Download JNTUK B-Tech 2020 R19 CE 1202 Mathematics III Model Question Paper

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[B19 BS 1202]

I B. Tech II Semester (R19) Regular Examinations

MATHEMATICS - 111

(Common to CE,CSE,ECE,EEE & IT)

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MODEL QUESTION PAPER

TIME : 3 Hrs. Max. Marks : 75M

Answer ONE Question from EACH UNIT

All questions carry equal marks

UNIT-I

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1. a) Find the Fourier series for the function f (t) =
   -1 , -p/2 < t < p/2
   0 , -p/2 < t < p/2
   1 , p/2 < t < p
2. b) Obtain Fourier series of the function f(x)=2x — X" in (0, 3) and hence deduce that 1/1² - 1/2² + 1/3² - ... = p²/12

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(OR)

1. a) Obtain a Fourier series for the function f(x) given by
   f(x) = 2x , if -p < x < 0
   1-= , if 0 < x < p
   and deduce that 1/1² + 1/3² + 1/5² + ... = p²/8

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2. b) Find the Half — Range cosine series for the function f(x) = x² in the range 0 < x < p

UNIT-II

1. a) Using the Fourier Sine Transform of e^-ax (a> 0), evaluate ? xsinkx / (a² +x²) dx
2. b) Using Fourier integral representation, show that ? wsin(wx)dw = e^-x, x > 0

(OR)

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1. a) Find the inverse Fourier sine transform f(x) of F_s (p) = 1/(1+p²)
2. b) Using Parseval’s Identity, prove that ? x² / (1+x²)² dx = p/4

UNIT-III

1. a) Express ? e^-x2 dx in terms of gamma function.
2. b) Express ? x^m (1 —xⁿ)^pdx in terms of Gamma functions and hence evaluate ? x⁷(1—x⁵)³dx

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(OR)

1. a) Apply change the order of integration and evaluate ? dy dx.
2. b) Obtain the volume of the tetrahedron bounded by x =0,y =0,z=0, x+y+z=1.

UNIT-IV

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1. a) Obtain the directional derivative of f = xy + yz + zx at A in the direction of AB where A= (1,2,-1), B=(5,6,8) .
2. b) Determine curl (curl F) where F = x²y i-2xz j+2yz k

(OR)

1. a) Show that the vector(x² — yz)i + (y² — zx)j + (z² — xy)k Is irrotational and find its scalar potential.
2. b) Determine the values of a and b such that the surface ax²—b yz=(a+2)x and 4 xzy +2z² =4 cut orthogonally at (1,-1, 2).

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UNIT-V

1. a) Determine the work done in moving a particle once round the circle x²+y²=9 in the xy- plane by the force F=(2x—-y—2)i+(x+y—2z)j+ (3x — 2y + 4z)k.
2. b) Evaluate the line integral by Stokes’s theorem for the vector function F = y²i+ x²j + (z + x)kand C is the triangle with vertices (0,0,0),(1,0,0) and (1,1,0).

(OR)

1. Verify Green’s theorem in the plane ? [(3x² — 8y²)dx + (4y — 6xy)dy], where C is boundary of the region defined by y= x,y=x²

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