Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 17 Scheme 17MAT11 Engineering Mathematics I Question Paper

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CBCS SCHEME

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17MAT11

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

Engineering Mathematics I

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. a. Find the nth derivative of sin 2x sin 3x. (06 Marks)
2. b. Find the angle between the two curves r = a/(1+ cos?) and r = b/(1-cos ?) (07 Marks)
3. c. Find the radius of curvature for the curve x³ + y³ = 3xy at (3/2, 3/2). (07 Marks)

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OR

1. a. If y = cos(m log x) then prove that x²y_n+2 + (2n + 1)xy_n+1 + (m² + n²)y_n = 0. (06 Marks)
2. b. With usual notation prove that tan f = r d?/dr (07 Marks)
3. c. Find the pedal equation of the curve r^m = a^m cos m? . (07 Marks)

Module-2

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1. a. Find the Taylor's series of log(cosx) in powers of (x – p/3) upto fourth degrees terms. (06 Marks)
2. b. If u = tan^-1((x³ + y³)/(x + y)), then prove that x(?u/?x) + y(?u/?y) = sin 2u by using Euler's theorem.(07 Marks)
3. c. if u= xy/z , V = yz/x, W = zx/y then find ?(u,v,w)/?(x,y,z). (07 Marks)

OR

1. a. Evaluate lim_x?0 (x - sin x)/x³. (06 Marks)

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2. b. Using Maclaurin's series, prove that e^sinx = 1+ x + x²/2 + x³/3 + x4/4 + ....(07 Marks)
3. c. If u = 4x - 3y, v = 3y - 4z, w = 2x - 4z then prove that (1/2)(?u/?x) + (1/3)(?u/?y) + (1/4)(?u/?z) = 0.(07 Marks)

Module-3

1. a. A particle moves along the curve r = (t³ - 4t)i + (t² + 4t)j + (8t² - 3t³)k. Find the components of velocity and acceleration in the direction of 2i - 3j + k at t = 0. (06 Marks)
2. b. Find the constant a and b such that F = (axy + z³)i + (3x² - z)j + (bxz² - y)k is irrotational and find scalar potential function f such that F = ?f. (07 Marks)

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3. c. Prove that curl(fA) = fcurlA + (?f) X A. (07 Marks)

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OR

1. a. Show that vector field F = (x i + y j)/(x² + y²) is both solenoidal and irrotational. (06 Marks)
2. b. If F = (x + y + 1)i + j - (x + y)k then prove that curl F = 0 . (07 Marks)

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3. c. Show that div(curl A) = 0 . (07 Marks)

Module-4

1. a. Obtain reduction formula for ? sinⁿ x dx (n > 0). (06 Marks)
2. b. Solve the differential equation dy/dx + y cot x = cos x (07 Marks)
3. c. Find the orthogonal trajectory of the curve r = a(1+ sin ?). (07 Marks)

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OR

1. a. Evaluate ?₀^p/2 sin⁷? cos⁵ ? d? (06 Marks)
2. b. Solve the differential equation : (2xy + y² - tan y)dx + (x² - x tan^-1 y + sec² y)dy = 0 . (07 Marks)
3. c. If the temperature of air is 30°C and the substance cools from 100°C to 70°C in 15 mins. Find when the temperature will be 40°C (07 Marks)

Module-5

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1. a. Find the rank of the matrix [ 1 2 1]
   [ 3 8 2]
   [ 4 7 6] by reducing to Echelon form. (06 Marks)
2. b. Find the largest eigen value and eigen vector of the matrix [ 2 -1 0]
   [-2 3 -1]
   

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   [ 2 -1 3] by taking initial vector as [ 1 1 1] 'by using Rayleigh's power method. Carry out five iteration. (07 Marks)
3. c. Reduce 8x² + 7y² + 3z² - 12xy + 4xz - 8yz into canonical form, using orthogonal transformation (07 Marks)

OR

1. a. Solve the system of equations 10x+y+ z = 12
   x+ 10y+z= 12
   

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   x+y+ 10z= 12 by using Gauss-Seidel method. Carry out three iterations. (06 Marks)
2. b. Diagonalise the matrix A = [5 4]
   [1 2]. (07 Marks)
3. c. Show that the transformation y1 = x1 + 2x2 + 5x3
   y2 = 2x1 + 4x2 + 11x3
   

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   y3 = -x2 + 2x3 is regular. Write down inverse transformation. (07 Marks)