Download AKTU B-Tech 1st Sem 2016-2017 RAS 103 Engineering Mathematics 1 Question Paper

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Printed Pages: 8

RAS-103

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

Paper ID: 2012439

Roll No.

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B.TECH.

Regular Theory Examination (Odd Sem - I),2016-17

ENGINEERING MATHEMATICS-I

Time: 3 Hours

Max. Marks: 70

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Note: The question paper contains three sections - A, B & C.

Read the instructions carefully in each section.

SECTION-A

Attempt all questions of this section. Each part carries 2 marks.

1. a) For what value of 'x', the eigen values of the given matrix A are real
   

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   A = $$\left[\begin{array}{ccc}x& 4& 2\\ 10& 5+i& -10\\ 2& 4& -5-3\end{array}\right]$$
2. b) For the given matrix A = $$\left[\begin{array}{cc}1& -3\\ 2& 0\end{array}\right]$$ prove that A³ = 19A+30I.
3. c) Find the maximum value of the function f(xyz) = (z - 2x² – 2y²) where 3xy - z + 7 = 0.
4. d) If the volume of an object expressed in spherical coordinates as following:
   V = ? r² sin f dr d? df. Evaluate the value of V.

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5. e) Find the condition for the contour on x - y plane where the partial derivative of (x² + y²) with respect to y is equal to the partial derivative of (6y + 4x) with respect to x.
6. f) The parabolic arc y = vx, 1=x=2 is resolved around x- axis. Find the volume of solid of revolution.
7. g) For the scalar field u = x²y² / (x4 + y4), Find the magnitude of gradient at the point (1, 3).

SECTION-B

Attempt any three parts of the following. Each part carries 7 marks.

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1. a) i) Express 2A-3A4+A²-4I as a linear polynomial in A where A= $$\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$$
   ii) Reduce the matrix P = $$\left[\begin{array}{ccc}1& 2& -2\\ 1& 2& 1\\ -1& -1& 0\end{array}\right]$$ to diagonal form.
2. b) i) If u = sin?¹( (x²/y² + y²/x²) ), then evaluate the value of x² (?²u/?x²) +2xy(?²u/?x?y) + y²(?²u/?y²).
   ii) Trace the curve x=a(?-sin?), y = a(1-cos?).
3. c) i) Find the relation between u, v, w for the values u = x + 2y + z; v=-x - 2y + 3z; w = 2xy - zx + 4yz - 2z².
   

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   ii) Divide a number into three parts such that the product of first, square of the second and cube of third is maximum.
4. d) i) Change the order of integration for I = ? xy dxdy and hence evaluate the same.
   ii) Evaluate the triple integral ?(xyz)dx dy dz where V is 0 < x < 1, 0 < y < 2, 0 < z < 2.
5. e) i) If F = (x² + y²)i-2xyj, then evaluate the value of ? F. dr.
   ii) Find the directional derivative of 1/r in the direction of r where r = ix + jy + kz.

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SECTION-C

Attempt all questions of this section, selecting any two parts from each question. All questions carry equal marks.

(5×7=35)

1. a) If I = dn/dxn (xn logx), show that I=nI+n-1.
2. b) If e^-z/(x²-y²) = x - y then show that yz + x = x²- y² .

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3. c) If w = vx² + y² + z² & x = v² + w², y = w² + u², z = u² + v² then show that (?(x,y,z))/(?(u,v,w)) =1.
4. a) If x = v² + w², y = w² + u², z = u² + v² then show that (?(x,y,z))/(?(u,v,w)) *(?(u,v,w))/(?(x,y,z)) =1.
5. b) Express the function f(x,y) = x² + 3y²-9x-9y+26 as Taylor's Series expansion about the point (1, 2).
6. c) Find the percentage error in measuring the volume of a rectangular box when the error of 1% is made in measuring the each side.
7. a) If A = $$\left[\begin{array}{cc}-3& 2\\ 4& 2\end{array}\right]$$ then evaluate the value of the expression (A +5I+2A?¹).

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8. b) Find the eigen value of the matrix $$\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 1& -3& =(2)\end{array}\right]$$ corresponding to the eigen vector $$\left[\begin{array}{c}10\\ 1\end{array}\right]$$
9. c) Show that A = 1/v3 $$\left[\begin{array}{ccc}1& 1& 1\\ 1& w& w²\\ 1& w²& w\end{array}\right]$$ is a unitary matrix, where w is complex cube root of unity.
10. a) Changing the order of integration in the double integral I=? f(xy)dy dx leads to the value I= ? f(xy)dxdy. What is the value of q?
11. b) Evaluate ? x²yz dx dy dz throughout the volume bonded by planes x =0,y=0,z=0 & x/a+y/b+z/c=1.
12. c) For the Gamma function, show that G(z+1) = z G(z).

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13. a) Verify Stokes theorem F = (2y+z, x-z, y-x) taken over the triangle ABC cut from the plane x + y + z = 1 by the coordinate planes.
14. b) Verify Gauss Divergence theorem for ?[(x² - yz)î - 2x²yj+2k].nds where S denotes the surface of cube bounded by the planes x = 0, x = a; y = 0, y = a; z = 0, z = a.
15. c) IfA = (xz²i + 2yj-3xzk) and B = (3xzi + 2yzj - z²k) Find the value of [AX(?B)] & [(Ax?)×B].