Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 2015 Jan 2014 Dec 10MAT21 Engineering Mathematics Question Paper

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10MAT21

Second Semester B.E. Degree Examination, Dec.2014/Jan.2015

Engineering Mathematics

Time: 3 hrs.

Max. Marks:100

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Note: 1. Answer any FIVE full questions, choosing at least two from each part.

2. Answer all objective type questions only on OMR sheet page 5 of the answer booklet.

3. Answer to objective type questions on sheets other than OMR will not be valued.

PART — A

1 a. Choose the correct answers for the following : (04 Marks)

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1. A differential equation of the first order but of higher degree, solvable for x, has the solution as:
   A) F(y, p, c) = 0 B) F(x, p, c) C) F(x, y, c) = 0 D) C(x, C2) = 0
2. If xy + C = C²x is the general solution of a differential equation then its singular solution is
   A) y = x B) y=-x C) 4x²y+1=0 D) 4x²y-1=0
3. The general solution of Clairant's equation is,
   

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   A) y = Cf (x) + f(C) B) y = Cx + f(C) C) x = Cf(y) + f(C) D) x = Cy +g(C)
4. The general solution of p² -7p +12 = 0 is,
   A) (? —3? — ?)(? —4x —c) = 0 B) (y - c)(x-c) = 0 C) (3x c)(4? — ?) D) (y +3x + c)(y —4x — ?) = 0

b. Solve: xp² (2x+3y)p +6y = 0. (05 Marks)

c. Solve : y=3x +logp , Clairant's equation. (05 Marks)

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2 a. Choose the correct answers for the following: (04 Marks)

1. The particular integral of (D ² + a²)y = sin ax is,
   A) xcosax / 2a B) -xcosax / 2a C) x sin ax / 2a D) - x sin ax / 2a
2. The solution of the differential equation (D 4-5D² +4)y = 0 is,
   A) y=C¡e' +C2e¯ +Cze²x+C4e^2x B) y= (C1 +C,x+C3x² +C,x³)e^2x C) y = C cos x +C2 sin x + C3 cos 2x + C4 sin 2x D) None of these

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3. The particular integral of (D — 1)² y = 3ex is,
   A) -3xex / 2 B) -- 3Xe^X / 2 C) 3x²ex / 2 D) -3x² ex
4. The roots of auxiliary equation of D' (D ² + 2D)² y = 0 2 are:
   A) 0,0,0,0,2,2 B) 0,0,0,0,-2,-2 C) 0,0,2,2,-2,-2 D) 2,2,2,2,0,0

b. Solve: (D²—2D +1)y = xe^x + x (05 Marks)

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c. Solve : (D² - 4D + 4)y = e^x + cos 2x + 4. (05 Marks)

d. Solve : dx/dt -7x + y = 0, dy/dt -2x —5y = 0. (06 Marks)

3 a. Choose the correct answers for the following : (04 Marks)

1. The complementary function of x 2 y" – 3xy' + 4y = x is:
   A) (C1 + C2 log x)x² B) (C1 + C2x)e²x C) C1/x + C2/x^x D) C+C2x

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2. By the method of variation parameters, the value of 'W' is called,
   A) Euler's function B) Wronskian of the function C) Demorgan's function D) Cauchy's function
3. The equation (2x+1)² y" –6(2x+1)y' +16y = 8(2x+1) 2 by putting z = log (2x +1) with D = d/dz reduces to
   A) (D² + 4D + 4)y = 3e^2z B) (D² – 4D + 4)y = 2e^2z C) (D² -4D + 4)y = 0 D) None of these
4. To find the series solution for the equation 2x²y" – xy' 4- (1– x² )y = 0, we assume the solution as,
   

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   A) y= S a_rx^r+1 (r=0 to infinity) B) y= S a_r+1x^r (r=1 to infinity) C) y= S a_rx^r (r=0 to infinity)

b. Solve : (D² +a²)y = sec ax by method of variation of parameters. (05 Marks)

c. Solve (3x + 2)² y" +3(3x + 2)y¹ –36y = 8x² + 2 + 4x + 1 (05 Marks)

d. Solve : y" + xy' + y = 0 in series solution. (06 Marks)

4 a. Choose the correct answers for the following (04 Marks)

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1. 2z = x²/a + y²/a, where 'a' arbitrary constants is a solution of:
   A) 2z=p²x+qy B) 2z =px+q²y C) 2z = px + qy D) None of these
2. The auxiliary equations of Lagrange's linear equation Pp + Qq = R are:
   A) dx/p = dy/q = dz/R B) dx/pq = dy/qR = dz/Rp C) dx/qR = dy/Rp = dz/pq D) dx/p = dy/q = dz/0
3. General solution of the equation ?²z/?x² = x + y is,
   

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   A) x³/6 + x²y/2 + f(y)+ g(y) B) x³/6 + x²y/2 + xf(y)+ g(y) C) x³/6 + x²y/2 + f(y) + yg(y) D) x³/6 + x²y/2 + xf(y) + yg(y)
4. Suitable set of multipliers to solve x ² (y - z)p+ y² (z – x)q = z² (x - y) is,
   A) (0, 1, 1) B) (x, y, z) C) (0, 1, 1) D) (1,1,1)

b. Form a partial differential equation by eliminating arbitrary function from the relation, f(xy+z²,x+y+z)=0 (05 Marks)

c. Solve : (y² +z²)p+ x(yq – z)= 0 (05 Marks)

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d. Solve by the method of separation of variables ?²z/?x² = ?z/?x + 2?z/?y (06 Marks)

PART— B

5 a. Choose the correct answers for the following : (04 Marks)

1. ?(X + y)dydx limits from 0 to 1 and 0 to 2x
   A) 3 B) 4 C) 5 D) 6

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2. ? xy²zdzdydx limits from oi 1
   A) 36 B) 16 C) 26 D) 46
3. The integral ?exp(-x²) dx from 0 to infinity
   A) G(1/2) B) G(-1/2) C) G(1/2) D) G(-1/2)
4. The value of p(5, 3) + p(3, 5) is:
   

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   A) 35 B) -3 C) 3 D) -35/4

b. Evaluate ? xy(x + y)dydx taken over the area between y=x and y = x. (05 Marks)

c. Evaluate : ?(x + y + z)dydxdz with integration limits -1 to 1 for y, 0 to x+z for x and x-z to x+z for z (05 Marks)

d. Show that ? (x sin T)/(sin T) dT = p/2 with integration limits 0 to p/2 (06 Marks)

6 a. Choose the correct answers for the following : (04 Marks)

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1. Gauss Divergence theorem is a relation between:
   A) a line integral and a surface B) a surface integral and a volume integral C) a line integral and a volume D) two volume integrals
2. ? Mdx + Ndy is also equal to
   A) ? (?M/?y - ?N/?x) dxdy B) ? (?M/?y + ?N/?x) dxdy C) ? (?N/?x - ?M/?y) dxdy D) ? (?N/?x + ?M/?y) dxdy
3. Using the following integral, work done by a force F can be calculated:
   

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   A) Surface integral B) Volume integral C) Both (A) and (B) D) Line integral
4. If F = x²i + xyj then the value of ?F. dr from (0, 0) to (1, 1) along the line y x is,
   A) 2/3 B) 3/2 C) 1/3 D) 1/2

b. Find the area between the parabolae, y ² = 4x and x² = 4y with the help of Green's theorem in a plane. (05 Marks)

c. Evaluate ?xydx + xy²dy by Stoke's theorem where C is the square in the x-y plane with vertices (1, 0), (-1, 0), (0, 1) and (0, -1) (05 Marks)

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d. Evaluate ?F.fids given F = xi + yj + zk over the sphere x ² + y² + z² = a² by using Gauss divergence theorem. (06 Marks)

7 a. Choose the correct answers for the following : (04 Marks)

1. If L{f(t)} = F(s) then L{tf(t)} is,
   A) ?F(s)ds B) -?F(s)ds C) -d/ds F(s) D) d/ds F(s)
2. If L{cos at – cos bt}/t = 1/2 log(s²+b²)/(s²+a²) then L{sin 2t}/t.
   

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   A) log(s²+4)/s² B) 1/2 log(s²+4)/s² C)log(s²+4)/s D) 2/(s² + 4)
3. L{e^3t H(t-4)}=
   A) e^-4s/(s+3) B) e^-4s/(s-3) C) e^-4s/(s +3) D) e^-4s/(s-3)
4. 4'd(t- a)}=
   A) H(-a) e^t B) H(a) e^as C) H(a) e^-as D) e^t

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b. Find the Laplace transform of tecosh3t. (05 Marks)

c. Find ? t sin 3t dt limits from 0 to t (05 Marks)

d. Given f(t) = E, 0<t<a/2 -E,- a/2<t<a where gt+a) - g0, show that L{f(t)}=E/s tanh(as/2). (06 Marks)

8 a. Choose the correct answer for the following : (04 Marks)

1. L^-1{26/(s-4)⁷}
   

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   A) t⁶e^4t B) e^4t/6 C) t⁶e^-4t D) t⁶e^4t/6
2. L^-1 {1/ (s² + 25)}
   A) sin 5t B) cos 5 t C) 1/5 cos5t D) cos,5 t
3. L^-1 {7/(s²+7s+12)}
   A) (e^-4t-e^-3t)/t B) (e^-3t-e^-4t)/t C) (e^-4t-e^-3t)/t D) (e^-4t+e^-3t)/t

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4. L^-1 {s/(s²+a²)}
   A) t sin at B) t cos at/2a C) t sin at / 2a D) tcosat

b. Find L^-1 {s/(s²+4)(s-4)} (05 Marks)

c. Find L^-1 {1/(s²+a²)} by using convolution theorem. (05 Marks)

d. Solve by using Laplace transform y"(t)+ y(t) = 0 ; y(0) = 2, y(p/ 2)=1 (06 Marks)

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