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(), , Printed Pages:4], , (ii) Questions, , Roll No. ...seseeeeseeeeeeeeeesee*e*eeeee, , 8], , Sub. Code:, , 0 5 42, , Exam.Code: 0 0 0 6, B.A./B.Sc.(General) 6th Semester, Examination, 1047, , MATHEMATICS, , Paper:1 (Analysis-1), [Max. Marks:30, , Time:3 Hours], , Note-Attempt five questions in all, selecting at least two, from each Unit. All questions can equal marks., Unit1., , (a) Let A = {(x,y):-1sxs 1,-1 sys 1}, , Andf:AR bedefined by:, y if x is rational, , fxy=o if ix irrational e r . i n, exists and the, Showthat inydy Jdx, x, , e, , other repeated integral is not defined., , (1)

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(b) Change the order of integration and hence evaluate, , evaluatef (x+y) dx dy., 2., , er., , (a) Find the area of the region bounded between, the parabolas y =4 ax and x'= 4 ay, where, a>0., , (b) Evaluate, , JJJ xyztx'+y+ zi) dx dydzover, , x*+y+z'=a, , in, , positive octant., , 3. (a) Showthat JJ x +y+2)'dx dy dz over the, regoin defined by x 2 0, y 2 0,z2 0, , x+y+z=1 is, , (b) Showthat F= (2 xy+z)i +x'j+3xz?kis, a conservative force field. Find the scalar potential., Also find the work done in movingan object in, thisfield from (1,-2, 1) to (3, 1,4)., 4., , er, , (a) State and prove Gauss's divergence theorem., (b) Verify Stokes' theorem for the vector point, function F=zitxj+yk,wherecurve is the unit, , circle in the XY plane bounding the, , semi-sphere z=y1 -x*- y'., (2), , 2, , 2

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Unit-ll, , 5., , in, , (a) Show that sequence {f,(X)} where, f(x), , =, , i s uniformly convergenton, X+n, , 0, k], where k is any positive real number but is, , not uniformly convergent on [0,o]., T, , (b) Show that the series 2, , CoS n x, , converges, , n, , n, , uniformly in (0, 27t)., 6., , (a) Test for uniform convergence and term by term, integration of the series:, , X, , (n+xT, , 0<xS1, , (b) Show that the series n?+nx, uniformly convergent fol all x, , S, , and.it can be, , differentiated term by term., e r . i n, , 7., , (a) Prove that, converges, , the series, for -1 <, , x<1., , ed, (3)

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(b) Prove that:, , sin x=X+23.5...2n-1), Per.in, xn+1, , easypa, , 2n+1 X E - 1 , ] ., , Hence deduce that=1 .3t24 5, 8., , (a) Find a series of sines and cosines of multiples of, , xwhich represents x +x in (-T, n), Hence show, , that 1 1+t, , 0sxs, (6) Iffix) =F, , 0;SxS, , xs, then show that:, , fix)= cos x, V3, , Hence deduce that, , ed, , cos5x Cos7x, , 2 5+7t, (4), , n, , ****