Download OU B.Sc 2016 April 3rd Year 2019E Mathematics Question Paper

Code No. 2019/

FACULTIES OF ARTS AND SCIENCE

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

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Subject : MATHEMATICS

Paper—II . Linear Algebra and Vector Calculus

Time : 3 hours

Max. Marks : 100

Part - A (6X6=36 Marks)

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Unit - I

1. Prove that the linear span L(S) of any subset S of a vector space V(F) is a subspace of V.
2. If U(F) and V(F) are vector spaces, define linear transformation.
3. Find the eigen roots and the corresponding eigen vectors of the matrix A= 1 & 4 \\ 2 & 3
4. Prove that S = { (1/3, 2/3, -2/3), (-2/3, 1/3, -2/3) } is an orthonormal set in R³ with standard inner product.

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Unit - II

5. Evaluate ∫ (2x² + y²)dx + (3y - 4x)dy around the triangle ABC whose vertices are A(0,0), B(2,0) and C(2, 1).
6. Evaluate ∫∫ x²y dx dy over [0,1;0, 1].

Unit - III

7. State Green's theorem.

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8. Show that ∫∫ (ax²+by²+cz²) dS = 4p(a +b+c) where S is the surface of the sphere x² + y² + z² = 9.

Part - B (4 X 16 = 64 Marks)

Unit - I

9. a) Prove that every non-empty subset of a Linearly Independent set of vectors is Linearly Independent.
   b) Prove that every Linearly Independent subset of a finitely generated vector space V(F) is either a basis of V or can be extended to form a basis of V.

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10. a) State and prove rank-nullity theorem.
    b) Prove that zero transformation is a linear transformation.

Unit - II

11. a) Find characteristic values and characteristic vectors of the matrix
    1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1

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12. a) Explain Gram-Schmidt orthonormalization process.
    b) In an inner product space V(F), prove |(a, ß)| = ||a|| ||ß|| for all a, ß ? V

Unit - III

13. a) Prove that every continuous function is integrable.
    b) Prove sufficient condition for the existence of the integral.

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14. a) Change the order of integration and evaluate ∫₀⁸ ∫₀⁸ e^-(1+x2+y2) dxdy.

Unit-IV

15. a) If F= (3xz² + 6y)i - 14yz j + 20xz² k evaluate ∫ F. dr along the straight line joining (0, 0, 0) to (1,0, 0) and then from (1, 0, 0) to (1, 1, 0) to (1, 1, 1).
    b) If F= (x + yz)i - 2xy j + 2yz k. Evaluate ∫∫ F. ds where S is the surface of plane 2x+y+2z=6 in the first octant.
16. a) State and prove Green's theorem in a plane.
    

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    b) Evaluate ∫ (cos x sin y - xy) dx + sinx cosy dy, by Green’s theorem where C is the circle X² + y² = 1.

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