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19th December 2014, 08:48 AM
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| Re: Allahabad University MSc Entrance Exam Maths Question Paper
As you want to get MSc Entrance Exam Maths Question Paper of Allahabad university so here I am giving you some questions of that paper: 1. If a, B and,? are the roots of 13 + asz +bt * c: 0 then the value of. a2 + 02 + l2 is tAl a2 - 2b. tBl b2 - 2c. tcl c2 + 2a. tDl b2 + 2c. 2. Let / : IR -+ lR and f (r) : l, - 1l + lr * 21. Let,s1 : {r l / is continuous at r} and 52: {c | / is differentiable at r}. Then tA] Sr : lR, Sz : lR. [B] & : In, Sz : R' \ {1,2}. [C] Sr:lR\{1,2}, Sz:lR' [D] Sr:R,\{1,2}, Sz:lR\{1,2}' 3. Consider the following statements 51: If / is Riemann integrable in [0,1] then /2 is Riemann integrable in [0,1]. ^92: If /2 is Riemann integrable in [0,1] then / is Riemann integrable in [0,1]. Then tA] ,5r is true but ^92 is false. tB] 51 is false but ,92 is true. tC] both,Sr and ,Sz are false. tD], both & and ,52 are true. 4. The tunction /(t) : sin(r) * cos(r) is [A] increasing in l0,rl2l. [B] decreasing in l},nl2l. [C] increasing in l},nlal and decreasing in ltrla,nl2l' [D] decreasing in l},nl4 and increasing in lnla,rl2l. 5. Let Gr and Gzbetwo finite groups with lGrl : 100 and lG2l :25.If f : Gt - Gz is a surjective group homomorphism, then tAl lKer(f)l:2. [B] lKer(f)l: a. tcl lKer(/)l :5. tDl lKer(f)l :10' 6. Let {p"} be a strictly increasing sequence of prime numbers and let r,,  -1)e"*'(t* 1)tn*' tAl ;j8"" : -112. tBl j*", - -1. tcl j$r,,: 1. [D] ,,!gr" does not exist. 7. Let V be avector space of dimension n and {rr,ur,... ,?r,} be a basis of V. Let o € Sn and ? :V *--*+V be a linear transformation definedby T(u6): uoe).Then lA] 7 is nilpotent. tB] ? is one-one but not onto. lC] T is onto but not one-one. tD] ? is an isomorphism. 8. Let G be agroup and a € G be aunique element of order n where n) 7. Let Z(G) denote the center of the group G. Then lAl o(G) : n. tBl o(z(G)) > 1. tcl z(G) :6. tDl G : Sz. oo 9. If the series | (sinr)" converges to the value (4 + 2\/3) for some value of r in (0,r12),then"t:fe value of r is tAl rl3. tBl rl4. ICI rl5. tDl rl1. 10. If rn and M arcrespectively the greatest lower bound and the least upper bound of the ser s: { Tl:. " > o} tr,"n lr*2' ) lAl m€S, IvI#5. tBl rn#S,M#5. tcl rn4S,MeS. [D] m€S,MeS. 11. The value "f lTe (cosr;(1/'i"'") is [A] exp(-l). IBI exp(l). tCl exp(-1l2). [D] exp(1/2). 12. The graphs of the real valued functions f (r):2log(r) and 9(r) : log(2r) [A] do not intersect. tBl intersect at one point only. tC] intersect at two points. tD] intersect at more than two points. 13. The points of continuity of the function / : R. -+ IR defined by r/_-\ f lr' - tl, if r is irrational J(t/ : t o, if s is rational are tA] tr:-L,tr:A,r:1. iB] n:-I,r:1. lcl r: -7, r:0. tD] ,r:0, r: t. 14. The smallest positive integer n such that 5n - 1 is divisible by 36 is lAl 2. tBl 3. [c] 5. tDl 6. 15. Let f(r) : r5 + a1r4 * a2n7 * a3r2. Suppose /(-1) ) 0 and /(l) < 0 then lA] / has at least 3 real roots. tB] / has at most 3 real roots. lC] / has at most 1 real root. tDl all roots of / are real. 16. Let {u,u} be a linearly independent subset of a real vector space V. Then which of the following is not a linearly independent set? [A] {r, u * u}. [B] {u + t/1u,, - ,/zr}. tcl {a,2u - ul2}. iDl {2u + u, -4u - 2u}. L7. Let V be avector space of 2x2 realmatrices. Let A: I t ' I ,n.r, the dimension L 1 -1 I of the subspace spanned by {4, A', A',Aa} is lAl 2. [B] 3. [c] 4. [D] 5. 18. Let A € MB(Q). Consider the statements P: Matrix A is nilpotent. Q: A3 :0. Pick up true statements from the following. iAl P+Q. tBl Q+PandP+A. [C] P # Q andQ + P. tDl None of [A], [B], [C] is true. 19. Consider the statements & : 1- 1+ 1 - 1+ 1* 1+...: *1. Sz, #- 1 - 2 +22 - 23 +.... Then tA] ,S1 is true but ,Sz is false. iB] s1 is false but ^g2 is true. lC] both ^9r and ,s2 are true. IDI both s1 and ^gz are false. 20. Let ts I n1 tA] there exists a unique continuous function / such that F'(r  : y5 for alI i.tB] there exists a unique differentiabie function / such that F(r : gi for all e.IC] there exists a unique n times differentiable function / such tltat F(r6) : y6 for all i. lD] there exists a unique polynomial function / of degree n such that F(r1) : yi for all i. 21. Solution of the differential equation g" - n (y,)' :0, subject to the boundary condi_ tions g(0) :0, g'(0): -1 is r:, ./ r \ tA] a : | --7 6n-t 16,)* b, where a and b are arbitrary constants. tBl y: -t/it*,-' (fr) lt ,/ r \ tc] y : \f l tan-' \d * b, where a and b are arbitrary constants. tDl a: -iran-l (Jr") 22. Let V be the vector space of all continuous functions on lR. over the field IR. Let S : {lsl, l" - 11, lx - Zli. tA] ,9 is linearly independent and does not span V. lB] ,9 is linearly independent and spans V. tC] ,S is linearly dependent and does not span V. tD] ,S is linearly dependent and spans I,/. 23. 10 red balls (all alike) and 10 blue balls (all alike) are to be arranged in a row. If every arrangement is equally likely, then the probability that the balls at two ends of the arrangement are of the same colour is tAl eaual to ]. iBl eoual to |. tcl less than ] IDI greater than ;. 24. 3 studeuts are to be selected to forrn a cornmittee from a class of 100 students. The chances that the tallest student is one among them is lAj less than 5%. tBl 6 to 10%. tcl r5Yo. tDl 50%. 25. Let f-be a smooth vector valued function of a real variable. Consider the two statements 51 : div curl/: g. ,91 : grad div /:0. Then lA] both Sr and ,92 are true. tC] ,91 is true but 52 is false, |
