JNTU
JNTU B.Tech II MATHEMATICS - II Previous Question Paper (November/December - 2018 - R13)
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
Find ∇²(xyz).
Answer
The Laplacian operator is defined as ∇² = ∂²/∂x² + ∂²/∂y² + ∂²/∂z². For f(x,y,z) = xyz, we have ∂²f/∂x² = 0, ∂²f/∂y² = 0, and ∂²f/∂z² = 0. Therefore, ∇²(xyz) = 0 + 0 + 0 = 0.
Evaluate by Green’s Theorem ∮(x²-y)dx+(x+y²)dy where C is x²+y²=1.
Answer
By Green's Theorem, ∮(P dx + Q dy) = ∬(∂Q/∂x - ∂P/∂y) dA. Here P = x² - y and Q = x + y². Thus, ∂Q/∂x = 1 and ∂P/∂y = -1. The integral becomes ∬(1 - (-1)) dA = 2 ∬ dA, where ∬ dA is the area of the unit circle x² + y² = 1, which is π(1)². The value of the integral is 2π.
Find constant term in the Fourier series of f(x) = sin x in [-π, π], f(x+2π)=f(x).
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If F{f(t)} = F(s), then prove that F{e^(iat)f(t)} = F(s-a).
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Evaluate Δ(1/x!), by taking step size equal to one.
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Write the three normal equations to fit the parabola y = a + bx + cx² by method of least squares.
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Find any two values of x between which the root of xe^x = cos x lies.
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Find LU Decomposition of [[2, 5], [2, 6]].
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Write the formula to evaluate ∫[a to b] f(x)dx by Trapezoidal rule.
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If dy/dx = xy, y(0)=1, find y(0.1) by Taylor’s series method taking upto 2 differentials.
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PART - B
Find the unit normal vector to the surface x³+y³+z³=3 at the point (1, -2, 1).
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Evaluate by Stoke’s theorem ∫(e^x dx + 2ydy - dz) where c is the curve x²+y²=9 and z=2.
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Verify divergence theorem for (x²-y²)i + z²j + x²k taken over the region in first octant of the cylinder x²+y²=9 and z=2.
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Find the Fourier series to represent f(x) = x², -π < x < π, f(x+2π)=f(x).
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Find the Fourier cosine transform of e^(-ax)/x.
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Expand f(x) = e^(-x), -1 < x < 1; f(x+2)=f(x) in Fourier series.
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Find the finite Fourier sine and cosine transform of f(x) = x³, 0 < x < π.
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| x | 20 | 25 | 30 | 35 | 40 | 45 |
|---|---|---|---|---|---|---|
| y | 204 | 231 | 260 | 291 | 332 | 352 |
Find y(42) from the data using Newton’s forward formula.
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Fit a straight line y = a + bx by the method of least squares.
| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| y | 14 | 33 | 40 | 63 | 76 | 85 |
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Apply Lagrange’s 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 a curve of the form y = ab^x by the method of least squares.
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| y | 1.0 | 1.2 | 1.8 | 2.5 | 3.6 | 4.7 | 6.6 | 9.1 |
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Solve the system of equations by expressing the coefficient matrix as a product of lower triangular and upper triangular matrices: x + 2y = 7, x - 3y - z = 4, 4y + 3z = 5.
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Solve the system of equations using Gauss seidal method: 9x + 2y + 4z = 20, x + 10y + 4z = 6, 2x - 4y + 10z = -15.
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Find y(0.1) and y(0.2) using Runge Kutta fourth order formula given that dy/dx = x²+y², y(0)=1.
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Evaluate ∫[0 to 1] (1/(1+x))dx by taking h=1/6 using: a) Simpson’s 1/3 rule b) Simpson’s 3/8 rule.
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