Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 2017 Jan 2016 Dec 15MAT11 Engineering Mathematics I Question Paper

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15MAT11

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First Semester B.E. Degree Examination, Dec.2016/Jan.2017

Engineering Mathematics - I

Time: 3 hrs.

Max. Marks: 80

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

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Module-1

1. 1. If y = e^x cos²x, find y_n. (06 Marks)
   2. Find the angle between the curves r = a / (1 + cos?) and r = b / (1 - cos?). (05 Marks)
   3. Find the radius of curvature of the curve x⁴ + y⁴ = 2 at the point (1, 1). (05 Marks)

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OR

1. 1. If x = tan(log y), find the value of (1+x²)y_n+2 + (2nx-1)y_n+1 + (n²+n)y_n. (06 Marks)
   2. Find the Pedal equation of 2a / r = 1+ cos ?. (05 Marks)
   3. Find the radius of curvature of the curve rⁿ = aⁿ cos n?. (05 Marks)

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Module-2

1. 1. Expand e^x / (1+ e^x) using Maclaurin's series upto and including 3rd degree terms. (06 Marks)
   2. State Euler's theorem and find x (?u/?x) + y (?u/?y) when u = tan^-1((x³+y³)/(x+y)). (05 Marks)
   3. Expand log(cos x) about the point x = p/3 upto 3rd degree terms using Taylor's series. (05 Marks)

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OR

1. 1. Find du / dt with x = at², y = 2at. Use Partial derivatives. (06 Marks)
   2. If u = x² + y² + z², v = xy + yz + zx, w = x + y + z , find the value of Jacobian J(u,v,w)/J(x,y,z). (05 Marks)
   3. If u = x₁ / (x₂x₃), v = x₂ / (x₁x₃), w = x₃/(x₁x₂) , find the value of Jacobian J(u,v,w)/J(x₁,x₂,x₃). (05 Marks)

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Module-3

1. 1. A particle moves on the curve x = 2t², y = t² – 4t, z = 3t – 5, where t is the time find the components of velocity and acceleration at time t = 1 in the direction of i – 3j + 2k. (06 Marks)
   2. Find the divergence and curl of the vector V = (xyz)i + (3x²y)j + (xz² – y²z)k at the point (2, -1, 1). (05 Marks)
   3. A vector field is given by A = (x² + xy²)i + (y² + x²y)j, show that the field is irrotational and find the scalar potential. (05 Marks)

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OR

1. 1. Find grad (f) when f = 3x²y - y³z² at the point (1, -2, -1). (06 Marks)
   2. Find a for which f = (x + 3y)i + (y - 2z)j + (x + az)k is solenoidal. (05 Marks)
   3. Prove that Div(curl V) = 0. (05 Marks)

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Module_4

1. 1. Obtain the reduction formula of ?sinⁿ x cos ^mx dx. (06 Marks)
   2. Evaluate ? xv(2ax - x²) dx. (05 Marks)
   3. Solve (2x log x - xy) dy + 2y dx = 0. (05 Marks)

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OR

1. 1. Obtain the reduction formula of ? cosⁿ x dx. (06 Marks)
   2. Obtain the Orthogonal trajectory of the family of curves rⁿ cos n? = aⁿ. Hence solve it. (05 Marks)
   3. A body originally at 80°C cools down at 60°C in 20 minutes, the temperature of the air being 40°C. What will be the temperature of the body after 40 minutes from the original? (05 Marks)

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Module_5

1. 1. Find the rank of the matrix A= [ 2 3 -1 -1 ; 1 -1 -2 -4 ; 3 1 3 -2 ; 6 3 0 -7 ]. (06 Marks)
   2. Solve by Gauss — Jordan method the system of linear equations 2x + y + z = 10, 3x + 2y + 3z = 18, x + 4y + 9z = 16. (05 Marks)
   3. Find the largest eigen value and the corresponding Eigen vector by power method given that A=[2 0 1; 0 2 0; 1 0 2] (Use [1; 1; 1] as the initial vector). (Apply 4 iterations). (05 Marks)

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OR

1. 1. Use Gauss Seidel method to solve the equations 20x + y - 2z = 17 ; 3x + 20y - z = 18 ; 2x - 3y + 20z = 25. Carry out 2 iterations with x₀ = y₀ = z₀ = 0. (06 Marks)
   2. Reduce the matrix A = [-1 2 -2 ; 1 2 1 ; -1 -1 0] to the diagonal form. (05 Marks)
   3. Reduce the quadratic form 3x² + 5y² + 3z² — 2yz + 2zx - 2xy to the canonical form. (05 Marks)

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