2010 Andhra University M Tech Automobile I B.Sc Maths -March 2010 Question paper
(Common Core Scheme)
Paper I- DIFFERENTIAL EQUATIONS AND SOLID GEOMETRY
(Revised Syllabus w.e.f. 2008-09)
(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.
(b) (i) Find the orthogonal trajactories of the family of cardioids r=a(1-cos?), where 'a' is the parameter.
(ii) Solve y2logy=xyp+p2
2. (a) (i) Solve (D2-3D+2)y=sin(e-x)
(ii) Solve D2-3D+2y=x3
(b) (i) Solve x2d2ydx2-3xdydx+4y=2x2
(ii) dxdt=2tx-y=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+y-2z=0, x-y=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+3z-8=0, 2x+3y+4z-11=0 are intersecting and find the point of their intersection. Find also the equation of the plane containing them.
(b) (i) Show that the two circles x2+y2+z2-y+2z=0, x-y+z=2 and x2+y2+z2+x-3y+z-5=0, 2x-y+4z-1=0 lie on the same sphere and find its equation.
(ii) Find the limiting points of the coaxial system of spheres x2+y2+z2-20x+30y-40z+29+?2x-3y+4z=0.
4. (a) (i) Find the enveloping cone of the sphere x2+y2+z2+2z-4y=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 2x-y-z=0.
(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: 3y-7x=7dx+7y-3x+3dy=0.
6. Solve: dydx+2x1+x2y=11+x22, given that y=0 when x=1.
7. Solve: D2+D+1y=sin2x.
8. Solve: d2ydx2-cotxdydx-(1-cotx)y=exsinx
9. Find the bisecting plane of the acute angle between the planes 3x-2y-6z+2=0, -2x+y-2z-2=0.
10. Find the equation to the plane containing the line x-12=y+1-1=z-34 and is perpendicular to the plane x+2y+z-2=0.
11. Show that the spheres x2+y2+z2-64=0, x2+y2+z2-12x+4y-6z+48=0 touch internally at the point 487,-167,247.
12. Find the equations to the tangent planes which pass through the line x3=y-3-3=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 D-12y=x2.
18. Find the equation to the plane through the line of intersection of x-y+3z+5=0 and 2x+y-2z+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 x-84=y1=-z-1 and the sphere x2+y2+z2-4x+6y-2z+5=0.
22. Define enveloping cylinder.
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