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Pune University - FY BSc MATHEMATICS - III (Old Course)

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Total No. of Questions : 5] P221 [Total No. of Pages : 4 [3717] - 53 F.Y. B.Sc. MATHEMATICS Analytical Geomery and Differential Equations (Paper - III) (Old Course) Time : 3 Hours] [Max. Marks : 80 Instructions to the candidates : 1) All questions are compulsory. 2) Figures to the right indicate full marks. Q1) Attempt all the subquestions: a) [16] Find the centre of the conic x2 + 12xy 4y2 6x + 4y + 9 = 0. b) Find the equation of the plane making intercepts 4, 2, 3 on the coordinate axes. c) Find the equation of the sphere described on the join of the points A(2, 4, 3) and B(1, 0, 2) as a diameter. d) Find the angle through which the axes should be rotated so as to remove the xy term in the equation 4 x 2 3xy + y 2 = 5 . e) Find the order and degree of the differential equation d 3y dy 2 + 4 dx = 3 + x . dx dx f) Solve p = log (px y). g) Find orthogonal trajectories of the family ex + e y = c where c i s a parameter. h) Find integrating factor of the differential equation (2x2y 3y3 + 6x3) dx + (3xy2 2x3)dy = 0 P.T.O. Q2) Attempt any four of the following: [16] a) The direction ratios of two lines are 1, 2, 1 and 3, 1, 2. Find the direction cosines of a normal to the plane containing them. b) If due to translation of axes, the expression 2 x2 3 y2 + 4 y + 5 i s transformed into 2 x 2 3y 2 + 4 x 8 y + 3 then find the co-ordinates of new origin with respect to old one. c) Find the equation of the tangent plane to the sphere x2 + y2 + z2 6x + 5y + 4z 5 = 0 at the point (2, 1, 3). d) Show that the line x l = y z = m n intersects the sphere x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 in atmost two points. e) Prove that a general equation of first degree in x, y and z represents a plane. f) Find the symmetric form of equations of the line x + y + z 1 = 0, 4x + y 2z + 2 = 0. Q3) Attempt any two of the following: [16] a) Prove that the general equation of second degree in x and y represents a conic. b) i) Find the direction ratios of two lines whose direction cosines are connected by the relations l m + n = 0, l2 + m2 5n2 = 0. ii) Find the equation of the plane to which the foot of the perpendicular from the origin is (3, 2, 5). i) Find the co-ordinates of foot of the perpendicular from the point x y+1 z 2 (1, 3, 2) to the line = . = 2 3 1 ii) Find the angle between the line c) plane ax + by + cz + d = 0. [3717] - 53 2 x x1 y y1 z z 1 and the = = l m n i) Find the equation of the sphere containing the circle x2 + y2 + z2 = 9, 2x + 3y + 4z = 5 and passing through the point (1, 2, 3). ii) d) Show that the two spheres x2 + y2 + z2 2x 6y 15 = 0 and 5x2 + 5y2+ 5z2 10x + 26y + 42z + 107 = 0 touch each other. Q4) Attempt any four of the following: [16] a) From the differential equation by eliminating arbitrary constants from y = ex (A cosx + B sinx). b) Define homogeneous differential equation and explain the method of solving it. c) Solve d) Solve (xy2 + 2x2y3)dx + (x2y x3y2)dy = 0 e) Explain the method of solving the differential equation dy x y = . dx x y + z pn + p1, pn 1 + p2 pn 2 + - - - - pn 1 p + pn = 0, dy and p1, p2, - - - - pn are functions of x and y, which is dx solvable for x. where p = f) A body is heated to 110 C and placed in air at 10 C. After one hour its temperature was noted 60 C. Find how much additional time will it require to cool to 30 C. Q5) Attempt any two of the following: [16] a) Define linear differential equation and explain the method of solving it. 2 dy Hence solve (1 + x ) + 2 xy 1 = 0 . dx b) i) Solve xp2 3yp + 9x2 = 0. ii) Solve xy(p2 + 1) + (x2 + y2)p = 0 [3717] - 53 3 Find the orthogonal trajectories of the family of curves x2 + 2y2 = c, where c is a parameter. The population of a town increases at a rate proportional to the population at that time. If the population increases from 40 lakhs to 60 lakhs in 40 years, wheat will be the population in another 40 years? i) Sole (y2 + 2xy + 6x) dx (2 2xy - x2) dy = 0 ii) d) i) ii) c) Solve 6y2dx x (2x3 + y)dy = 0. Y [3717] - 53 4

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Additional Info : F.Y.B.Sc. MATHEMATICS Analytical Geomery and Differential Equations (Paper - III) (Old Course), Pune University
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