Download OU B.Sc 2019 June-July 6th and 7th Semester 3318 Mathematics Question Paper

Note: Answer any FIVE of the following (Short Answer Type) (5 x 9 = 45 Marks)

1. If F = (y, x, z) and C is the curve given by x = cos(t), y = sin(t), z = 0, then evaluate ? F · dr
2. Define conservative vector field with example.

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3. Show that ? x(1-x/a)(1-y/b) dz dy dx = abc/6, where the limits are 0 to a, 0 to b, 0 to c respectively.
4. Find the angle between the surfaces of the sphere x² + y² + z² = 2 and the cylinder x² + y² = 1 at a point where they intersect.
5. If f = grad (x³ + y³ + z³ - 3xyz), find curl f.
6. Find unit normal to the surface y = x + z² at the point (1, 2, 1).
7. Give the physical interpretation of curl.

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8. Show that curl (grad f) = 0 if f is a scalar field.

Note: Answer ALL from the questions. (Essay Answer Type) (3 x 15 = 45 Marks)

9. (a) Evaluate the surface integral of u = (y², x², z²) over the surface S where S is the triangular surface on x = 0 with y > 0, z > 0, y + z < 1, with the normal n directed in the positive direction of x-axis.
   OR
   (b) Evaluate the line integral ? F · dr where F is the vector field (y, x, z) along the curve y = x, z = sin x, z = 0 between x = 0 and x = p.

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10. (a) A cube has a variable density given by ? = 1 + x + y + 2z. What is the mass of the cube?
    OR
    (b) Find the volume integral of the scalar field f = x² + y² + z² over the region specified by 0 < x < 1, 1 < y < 2, 0 < z < 3.
11. (a) Show that u = (y²z - 2zsiny + 2xyz, 2zcosy + y²x) is irrotational. Find the corresponding potential function.
    OR
    

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    (b) Find the gradient and Laplacian of r = sin(r).

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