Download VTU BE 2020 Jan Question Paper 18 Scheme 8MAT11 Calculus and Linear Algebra First And Second Semester

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1H1KOD18MAT11

First Semester B.E. Degree Examination, Dec.2019/Jan.2020

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-1

1. 1. With usual notations prove that tan f = r (dr/de) (06 Marks)
   2. Find the angle between the curves r = sin? + cos? and r = 2 sin? (06 Marks)
   3. Show that the radius of curvature for the catenary of uniform strength y = a log sec (x/a) is a sec (x/a). (08 Marks)

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2. OR

   1. Show that the pairs of curves r = a(1 + cos?) and r = b(1 - cos?) intersect each other Orthogonally. (06 Marks)
   2. Find the pedal equation of the curve rⁿ = aⁿ cos n?. (06 Marks)
   3. Show that the evolute of y² = 4ax is 27ay² = 4(x + a)³ (08 Marks)

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Module-2

3. 1. Find the Maclaurin's series for tanx upto the term x³ (06 Marks)
   2. Evaluate lim _x->0 (a^x + b^x + c^x)/3 (07 Marks)
   3. If u = f(x-y, y-z, z-x), prove that (?u/?x) + (?u/?y) + (?u/?z) = 0 (07 Marks)

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4. OR

   1. Expand log (sec x) upto the term containing x⁴ using Maclaurin's series. (06 Marks)
   2. Find the extreme values of the function f(x, y) = x³ + y² - 3x - 12y + 20. (07 Marks)
   3. Find ?(u,v)/?(x,y,z) where u = x² + y² + z², v = xy + yz + zx, w = x + y + z. (07 Marks)

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Module-3

5. 1. Evaluate ? xyz dzdydx where x: 0->1, y: v1-x² -> v1-x and z: 1-x->1-x². (06 Marks)
   2. Evaluate ?¹₀ ?²_-2 (2-x)dydx by changing the order of integration. (07 Marks)
   3. Prove that ß(m, n) = G(m). G(n) / G(m+n) (07 Marks)

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6. OR

   1. Evaluate ? ydx dy over the region bounded by the first quadrant of the ellipse x²/a² + y²/b² = 1. (06 Marks)
   2. Find by double integration the area enclosed by the curve r = a (1 + Cos?) between ? = 0 and ? = p. (07 Marks)
   3. Show that ?^p/2₀ Sin^n-1? d? = vp G(n/2) / G((n+1)/2) . (07 Marks)

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Module-4

7. 1. Solve dy/dx cos x + sin y + y = 0 / sin x + x cosy + x (06 Marks)
   2. Solve rSin? dr/d? - Cos? = r² (07 Marks)
   3. A series circuit with resistance R, inductance L and electromotive force E is governed by the differential equation L di/dt + Ri = E, where L and R are constants and initially the current i is zero. Find the current at any time t. (07 Marks)

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8. OR

   1. Solve (4xy + 3y² – x)dx + x (x + 2y)dy = 0. (06 Marks)
   2. Find the orthogonal trajectories of the family of parabolas y² = 4ax. (07 Marks)
   3. Solve p² + 2py cotx = y². (07 Marks)

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Module-5

9. 1. Find the rank of the matrix [[1, 2, 3, 2], [2, 3, 5, 1], [3, 4, 5, 4]] by elementary row transformations. (06 Marks)
   2. Apply Gauss-Jordan method to solve the system of equations 2x₁ + x₂ + 3x₃ = 1, 4x₁ + 4x₂ + 7x₃ = 1, 2x₁ + 5x₂ + 9x₃ = 3. (07 Marks)
   3. Find the largest Eigen value and the corresponding Eigen vector of the matrix A = [[2, 0, 1], [0, 2, 0], [1, 0, 2]] by power method. Using initial vector (1 0 0)^T. (07 Marks)

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10. OR

    1. Solve by Gauss elimination method x - 2y + 3z = 2, 3x - y + 4z = 4, 2x + y - 2z = 5 (06 Marks)
    2. Solve the system of equations by Gauss-Seidal method 20x + y - 2z = 17, 3x + 20y - z = -18, 2x - 3y + 20z = 25 (07 Marks)
    3. Reduce the matrix A = [[-1, 1, 2], [-1, 2, 1], [-1, 1, 2]] to the diagonal form. (07 Marks)

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