JNTU
JNTU B.Tech II MATHEMATICS-II Previous Question Paper (April - 2018 - R13)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit.
PART A
If a = a1i + a2j + a3k and r = xi + yj + zk, then find ∇(a · r).
Answer
Given a = a1i + a2j + a3k and r = xi + yj + zk, the dot product is a · r = a1x + a2y + a3z. Applying the gradient operator ∇ = i(∂/∂x) + j(∂/∂y) + k(∂/∂z):
State Divergence theorem.
Answer
The Divergence theorem states that the surface integral of a vector field F over a closed surface S is equal to the volume integral of the divergence of F over the volume V enclosed by S:
If f(x) is defined as f(x) = -x in -π < x < 0 and f(x) = x in 0 < x < π, find a0 in Fourier series.
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Write the Euler’s formulae for Fourier series in the interval (-λ, λ).
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If h = 1, evaluate Δ(1/x!).
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Find the two values between which the root of x3 - cos x - 1 = 0 lies.
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If A = [[3, -4], [6, -16]], find LU decomposition.
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If dy/dx = x - y and y(0) = 1 then find first approximation up to 2nd degree terms y(x) using Picards method.
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y'' = x - y, with h = 0.25 then find the recurrence formula relating yi-1, yi, yi+1.
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PART B
Find the directional derivative of f(x,y,z) = xy2 + yz3 at the point (2, -1, 1) in the direction of the vector i + 2j + 2k.
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Evaluate ∫(yz dx + xz dy + xy dz) over a helix x = a cos t, y = a sin t, z = kt as t varies from 0 to 2π.
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Use Stoke’s Theorem to evaluate ∫((x+y)dx + (2x-z)dy + (y+z)dz) where C is the boundary of the triangle with vertices (2,0,0), (0,3,0) and (0,0,6).
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Obtain Fourier series for f(x) = x2 + x in -π < x < π.
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Find the Fourier Transform of f(x) = cos x for 0 < x < a and f(x) = 0 for x ≥ a.
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Find the fourier sine transform of 1 / (x2 + a2).
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Find y(42) from the following data using Newton’s forward interpolation formula.
| x | 20 | 25 | 30 | 35 | 40 | 45 |
|---|---|---|---|---|---|---|
| y | 352 | 332 | 291 | 260 | 231 | 204 |
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Apply Lagranges formula to find f(5) given that f(1) = 2, f(2) = 4, f(3) = 8, f(4) = 16 and f(7) = 128.
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Fit an equation of the form y = abx to the following data.
| x | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|
| y | 144 | 172.8 | 207.4 | 248.8 | 298.5 |
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Find a root of the equation sin x = 1 - x using Newton Raphson method.
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Find a real root of 10x + log10 x - 1.2 = 0 correct to four decimal places using Regula falsi method.
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Solve the system of equations using Gauss Seidal method:
x + 5y + 2z = 7
7x - y + z = 2
x + 2y + 5z = 9
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Find y(0.1) and y(0.2) using Runge Kutta method given that y' = x2 - y, y(0) = 1.
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Solve the boundary value problem y'' = y, with h = 0.5 given that y(0) = 0, y(2) = 2 by finite difference method.
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