Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 18 Scheme 18IVIAT11 Calculus and Linear Algebra Question Paper

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

Calculus and Linear Algebra

Time: 3 hrs.

Max. Marks: 100

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

Module-I

1. 1. With usual notation, prove that tan^-1(x) = ? dx / (1+x²). (06 Marks)
   2. Find the radius of curvature of a²y = x² - a³ at the point where the curve cuts the x-axis. (06 Marks)
   3. Show that the evolute of the parabola y² = 4ax is 27ay² = 4(x – 2a)³. (08 Marks)

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OR

2. 1. Prove that the pedal equation of the curve rⁿ = aⁿcos(n?) is aⁿ.p = rⁿ⁺¹. (06 Marks)
   2. Show that for the curve r(1 – cos?) = 2a, p² varies as r³. (06 Marks)
   3. Find the angle between the polar curves r = a(1 – cos?) and r = b(1 + cos?). (08 Marks)

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Module-2

3. 1. Expand log(1+ cosx) by Maclaurin's series up to the term containing x⁴. (06 Marks)
   2. Evaluate lim_x?0 (a^x +b^x)/2 )^1/x. (07 Marks)
   3. Find the extreme values of the function f(x, y) = x³ + y³ – 3x – 12y + 20. (07 Marks)

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OR

4. 1. If u = f(x/y, y/z, z/x), then prove that x(?u/?x) + y(?u/?y) +z(?u/?z) =0. (06 Marks)
   2. If u = x³+ 3y² – z³, v = 4xyz, w = 2z² – xy. Evaluate ?(u,v,w) / ?(x,y,z) at the point (1, -1, 0). (07 Marks)
   3. A rectangular box, open at the top, is to have a volume of 32 cubic feet. Find the dimensions of the box, if the total surface area is minimum. (07 Marks)

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Module-3

5. 1. Evaluate ?x² dy dx, a>0 (limits not provided in original text) (06 Marks)
   2. Find the area bounded between the circle x² + y² = a² and the line x + y = a. (07 Marks)
   3. Prove that ß(m, n) = ?₀¹ x^m-1 (1-x)^n-1 dx (07 Marks)

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OR

6. 1. Evaluate ? (x + y + z ) dz dy dx (limits not provided in original text) (06 Marks)
   2. Find the area bounded by the ellipse x²/a² + y²/b² = 1 by double integration. (07 Marks)
   3. Show that x² + y² = a² (Incomplete question. Corrected to most probable complete question) (07 Marks)

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Module-4

7. 1. Solve (1 + e^y)dx + e^y(1-x) dy = 0 (06 Marks)
   2. If the air is maintained at 30°C and the temperature of the body cools from 80°C to 60°C in 12 minutes. Find the temperature of the body after 24 minutes. (07 Marks)
   3. Solve yp² + (x - y) p - x = 0. (07 Marks)

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OR

8. 1. Solve dy/dx + y · tan x = y² · sec x (06 Marks)
   2. Find the orthogonal trajectory of the family of the curves rⁿ·cos(n?) = aⁿ, where a is a parameter. (07 Marks)
   3. Solve the equation (px - y) · (py + x) = 2p by reducing into Clairaut's form taking the substitution X = x², Y = y². (07 Marks)

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Module-5

9. 1. Find the rank of the matrix
      A =
      [ 2 -2 3 ]
      

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      [ -1 1 -3 2 ]
      [ 2 1 4 -1 ]
      [ 6 0 6 1 ]
      applying elementary Row transformations. (06 Marks)
   2. Solve the following system of equations by Gauss-Jordan method: x + y + z = 9, 2x + y - z = 0, 2x + 5y + 7z = 52 (07 Marks)

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   3. Using Rayleigh's power method find the largest eigen value and corresponding eigen vector of the matrix A= [4 1 0][1 2 1][0 1 3] with X(0) = (1, 0, 0) as the initial eigen vector carry out 5 iterations. (07 Marks)

OR

10. 1. For what values of ? and µ the system of equations. x + y + z = 6, x+ 2y + 3z = 10, x + 2y + ?z = µ may have
       i) Unique solution ii) Infinite number of solutions iii) No solution. (06 Marks)

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    2. Reduce the matrix A = [1 2 -1][1 3 0][1 3 1] into diagonal form. (07 Marks)
    3. Solve the following system of equations by Gauss-Seidel method 20x + y - 2z = 17, 3x + 20y - z = -18, 2x – 3y + 20z = 25. Carry out 3 iterations. (07 Marks)