Download VTU BE 2020 Jan Question Paper 17 Scheme 17MAT11 Engineering Mathematics I First And Second Semester

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CBCS SCHEME

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USN

17MAT11

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

Engineering Mathematics I

Time: 3 hrs.

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Max. Marks: 100

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

Module-I

1. a. Find the nth derivative of sin 2x Cos x. (06 Marks)
2. b. Prove that the following curves cuts orthogonally r = a(1+ sin ?) and r = a(1— sin ?). (07 Marks)

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3. c. Find the radius of the curvature of the curve r = a sin n? at the pole. (07 Marks)

OR

1. a. If tan y = x, prove that (1 + x²)y_n+2 + 2(n+1)xy_n+1 + n(n + 1)y_n = 0. (06 Marks)
2. b. With usual notations, prove that tan(?) = r(d?/dr). (07 Marks)
3. c. Find the radius of curvature for the curve x² + y² = a(x² - y²) at (-2a, 2a). (07 Marks)

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Module-2

1. a. Using Maclaurin's series prove that v(1 + sin 2x) = 1 + x - (x²/2) - (x³/6) + (x⁴/24) + ... (06 Marks)
2. b. If U = cot^-1((x³ + y³ + z³)/(xy + yz + zx)), prove that x(?U/?x) + y(?U/?y) + z(?U/?z) = -sin 2U. (07 Marks)
3. c. Find the Jacobian of u=x² + y² + z², v = xy + yz +zx, w = x+y+z. (07 Marks)

OR

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1. a. Evaluate lim_x?0 (e^2x - 2x -1)/x² (06 Marks)
2. b. Find the Taylor's series of log(cos x) about the point x = 0 upto the third degree. (07 Marks)
3. c. If u = f((x/y),(y/z),(z/x)), prove that x(?u/?x) + y(?u/?y) + z(?u/?z) = 0. (07 Marks)

Module-3

1. a. If x = t² + 1, y = 4t - 3, z = 2t² - 6t represents the parametric equation of a curve then, find velocity and acceleration at t = 1. (06 Marks)

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2. b. Find the constants a and b such that F = (axy + z³)i+(3x² - z)j+(bxz²- y)k is irrotational. Also find a scalar function f such that F = ?f. (07 Marks)
3. c. Prove that div(curl A) = 0. (07 Marks)

OR

1. a. Find the component of velocity and acceleration for the curve r = (2t³)i + (t² - 4t)j+(3t -5)k at the points t = 1 in the direction of i -3 j+ 2k. (06 Marks)
2. b. If f = v(xyz²), find div f and curl f at the point (1, -I, 1). (07 Marks)

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3. c. Prove that curl(grad f) = 0. (07 Marks)

Module-4

1. a. Prove that ?₀^p/2 sin^m ? cosⁿ ? d? = (3p/16) using reduction formula. (06 Marks)
2. b. Solve (x² + y + x)dx + xydy = 0. (07 Marks)
3. c. Find the orthogonal trajectory of rⁿ = a sin n?. (07 Marks)

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OR

1. a. Find the reduction formula for ? cosⁿ x dx and hence evaluate ?₀^p/2 cos⁴ xdx (06 Marks)
2. b. Solve ye^x dx +(e^x + 2y)dy = 0. (07 Marks)
3. c. 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. (07 Marks)

Module-5

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

   1	-1	2	-1
   1	1	-1	1
   1	-1	1	-1

   by reducing to row echelon form. (06 Marks)
2. b. Find the largest eigen value and the corresponding eigen vector for

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

   by taking the initial approximation as [1, 0.8, -0.8] by using power method. Carry out four iterations. (07 Marks)
3. c. Show that the transformation y₁ = 2x₁ -2x₂ -x₃, y₂ = -4x₁ + 5x₂ + 3x₃, y₃ = x₁ - x₂ -x₃ is regular. Find the inverse transformation. (07 Marks)

OR

1. a. Solve the equations 5x + 2y + z =12, x + 4y + 2z=15, x + 2y + 5z = 20 by using Gauss Seidal method. Carryout three iterations taking the initial approximation to the solution as (1, 0, 3). (06 Marks)

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2. b. Diagonalize the matrix A =

   3	-1	1
   -1	5	-1
   1	-1	3

   . (07 Marks)
3. c. Reduce the quadratic form 8x² + 7y² +3z² -12xy + 4xz -8yz into canonical form by orthogonal transformation. (07 Marks)

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