Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 2014 June-July 10MAT11 Engineering Mathematics I Question Paper

Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 2014 June-July 10MAT11 Engineering Mathematics I Question Paper

This post was last modified on 01 January 2020

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VTU B.Tech 1st Year Last 10 Years 2011-2021 Question Papers

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First Semester B.E. Degree Examination, June / July 2014

Engineering Mathematics - I

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10MAT11

Time: 3 hrs.

Max. Marks:100

Note: 1. Answer any FIVE full questions, choosing at least two from each part.

2. Answer all objective type questions only on OMR sheet page 5 of the answer booklet.

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3. Answer to objective type questions on sheets other than OMR will not be valued.

PART - A

1. Choose the correct answers for the following : (04 Marks)
  1. If y = (Acosx + Bsinx)e^x then y₄ is,
    1. e^x cosx
    2. e^2x sin 3x
    3. e^x cos x
    4. None of these
  2. sin x = x - x³/3! + x⁵/5! - x⁷/7! + ... is,
    1. Taylor's series
    2. Exponential series
    3. Maclaurin's series
    4. None of these
  3. In Rolle's theorem if F'(c) = 0 then the tangent at the point x = c is,
    1. parallel to y-axis
    2. parallel to x-axis
    3. parallel to both axes
    4. None of these
  4. If y = 3^x then y_n =
    1. (log x)3ⁿ
    2. 3(logx)ⁿ
    3. 3ⁿ log 3
    4. 3^x (loge 3)ⁿ
2. If x = sin t, y = sinpt prove that, (1-x²)y_n+2 - (2n+1)xy_n+1 + (p²-n²)y_n = 0. (04 Marks)
3. State and prove Cauchy's mean value theorem in [0, 16]. (06 Marks)
4. Expand 4 + sin 2x by using Maclaurin's expansion. (06 Marks)
1. Choose the correct answers for the following : (04 Marks)
  1. The value of Lt _x?0 (1 + x)^1/x is,
    1. e²
    2. 1
    3. 1/e
    4. 0
  2. The angle between two curves r = ae^? and r e^-? = b is,
    1. p/2
    2. p/4
    3. 0
    4. p
  3. v((dx/dt)² + (dy/dt)²) dt is,
    1. Polar form
    2. Parametric form
    3. Cartesian form
    4. None of these
  4. Lt _x?0 (a^x - b^x)/x is,
    1. 1
    2. 0
    3. 2
    4. None of these
2. Find a & b, if Lt _x?0 (a sin x - b sinx) / x(1+ a cosx) = 1 (04 Marks)
3. Find the pedal equation of the curve r² = a² cos2? (06 Marks)
4. Find the radius of curvature at any point t of the curve x = a(t +sin t) and y = a(1— cost). (06 Marks)
1. Choose the correct answer (04 Marks)
  1. If u = (x - y)² + (y - z)² + (z - x)² then ?u/?x + ?u/?y + ?u/?z is,
    1. 1
    2. 24
    3. 2(x + y + z)
    4. 0
  2. ex cosy = p [1+ (x - 1)-(y-p/4)+...+1/2!((x-1)(y---) +...
    1. (1,p/4)
    2. (0, 0)
    3. (1, 1)
    4. (4,1)
  3. At (a, b) = A, ?²u/?x² = B and ?²u/?y² = H and if AB - H² < 0 the F.Such a point is called,
    1. Maximum
    2. Minimum
    3. Saddle
    4. Extremum
  4. If J = ?(u, v) / ?(x, y) J¹ = ?(x, y) / ?(u, v) then JJ¹ is,
    1. 0
    2. 2
    3. 1/2
    4. 1
2. If u=f(x/y,y/z,z/x) then prove that x ?u/?x + y ?u/?y + z ?u/?z = 0 (04 Marks)
3. If u = x² + y² + z², v = xy + yz + zx, w = x + y + z then show that ?(U, V, W) / ?(x, y, z) = 0 (06 Marks)
4. For the kinetic energy E = 1/2 mv² find approximately the change in E as the mass m changes from 49 to 49.5 and the velocity v changes from 1600 to 1590. (06 Marks)
1. Choose the correct answers for the following (04 Marks)
  1. The value of V x Vf is,
    1. 0
    2. R
    3. 9
    4. 3
  2. Any motion in which the curl of the velocity vector is zero, then the vector v is said to be,
    1. Constant
    2. Solenoidal
    3. Vector
    4. Irrotational
  3. In orthogonal curvilinear co-ordinates the Jacobian J = ?(x, y, z) / ?(u, v, w) is,
    1. h₁/h₂h₃
    2. h₁h₂/h₃
    3. h₁h₂h₃
    4. h₁h₂h₃
  4. A gradient of the scalar point function f, Vf is,
    1. Scalar function
    2. Vector function
    3. f
    4. zero
2. Find the value of the constant a such that the vector field, F = (axy - z³)i + (a - 2)x²j + (1-a)xz²k is irrotational and hence find a scalar function f such that F = Vf. (04 Marks)
3. Prove that curl(curl A) = V(V.A) - V²A . (06 Marks)
4. Express V²iv in orthogonal curvilinear co-ordinates. (06 Marks)

PART - B

1. Choose the correct answers for the following : (04 Marks)
  1. The value of ?₀^p/2 cos³ (4x)dx is,
    1. 3/16
    2. 3/8
    3. 1/6
    4. 0
  2. If the equation of the curve remains unchange after changing ? to – ? the curve r = f(?) is symmetrical about,
    1. A line perpendicular to initial line through pole
    2. Radially symmetric about the point pole.
    3. Symmetry does not exist
    4. Initial line
  3. The volume of the curve r = a(1+ cos?) about the initial line is,
    1. 4pa³/3
    2. 2pa³/3
    3. 8pa³/3
    4. pa³/3
  4. The asymptote for the curve x³ + y³ = 3axy is equal to,
    1. x+y+a= 0
    2. x-y-a=0
    3. No Assymptote
    4. x+y-a=0
2. Evaluate ?₀^p/2 log(1 + cos x)dx. (04 Marks)
3. Evaluate ?₀^2a v(2ax - x²) dx. (06 Marks)
4. Find the area of surface of revolution about x-axis of the astroid x^2/3 + y^2/3 = a^2/3. (06 Marks)
1. Choose the correct answers for the following (04 Marks)
  1. In the homogeneous differential equation, dy/dx = f₁(xy) / f₂(xy) the degree of the function, f₁(xy) and f₂(xy) are,
    1. Different
    2. Relatively prime
    3. Same
    4. None of these
  2. The integrating factor of the differential equation, dy/dx + cot xy = cos x is,
    1. e^{- sin x}
    2. sin x
    3. – sin x
    4. cot x
  3. Replacing dy/dx by -dx/dy in the differential equation f(x, y, dy/dx) = 0 we get the differential equation of,
    1. Polar trajectory
    2. Orthogonal trajectory
    3. Parametric trajectory
    4. Parallel trajectory.
  4. Two families of curves are said to be orthogonal if every member of either family cuts each member of the other family at,
    1. Zero angle
    2. Right angle
    3. p
    4. 2p
2. Solve (1+e^x/y)dx + e^x/y(1-x/y)dy = 0. (04 Marks)
3. Solve dy/dx + x sin 2y = x³ cos² y. (06 Marks)
4. Find the orthogonal trajectories of r = a² cos² ?. (06 Marks)
1. Choose the correct answer (04 Marks)
  1. A = [7 0 0; 0 7 0; 0 0 7] is called,
    1. Scalar matrix
    2. Diagonal matrix
    3. Identity matrix
    4. None of these
  2. If r = n and x = y = z = 0. The equations have only solution.
    1. Non trivial
    2. Trivial
    3. Unique
    4. Infinite
  3. In Gauss Jordan method, the coefficient matrix can be reduced to,
    1. Echelon form
    2. Unit matrix
    3. Triangular form
    4. Diagonal matrix
  4. The inverse square matrix A is given by,
    1. A^-1 = adjA / |A|
    2. |A|
    3. adjA
    4. adjA/|A|
2. Find the Rank of the matrix, [1 2 3 2; 2 3 5 1; 1 3 4 5] (05 Marks)
3. Investigate the values of ? and µ such that the system of equations, x +y+z=6, x + 2y + 3z =10, x+2y + ?z = µ may be i) Unique solution ii) Infinite solution iii) No solution. (06 Marks)
4. Using Gauss elimination method solve, 2x₁-x₂+3x₃ =1, -3x₁+4x₂ -5x₃ = 0, x₂-5x₃ = 5 (05 Marks)
1. Choose the correct answers for the following : (04 Marks)
  1. A square matrix A of order 3 has 3 linearly independent eigen vectors then a matrix P can be found such that P^-1 AP is a,
    1. Diagonal matrix
    2. Unit matrix
    3. Singular matrix
    4. Symmetric matrix
  2. The eigen values of matrix [2 2; 1 3] are,
    1. x=y=z=0
    2. 2 ±v2
    3. 2
    4. None of these
  3. Solving the equations x - 2y + 3z = 0, 3x + 4y + 4z = 0, 7x +10y +12z = 0 . ). x, y and z values are,
    1. x=y=z=0
    2. x=y=z= 1
    3. x=y=z
    4. None of these
  4. The index and significance of the quadratic form, x² + 5y² + 3z² and 2x - 3x are respectively
    1. Index = 1, Signature = 1
    2. Index = 1, Signature = 2
    3. Index = 2, Signature = 1
    4. None of these.
2. Find all the eigen values and the corresponding eigen vectors of the matrix, A = [8 -6 2; -6 7 -4; 2 -4 3] (04 Marks)
3. Reduce the matrix A = [1 -1 4; 7 4 -2; 2 4 3] into a diagonal matrix. (06 Marks)
4. Reduce the quadratic form 3x² + 5y² + 3z² - 2yz + 2zx - 2xy to the canonical form. (06 Marks)

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This post was last modified on 01 January 2020

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