Download AKTU B-Tech 1st Sem 2015-2016 NAS 301 Engineering Mathematics I Question Paper

Section-A

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

1. If y=esin^-1x, find the value of (1-x²)y2 - xy1 - a²y.
2. If V = (x² + y² + z²)^-1/2, then find ?V/?x + ?V/?y + ?V/?z.

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3. If f(x,y,z,w)=0, then find ?x/?y . ?y/?z . ?z/?w . ?w/?x.
4. If pv² = k and the relative errors in p and v are respetively 0.05 and 0.025, show that the error in k is 10%.
5. Examine whether the vectors x1 =[3,1,1], x2=[2,0,-1], x3=[4,2,1] are linearly independent.
6. If A =

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

   find the eigen values of A².
7. Evaluate ? dxdy / v(1-x).

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8. Find the value of integral ?e^xdx.
9. Find the curl of F = xyi + y²j+xzk at (-2,4,1)
10. State Stoke's theorem.

Section-B

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

2. If cos(mx) = log(y), then show (1-x²)y_n+2 - (2n+1)xy_n+1-(n²+m²)y_n = 0 and hence Calculate Y_n when x = 0.

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3. If u, v, w are the roots of the equation (1-x)²+(1-y)²+(1-z)²=0 find ?(x,y,z)
4. Using the Lagrange's method find the dimension of rectangular box of maximum capacity whose surface area is given when (a) box is open at the top (b) box is closed.
5. Find the characteristic equation of the matrix A=

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

   and verify Cayley Hamilton theorem. Also evaluate A⁸ -6A⁷ +9A⁶ -2A⁵-12A² + 23A-9I.
6. Prove that ? x^l-1y^m-1z^n-1 e^-(x+y+z) dx dy dz = G(l)G(m)G(n) the integral being extended to all positive values of the variables for which the expression is real.
7. Verify the Green's theorem to evaluate the line integral ?(2y²dx + 3xdy), where C is the boundary of the closed region bounded by y = x and y = x².

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8. Determine the values 'a' and 'b' for which the following system of equation has. x+y+z=6, x+2y+3z = 10, x + 2y + az = b
   1. No solution
   2. A unique solution
   3. Infinite no of solutions.
9. Find the mass of a solid (x/a)²+(y/b)²+(z/c)² = 1, the density at any point being ? = kx^l-1y^m-1z^n-1 where x, y, z are all positive.
   

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   Change the order of Integration in I= ?₀²?₀^xxy dxdy and hence evalute.
   Find the rank of the matrix by reducing to normal form.

   3	2	-1
   4	2	6
   7	4	5

10. A fluid motion is given by v = (y + z) i + (z + x)j+(x+y) k. Show that the motion is irrotational and hence find the velocity potential.
    If x+y+z = u, y + z = uv, z = uvw then find ?(x, y, z) / ?(u, v, w)

Section-C

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

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10. a) If u = f(r) where r² =x²+y², show that ?²u/?x² + ?²u/?y² = f''(r) + (1/r)f'(r).
    Prove that, for every field F; div curl F =0.
11. a) Evaluate ? (x + y + z)dx dy dz where R: 0 = x = 1; 1 = y = 2; 2 = z=3.
    Trace the curve y²(2a-x) = x².
12. Verify Euler's theorem for the function z = (x+y) / (x³ + y³) - 1 / (x² + y²)

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