Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 18 Scheme 18INIAT11 Calculus and Linear Algebra Question Paper

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

Calculus and Linear Algebra

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

Module-1

1. a. With usual notation, prove that tan f = r d?/dr (06 Marks)

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2. b. Find the radius of curvature of a²y = x³ at the point where the curve cuts the x-axis. (06 Marks)
3. c. Show that the evolute of the parabola y² = 4ax is 27ay² = 4(x – 2a)³. (08 Marks)

OR

1. a. Prove that the pedal equation of the curve rⁿ = aⁿcos(n?) is aⁿp = rⁿ⁺¹ (06 Marks)
2. b. Show that for the curve r(1 – cos?) = 2a, p² varies as r³. (06 Marks)

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3. c. Find the angle between the polar curves r = a(1+cos?) and r = a(1 – cos?). (08 Marks)

Module-2

1. a. Expand log(1 + cosx) by Maclaurin's series up to the term containing x⁴ (06 Marks)
2. b. Evaluate lim_x?0 (a^x + b^x + c^x)/3^1/x (07 Marks)
3. c. Find the extreme values of the function f(x, y) = x² + y² – 6x – 12y + 20. (07 Marks)

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OR

1. a. If u = f(x, y, z) where x³ + y³ + z³ + 3xyz = c, then prove that ?u/?x + ?u/?y + ?u/?z = 0 (06 Marks)
2. b. If u = x + 3y, v = 4x - 2y, w = 2z - xy. Evaluate ?(u, v, w)/?(x, y, z) at the point (1, -1, 0). (07 Marks)
3. c. 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)

Module-3

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1. a. Evaluate ?₀^a ?_x^a x dy dx (06 Marks)
2. b. Find the area bounded between the circle x² + y² = a² and the line x + y = a. (07 Marks)
3. c. Prove that ß(m, n) = ?₀¹ x^m-1(1-x)^n-1 dx. (07 Marks)

OR

1. a. Evaluate ?_-a^a ?_-b^b ?_-c^c (x² + y² + z²) dz dy dx (06 Marks)

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2. b. Find the area bounded by the ellipse x²/a² + y²/b² = 1 by double integration. (07 Marks)
3. c. Show that ?₀^p/2 d?/v(sin ?) * ?₀^p/2 v(sin ?) d? = p (07 Marks)

Module-4

1. a. Solve (1 + e^x)dx + e^x dy = 0 (06 Marks)
2. b. 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)

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3. c. Solve yp² + (x - y) p - x = 0. (07 Marks)

OR

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

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Module-5

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

   1	2	-2	3
   2	4	-1	6
   3	6	-3	5
   -1	-2	2	-2

   applying elementary Row transformations. (06 Marks)
2. b. Solve the following system of equations by Gauss-Jordan method: x + y + z = 9, 2x + y - z = 0, 2x + 5y + 7z = 52 (07 Marks)
3. c. Using Rayleigh's power method find the largest eigen value and corresponding eigen vector of the matrix A =

   1	0	1
   0	2	0
   1	0	2

   with X⁽⁰⁾ = (1, 0, 0)^T as the initial eigen vector carry out 5 iterations. (07 Marks)

OR

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1. a. 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)
2. b. Reduce the matrix A =

   8	-6	2
   -6	7	-4
   2	-4	3

   into diagonal form. (07 Marks)
3. c. 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)

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