Download JNTUK B-Tech 2020 R19 CSE B19 BS 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

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Answer ONE Question from EACH UNIT
All questions carry equal marks
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UNIT-I Cco KL | Marks
1. a) Find the Fourier series for the function \( f (t) = \begin{cases} -1, & -\pi/2 < t < \pi/2 \\ 0, & \pi/2 < t < \pi \end{cases} \) CO1 K2 7

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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 CO1 K2 8
(OR)
2. 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 1/1² + 1/3² + 1/5² + ... = p²/8 CO1 K2 8
b) Find the Half — Range cosine series for the function f(x) = x~ in the range 0 < x < p CO1 K3 7
UNIT-II

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3. a) Using the Fourier Sine Transform of e^-ax (a> 0), evaluate ?₀⁸ (x sin(kx) / (a² + x²)) dx CO2 K3 7
b) Using Fourier integral representation, show that ?₀⁸ (? sin(?x) d?) = \( \begin{cases} \frac{\pi}{2}, & x > 0 \\ 0, & x < 0 \end{cases} \) CO2 K3 8
(OR)
4. a) Find the inverse Fourier sine transform f(x) of F_s(p) = 1/(1+p²) CO2 K2 8
b) Using Parseval’s Identity, prove that ?₀⁸ (x² / (1 + x⁶)) dx = p/4 CO2 K3 7

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UNIT-III
5. a) Express ?₀⁸ e^-x2 dx in terms of gamma function. CO3 K2 7
b) Express ?₀¹ x^m (1 — xⁿ)^p dx in terms of Gamma functions and hence evaluate ?₀¹ x⁷(1—x⁵)⁸dx CO3 K2 8
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(OR)
6. a) Apply change the order of integration and evaluate ?₀² ?_x² dy dx. CO3 K3 8
b) Obtain the volume of the tetrahedron bounded by x =0, y =0, z=0, x+y+z=1. CO3 K3 7
UNIT-IV

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7. 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) . CO4 K2 8
b) Determine curl (curl F) where F = x²y i - 2xz j + 2yz k CO4 K2 7
(OR)
8. a) Show that the vector (x² — yz)i + (y² — zx)j + (z² — xy)k is irrotational and find its scalar potential. CO4 K2 8
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). CO4 K2 7

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UNIT-V
9. 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. CO5 K2 7
b) Evaluate the line integral by Stokes’s theorem for the vector function F = y²i + x²j + (z + x)k and C is the triangle with vertices (0,0,0),(1,0,0) and (1,1,0). CO6 K3 8
(OR)
10. Verify Green’s theorem in the plane ? [(3x² — 8y²)dx + (4y — 6xy)dy], CO6 K3 15 where C is boundary of the region defined by y= x, y=x²

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