2010 Andhra University M Tech Automobile I B.Sc Maths March 2010 Question paper
[BS183/B182] PART IIMATHEMATICS (Common Core Scheme) Paper I DIFFERENTIAL EQUATIONS AND SOLID GEOMETRY (Revised Syllabus w.e.f. 200809) (Common for Students studying I B.Sc./ I B.A.) Time: Three hours Maximum : 100 marks Follow the instructions carefully, given in each Section. SECTION A – (4x15 = 60 marks) Answer all the FOUR questions. Each question carries 15 marks. 1. (a) (i) Solve xy3+ydx+2x2y2+x+y4dy=0. (ii) Solve d2ydx2+dydx tanx=secx+cosx. Or (b) (i) Find the orthogonal trajactories of the family of cardioids r=a(1cos?), where 'a' is the parameter. (ii) Solve y2logy=xyp+p2 2. (a) (i) Solve (D23D+2)y=sin(ex) (ii) Solve D23D+2y=x3 Or (b) (i) Solve x2d2ydx23xdydx+4y=2x2 (ii) dxdt=2txy=1, tdydt+x+5y=t2 3. (a) (i) P is a point such that the sum of the squares of its distances from the planes x+y+z=0, x+y2z=0, xy=0 is 5. Show that the locus of P is x2+y2+z2=5. (ii) Prove that the lines x+11=y+12=z+13 and x+2y+3z8=0, 2x+3y+4z11=0 are intersecting and find the point of their intersection. Find also the equation of the plane containing them. Or (b) (i) Show that the two circles x2+y2+z2y+2z=0, xy+z=2 and x2+y2+z2+x3y+z5=0, 2xy+4z1=0 lie on the same sphere and find its equation. (ii) Find the limiting points of the coaxial system of spheres x2+y2+z220x+30y40z+29+?2x3y+4z=0. 4. (a) (i) Find the enveloping cone of the sphere x2+y2+z2+2z4y=0 with its vertex at (1,1,2). (ii) Find the equation of the quadratic cone which touches the three coordinate planes and the planes x+y+z=0 and 2xyz=0. Or (b) (i) Prove that the necessary and sufficient condition that the planelx+my+nz=p may be a tangent plane to the conicoid lx+my+nz=p may be a tangent plane to the conicoid ax2+by2+cz2=cz2=1(abc?0) is p2=l2a+m2b+n2c>0. (ii) Find the locus of the point from which three mutually perpendicular tangent lines can be drawn to the paraboloid ax2+by2=2z. SECTION B(5X4=20 marks) Answer any FIVE out of Eight questions. Each question carries 4 marks. 5. Solve: 3y7x=7dx+7y3x+3dy=0. 6. Solve: dydx+2x1+x2y=11+x22, given that y=0 when x=1. 7. Solve: D2+D+1y=sin2x. 8. Solve: d2ydx2cotxdydx(1cotx)y=exsinx 9. Find the bisecting plane of the acute angle between the planes 3x2y6z+2=0, 2x+y2z2=0. 10. Find the equation to the plane containing the line x12=y+11=z34 and is perpendicular to the plane x+2y+z2=0. 11. Show that the spheres x2+y2+z264=0, x2+y2+z212x+4y6z+48=0 touch internally at the point 487,167,247. 12. Find the equations to the tangent planes which pass through the line x3=y33=z1 and touching x26+y23+z22=1. SECTION C(10X2=20 marks) Answer all the TEN questions. Each question carries 2 marks. 13. Solve: xex2+y=ydydx. 14. Solve: xdy=y+xcos2yxdx. 15. Solve: y=px+1+p2. 16. Solve: D2+1y=0. 17. Find the particular integral D12y=x2. 18. Find the equation to the plane through the line of intersection of xy+3z+5=0 and 2x+y2z+6=0 and passing through (3,1,1). 19. Find the area of the triangle with vertices (1,1,1),(2,1,1),(1,2,3). 20. Find the equation to the sphere with centre x1,y1,z1 and radius 'a'. 21. Find the point of intersection of the line x84=y1=z1 and the sphere x2+y2+z24x+6y2z+5=0. 22. Define enveloping cylinder.
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