Download AKTU B-Tech 1st Sem 2017-2018 RAS 103 Engineering Mathe Matics Question Paper

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Sub Code: RAS-103

B. TECH

(SEM I) THEORY EXAMINATION 2017-18

ENGINEERING MATHEMATICS

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Total Marks: 70

Note: 1. Attempt all Sections. If require any missing data; then choose suitably.

SECTION A

1. Attempt all questions in brief. 2 x 7 = 14

1. a. Find the nth derivative of X^n-1log x.

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2. b. Evaluate ? x² exy dxdy.
3. c. If x² = au + bv, y² = au - bv, then find (?u/?x)_v (?x/?u)_v
4. d. Evaluate the area enclosed between the parabola y = x² and the straight line y = x.
5. e. What error in the logarithm of a number will be produced by an error of 1% in the number?
6. f. Find the value of m if f = mxi – 5yj + 2zk is a solenoidal vector.

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7. g. Reduce the matrix $$\left[\begin{array}{ccc}1& 1& 1\\ 3& 1& 1\\ 3& 1& 1\end{array}\right]$$ in to normal form and find its rank.

SECTION B

2. Attempt any three parts of the following: 7 x 3 = 21

1. i) If u = sin^-1 ((x³+y³+z³)/xyz), prove that x (?u/?x)+y(?u/?y)+z(?u/?z) = 2 tan u.
2. i) If u = sinⁿx + cosⁿx, then prove that u_r ={n^r + (-1)ⁿ sin²nx}^1/2,where u_r is the rth differential coefficient of u w.r.t.x.

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3. i) Using elementary transformations, find the rank of the following matrix: $$A=\left[\begin{array}{cccc}2& -1& 3& -1\\ 1& 2& -3& -1\\ 1& 0& 1& 1\\ 0& 1& 1& -1\end{array}\right]$$
4. ii) Compute the inverse of the matrix $$\left[\begin{array}{ccc}1& 2& 3\\ 1& -1& -1\\ 2& 1& -1\end{array}\right]$$ by employing elementary row transformation.
5. i) If u₁ = (x₂x₃)/x₁, u₂ = (x₃x₁)/x₂ and u₃ = (x₁x₂)/x₃ find the value of ?(u₁,u₂,u₃)/?(x₁,x₂,x₃)
6. ii) If u = f ( r, s, t ), where r = x/y, s = y/z, t=z/x, show that x (?u/?x)+y(?u/?y)+z(?u/?z) = 0.
7. i) Change the order of integration in I = ?02 ?x22-x f(x, y)dy dx.

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8. ii) Prove that: ß (m,n) = (GmGn)/G(m+n), m>0, n>0.
9. i) Determine the value of constants a, b, c if, F= (x + 2y + az)î + (bx – 3y – z)j + (4x + cy + 2z)k is irrotational.
10. ii) If A = (x - y)î + (x + y)j, evaluate § A. dr around the curve C consisting of y = x² and y² = x.

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3. Attempt any two parts of the following: 7 x 5 = 35

1. If y = e^{tan-1 x}, then prove that (1 + x²)y₂+(2x - 1)y₁ = 0 and (1 + x²) y_n+2+ [2(n + 1)x - 1] y_n+1 + n(n + 1)y_n = 0.
2. If u = x² tan^-1 (y/x)-y² tan^-1(x/y); xy ? 0 prove that ?²u/?x?y = (x²-y²)/(x²+y²)
3. Trace the curve:y²(a + x) = x²(3a – x).

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4. Attempt any two parts of the following: 7 x 5 = 35

1. A balloon in the form of right circular of radius 1.5m and length 4m is surmounted by hemispherical ends. If the radius is increased by 0.01m find the percentage change in the volume of the balloon.
2. Using Lagrange's method of Maxima and Minima, find the shortest distance from the point (1,2, -1) to sphere x² + y² + z² = 24.
3. Express the function f(xy) = x² + 3y² – 9x – 9y + 26 as Taylor's Series expansion about the point (1, 2).

5. Attempt any two parts of the following: 7 x 5 = 35

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1. Investigate for what values of ? and µ, the system of equations x + y + z = 6, x + 2y + 3z = 10 and x + 2y + ?z = µ, has:
   1. (i) No solution
   2. (ii) Unique solution
   3. (iii) Infinite number of solutions.
2. Verify Cayley–Hamilton theorem for the matrices A = $$\left[\begin{array}{ccc}2& 1& 1\\ -1& 1& -1\\ 1& 2& -1\end{array}\right]$$

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3. Show that the matrix $$\left[\begin{array}{cc}a+i?& ß+id\\ -ß+id& a-i?\end{array}\right]$$ is unitary if a² + ß² + ?² + d² = 1.

6. Attempt any two parts of the following: 7 x 5 = 35

1. Find the mass of a plate which is bounded by the co-ordinate planes and the plane x/a + y/b + z/c = 1, the density is given by ? = k xyz.
2. Evaluate I = ?₀¹ x³/(v(2-x)) dx
3. Evaluate ?_R (x + y + z) dxdydz where R: 0 < x < 1, 1= y = 2, 2 < z < 3.

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7. Attempt any two parts of the following: 7 x 5 = 35

1. Verify Green's theorem, evaluate ? (x² + xy) dx + (x² + y²)dy where c is square formed by lines x = +1, y = + 1
2. Verify Stoke's theorem for F = (x² + y²)î -2xyj taken round the rectangle bounded by the lines x = + a, y = 0, y = b .
3. If all second order derivatives of f and ? are continuous, then show that
   1. (i) Curl (grad f) = 0
   2. (ii) div (curl ? ) = 0.

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