Download PTU B.Tech 2020 March Automation-And-Robotics Question Paper 3rd Sem BTAR 303 18 Mathematics Iii

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Roll No. | [ | [ 1] [ | ‘ Total No. of Pages : 02
Total No. of Questions : 09
B.Tech.(Automation & Robotics) (2018 Batch) (Sem.-3)
MATHEMATICS-III
Subject Code : BTAR-303-18

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students have to attempt any FOUR questions.

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3. SECTION-C contains THREE questions carrying TEN marks each and students have to attempt any TWO questions.

SECTION-A

1. Write briefly :
   1. Explain the Dirichlet conditions for a function to be expressed in terms of Fourier series.
   2. Find Laplace Transform of e cos.
   3. Find Laplace Inverse Laplace transform of
      

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      s²+4s+1
   4. Define ordinary point and regular singular point.
   5. Express f(x)= 2x² + 1 in terms of Lagendre function.
   6. Form a partial differential equation from f (x+y, z — xy) = 0.
   7. Solve the partial differential equation y ?z/?x — z + x ?z/?y — 2xy.

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   8. Evaluate ? c z²-z+5/(z+2) dz.
   9. Show that sin z is analytic function
   10. Define even function and write fourier series for an even function in the interval (—l, l), provided it satisfies all the conditions.

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SECTION-B

1. Find the Fourier series for f (x) in the interval (-p, p) when f(x) = { p+x, -p < x < 0 ; p-x, 0 < x < p }

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2. Solve the differential equation using Method of Laplace transform
   d²y/dt² +4 dy/dt +5y =sin 5t, y (0)=0, y'(0)=0
3. Prove that d/dx[xnJ?(x)]=xnJ?1(x)
4. Expand f(z) = 1/((z+1)(z+3)) in Laurents series, valid for (i) 1 < |z| < 3
5. Solve the Partial differential equation

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SECTION-C

1. 1. Find half-range cosine series for f(x) = x in the interval [0, p]
   2. Define Unit step function and find its Laplace transform.
2. Evaluate ? cos(x)/x dx by contour integration.

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3. A homogeneous conducting rod of length 100 cm has its ends kept at zero temperature and temperature initially is
   u(x,0) = { x, 0 < x < 50 ; 100-x, 50 < x < 100 }
   Find the temperature u (x, t) at any time t.

NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.

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