Download OU B.Sc 2016 April 1st Year 2017E Mathematics Question Paper

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Code No. 2017 / E

FACULTIES OF ARTS AND SCIENCE

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B.A./B.Sc. I- Year Examination, March / April 2016

Subject : MATHEMATICS

Paper — I : Differential Equations and Solid Geometry

Time : 3 hours Max. Marks : 100

Note : Answer Six questions from Part-A & Four questions from Part-B. Choosing at least one from each Unit. Each question in Part-A carries 6 marks and in Part-B carries 16 marks.

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Part— A (6 X 6 = 36 Marks)

Unit - I

1. Solve sec²y dy/dx + 2xtany = x³.

Unit - II

3. Solve y"+3y'+2y =12e^x

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4. Solve (D² -3D+2)y =3sin2x.

Unit - III

5. Find the equation of the plane passing through the points (1, 0, -1), (3, 2, 2), (1, 1, -1).
6. Find the point where the line joining (2, -3, 1), (3, -4, -5) cuts the plane 2x + y+z=7.

Unit - IV

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7. Find the equation of the cone whose vertex is at the origin and the direction cosines of whose generators satisfy the relation 3l² —4m² +5n² =0.
8. Find the equation of the cylinder whose generators are parallel to the line x/1 = y/2 = z/3 and whose guiding curve is the ellipse x² + 2y² =1, z =0.

Part— B (4 X 16 = 64 Marks)

Unit - I

9. a) Prove that the integrating factor of non-exact differential equation Mdx+Ndy=0 is 1/Mx+Ny if the differential equation is homogeneous and Mx + Ny ? 0.

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10. b) Solve (1+y²)dx = (tan^-1y — x)dy.

10. a) Explain the method of solving Clairaut’s equation y = px + f(p)
11. b) Solve (x²+y²+2x)dx+2ydy=0

Unit - II

11. a) Explain the method of solving second order Cauchy Euler equation a₀x² d²y/dx² + a₁x dy/dx + a₂y = Q(x) where a₀, a₁ and a₂ are constants which are non-zero.

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12. b) Solve (D² —3D +2)y = xe^x +sinx.

12. a) Solve (D² + 4D+4) y = 4x² + 6e^x by undetermined coefficients.
13. b) Apply method of variation of parameters to solve (D²-2D)y = e^x sinx.

Unit - III

13. a) 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

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14. b) Find the shortest distance between the lines x-1/2 = y-2/3 = z—3/3 and x—2/3= y-3/4 = z—4/5

14. a) Find the equation of the sphere which passes through the points (0,0,0), (0,1,-1), (-1,2,0) and (1,2,3).
15. b) Find the equation of the sphere which passes through the circle x²+y²+z²=5,x+2y+3z= 3 and touch the plane 4x + 3y = 15.

Unit - IV

15. a) Prove that 2x² +2y² +7z² -10yz -10zx +2x +2y +26z -17 = 0 represents a cone with vertex at (2,2,1).

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16. b) Find the angle between the lines of intersection of 4x -y- 5z =0 and 8yz + 3zx - 5xy = 0.

16. a) Find the equation of the cylinder whose generators touch the sphere x²+y²+z²= a² and are parallel to the line x/l = y/m = z/n

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