JNTU
JNTU B.Tech II Year I Semester MATHEMATICS-IV Previous Question Paper (April/May - 2018 - R16)
Note: This question paper contains two parts A and B. Part A is compulsory. Part B consists of 5 Units. Answer any one full question from each unit.
Part A
Show that
Answer
A function is harmonic if it satisfies Laplace's equation:
- uₓ = (y² - x²) / (x² + y²)²
- uₓₓ = (2x³ - 6xy²) / (x² + y²)³
- uᵧ = -2xy / (x² + y²)²
- uᵧᵧ = (6xy² - 2x³) / (x² + y²)³
Summing these,
Write Cauchy-Riemann equations in polar form.
Answer
For a complex function
Expand
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Find residue of
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Find image of the circle
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Determine the region of w-plane into which the region is mapped by the transformation
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Find the value
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Find the Fourier sine transformation of
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Classify the equation
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Write the one dimensional Heat equation in steady state.
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Unit I
Discuss the continuity of f(x,y) = (xy)/(x²+y²) at (0,0)
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Construct the analytic function f(z), whose real part is eˣcos y.
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If f(z) = u+iv is an analytic function and u-v = eˣ(cos y - sin y), find f(z) in terms of z.
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If u and v are harmonic, prove (∂u/∂y - ∂v/∂x) + i(∂u/∂x + ∂v/∂y) is analytic.
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Unit II
Evaluate integral of (x-y)dx - (x+y)dy over the first quadrant of x²+y²=4.
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Evaluate integral dz/(z²+1) where C is |z|=2.
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Obtain the expansion for sin(1/(z-1)) valid in 1 < |z| < infinity.
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Evaluate integral dz/(z²(z²+8z+3)) over unit circle C.
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Unit III
Prove that integral (0 to 2π) sin²θ / (a + b cos θ) dθ = (2π/b²)(a - sqrt(a²-b²)) for a > b > 0.
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Find the bilinear transformation that maps the points 1, i, -1 into 2, i, -2 respectively.
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Unit IV
Obtain the Fourier series for f(x) = sin x in (-π, π).
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Find Fourier sine transform of e⁻ˣ and evaluate integral sin(αx)/(1+x²) dx.
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Find the Half range cosine series for f(x) = x(2-x) in 0 ≤ x ≤ 2.
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Find the inverse Fourier sine transform of F(p) = e⁻ᵃᵖ/p.
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Unit V
Solve the heat equation ∂u/∂t = α²(∂²u/∂x²) for a rod with conditions: u bounded at t → ∞, ∂u/∂x=0 at x=0,l and u=lx-x² at t=0.
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Solve by separation of variables: ∂²z/∂x² - 2(∂z/∂x) + ∂z/∂y = 0.
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Solve 4(∂u/∂x) + ∂u/∂y = 3u, given u = 3e⁻ʸ - e⁻⁵ʸ when x=0.
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