MATHEMATICS-I (utu previous year question papers)

MATHEMATICS-I
SEM-I, 2012-13
B.TECH EXAMINATION
UTTARAKHAND TECHNICAL UNIVERSITY
utu previous year question papers

Time: 3 hours

Total marks: 100

Attempt any four parts of the following:

1. Reduce the following matrix to row echelon form and finds its rank:
   
2. Examine whether the following set of vectors is linearly independent and also find the dimension and the basis of the vectors:
   (1, 1, 0, 1), (1, 1, 1, 1), (-1, 1, 1, 1), (1, 0, 0, 1)
3. Check whether the following system is consistent, if it is then find its solution.
   
4. Verify Cayley-Hamilton theorem for the matrix
   
   Also obtain A^-1
5. Find the eigenvalues and corresponding eigenvectors of the given matrix:
   
6. The eigenvectors of a 3x3 matrix A corresponding to the eigenvalues 1, 1, 3 are [1, 0, -1]^T, [0, 1, -1]^T and [1, 1, 0]^T respectively. Find the matrix A.

Attempt any two parts of the following:

1. Find dy/dx, where y = (sinx)^(cosx) + (cosx)^(sinx)
2. If y = sin(m sin^-1x), prove that
   (1-x²) y⁽ⁿ⁺²⁾ - (2n + 1)xy⁽ⁿ⁺¹⁾ + (m² – n²)y⁽ⁿ⁾ = 0
3. (a) If 
   show that
   (b) Expand y^x at (1, 1) up to second term.

Attempt any two parts of the following:

1. (a) Find the value of Jacobian,
    where u = x² + y², v = 2xy and x = rcosθ, y = rsinθ
   (b) If x + y + z =u, y + z =uv, z = uvw,
   show that
   
2. Show that the function
   f(x, y) = x³ + y³   ̶ 63(x + y) + 12xy is maximum at (-7, -7) and maximum at (3, 3).
3. Use the method of Lagrange's multipliers to find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid
   

Attempt any two parts of the following:

1. Evaluate ∬(x + y) dxdy over the area bounded by the ellipse
   
2. (a) Find the value of Г(½)
   (b) Evaluate x^{α- 1} y^β-1 z^γ-1 dx dy dz where V is the region in the first octant bounded by sphere x² + y² + z² = 1 and the coordinate planes.
3. Evaluate
   using the substitutions
   where A is bounded by x² + y² - x = 0, y = 0, y >0

Attempt any two parts of the following:

1. (a) Find the normal vector and the equation of the tangent plane to the surface
   at the point (3, 4, 5)
   (b) If r = xi + yj + zk and r = |r|, show that iv(r/r³)=0
2. State and prove Green's theorem.
   Using Green's theorem evaluate ∮_C(x² + y²)dx + (y + 2x)dy, where C is the boundary of the region in the first quadrant that is bounded by the curves x² = y and y² = x.
3. Verify the Divergence Theorem by direct computation of the surface integral and the triple integral, where  F = 7xi - zk and s is the surface x² + y² + z² =4

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