JAIPUR NATIONAL UNIVERsITY, JAIPUR Total Printed Pages: Roll No.: BSC 103 sCHOOL OF ENGINEERING AND TECHNOLOGY B.Tech. (All Branches except BT/FT) ISEMESTER END EXAMINATION: DECEMBER 2019 ENGINEERING MATHEMATICS II Max. Marks: 70 Time: 3 Hrs. Attempt all questions & each carries equal marks. Q.1 a) Show that the four asymptotes of the curve: (x-y) (y4x)+ 7 6x-5x'y-3xy +2y- x + 3xy - I = 0, cut the curve again in eight points which lie on the circlex+y'= 1. 1 b) Trace the curve: x+y=3axy. OR 1 a) Find the equation of the cubic curve whose asymptotes are xta-0, y-a=0, and x+yta=0 & which touches the axis of x at the origin & passes through the point (-2a, -2a). b Show that the points of inflexion of the curve: yf=(x-a)(x-b) lie on the straight line 3x+a=4b. 0.2 a) State & prove the Euler's theorem for second derivatives. 1 7 b) Ifu = log (x+y+z-3xyz). show that Determine the optimal solutíon for the following NLPP & b) um -s check whether it maximizes or minimizes the objective (+y+z)2 OR function: Ifu-fx,y), where x =r cos 6 & y= rsin 6, prove that the Optimize Z=x- 10x t xg* - 6x2 + x - 4x. 14 equation 0. transferred into 19 Subject to Xj +X2+X3 7 rarr2 a82 0 X1, X2, X3 20 The period of a simpie penauum with small oscillations a) oiven by T= 2T V/9). Ul is computed using I = 9m, g = 1 0.4 a) Find the volume of the solid generated by revolving the al2 the approximate error in T if the true values are I = astroid x + y8 = 1 about x-axis. 907 m &g = 9.81 m/s". Find also the percentage error in T. Solve by change of order of integration: b) b) Write the short note on: dxdy Error approximations OR ii) Positive & Negative definite matrix a) ca(1+cos 0) rdedv Evaluate: i) ii) Necessary & sufficient condition for existance of c1+x2 dxdy 1+x2+y2 i) Maxima & Minima 1 b Prove the relations: iv) Hessian matrix (m+1(n+1 i) Scosme sin" @de = 2( 2 OR a) Find the extreme points of the function f(x.y) = x' +y' +3x* ii) n) (n)=T_ sin n t +12y+14 & determine their nature also. Q.5 Verify Green's theorem in plane for 14 .Cr3-2xy)dx + (x*y + 3)dy Where c is the boundary of the region defined by y* =8x &x = 2. OR 14 Verify Stoke's theorem for F= (x2+y2)i - 2xy taken round the rectangle bounded by the lines x =ta, y =0 &y=b.