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

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I B. Tech II Semester (R19) Regular Examinations

MATHEMATICS - III

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(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

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All questions carry equal marks

UNIT-I

1. a) Find the Fourier series for the function \( f(x) = \begin{cases} -1, & -\pi < x < -\pi/2 \\ 0, & -\pi/2 < x < \pi/2 \\ 1, & \pi/2 < x < \pi \end{cases} \)
2. b) Obtain Fourier series of the function \( f(x) = 2x - x^2 \) in (0, 3) and hence deduce that \( \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + ... = \frac{\pi^2}{12} \)

(OR)

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1. a) Obtain a Fourier series for the function f(x) given by \( f(x) = \begin{cases} 1 + \frac{2x}{\pi}, & -\pi < x < 0 \\ 1 - \frac{2x}{\pi}, & 0 < x < \pi \end{cases} \) and deduce that \( \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + ... = \frac{\pi^2}{8} \)
2. b) Find the Half — Range cosine series for the function \( f(x) = x^2 \) in the range \( 0 < x < \pi \)

UNIT-II

1. a) Using the Fourier Sine Transform of \( e^{-ax} \) (a>0), evaluate \( \int_0^\infty \frac{x \sin kx}{a^2 + x^2} dx \)
2. b) Using Fourier integral representation, show that \( \int_0^\infty \frac{\omega \sin(x\omega)}{1 + \omega^2} d\omega = \frac{\pi}{2} e^{-x}, x > 0 \)

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(OR)

1. a) Find the inverse Fourier sine transform f(x) of \( F_s(p) = \frac{1}{1+p^2} \)
2. b) Using Parseval’s Identity, prove that \( \int_0^\infty \frac{x^2}{(1+x^2)^2} dx = \frac{\pi}{4} \)

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UNIT-III

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1. a) Express \( \int_0^\infty e^{-x^2} dx \) in terms of gamma function.
2. b) Express \( \int_0^1 x^m (1 - x^n)^p dx \) in terms of Gamma functions and hence evaluate \( \int_0^1 x^7 (1 - x^2)^8 dx \)

(OR)

1. a) Apply change the order of integration and evaluate \( \int_0^1 \int_x^1 7 dy dx \)
2. b) Obtain the volume of the tetrahedron bounded by x = 0, y = 0, z = 0, x+y+z=1.

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UNIT-IV

1. a) Obtain the directional derivative of \( \phi = xy + yz + zx \) at A in the direction of AB where A= (1,2,-1), B=(5,6,8) .
2. b) Determine curl (curl F) where \( F = x^2y \mathbf{i} - 2xz \mathbf{j} + 2yz \mathbf{k} \)

(OR)

1. a) Show that the vector \( (x^2 - yz)\mathbf{i} + (y^2 - zx)\mathbf{j} + (z^2 - xy)\mathbf{k} \) is irrotational and find its scalar potential.

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2. b) Determine the values of a and b such that the surfaces \( ax^2 - byz = (a+2)x \) and \( 4x^2y + z^3 = 4 \) cut orthogonally at.(1,-1, 2).

UNIT-V

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

(OR)

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1. Verify Green’s theorem in the plane \( \oint_C [(3x^2 - 8y^2)dx + (4y - 6xy)dy] \) where C is boundary of the region defined by \( y = \sqrt{x}, y = x^2 \)

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