Download JNTUK B-Tech 2020 R19 IT 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 \( f (t) = \begin{cases} -1, & -\pi/2 < t < \pi/2 \\ 0, & \pi/2 < t < \pi \end{cases} \)
2. b) Obtain Fourier series of the function \(f(x) = 2x - x^2\) in (0, 3) and hence deduce that \( \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - ... = \frac{\pi^2}{12} \)

(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 \( \int_0^\infty \frac{\omega \sin(x\omega)}{1 + \omega^2} d\omega = \begin{cases} 0, & x < 0 \\ \frac{\pi}{2}e^{-x}, & x > 0 \end{cases} \)

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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 \( \int_0^\infty \frac{x^2}{(1 + x^2)^2} dx = \frac{\pi}{4} \)

UNIT-III

1. a) Express \( \int_0^\infty \sqrt{x} e^{-x} dx \) in terms of gamma function.

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2. b) Express \( \int_0^1 x^m (1 - x^n)^p dx \) in terms of Gamma functions and hence evaluate \( \int_0^1 x^7 (1 - x^5)^3 dx \)

(OR)

1. a) Apply change the order of integration and evaluate \( \int_0^2 \int_x^2 dy dx \)
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 \( \phi = xy + yz + zx \) at A in the direction of AB where A = (1, 2, -1), B = (5, 6, 8).
2. b) Determine curl (curl F) where \( F = x^2y i - 2xz j + 2yz k \)

(OR)

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

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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 \(F = (2x - y - z)i + (x + y - z^2)j + (3x - 2y + 4z)k\).
2. b) Evaluate the line integral by Stokes’s theorem for the vector function \(F = y^2 i + x^2 j + (z^2 + x)k\) and 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 \( \oint [(3x^2 - 8y^2)dx + (4y - 6xy)dy] \), where C is boundary of the region defined by \(y = x, y = x^2\)

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