Download AKTU B-Tech 1st Sem 2015-2016 EAS 103 Mathematics I Question Paper

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Printed Pages: 5

175

EAS-103

(Following Paper ID and Roll No. to be filled in your Answer Book)

Paper ID :199123 Roll No.

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B.Tech.

(SEM. I) THEORY EXAMINATION, 2015-16

MATHEMATICS-I

[Time:3 hours]

[Total Marks:100]

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Section-A

1. Attempt all parts. All parts carry equal marks. Write answer of each part in shorts. (10×2=20)

   1. If u = log(x² / y) then value of x ?u/?x + y ?u/?y = ?
   2. If z = xy show that x ?z/?x + y ?z/?y = 2z.
   3. Apply Taylor's series find expansion of f(x, y) = x² + xy² about point (2,1), upto first degree term.

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   4. If x = u - v, y = u² - v², find the value of ?(x, y) / ?(u, v)
   5. Find all the asymptotes of the curve xy² = 4a²(2a-x).
   6. Find the inverse of the matrix by using elementary row operations. A =

       1 -1 0 -1 0 0 

   7. If A=

       2 -3 0 1 -1 3 4 -2 1 

      find the eigen values of A².
   8. Evaluate

       ? xyz dx dy dz, limits 1 to 4, 1 to -2, 1 to 1. 

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   9. If f(x, y, z) = x²y + y²x + z² find ?f at the point (1,1,1).
   10. Evaluate G(8/3) / G(2/3)

Section-B

Note: Attempt any five Questions from this section: (5x10=50)

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2. If x = sin^-1(sin^-1 y) find the value of y_n at x = 0.
3. If u, v, w are the roots of the equation (1-x)^-1+(1-y)^-1+ (1-z)^-1 = 0 find ?(u, v, w) / ?(x,y,z)
4. If r is the distance of a point on Conic ax² + by² + cz² = 1, lx + my + nz = 0 from origin, then that the stationary values of r are given by the equation l² / (1-ar²) + m² / (1-br²) + n² / (1-cr²) = 0.
5. Find the Eigen values and corresponding Eigen vectors of A=

    6 -2 2 -2 3 -1 2 -1 3 

6. The plane x/a + y/b + z/c =1 meets the axes in A, B, and C. Apply Dirichlet's integral to find the volume of the tetrahedron OABC. Also find its mass if the density at any point is kxyz.

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7. Change the order of Integration in I = ? xy dxdy limits 0 to 1, 0 to 2-x and hence evaluate the same.
8. Verify gauss's divergence theorem for the function F = x²i + zj + yzk, taken over the cube bounded by x = 0, x = 1, y = 0, y = 1 and z = 0, z = 1.
9. Show that the Vector field F is irrotational as well as solenoidal. Find the scalar potential.

Section-C

Attempt any two questions from this section: (2×15=30)

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10. 1. Expand e^x cos(by) in powers of the powers of x and y as terms of third degree.
    2. Reduce the matrix in to normal form and hence find its rank

        1 2 1 0 -2 -4 3 0 1 0 2 -8 

    3. Examine the following vectors for linearly dependent and find the relation between them, if possible, X1 = (1,1,-1,1), X2 = (1,-1,2,-1), X3 = (3,1,0,1).
11. 1. Define Beta and Gamma function and Evaluate ? dx / v(1+x) limits 0 to 8

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    2. Find the area between the parabola y² = 4ax and x² = 4ay.
    3. If y1 = x2x3/x1, y2 = x3x1/x2, y3 = x1x2/x3 find ?(y1,y2,y3) / ?(x1,x2,x3)
12. 1. Evaluate ? dx / (xv(x²-a²))
    2. Determine the constant a and b such that the curl of vector A = (2xy+3xz) i +(x² + axz-4z²)j-(3xy+byz)k is zero.

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    3. If u = (y-x)/xy , v = (z-x)/xz , w = (z-y)/yz show that x² ?u/?x +y² ?u/?y +z² ?u/?z = 0

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