Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 15 Scheme 15MAT11 Engineering Mathematics I Question Paper

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15MAT11

First Semester B.E. Degree Examination, June/July 2019

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Engineering Mathematics - I

Time: 3 hrs.

Max. Marks: 80

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Note: Answer any FIVE full questions, choosing ONE full question from each module.

Module-1

1. 1. Find the nth derivative of ^{(7x + 6)}/_{(2x² + 7x + 6)} (05 Marks)

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   2. Find the angle between the radius vector and the tangent for the curve r^m = a^m (cos(m?) + sin(m?)). (05 Marks)
   3. Show that the radius of curvature at any point ? on the cycloid x = a (? + sin?), y = a(1 - cos ?) is 4 acos(?/2) (06 Marks)

2. OR

   1. If x = sint and y = cosmt, prove that (1-x²)y_n+2 - (2n+1)xy_n+1 + (m²-n²)y_n = 0 (05 Marks)

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   2. Find the pedal equation of the curve r = a sec 2?. (05 Marks)
   3. Prove with usual notation tan f = r d?/dr (06 Marks)

Module-2

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3. 1. Expand e^x using Maclaurin's series upto third degree term. (05 Marks)
   2. Evaluate lim _x->0 [^{(1 - cos x)}/_{x sin²x}] (05 Marks)
   3. If u = e^(ax+by) f(ax - by), prove that ^?u/_?x + a ^?u/_?y = 2abu (06 Marks)

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4. OR

   1. Expand sin x in ascending power of x upto the term containing x⁴. (05 Marks)
   2. If u = tan^-1(^xy/_x+y), show that x u_x + y u_y = sin2u. (05 Marks)
   3. If u = ^xy/_z, v = ^yz/_x, w = ^zx/_xy Find ^{?(u, v, w)}/_?(x,y,z) (06 Marks)

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Module-3

5. 1. Find the angle between the surfaces x² + y² + z² = 9 and x² + y² - z = 3 at the point (2, -1, 2). (05 Marks)
   2. Show that F = (y + z)i + (x + z)j + (x + y)k is irrotational. Also find a scalar function f such that F = ?f. (05 Marks)

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   3. Prove that ? · (fA) = f(? · A) + (?f) · A. (06 Marks)

6. OR

   1. Prove that Curl (fA) = f(Curl A) + grad f x A. (05 Marks)
   2. A particle moves along the curve C ; x = t³ - 4t, y = t² + 4t, z = 8t² - 3t³ where t denotes the time. Find the component of acceleration at t = 2 along the tangent. (05 Marks)

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   3. Show that F = (2xy² + yz)i + (2x²y + xz + 2yz² )j + (2y²z + xy)k is a conservative force field. Find its scalar potential. (06 Marks)

Module-4

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7. 1. Obtain the reduction formula for ? sinⁿ x dx (05 Marks)
   2. Solve (y²e^x-1y + 4x³ )dx +(2xye^x-1y -3y² )dy = 0. (05 Marks)
   3. Find the orthogonal trajectories of r = a (1+sin?). (06 Marks)

8. OR

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   1. Evaluate ? x³vx²-1 dx (05 Marks)
   2. Solve (y³-3x²y)dx — (x³-3xy² )dy = 0. (05 Marks)
   3. A bottle of mineral water at a room temperature of 72°F is kept in a refrigerator where the temperature is 44°F. After half an hour, water cooled to 61°F
      1. What is the temperature of the mineral water in another half an hour?
      2. How long will it take to cool to 50°F?
      (06 Marks)

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Module-5

9. 1. Find the rank of the matrix A =

        | -1 -3 -1 1 | | 2 3 -1 0 | | 1 0 1 1 | | 1 0 1 -1 |  

      (05 Marks)

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   2. Find the largest eigen value and corresponding eigenvector of the matrix A =

        | 6 -2 2 | | -2 3 -1 | | 2 -1 3 |  

      by power method taking X₁ = [1, 1, 1]^T (05 Marks)
   3. Reduce the matrix A =

        | -1 3 | | -2 4 |  

      to the diagonal form. (06 Marks)

10. OR

    1. Use Gauss elimination method to solve

         2x + y + 4z = 12 4x + 11y - z = 33 8x - 3y + 2z = 20  

       (05 Marks)

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    2. Find the inverse transformation of the following linear transformation.

         y1 = x1 + 2x2 + 5x3 y2 = 2x1 + 4x2 + 11x3 y3 = -x1 + 2x2  

       (05 Marks)
    3. Reduce the quadratic form 2x1² + 2x2² + 2x3² + 2x1x2 to the Cannonical form. (06 Marks)

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