Download AKTU B-Tech 2nd Sem 2017-2018 RAS203 Engineering Mathematics II Question Paper

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Sub Code: RAS203

Paper Id: 199223

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B. TECH

(SEM-II) THEORY EXAMINATION 2017-18

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ENGINEERING MATHEMATICS - II

Time: 3 Hours

Total Marks: 70

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

SECTION A

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1. Attempt all questions in brief. 2 x 7 = 14
   1. Determine the differential equation whose set of independent solutions is {ex, xex, x2ex}.
   2. Solve: (D+1)³ y = 2e^-x .
   3. Prove that: P_n(-x) = (-1)ⁿ P_n(x).
   4. Find inverse Laplace transform of (s+8)/(s²+4s+5).
   5. If L{?(v?)} = e^-1/s / s , find L{e^t?(3v?)}.

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   6. Solve: (?² + 4?' + 5?'²)? = 0, where ? = ?/? and ?' = ?/? .
   7. Classify the equation: z_xx + 2xz_xy +(1-y²)z_yy = 0.

SECTION B

2. Attempt any three of the following: 7 x 3 = 21
   1. Solve (?² - 2? + 4)? = ?^x cos ? + sin ? cos 3?.

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   2. Prove that: ?_5/2(?) = v(2/?) [(3-?²)/?² sin? - (3/?) cosx].
   3. Draw the graph and find the Laplace transform of the triangular wave function of period 2p given by
      F(?) = { ? , 0 < ? = ?
                -?, ? < ? < 2p .
   4. Obtain half range cosine series for the function ?(?) = { 2?, 0 < ? < 1
      

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                 2(2-?),1 < ? < 2
   5. Solve by method of separation of variables: ?/? = ?/? - 2?; ?(?, 0) = 10?^-? - 6?^-4? .

SECTION C

3. Attempt any one part of the following: 7 x 1 = 7
   1. Solve the simultaneous differential equations:
      

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      d²x/dt² - 4 dx/dt + 4? = ? and d²y/dt² + 4 dy/dt + 4? = 25? + 16?^t.
   2. Use variation of parameter method to solve the differential equation x²y'' + xy' – y = x²e^x.
4. Attempt any one part of the following: 7 x 1 = 7
   1. State and prove Rodrigue's formula for Legendre's polynomial.
   2. Solve in series: 2x(1-x)y'' + (1 - x)y' + 3y = 0.

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5. Attempt any one part of the following: 7 x 1 = 7
   1. State convolution theorem and hence find inverse Laplace transform of 1/((s²+a²)(s²+b²)).
   2. Solve the following differential equation using Laplace transform d³y/dt³ - 3 d²y/dt² + 3 dy/dt - ? = ?²?^t
      where y(0) = 1, y'(0) = 0 and y''(0) = -2 .

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6. Attempt any one part of the following: 7 x 1 = 7
   1. Obtain Fourier series for the function ?(?) = { -?, -p < x =0
                -?, 0 < x < p
      and hence show that 1/1² + 1/3² + 1/5² = ?²/8.
   2. Solve the linear partial differential equation: ?²?/?² - ?²?/? = sinxcos2y.

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7. Attempt any one part of the following: 7 x 1 = 7
   1. A string is stretched and fastened to two points l apart. Motion is started by displacing the string
      in the form y = Asin(p?/?) from which it is released at time t = 0. Find the displacement of any
      point at a distance x from one end at any time t.
   2. A rectangular plate with insulated surfaces is 8 cm wide and so long compared to its width that it
      may be considered infinite in length without introducing an appreciable error. If the temperature
      

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      along one short edge y = 0 is given by ?(?,0) =100sin(p?/8) , 0 < x < 8
      while the two long edges x = 0 and x = 8 as well as the other short edge are kept at 0 °C. Find the
      temperature u(x, y) at any point in steady state.

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