true but Q is false P स य ह ल कन Q अस य ह P and Q both are true P तथ Q द न स य ह P is false while Q is true P अस य ह जब क Q स य ह P and Q both are false P तथ Q द न अस य ह nswer Key Q19 onsider the six vector spaces Q(Q), R(Q), (Q), R(R), (R) and (), where Q, R and denotes the fields of rational numbers, real numbers and complex numbers respectively. Here the number of finite dimensional vector spaces is छ र खक सम टय Q(Q), R(Q), (Q), R(R), (R)तथ ()पर वच र क जय, जह Q, R तथ मश प रम य स य ओ, व त वक स य ओ तथ स म स य ओ क क द श त करत ह यह प र मत वम ओ क र खक सम टय क स य ह Three त न Four च र Five प च Six छ nswer Key

10 Q20 Let V be the vector space of all real polynomials of degree upto 2 and let S = {1, x 2 + x, x 2 - x} and T = {x, x 2 + 1, 3x 2-2x + 3} hoose your answer as म नय क V द घ त तक क सभ व त तवक बह पद क र खक सम ट ह तथ म नय क S = {1, x 2 + x, x 2 - x} तथ T = {x, x 2 + 1, 3x 2-2x + 3} अपन ऐस उ तर च नय S and T both are bases of V S तथ T, V क लय आध र ह None of S and T is a basis for V S तथ T म स क ई V क लय आध र न ह S is a basis, but T is not a basis for V S एक आध र ह ल कन T, V क लय आध र न ह S is not a basis but T is a basis for V S एक आध र न ह ल कन T, V क लय आध र ह nswer Key Q21 Let V denote the vector space of all infinitely differentiable functions and let operator. onsider the sets S1 = {1, x, x 2 } S2 = {x, x 2, x 3 } S3 = {1+x, x+x 2, 1+x 2 } The sets which span the solution space of the problem 3 (f) = 0 are म न क V सभ अन त य अवकलन य फलन क र खक सम ट क द श त करत ह तथ स क रक ह सम चय S1 = {1, x, x 2 } S2 = {x, x 2, x 3 } S3 = {1+x, x+x 2, 1+x 2 } पर वच र क जय व सम चय, ज सम य 3 (f) = 0 क हल सम ट क प ण व त र करत ह, ह ग be the differential अवकल य ll the three S 1, S 2 and S 3 सभ त न S 1, S 2 and S 3 S 1 and S 2 S 1 त थ S 2 S 2 and S 3

11 S 2 तथ S 3 S 3 and S 1 S 3 तथ S 1 nswer Key Q22 If V be a vector space over the field of real numbers and a linear transformation by, x V. Then the nullity of T य द V व त वक स य ओ क पर बन क ई र खक सम ट ह तथ क ई र खक is defined प तरण, x V स प रभ षत ह तब T क श यत is zero श य ह ग is one एक ह ग is infinite अन त ह ग annot be determined unless V is known नह नक ल ज सकत जब तक क V त न ह nswer Key Q23 x 1 < x 2 x 1 < x 2 x 2 < x 1 x 2 < x 1 Let x 1 = 3 1/ /7 and x 2 = 5 1/ /7. Then which of the following is true statement? म न x 1 = 3 1/ /7 तथ x 2 = 5 1/ /7 तब न न म स क न स स य कथन ह f(x) = (6-x) 1/7 + (6+x) 1/7 is an increasing function in [1,3] f(x) = (6-x) 1/7 + (6+x) 1/7 अ तर ल [1,3] म वध म न फलन ह None of these are correct क ई वक प सह नह nswer Key

12 Q24 If f(z) is analytic throughout the complex plane and its modulus is uniformly bounded, then f ' (z) is identically zero The above result is known by the name of य द f(z) स प ण स म तल म वशल षक ह तथ इसक म प क एकसम न प रब ह त f ' (z) सव थ सम श य ह त ह उपर त प रण म क कसक न म स ज न ज त ह Morera म र र auchy क श Liouville लय वल Rouche र च nswer Key Q25 Let f(z) be an analytic function whose value lie on a straight line in the complex plane. Then म न क कस वशल षक फलन f(z) क म न स म तल म एक सरल र ख पर थत ह त ह तब f(z) is identically equal to zero f(z) सव थ सम श य क बर बर ह f(z) is constant f(z) अचर ह f(z) is a linear map f(z) क ई र खक फलन ह f(z) is a bilinear transformation f(z) क ई प तरण ह nswer Key Q26 The image of line x = c under the conformal transformation w = z 1/2 is a अन क ण प तरण w = z 1/2 क अ तग त र ख x = c क त ब ब ह ircle व त

13 Hyperbola अ तपरवलय Parabola परवलय None of these are correct क ई वकलप सह नह nswer Key Q27 Under what assumptions on f does auchy's theorem imply that circle? कन प रक पन ओ क अ तग त f पर क श म य स यह नष कष नकलत ह क इक ई व त ह? where c is the unit जह c एक f is analytic outside c f, c क ब हर वशल षक ह f is analytic at every point of c c क य क ब द पर f वशल षक ह f is analytic in a region containing the closed unit disc f उस प र म, जसम स व त इक ई ड क अ त व षट ह, वशल षक ह f is analytic in the annulus containing c c य त वलयक र प र म, f वशल षक ह nswer Key Q28 Let f म न क f be an entire function such that, f(1/n) = 1/n 2, for all positive integers n. Then इस क र क सव फलन ह क सभ धन तमक प ण क n क लए f(1/n) = 1/n 2 ह, तब Such a function does not exist ऐस फलन क अ ततव नह ह त f should be bounded f क प रब ह न च हए f has a pole at 0 f, 0 पर व रखत ह f(z) = z 2 for all z in

14 म तय क z क लए, f(z) = z 2 nswer Key Q29 Let G be a group of order 99. onsider the following three statements (i) G has a normal sub group of order 3 (ii) G has a subgroup of order 11 (iii) the group G is abelian. Here the number of true statements is म न क G क ट 99 क एक सम ह ह न न ल खत त न कथन पर वच र क जय क G क ट 3 क एक स म य उपसम ह रखत ह ख G क ट 11 क एक उपसम ह रखत ह ग G एक आब ल सम ह ह यह स य कथन क स य ह zero श य one एक two द three त न nswer Key Q30 " Every finite group G is isomorphic to a permutation group". This statement is '' य क प र मत सम ह G मचय क सम ह स त यक र ह '' यह कथन ह - ayley's theorem क ल क म य Lagrange's theorem ल गर ज क म य Lioville's theorem लय ब ल क म य None of these are correct क ई वक प सह नह

15 nswer Key Q31 Let F be a field of 32 elements and = { x F x 31 = 1 and x k 1 for all natural numbers k < 31}. Then the number of elements in is म न क F, 32 अवयव क एक ह तथ = { x x k 1}. तब म अवयव क स य ह F x 31 = 1 एव सभ क त स खय ओ k < 31 क लए 1 एक 2 द 30 त स 6 छ nswer Key Q32 H H Let G be a group of order 36 and H be its subgroup of order 4. If Z(G) denotes the centre of G. Then म न क G क ट 36 क एक सम ह ह तथ H इसक क ट 4 क उपसम ह ह य द Z(G), G क क क द श त करत ह तब Z(G) Z(G) H = Z(G) H = Z(G) H is normal in G H, G म स म य ह H is ableian subgroup H एक आब ल उपसम ह ह nswer Key Q The number of non-trivial topologies on a three elements set X = {a,b,c} is एक त न अवयव क सम चचय X = {a,b,c} पर अत चछ स थ त कय क स खय ह

16 nswer Key Q34 The cofinite topology on an infinite set is एक अन त सम चय पर सह-प र मत स थ तक ह त ह T 0, compact but not T 2 T 0, स हत ल कन T 2 नह T 1, connected but not T 2 T 1, स ब ल कन T 2 नह T 2, but not T 3 T 2, ल कन T 3 नह T 2, compact but not T 4 T 2, स हत ल कन T 4 नह nswer Key Q35 Let be the real line and x be the plane as a topological space. onsider the map defined by f(x,y)=(x,0). Then the map f is म न क R व त वक र ख ह तथ तल x क एक स थ तक सम ट क तरह समझ फलन. f(x,y)=(x,0) व र प रभ षत ह तब फलन f ह losed and continuous स व त तथ सतत ontinuous and open सतत तथ वव त Open and closed वव त तथ स व त losed, open and continuous स व त, वव त तथ सतत

17 nswer Key Q36 Let g be a differentiable function satisfying for all x 0. Then the value of is equal to म नय क g एक अवकलन य फलन ह ज सभ x 0 क लय क स त ट करत ह तब क म न ह /6 /6 /3 /3 /4 /4 /2 /2 nswer Key Q37 If y =e -x is a solution of OE,, then the correct relation between P and Q is य द y =e -x स. अ. सम., क एक हल ह त P तथ Q क म य सह स ब ध ह 1 + P + Q = P + Q = 0 P = 1 + Q P = 1 + Q Q = 1 + P Q = 1 + P P + Q = 1 P + Q = 1

18 nswer Key Q38 Given that y = x is one solution of OE x 2 y" + xy' - y = 0, a second linearly independent solution will be y = x स.अ.सम. x 2 y" + xy' - y = 0 क एक हल दय गय ह त द सर र खक वत हल ह ग nswer Key Q39 The region in which the PE is hyperbolic, is वह जसम आ. अ. सम. अ तपरवल य ह, ह xy > 1 xy > 1 xy 1 xy 1 xy 0 xy 0 xy > 0 xy > 0 nswer Key

19 Q40 The differential equation is अवकल सम करण ह Linear and homogeneous र खक तथ समघ त non-linear and homogeneous अर खक तथ समघ त linear and non-homogeneous र खक तथ असमघ त Non-linear and non-homogeneous अर खक तथ असमघ त nswer Key Q41 The partial differential equation of the family of surfaces z = (x+y) + xy is प ट z = (x+y) + xy क क ल क आ शक अवकल सम करण ह ग xp - yq = 0 xp - yq = 0 xp - yq = x - y xp - yq = x - y xp + yq = x + y xp + yq = x + y xp + yq = 0 xp + yq = 0 nswer Key Q42 The function satisfying the integral equation सम कल सम करण क स त ट करन व ल फलन ह

20 nswer Key Q43 The integral equation is equivalent to सम कल सम करण न न म स कसक त य ह y" - y = 0; y(0) = 0, y(1) = 0 y" - y = 0; y(0) = 0, y(1) = 0 y" - y = 0; y(0) = 0, y'(1) = 0 y" - y = 0; y(0) = 0, y'(1) = 0 y" + y = 0; y(0) = 0, y(1) = 0 y" + y = 0; y(0) = 0, y(1) = 0 y" + y = 0; y(0) = 0, y'(0) = 0 y" + y = 0; y(0) = 0, y'(0) = 0 nswer Key Q44 The variational problem of extremizing the functional has फलनक क इ टतम करन क वचरण य सम य क a unique solution

21 एक अ त य हल ह त ह exactly two solutions ठ क द हल ह त ह an infinite number of solutions अन त स य म हल ह त ह no solution क ई हल नह ह त ह nswer Key Q45 The Newton - Raphson iteration formula can be used to compute the य टन र फश न प नर व त, स क कस गणन क लय य ग कय ज सकत ह square of क वग क लय reciprocal of क य म क लय square root of क वग म ल क लय logarithm of क लघ गणक क लय nswer Key Q46 5/12 5/12 3/8 3/8 19/24 One bag contains 5 white balls and 7 black balls, another bag contains 3 white balls and 5 black balls. If a bag is chosen at random and a ball is drawn from it, the probability that it is a white ball is एक ब ग म 5 सफ द ग द तथ 7 क ल ग द ह, द सर ब ग म 3 सफ द ग द तथ 5 क ल ग द ह. य द य छय एक ब ग क च न ज त ह तथ उसम स एक ग द नक ल ज त ह त उस ग द क सफ द ह न क यकत ह ग

22 19/24 19/48 19/48 nswer Key Q47 The function f(x) = ce -x, 0 x is a probability density function for फलन f(x) = ce -x, 0 x एक यकत घन व फलन ह all values of c c क सभ म न क लय all values of c > 1 सभ c > 1 क लय all values of c < 1 सभ c < 1 क लय c = 1 only क वल c = 1 क लय nswer Key Q For a fixed t, consider the linear programming problem maximize z = 3x + 4y, subject to x + y 100, x + 3y t and x,y 0 The maximum value of z is 400 for t equal to एक न चत t क लय, र खक मन सम य अ धकतम कर z = 3x + 4y, तब ध x + y 100, x + 3y t तथ x,y 0 तब t क कस म न क लय z क अ धकतम म न 400 ह ग

23 50 nswer Key Q49 Let X(t) = number of customers in the system at time t in an M/M/ queuing model with = 3, arrival rate > 0 and service rate > 0. onsider the following statements P and Q P If X(t) has a stationary distribution then < 3 Q The number of customers undergoing service at time t is min{x(t),3}. Then = 3,आगमन दर > 0 तथ स व ग त > 0 क स थ एक M/M/ कत र म डल म समय t पर नक य म हक क स य X(t) ह न न ल खत कथन P तथ Q पर वच र क जय P य द X(t) क एक त ध ब टन ह त < 3 ह Q समय t पर स व प न व ल हक क स य य न {X(t),3} ह तब P is true but Q is false P स य ह ल कन Q अस य ह both P and Q are true P तथ Q द न स य ह None of P and Q is true P तथ Q म स क ई भ स य नह ह Only Q is true क वल Q स य ह nswer Key Q50 onsider the linear programming problem Maximize z = 2x 1 + 3x 2-4x 3 + x 4 subject to x 1 + x 2 + x 3 = 2 x 1 - x 2 + x 3 = 2 2x 1 + 3x 2 + 2x 3 - x 4 = 0 and x 1, x 2, x 3, x 4 > 0 Which of the following is true? र खक मन सम य अ धकतम कर z = 2x 1 + 3x 2-4x 3 + x 4 तब ध x 1 + x 2 + x 3 = 2 x 1 - x 2 + x 3 = 2 2x 1 + 3x 2 + 2x 3 - x 4 = 0 तथ x 1, x 2, x 3, x 4 > 0 पर वच र क जय तब न न ल खत म स क न स सत य ह? (1,0,1,4) is a basic feasible solution but (2,0,0,4) is not (1,0,1,4) एक आध र स स गत हल ह ल कन (2,0,0,4) नह

24 Neither (1,0,1,4) nor (2,0,0,4) is a basic feasible solution (1,0,1,4) तथ (2,0,0,4) म स क ई भ आध र स स गत हल नह ह (1,0,1,4) in not a basic feasible solution but (2,0,0,4) is (1,0,1,4) एक आध र स स गत हल नह ह ल कन (2,0,0,4) ह oth (1,0,1,4) and (2,0,0,4) are basic feasible solutions द न (1,0,1,4) तथ (2,0,0,4) आध र स स गत हल ह nswer Key Part- Q1 Let म न क be a convergent series of positive terms. Then घन मक पद क एक अ भस र ण ह, तब is always divergent सद व अपस र ह may be divergent अपस र ह सकत ह may be divergent अपस र ह सकत ह all the series are necessarily convergent सभ णय क अ भस र ह न आवशयक ह nswer Key Q2 For a function which one of the following is true? कस फलन क लय न न ल खत म स क न स एक स य ह

25 is Riemann integral on [0,1] is Riemann integrable [0,1] पर र म न सम कलन य ह र म न सम कलन य ह ग is Riemann intergable on [0,1] is Riemann integrable [0,1] पर र म न सम कलन य ह र म न सम कलन य ह ग is Riemann integrabe on [0,1] is continuous [0,1] पर र म न सम कलन य ह सतत ह ग None of these इनम स क ई नह nswer Key Q3 The improper integral converges, if अन चत सम कल अ भस र ह, य द m > 0 and n < 0 m > 0 तथ n < 0 m > 0 and n > 0 m > 0 तथ n > 0 m < 0 and n > 1 m < 0 तथ n > 1 m < 0 and n < 0 m < 0 तथ n < 0 nswer Key Q4 Let Q be the set of all rational numbers and T be the set of all irrational numbers, then म न ल जय क Q सभ प रम य स य ओ तथ T सभ अप रम य स य ओ क सम चय ह, तब oth Q and T are countable द न Q तथ T गणन य ह Q is countable but T is Uncountable

26 Q गणन य ह पर त T गणन य नह ह T is countable but Q is Uncountable T गणन य ह पर त Q गणन य नह ह oth Q and T are uncountable द न Q तथ T गणन य नह ह nswer Key Q5 Which of the following is NOT true? न न ल खत म स क न स अस य ह? bounded monotonic function is a function of bounded variation क ई प रब द एक द ट फलन प रब द वचरण व ल फलन ह continuous function is necessarily a function of bounded variation क ई स तत फलन न चत ह प रब द वचरण व ल फलन ह function of bounded variation is necessarily bounded प रब द वचरण व ल क ई भ फलन न चत ह प रब द ह त ह The sum of two functions of bounded variation is also of bounded variation द प रब द वचरण व ल फलन क य ग भ प रब द वचरण क ह त ह nswer Key Q6 Let be a Riemann integrable function on [a, b] and let. Then म नय क अ तर ल [a, b] पर, एक र म न सम कलन य फलन ह तथ य द ह, तब (x) is continuous on [a, b] फलन (x) अ तर ल [a,b] पर सतत ह (x) is continuous on [a, b] except at finite number of points of [a,b] [a,b] क क वल क छ प र मत ब द ओ क छ ड़कर (x) अ तर ल [a,b] पर सतत ह (x) may be discontinuous at an infinite number of points in [a, b] फलन (x) अ तर ल [a,b] क अन त ब द ओ पर असतत ह सकत ह Nothing can be said about continuity of (x)

27 फलन (x) क स ततयत क वषय म क छ नह कह ज सकत ह nswer Key Q7 onsider the following statements P and Q P is convergent Series Q. Then न न कथन P तथ Q पर वच र क जए P एक अ भस र ण ह Q P Q P Q Q P Q P P Q P Q None of these इनम स क ई नह nswer Key Q8 For a subset of a metric space, which of the following implies the other three? एक द रक सम ट क उपसम चय क लय, न न ल खत म स क न स अ य त न क स चत करत ह is closed स व त ह is bounded प रब द ह losure of is compact for every य क क लय, क स वरक, स हत ह is compact

28 स हत ह nswer Key Q9 For the series and the sequence < 1/n >, which of the following is true? ण तथ अन म < 1/n > क लय, न न म स क न स स य ह oth द न and < 1/n > are convergent तथ < 1/n > अ भस र ह is convergent but < 1/n > is not convergent अ भस र ह ल कन < 1/n > अ भस र नह ह is not convergent but < 1/n > is convergent अ भस र नह ह ल कन < 1/n > अ भस र ह as well as < 1/n > both are not convergent nswer Key स थ ह स थ < 1/n > द न अ भस र नह ह Q10 The function is both continuous and differentiable at x = 0 if फलन x = 0 पर स तत तथ अवकलन य द न ह, य द - 1 < P < 0-1 < P < 0 0 P 1

29 0 P 1 1 < P 1 < P None of these इनम स क ई नह nswer Key Q11 The series is divergent for all x such that ऐस सभ x, जनक लय ण अपस र ह त ह 0 < x < e 0 < x < e 0 < x e 0 < x e X X X > e only क वल X > e e e nswer Key Q12 Let < f n > be a sequence of real valued continuous functions defined on [0, 1] and suppose that. Then म न ल जय क < f n > अ तर ल [0,1]पर प रभ षत व त वक म न क सतत फलन क अन म ह तथ म न ल जय क तब f is continuous f स तत ह त ह f is continuous provided < f n (x) > is a uniformly bounded sequence f स तत ह त ह य द < f n (x) > एक सम न प रब द अन म दय ह f is continuous provided < f n (x) > is a decreasing sequence

30 f स तत ह त ह य द < f n (x) > एक सम न अन म दय ह f is continuous provided < f n (x) > is convergent for some x [0, 1] f स तत ह त ह य द < f n (x) > क छ x [0, 1] क लय, अ भस र दय ह nswer Key Q13 If, which of the following theorems enables you to conclude that c 0 +c 1 x+ +c n x n = 0 for some x ]0,1[? य द करत ह क c 0 +c 1 x+ +c n x n = 0, क छ x ]0,1[ क लए?, त न न म य म स क न स आपक न कष नक लन म समथ Rolle's Theorem र ल क म य Intermediate value Theorem मधयवत म न म य Fundamental Theorem of algebra ब ज ग णत क आध र भ त म य Mean value Theorem of integral calculus सम कल ग णत क म य म न म य nswer Key Q14 Let X be a connected subset of real numbers. If every element of X irrational, then the cardinality of X is य द X व त वक स य ओ क स ब उपसम चय ह तथ X क य क अवयव अप रम य स य ह त X क क ड नल स य ह ग Infinite अन त ountably infinite गणन य अन त

31 nswer Key Q15 Far all x ]0,1[, which of the following is true? सभ x ]0,1[ क लय, न न ल खत म स क न स स य ह? log e (1+x) < x log e (1+x) < x ex < 1+x e x < 1+x x < sinx x < sinx x < log x e x x < log e nswer Key Q16 Which of the following sets has a non-zero measure? न न ल खत सम चय म स कसक म प (म जर) अश य ह त ह Set of all rational numbers सभ प रम य स य ओ क सम चय Set of all irrational numbers सभ अप रम य स य ओ क सम चय Set of all integers सभ प ण क क सम चय Set with finite number of elements सम चय जसम अवयव क स य प र मत ह nswer Key Q17 Let f X X such that f(f(x)) = x all x X. Then म न क f X X इस क र ह क f(f(x)) = x, सभ x X क लय ह तब f is one-one and onto f एक क तथ आ छ दक ह f is one-one but not onto f एक क ह, पर त आ छ दक नह ह

32 f is onto but not one-one f आ छ दक ह पर त एक क नह ह f is neither one-one nor onto f न त एक क ह और न ह आ छ दक nswer Key Q18 Let d 1, d 2 and d 3 be metrics on a set X with at least two elements. Which of the following is NOT a metric on X? य द d 1,d 2 तथ d 3 कम स कम द अवयव स हत सम चय X पर म स ह त न न ल खत म स क न एक X पर म क नह ह? min{d 1, 2} य न {d 1, 2} max{d 2, 2} अ धक {d 2, 2} nswer Key Q19 Let f य द f be a monotone function. Then एक एक द ट फलन ह, तब f is always continuous f सद व सतत ह त ह f has only finitely many discontinuities f क क वल प र मतत अस त य ह त ह f can have at most countably many discontinuities f क अ धक स अ धक गणन यत अस त य ह सकत ह

33 f can have uncountably many discontinuities f क अगणन यत अस त य ह सकत ह nswer Key Q If and are 5x5 matrices of rank 4 and 2 respectively, then rank of is य द तथ, 5x5 क द आ य ह ह जसक ज त मश 4 तथ 2 ह तब क ज त ह ग nswer Key Q21 The system of equations -2x + y + z = a x 2y + z = b x + y -2 z = c is consistent if सम करण क नक य -2x + y + z = a x 2y + z = b x + y -2 z = c स गत ह, य द a + b + c = 0 a + b + c = 0 a + b - c = 0 a + b - c = 0 a - b + c = 0

34 a - b + c = 0 a + b + c 0 a + b + c 0 nswer Key Q22 is a square matrix of odd order and t is its transpose then - t is always य द एक वषम क ट क वग आ य ह ह तथ t इसक प रवत ह त - t सद व ह त ह एक Singular symmetric matrix व च सम मत आ य ह Singular Skew-Symmetric matrix व च त सम मत आ य ह Non singular Skew-Symmetric matrix अ व च त सम मत आ य ह Non-singular matrix अ व च आ य ह nswer Key Q23 If, then the trace of 102 is य द त 102 क अन र ख ह ह

35 3 3 nswer Key Q24 Let be a non-zero linear transformation on a real vector space V of dimension n. Let the subspace V 0 V be the image of V under. Let K = dim V 0 < n and suppose that for some, 2 =. Then म न क एक n वम य व त वक र खक सम ट V पर अश य र खक प तरण ह म न क उपसम ट V 0 V, क अ तग त V क त ब व ह म न क K = dim V 0 < n एव म न क क छ क लय 2 = तब = 1 = 1 det = n स र णक = n is the only Eigen value of, क एकम अ भल णक म न ह There is a non-trivial subspace V 1 V such that x = 0 all x V 1 एक अत चछ उपसम ट V 1 V इस क र ह क सभ x V 1 क लय x = 0 nswer Key Q25 Let be a non-zero 3x3 matrix with the property 2 = 0. onsider the following three statements I. is not similar to a diagonal matrix II. has one non-zero Eigen vector III. can have at most two linearly independent Eigen vectors Then, the number of true statements is म न ल जय क एक 3x3 क श य ततर आ य ह ह जसम ग ण 2 = 0 ह न न ल खत त न कथन पर वच र क जय I. एक वकण आ य ह क सम प नह ह II. क एक श य तर अ भल णक स दश ह III. अ धक स अ धक द र खक वत अ भल णक स दश क रख सकत ह

36 तब, सतय कथन क स खय ह nswer Key Q26 The Eigen vector of the matrix are आ य ह क अ भल णक स दश ह ग None of these इनम स क ई नह nswer Key Q27 Let is given by म नय क क म न ह ग be a linear transformation with T(1,-1,1) = (1,0) and T(1,1,1) = (0,1). Then T(a,-b,a) एक र खक प तरण ह जह T(1-,1,1) = (1,0) तथ T(1,1,1) = (0,1) तब T(a,-b,a)

37 (a+b, a-b) (a+b, a-b) None of these इनम स क ई नह nswer Key Q28 Let T 3 म नय क T 3 be defined by T(x 1,x 2,x 3 ) = x 1 +2x 2 +3x 3. Then Ker(T) is a इस क र प रभ षत ह क T(x 1,x 2,x 3 ) = x 1 +2x 2 +3x 3 तब अ ट (T) ह ग Plane समतल ircle व त Straight line सरल र ख one श क nswer Key Q The rank of the linear transformation T T(x 1,x 2,x 3 ) = (x 2,0,x 3 ) is 3 3 defined by T(x 1,x 2,x 3 ) = (x 2,0,x 3 ) व र प रभ षत र खक प तरण T 3 3 क ज त ह ग

38 1 0 0 nswer Key Q30 hoose the correct statement from the following न न ल खत म स स य कथन क च नय imension of a subspace of a finite dimensional vector space divides the dimension of the vector space कस प र मत वम य र खक सम ट क कस उपसम ट क वम उस र खक सम ट क वम क वभ जत करत ह The trace of 5x5 identity matrix with entries in 5 is non-zero 5 क अवयव स बन 5x5 त समक आ य ह क अन र ख अश य ह त ह For real square matrices and, the matrix - can never be the identity matrix व त वक वग आ य ह तथ क लय, आ य ह - कभ भ त समक आ य ह नह ह सकत Every complex square matrix can be diagonalised य क स म वग आ य ह क वकण क त कय ज सकत ह nswer Key Q31 For an analytic function f(z) in a domain, consider the following three conditions P Re(f(z)) is constant QIm(f(z)) is constant Rrg (f(z)) is constant Which of these conditions imply that f(z) is constant? एक ड म न म कस वशल षक फलन f(z)क लय, न न त न थ तय पर वच र क जय P व त वक (f(z)) अचर ह Q क प नक (f(z)) अचर ह R क ण क (f(z)) अचर ह इनम स क न - क न स थ तय स चत करत ह क f(z) अचर ह? P and Q but not R

39 P तथ Q ल कन R नह P and R but not Q P तथ R ल कन Q नह Q and R but not P Qतथ R ल कन P नह ll P, Q and R सभ P, Q तथ R nswer Key Q32 onsider the following two statements P if f(z) and are analytic in a domain then f is constant. Q is analytic then so is. Then which of the following is correct? न न ल खत द कथन पर वच र क जय P य द f(z) तथ कस ड म न म वश ल षक ह त f अचर ह त ह Q वश ल षक ह त भ ऐस ह त ह तब न न म स क न स सह ह P is true but Q is false P स य ह ल कन Q अस य ह P is false but Q is true P अस य ह ल कन Q स य ह oth P and Q are true द न P तथ Q स य ह oth P and Q are false द न P तथ Q असतय ह nswer Key

40 Q33 The value of the integral Where c is positively oriented circle z = 2, is सम कल जह c घन मक अ भ व य त व तत z = 2 ह, क म न ह 0 0 nswer Key Q34 The mobius transformation maps z < 2 onto म बयस प तरण, z < 2 क आच छ दक त च त करत ह It self अपन पर ह w > 2 w > 2 पर w > 1/2 w > 1/2 पर w > 4 w > 4 पर nswer Key

41 Q35 The value of where c is oriented circle z = 2, is, जह c घन मक अ भ व य त व त z = 2 ह, क म न ह ग 0 0 i i nswer Key Q36 The singularity of f(z) = e sinz at z = z = पर f(z) = e sinz क व च त ह is a pole एक व a removable singularity एक अपन य व च त a non-isolated essential singularity एक अ वय त अ नव य व च त an isolated essential singularity एक वय त अ नव य व च त nswer Key Q37 The function f(z) = x 2 + y 2 + ixy, where z = x + iy, is analytic फलन f(z) = x 2 + y 2 + ixy, जह z = x + iy वशल षक ह त ह at z = 0 only

42 क वल z = 0 पर at infinity many points प र मतत कई ब द ओ पर every where सव nowhere कह भ नह nswer Key Q38, defined for z. Then which of the following is false?, z क लय प रभ षत ह तब न न म स क न स अस य ह f is entire function f सव वशल षक ह only singularities of f are poles f क क वल व च त य अन तक ( व) ह? f has infinitely many poles on the imaginary axis अ धक पत अ पर f क अन तत कई अन तक ह each pole of f is simple f क य क अन तक स ध रण ह त ह nswer Key Q39 Let f be a meromorphic function analytic at 0 satisfying for n 1. Then which of the following is wrong? म न क f एक अन तक फलन 0 पर वश ल षक ह ज n 1 क लय क स त ट करत ह तब न न म स क न स अस य (गलत) ह? f(0) = 1/2 f(0) = 1/2 f has a simple pole at z = -2

43 f पर z = -2 स ध रण अन तक रखत ह f(2) = 1/4 f(2) = 1/4 no such meromorphic function exists ऐस क ई अन तक फलन क अ त व नह ह त ह nswer Key Q40 Let G be a group of order 49. onsider the statements P & Q P The normalize of an element x( e) of G is whole group G Q The centre of G is whole Of G Then which of the following is correct? म नय क G एक 49 क ट क सम ह ह,कथन P तथ Q पर वच र क जय P G क एक अवयव x( e) क स म यक प ण सम ह G ह Q G क क प ण सम ह G ह न न म स क न स सह वक प ह oth P and Q are true द न P तथ Q स य ह P is true but Q is false P स य ह,ल कन Q अस य ह Q is true but P is false Q स य ह,ल कन P अस य ह oth P and Q are false द न P तथ Q अस य ह nswer Key Q41 Let Q be the field of rational numbers and let Q(a) be the field obtained by adjoining a Q to Q. onsider the statements P and R are field isomorphic. hoose your answer as

44 म न क Q एक प रम य स य ओ क ह तथ म न क Q(a) क a कथन P तथ Q पर वच र क जय Q म स ल न करक त ह आ ह त य क र ह त ह अपन ऐस उत तर च नय क If P and R are both true य द P तथ R द न स य ह If P is true, R is false य द P स य ह, R अस य ह If P is false, R is true य द P अस य ह, R स य ह If P and T are both false य द P तथ R द न अस य ह nswer Key Q onsider the following statements 1. Every finite integral domain is a field 2. Every finite division ring (Skew field) is a field 3.Every integral domain which is also a division ring is a field Here the number of true statements is न न कथन पर वच र क जय 1. य क प र मत प ण क य त एक ह त ह 2. य क प र मत वभ जन वलय ( बषम तल य ) एक ह त ह 3. य क प ण क य त ज क एक वभ जन वलय भ ह, एक ह त ह यह स य कथन क स य ह

45 nswer Key Q The number of group homomorphisms from the symmetric group S 3 to the additive group Z/6Z is. सम मत सम ह S 3 स घन मक सम ह Z/6Z पर सम ह सम क रत ओ क स य ह ग nswer Key Q44 Let G be a simple group of order 60. Then म न क G क ट 60 क एक सरल सम ह ह तब G has six sylow-5 subgroups G छ सल -5 उपसम ह क रखत ह G has four sylow-3 subgroups G च र सल -3 उपसम ह क रखत ह G has cyclic subgroup of order 6 G क ट 6 क एक च य उपसम ह रखत ह G has a unique element of order 2 G म क ट 2 क एक क अ त य अवयव ह nswer Key Q The total number of non- isomorphic groups of order 122 is क ट 122 क अत य क र सम ह क क ल स य ह त ह

46 nswer Key Q46 For a field F of integers modulo 11, consider the following three statements P, Q and R P x 2 +1 and x 2 +x+4 are irreducible over F Q and are isomorphic fields R Number of elements in is 121 Then choose the correct option below प ण क म डय ल 11 क F क लय, न न त न कथन P, Q तथ R पर वच र क जय P F पर x 2 +1 तथ x 2 +x+4 लघ करण य ह Q तथ त य क र ह त ह R म अवयव क स य 121 ह त ह तब नम न म सह वक प क च न only one of the statements is true दय कथन म स क वल एक स य ह only two statements are true क वल द कथन स य ह ll P, Q and R are true statements सभ P, Q तथ R स य कथन ह P and Q are true but R is false P तथ Q स य ह ल कन R अस य ह nswer Key Q47 Let X = {a, b, c} and = {,{a},{b},{a,b}, X} be a topology on X. Then म न क X = {a, b, c} तथ = {,{a},{b},{a,b},x}, X पर स थ तक ह तब (X, (X, ) is T 2 and regular ), T 2 तथ र ग लर ह त ह (X, ) is regular but not normal

47 (X, (X, (x, (X, (X, ) र ग लर ह ल कन न म ल नह ह ) is not normal but connected ) न रमल नह ह तथ स ब ह ) is not connected ) स ब नह ह nswer Key Q48 Which of the following property is not hereditary in a topological space कस स थ तक य सम ट म न न म स क न स ग ण आन व शक नह ह त ह Regularity र ग ल रट Normality न म लट Hausdorff property ह उसड फ ग ण None of these इनम स क ई नह nswer Key Q49 onsider usual topology and lower limit topology ' on the set of real numbers. Then choose the correct option below व त वक स य ओ क सम चय पर स ध रण स थ तक तथ न न स म स थ तक ' पर वच र क जय एव नम न म स सह वक प क च नय Identity mapping त समक फलन Identity mapping त समक फलन is continuous स तत ह त ह is continuous सतत ह त ह oth (Identity mapping is continuous & Identity mapping is continuous) are true (त समक फलन स तत ह त ह ) तथ (त समक फलन सतत ह त ह ) द न सह ह

48 None of these इनम स क ई नह nswer Key Q50 Let be defined by Then म न क, व र प रभ षत ह तब f is one one, continuous but not onto f एक क,सतत ह ल कन आ छ दक नह f is onto, continuous but not one one f आ छ दक, सतत ह ल कन एक क नह f is one one,onto and continuous f एक क, आ छ दक तथ सतत ह f is one one, onto but f -1 is not continuous f एक क तथ आ छ दक ह ल कन f -1 स तत नह ह nswer Key Q51 Let P be a continuous function on y 1 and y 2 of the OE and W, the Wronskian of two linearly independent solutions Let W(1) = a, W(2) = b and W(3) = c, then म न क P, पर एक स तत फलन ह तथ W स.अ.सम. क द र खक य वत हल y 1 तथ y 2 क र स कयन ह, य द W(1) = a, W(2) = b and W(3) = c, ह त a < 0 and b > 0 a < 0 तथ b > 0 a < 0, b < 0 and c > 0 a < 0, b < 0 तथ c > 0

49 a < b < c or a > b > c a < b < c अथव a > b > c 0 < a < b and b > c > 0 0 < a < b तथ b > c > 0 nswer Key Q52 Solution of OE ysin2x dx + (-y 2 cos 2 x)dy = 0 is स.अ.सम. ysin2x dx + (-y 2 cos 2 x)dy = 0 क हल ह 3ycos 2 x + y 3 = c, c is constant 3ycos 2 x + y 3 = c, जह c अचर ह 3ycos2x + 2y 3 = c, c is constant 3ycos2x + 2y 3, जह c अचर ह 3y 2 sin2x -2y 3 = c, c is constant 3y 2 sin2x -2y 3,जह cअचर ह 3ycos2x + y 3 = c, c is constant 3ycos2x + y 3 = c, जह c अचर ह nswer Key Q53 If y 1 (x) and y 2 (x) are solution of OE y + xy + (1-x 2 )y = sinx then which of the following is also its solution य द y 1 (x) तथ y 2 (x) स.अ. सम.OE y + xy + (1-x 2 )y = sinx क हल ह त न न म स भ क न इसक हल ह ग y 1(x) + y 2 (x) y 1 (x) + y 2 (x) y 1(x) - y 2 (x) y 1 (x) - y 2 (x) 2y 1(x) - y 2 (x) 2y 1 (x) - y 2 (x) y 1(x) - 2y 2 (x) y 1 (x) - 2y 2 (x) nswer Key Q54 If y 1 (x) and y 2 (x) are solutions of y" + x 2 y' + (1-x)y = 0 such that y 1 (0) = 0, y 1 '(0) = -1 and y 2 (0) = -1,

50 y 2 '(0) = 1, then the Wronskian W(y 1,y 2 ) on य द y" + x 2 y' + (1-x)y = 0 क हल y 1 (x) तथ y 2 (x) इस क र ह क y 1 (0) = 0, y 1 '(0) = -1 तथ y 2 (0) = -1, y 2 '(0) = 1 तब र स कयन W(y 1,y 2 ), पर is never zero कभ श य नह ह त ह is identically zero सम न प स श य ह त ह is zero only at finite number of points क वल प र मतत स य क ब द ओ पर श य ह त ह is zero at countably infinite number of points गणन य अन त ब द ओ पर श य ह त ह nswer Key Q55 The PE y 3 u xx (x 2 + 1)u yy = 0 is आ.अ.सम. y 3 u xx (x 2 + 1)u yy = 0 ह Parabolic in {(x,y) x < 0} {(x,y) x < 0} म परवल य hyperbolic in {(x,y) y > 0} {(x,y) y > 0} म अ तपरवल य elliptic in 2 2 म द घ व तत य parabolic in {(x,y) x > 0} {(x,y) x > 0} म परवल य nswer Key Q56 If (x) be a solution of OE, xy" + y' + xy = 0, then which of the following differential equation is satisfied by the function x 1/2 (x) य द (x) स.अ.सम. xy" + y' + xy = 0 क हल ह त फलन x 1/2 (x), नमन म स क न स अवकल सम करण क स त ट करत ह. x2 y" + (4x 2 + 1)y = 0 x 2 y" + (4x 2 + 1)y = 0 x 2 y" + (4x 2-1)y = 0

51 x 2 y" + (4x 2-1)y = 0 x2 y" + (x 2 + 1/4)y = 0 x 2 y" + (x 2 + 1/4)y = 0 x2 y" + (x 2 1/4)y = 0 x 2 y" + (x 2 1/4)y = 0 nswer Key Q57 particular solution of the PE is आ.अ.सम क एक वश ष हल ह nswer Key Q Using Euler's method taking step size = 0.1, the approximate value of y obtained corresponding to x = 0.2 for the initial value problem dy/dx = x 2 + y 2, y(0) = 1 is पगन प = 0.1 ल कर आयलर क व ध क य ग कर र भक म न सम य dy/dx = x 2 + y 2, y(0) = 1 क लय x = 0.2 क स गत y क स नकट म न ह

52 nswer Key Q function y(x) is given as y(1) = 3.24, y(6) = 8.43 and y(10) = Using Lagrange s interpolation, the approximate value of y(4) is फलन y(x) क लय दय ह क y(1) = 3.24, y(6) = 8.43 तथ y(10) = 24.6, तब ल ज अ तव शन क य ग करन पर y(4) क स नकट म न ह nswer Key Q60 function f(x) is given as f(1.0) = 2.32, f(1.2) = 4.68 and f(1.4) = Using Simpson's rule, the value of is एक फलन f(x) क लय, f(1.0) = 2.32, f(1.2) = 4.68 तथ f(1.4) = 6.48 दय गय ह स पसन क नयम क य ग करक क म न ह ग

53 nswer Key Q61 onsider the functional I, y(0) = 1, where y 2 ([0,1]). If y extremizes I, then फलनक I, y(0) = 1, जह y 2 ([0,1]) पर वच र क जए,य द I क y चर मत करत ह,तब nswer Key Q62 If 1, 2 be the characteristic numbers and f 1 and f 2 the corresponding Eigen functions for the homogeneous integral equation statement, consider the following Then य द समघ त सम कल सम करण क अ भल णक म न 1 एव 2 ह तथ f 1 एव f 2 इनक अ भल णक फलन ह त न न कथन पर वच र क जय

54 both P and Q are true द न P तथ Q स य ह P is true, Q is false P स य ह, Q अस य ह P is false, Q is true P अस य ह, Q स य ह oth P and Q are false द न P तथ Q अस य ह nswer Key Q63 The integral equation has सम कल सम करण रखत ह no solution क ई हल नह unique solution अ दत य हल infinitely many solution अन तत कई हल exactly two solutions ठ क द हल nswer Key Q64 If L and V are respectively the Lagrangian function and potential energy of a conservative holonomic dynamical system, then य द कस स र ह ल न म ग तक य त क ल ज फलन तथ थ तज ऊज मश L तथ V ह त L = constant

55 L = अचर L + V = constant L + V = अचर L + 2V = constant L + 2V = अचर L V = constant L V = अचर nswer Key Q65 onsider the following three statements i. Hamilton's principle follows from the 'lembert's principle ii. Hamilton's principle follows from Lagrange's equations. iii. Newton's second law of motion follows from the Hamilton's principle Then the number of true statements is न न त न कथन पर वच र क जय i. ह म टन स त डएल वट स त क अन ग म ह ii. ह म टन स त ल ज सम करण क अन ग म ह iii. य टन क वत य नयम ह म टन स त क अन ग म ह तब स य कथन क स य ह nswer Key

56 Q66 For a simple pendulum of length l and mass m, consider the following statements i. The Lagrangian function is ii. The equation of motion is here, choose ल ब ई l तथ यम न m क एक स ध रण द लक क लय न न कथन पर वच र क जय i. इसक ल जयन फलन ह ii. यह च नय इसक ग त क सम करण ह both (i) and (ii) are true द न (i) तथ (ii) स य ह (i) is true but (ii) is false (i) स य ह ल कन (ii) अस य ह (i) is false but (ii) is true (i) अस य ह ल कन (ii) स य ह (i) and (ii) are both false (i) तथ (ii) द न अस य ह nswer Key Q67 If x is a Poisson random variate with mean 3, then P ( x 3 <1) will be य द x म य 3 क स थ एक व स य छक वचर ह, त P ( x 3 <1) ह ग

57 nswer Key Q68 The mean and variance of the number of defective items drawn randomly one by one with replacement from a lot are found to be 10 and 6 respectively. Then distribution of the number of defective items is य द ल ट स य छकत प न सथ पत करत ह य एक-एक करक नक ल गय द ष य त मद क स य क म य तथ वचरण मश 10 व 6 ह त द ष य त मद क स य क ब टन ह Poisson with mean 10 व स क म य 10 क स थ inomial with n = 25 and p = 0.4 n = 25 तथ p = 0.4 क स थ वपद Normal with mean 10 and variance 6 म य 10 तथ वचरण 6 क स थ स म य None of these इनम स क ई नह nswer Key Q69 3/216 3/216 6/216 6/216 9/216 9/216 From the six letters,,,, E and F, three letters are chosen at random with replacement. What is the probability that either the word or the word can be formed from the chosen letters? छ अ र,,,, E तथ F म स य छकत त न अ र प न थ पन क स थ च न ज त ह च न गय अ र स श द य श द क रचन कर सकन क यकत य ह

58 12/216 12/216 nswer Key Q t a doctor's clinic patients arrive at an average rate of 10 per hour. The consultancy time per patient is exponentially distributed with an average of 6 minutes per patient. The doctor does not admit any patient if at any time 10 patients are waiting. Then at the steady state of this M M 1 R queque the expected number of patients waiting is एक ड टर क ल नक म 10 त घ ट क म य दर स र गय क आगमन ह त ह तर ग पर मश क ल म ध य 6 म नट तर ग य त चर घ त कत ब टत ह कस भ समय य द 10 र ग त कर रह ह त ड टर कस अ य र ग क व श क अन म त नह द त ह, त थर अव थ म इस M M 1 R कत र म य शत र गय क स य ह nswer Key Q71 Suppose X has density, where > 0 is unknown. efine Y as follows Y= k if k X < k + 1, k = 0, 1, 2,..., then the distribution of Y is म न क X क घन व ह जह > 0 अ त ह Y क प रभ ष न नवत कर Y= k if k X < k + 1, k = 0, 1, 2,..., त Y क ब टन ह Normal स म य inomial वपद Poisson व स

59 Geometric य मत य nswer Key Q72 If a primal linear programming problem has degenerate optimal solution then its dual problem has य द एक आ य र खक मन सम य अप ट इ टतम हल रखत ह त उसक व त सम य क हल ह ग lternative optima solution वक प य इ टतम हल degenerate optimal solution अप ट इ टतम हल no feasible solution क ई स स गत हल नह no optimal solution क ई इ टतम हल नह nswer Key Q73 If (1, 0, 3) is an optimal solution of LPP Maximize z = c 1 x + c 2 y + c 3 z Subject to x + y + z 4, x 2, z 3, 3x + z 7, x, y, z 0. Then य द र खक मन सम य अ धकतम कर z = c 1 x + c 2 y + c 3 z तब ध x + y + z 4, x 2, z 3, 3x + z 7, x, y, z 0 क (1, 0, 3) एक इ टतम हल ह त c 1 c 2 c 3 c 1 c 2 c 3

60 c 3 c 2 c 1 c 3 c 2 c 1 c 2 c 1 c 3 c 2 c 1 c 3 c 2 c 3 c 1 c 2 c 3 c 1 nswer Key Q74 onsider the LPP Maximize x 1 + x 2 Subject to x 1-2x 2 12, x 2-2x 1 12, x 1, x 2 0. Then र खक मन सम य अ धकतम कर x 1 + x 2 तब ध x 1-2x 2 12, x 2-2x 1 12, x 1, x 2 0.तब The LPP admits an optimal solution र खक मन सम य, इषटतम (ऑ टमल) हल रखत ह The LPP admits no feasible solution र खक म सम य, क ई स स गत हल नह रखत The LPP is unbounded र खक मन सम य अप रब ह None of these इनम स क ई नह nswer Key Q75 The optimal table for the primal LPP Maximize z = 6x x x 3-6x 4 Subject to x 1 + x 2 + x 3 = 4, x 1 + 4x 2 + x 4 = 8 and x 1, x 2, x 3, x 4 0 is

61 If y 1 and y 2 are the dual variables corresponding to the first and second constraints then their values in the optimal solution of the dual problem are respectively आ य र खक मन सम य अ धकतम कर z = 6x x x 3-6x 4 तब ध x 1 + x 2 + x 3 = 4, x 1 + 4x 2 + x 4 = 8 तथ x 1, x 2, x 3, x 4 0 क इष टतम त लक ह - य द थम तथ वत य आ य तब ध क स गत व त सम य क चर y 1 तथ y 2 ह त व त सम य क इ टतम हल म उनक म न मश ह ग 0 and 6 0 तथ 6 12 and 0 12 तथ 0 6 and 3 6 तथ 3 4 and 4 4 तथ 4 nswer Key

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