Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 18 Scheme 18MAT11 Calculus and Linear Algebra Question Paper

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18MAT11

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First Semester B.E. Degree Examination, Dec.2018/Jan.2019

Calculus and Linear Algebra

Time: 3 hrs.

Max. Marks: 100

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

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Module-1

1. a. Show that the curves rⁿ = aⁿ cos n? and rⁿ = bⁿ sin n? intersect orthogonally. (06 Marks)
2. b. Find the radius of curvature of the curve y = a log(sec(x/a)) at any point (x, y). (06 Marks)
3. c. Show that the evolute of the parabola y² = 4ax is 27ay² = 4(x-2a)³ (08 Marks)

OR

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1. a. With usual notation, prove that tan(f) = r (d?/dr) . (06 Marks)
2. b. Find the pedal equation of the curve r = a e^{cot ?}. (06 Marks)
3. c. Find the radius of curvature for the curve r = a(1 + cos?). (08 Marks)

Module-2

1. a. Using Maclaurin's expansion, prove that tan^-1(sin 2x) = 2x - (8x³/6) + (64x⁵/120) - ... (06 Marks)

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2. b. Evaluate lim_x?0 (a^x + b^x + c^x / 3)^1/x (07 Marks)

Module-3

1. a. Evaluate ? e^-(x2+y2) dxdy, by changing into polar coordinates. (06 Marks)
2. b. Find the volume of the tetrahedron bounded by the planes: x = 0, y = 0, z = 0, x/a + y/b + z/c = 1. (07 Marks)
3. c. Prove that ß(m,n) = G(m)G(n) / G(m+n) (07 Marks)

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OR

1. a. Evaluate ?₀¹ ?_x^vx xy dy dx by change of order of integration. (06 Marks)
2. b. Evaluate ? (x + y + z) dy dx dz with limits -1 to 1 for y, 0 to x for x and x-z to x+z for z. (07 Marks)
3. c. Prove that ?₀^p/2 (vsin ?) d? * ?₀^p/2 (1/vsin ?) d? = p. (07 Marks)

Module-4

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1. a. A body, in air at 25°C cools from 100°C to 75°C in 1 minute. Find the temperature of the body at the end of 3 minutes. (06 Marks)
2. b. Solve dy/dx = -(y cos x + sin y + y) / (sin x + x cos y + x) (07 Marks)
3. c. Solve xyp² - (x² + y²)p + xy = 0. (07 Marks)

OR

1. a. Solve dy/dx + y tan x = y² sec x. (06 Marks)

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2. b. Show that the family of parabolas y² = 4a(x + a) is self orthogonal. (07 Marks)
3. c. Find the general solution of the equation (px - y)(py + x) = 0 by reducing into Clairaut's form, taking the substitution X = x², Y = y². (07 Marks)

Module-5

1. a. Find the rank of the matrix:
   A =
   

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   [1 2 -2 3]
   [2 5 -4 6]
   [-1 -3 2 -2]
   [2 4 -1 6]
   (07 Marks)

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2. b. Solve the system of equations:
   12x + y + z = 31
   2x + 8y - z = 24
   3x + 4y + 10z = 58
   By Gauss-Seidel method. (07 Marks)

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3. c. Diagonalize the matrix:
   A = [5 -4]
   [-2 1]
   (06 Marks)

OR

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1. a. For what values of ? and µ the system of equations:
   x + 2y + 3z = 6
   x + 3y + 5z = 9
   2x + 5y + ?z = µ
   has i) no solution ii) a unique solution iii) infinite number of solutions. (07 Marks)

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2. b. Find the largest eigenvalue and the corresponding eigenvector of:
   A = [6 -2 4]
   [-2 3 -2]
   [2 -1 3]
   by Rayleigh's power method, use [1 1 1] as the initial eigenvector (carry out 6 iterations). (07 Marks)

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3. c. Solve the system of equations:
   x + y + z = 9
   2x + y - z = 0
   2x + 5y + 7z = 52
   By Gauss elimination method. (06 Marks)

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