Download PTU B.Tech 2020 March ICE 3rd Sem Applied Mathematics III Question Paper

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Roll No. Total No. of Pages : 02

Total No. of Questions : 09

B.Tech.(Instrumentation & Control Engg.) (Sem.-3)

APPLIED MATHEMATICS - II

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Subject Code : AM-201

M.Code : 54501

Time : 3 Hrs. Max. Marks : 60

INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.

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

SECTION-A

1. Write briefly :
   1. Evaluate, ?_C z² dz along the circle C : |z| = 1.
   2. Find L (sin² 3t).

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   3. Solve (x² —yz) p + (y² —zx) q = z²—xy.
   4. Show that an analytic function with constant modulus is constant.
   5. Write half range sine series of the function f(x) = x in the interval 0 < x < 2.
   6. Write the sufficient conditions for the existence of Laplace transform.
   7. Find solution of homogeneous partial differential equation 4r — 12s + 9t = 0.

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   8. Show that nP_n(x) = xP'_n(x)- P'_n-1 (x).
   9. If f(x) is an odd function in (-l, l), then what are the values of a₀ and a_n?
   10. Find the bilinear transformation that map the points z = 1, i, —1 into the points w = 1, i, —1.

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SECTION-B

2. Find a Fourier series to represent e^-x from x = -l to x = l
3. A tightly stretched string with fixed end points x = 0 and x = l is initially in a position given by y = y₀ sin² (px). If it is released from rest from this position, find the displacement y (x, t).
4. Show that function f (z) defined by f(z) = xy²/(x²+y⁴), z ? 0, f(0) = 0, is not analytic at the origin even though it satisfies Cauchy-Riemann equations.
5. Show that J_-1/2(x)= v(2/(px)) cos x

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6. Evaluate ?₀⁸ dx/(1+x⁴)

SECTION-C

6. Use the concept of residues to evaluate ?₀^2p dx/(5 —4sinx)
7. Solve the equation using Laplace transformation : d²x/dt² +2dx/dt +5x = e^-t sin t, x (0)=0, x'(0)=1
8. Find the power series solution about the origin of the equation : (1-x²)y" -2xy' +6y=0.

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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.