MATHEMATICS PAPER I SYLLABUS DIFFERENTIAL EQUATIONS & ANALYTICAL SOLID GEOMETRY Differential Equations Unit - 1 Differential Equations of First Order and First Degree Linear Differential Equations Differential Equations Reducible to Linear Form Exact Differential Equations Integrating Factors Change of Variable Total Differential Equations Simultaneous Total Differential Equations Equations of the form dx/P = dy/Q = dz/R ( Ii ) Method of Grouping ( ii ) Method of Multipliers Differential Equations of the First Order but not of the First Degree Equations Solvable for p Equations Solvable for y Equations Solvable for x Equations that do not contain x ( or y) Equations Homogeneous in x and y Equations of the First Degree in x and y - Clairout's Equation Unit - 2 Higher Order Linear Differential Equations Solution of Homogeneous Linear Differential Equations of Order n with Constant Coefficients Solution of Non-homogeneous Linear Differential Equations with Constant Coefficients by means of Polynomial Operators. ( ii ) When Q(x) = bxk and P(D) = D - a0, a0 0. ( ii ) When Q(x) = bxk and P(D) = anDn + an-1 Dn-1 + …. + a1 D. ( iii ) When Q(x) = beax ( iv ) When Q(x) = b sinax or bcosax ( v ) When Q(x) = eax V, where V is a function of x ( vi ) When Q(x) = beax and P(a) = 0 ( vii ) When Q(x) = x V, where V is a function of x Method of Undetermined Coefficients Method of Variations of Parameters Linear Differential Equations with Non-constant coefficients The Cauchy-Euler Equation System of Linear Diffrential Equations Solution of a System of Linear Equations with Constant Coefficients An Equivalent Triangular System Degenerate Case: p1(D)p4(D) - p2(D) p3(D) = 0 (Scope as in Differential Equations and their Applications by Zafar Ahsan, published by Prentics-Hall of India Private Limited, New Delhi.) Solid Geometry Unit - 3 The Plane Equation of a plane in terms of its intercepts on the axes Equation of a plane through the given points Systems of planes,two sides of a plane,length of the perpendicular Bisectors of angles between two planes Combined equation of two planes Orthogonal projection on a plane The Right Line Equation of a line Angle between a line and a plane The condition that given line may lie in a given plane The condition that the two given lines are coplanar Number of arbitrary constants in the equations of straight line,Sets of conditions which determine a line Shortest distance between the two lines,The length and the equations of the line of shortest distance between two straight lines Length of the perpendicular from a given point to a given line Intersection of three planes,Triangular Prism The Sphere Definition and equation of the sphere Equation of the sphere through four given points Plane sections of a sphere,Intersection of two spheres Equation of a circle,Sphere through a given circle Intersection of a sphere and a line,power of a point Tangent plane, plane of contact,polar plane, pole of a plane Angle of intersection of two spheres,Conditions for the two spheres to be orthogonal Radical plane,Coaxial system of spheres,Simplified form of the equation of two spheres Unit - 4 Cones Definitions of a cone,vertex,guidingcurve,generators,Equation of a cone with a given vertex and guiding curve Enveloping cone of a sphere, equations of a cone with vertex at origin are homogeneous Condition that the general equation of second degree should represent a cone Condition that a cone may have three mutually perpendicular generators Intersection a line and a quadric cone Tangent lines andTangent plane at a point,Condition that a plane may touch a cone Reciprocal cones,Intersection of two cones with common vertex Right circular cone-Equation of Right circular cone with a given vertex,axis and semivertical angle. Cylinders &Conicoids Definition and equation of the cylinder whose generators intersect a given conic and are parallel to a given line. Enveloping cylinder of a sphere Right circular cylinder-Equation of Right circular cylinder with a given axis and radius. The conicoid-Nature of hyperboloid of one sheet (Scope as in Analytical Solid Geometry by Shanti Narayan, published by S.Chand& Company Ltd. I YEAR PRACTICAL QUESTION BANK Unit – I: Differential equations of first order and first degree: 2 dy = y 3e − x . dx dy y 2. Solve: + = y 2 sin x. dx x x x/ y x/ y 1 − dy = 0. 3. Solve: 1 + e dx + e y 4. Solve: (xy sin xy + cos xy ) ydx + ( xy sin xy − cos xy )xdy = 0. 5. Solve: x 2 + y 2 + 2 x dx + 2 ydy = 0. 1. Solve: xy − ( ) ( ) dy = 0. dx 3 2 2 4 7. Solve: xy + y dx + 2 x y + x + y dy = 0. 8. Solve: x 2 ydx − x3 + y 3 dy = 0. 6. Solve: 3e x tan y + 1 − e x sec 2 y ( ) ( ( ) ) Equations of the first order but not of the first degree: 9. Solve: p 3 (x + 2 y ) + 3 p 2 ( x + y ) + ( y + 2 x ) p = 0. Solve ) ( 10. Solve: x 2 p 2 − 2 xyp + 2 y 2 − x 2 = 0. 11. Solve: xp 2 − 2 yp + ax = 0. ( ) 12. Solve: y = 2 px + tan −1 xp 2 . . ( ) 13. Solve: x 2 = a 2 1 + p 2 . 14. Solve: y = 2 p + 3 p 2 . 15. Solve: ( x − a ) p 2 + ( x − y ) p − y = 0. 16. Solve: sin px cos y = cos px sin y + p. Applications of first order differential equations: 17. Find the orthogonal trajectories of x 2 + y 2 = cx. 18. Find the orthogonal trajectories of r = c (1 − sin θ ). 1 19. Find the orthogonal trajectories of y = c e x . 1 20. Find the orthogonal trajectories of x 2 3 + y 2 3 = c . 1 Unit – II: Higher order differential equations: ( ) 22. Solve: (D 2 + 1)y = e − x + cos x + x3 + e x cos x. x 3x 23. Solve: (D 2 + 1)(D 2 + 4 )y = cos cos . 2 2 24. Solve: (D 2 + 1)y = cos x + xe 2 x + e x sin x. 21. Solve: D 2 + a 2 y = tan ax. 25. Solve d2y dy + 3 + 2 y = sin x by the method of undetermined coefficients 2 dx dx d 2 y dy 26. Solve + + y = x 2 by the method of undetermined coefficients 2 dx dx ( 2 ) 27. Solve D − 2 D − 8 y = 9 xe ( ) − x + 10e − x by the method of undetermined coefficients 28. Solve D 2 − 3D y = 2e 2 x sin x by the method of undetermined coefficients 29. Solve: y"+3 y '+2 y = 12e x using the method of variation of parameters. 2 − x using the method of variation of parameters. 30. Solve: y"+2 y '+ y = x e 31. Solve: y "+ y = 4 x sin x using the method of variation of parameters. x 32. Solve: y"−2 y '+ y = e log x using the method of variation of parameters. 2 2 y '+ y = 0 by reduction of order method, given y1 = x. x x2 Solve 2 x 2 + 1 y"−4 xy '+4 y = 0 by reduction of order method, given y = x. 1 2 2 Solve y"− y '+ y = x log x by reduction of order method, given y1 = x. x x2 Solve x 2 y"+ xy '− y = x 2e − x by reduction of order method, given y = x. 1 3 2 d y d y dy Solve: x 4 + 2 x3 − x2 + xy = 1. 3 2 dx dx dx 2 2 d y − x dy + 2 y = x log x. Solve: x dx dx 2 2 dy 2d y Solve: (2 x + 3) − 2(2 x + 3) − 12 y = 6 x. dx dx 2 2 dy 2d y Solve: (x + 3) − 4( x + 3) + 6 y = log( x + 3). dx dx 2 33. Solve y"− 34. 35. 36. 37. 38. 39. 40. ( ) Unit – III: Planes: 41. A variable plane is at a constant distance 3p from the origin and meets the axes in A, B and C. Show that the locus of the centroid of the triangle ABC is x −2 + y −2 + z −2 = p −2 . 42. A variable plane passes through a fixed point (a,b,c) and meets the co-ordinate axes in A,B,C. Show that the locus of the point common to the planes through A,B,C parallel to the coordinate planes is a x + b y + c z = 1. 43. Show that the equation 12 x 2 − 2 y 2 − 6 z 2 − 2 xy + 7 yz + 6 zx = 0 represents a pair of planes and also find the angle between them. Right Lines: 44. Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2 x + 3 y − 4 z + 1 = 0; also find the coordinates of the point, which is the image of the origin in the plane. 45. Find the equation of the plane through the point (1,1,1) and perpendicular to the line x − 2 y + z = 2,4 x + 3 y − z + 1 = 0. 46. A square ABCD of diagonal 2a is folded along the diagonal AC, so the planes DAC, BAC and at right angles. Show that the shortest distance between DC and AB is then 2a/√3 . 47. Find the magnitude and the equations of the line of shortest distance between the two lines: x − 3 y + 15 z − 9 x + 1 y − 1 z − 9 = = ; = = . −7 −3 2 5 2 1 48. Find the length and the equations of the shortest distance line between 5 x − y − z = 0, x − 2 y + z + 3 = 0; 7 x − 4 y − 2 z = 0, x − y + z − 3 = 0. 49. Find the magnitude and the equations of the line of the shortest distance between the lines x y +1 z − 2 = = ;5 x − 2 y − 3 z + 6 = 0 = x − 3 y + 2 z − 3. 4 3 2 50. Obtain the coordinates of the points where the shortest distance line between the lines x − 23 y − 19 z − 25 x − 12 y − 1 z − 5 = = ; = = . meet them. −6 −4 3 −9 4 2 Spheres: 51. A variable plane through a fixed point (a,b,c) cuts the co-ordinate axes in the point A,B,C show that the locus of the centers of the spheres OABC is a x + b y + c z = 2. 52. Find the equation of the sphere through the four points (0,0,0), (–a,b,c,), (a, –b,c,), (a,b, –c,) and determine its radius. 53. Obtain the equation of the sphere which passes through the three points (1,0,0), (0,1,0), (0,0,1) and has its radius as small as possible. 54. Find center and the radius of the circle x 2 + y 2 + z 2 − 2 y − 4 z = 11, x + 2 y + 2 z = 15. 55. Obtain the equation of the sphere having the circle x 2 + y 2 + z 2 + 10 y − 4 z − 8 = 0, x + y + z = 3 as the great circle. 56. Show that the plane 2 x − 2 y + z + 12 = 0 touches the sphere x 2 + y 2 + z 2 − 2 x − 4 y + 2 z = 3 and find the point of contact. 57. Obtain the equations of the sphere which pass through the circle x 2 + y 2 + z 2 − 2 x + 2 y + 4 z − 3 = 0,2 x + y + z = 4 and touches the plane 3 x + 4 y = 14. 58. Show that the polar line of is the line x +1 y − 2 = = z + 3 w.r.t to the sphere x 2 + y 2 + z 2 = 1 2 3 7x + 3 2 − 7 y z . = = 11 5 −1 59. Find the equation of the sphere that passes through the circle x 2 + y 2 + z 2 − 2 x + 3 y − 4 z + 6 = 0,3x − 4 y + 5 z − 15 = 0 and cuts the sphere x 2 + y 2 + z 2 + 2 x + 4 y − 6 z + 11 = 0. 60. Find the limiting points of the co-axial system of spheres x 2 + y 2 + z 2 − 20 x + 30 y − 40 z + 29 + λ ( 2 x − 3 y + 4 z ) = 0. Unit – IV: CONES AND CYLINDERS: 61. Find the equation of the cone whose vertex is the point (1,1,0) and whose guiding curve is y = 0, x 2 + z 2 = 4. 62. Find the equation of the cone with vertex at (1,2,3) and guiding curve x 2 + y 2 + z 2 = 4, x + y + z = 1. 63. Find the enveloping cone of the sphere x 2 + y 2 + z 2 − 2 x + 4 z = 1 with its vertex at (1,1,1). 64. Find the equation of the cone whose vertex is at the origin and the direction cosines of whose generators satisfy the relation 3l 2 − 4m 2 + 5n 2 = 0. 65. Find the equation of the cone which passes through the three co-ordinates axes as well as the x y z x y z = = ; = = . 1 − 2 3 3 −1 1 66. Show that the equation x 2 − 2 y 2 + 3 z 2 − 4 xy + 5 yz − 6 zx + 8 x − 19 y − 2 z − 20 = 0 two lines represents a cone with vertex (1,–2,3). 67. Find the angle between then lines of intersection of x + y + z = 0 and x 2 + yz + xy − 3 z 2 = 0. 68. If the plane 2 x − y + cz = 0 cuts the cone yz + xy + zx = 0 in perpendicular lines, find the value of c. 69. Find the equations of the lines in which the plane 2 x + y − z = 0 cuts the cone 4 x 2 − y 2 + 3 z 2 = 0. 70. Show that the locus of mid-points of chords of the cone ax 2 + by 2 + cz 2 + 2 fyz + 2 gxz + 2hxy = 0 drawn parallel to the line x l = y m = z n is the plane x (al + hm + gn) + y (hl + bm + fn ) + z ( gl + fm + cn) = 0. 71. Find the plane, which touches the cone x 2 + 2 y 2 − 3 z 2 + 2 yz − 5 zx + 3 xy = 0 along the generator whose direction ratios are (1,1,1). 72. Prove that the cones fyz + gzx + hxy = 0; fx ± gy ± hz = 0 are reciprocal. 73. Find the equation of the right circular cone whose vertex is (1,–2, –1) , axis the line x −1 y + 2 z +1 and semi-vertical angle 60°. = = 3 4 5 74. Find the equation to the cylinder whose generators are parallel to x y z = = and guiding 1 2 3 curve is x 2 + y 2 = 16, z = 0. 75. Find the equation of the enveloping cylinder of the conicoid generators are parallel to the line x2 y2 z2 + + = 1 whose a 2 b2 c2 x y z = = . l m n 76. Obtain the equation of a cylinder whose generators touch the sphere x 2 + y 2 + z 2 + 2ux + 2vy + 2 wz + d = 0 whosegenerators are parallel to the line x y z = = . l m n 77. Obtain the equation of the right circular cylinder whose guiding curve is the circle through the points (1,0,0), (0,1,0), (0,0,1). 78. Find the equation of the right circular cylinder of radius 2 whose axis is x −1 y − 2 z − 2 = = . 2 2 2 79. Find the equation of the right circular cylinder whose axis is x −1 y −1 z = = and passes 2 1 3 through (0,0,3). 80. Prove that the right circular cylinder whose one section is the circle x 2 + y 2 + z 2 − x − y − z = 0, x + y + z = 1, is x 2 + y 2 + z 2 − yz − zx − xy = 1.

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