JNTU
JNTU B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (August - 2018 - R15)
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. Each question carries 10 marks and may have a, b, c as sub questions.
Part A
Construct the forward difference table for the following data.
| x | 1 | 2 | 3 | 4 | 5 |
| y | 4 | 13 | 34 | 73 | 136 |
Answer
| x | y | Δy | Δ2y | Δ3y | Δ4y |
|---|---|---|---|---|---|
| 1 | 4 | ||||
| 9 | |||||
| 2 | 13 | 12 | |||
| 21 | 6 | ||||
| 3 | 34 | 18 | 0 | ||
| 39 | 6 | ||||
| 4 | 73 | 24 | |||
| 63 | |||||
| 5 | 136 |
Write the normal equations to fit a curve of the form
Answer
The normal equations for the curve y = a + bx + cx2 are obtained by minimizing the sum of squares of residuals:
Derive the Newton-Raphson iterative formula to find
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Evaluate
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Determine the Fourier coefficient a0 in the Fourier series of
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Find the Fourier sine transform of
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Obtain a partial differential equation by eliminating the arbitrary function ‘f’ from
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Solve
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Find the unit normal vector to the surface
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State Stoke’s theorem.
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Part B
Find the cubic polynomial which takes the values y(0)=1, y(1)=0, y(2)=1 and y(3)=10.
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Using Lagrange’s formula to find y(10) from the data given below.
| x | 5 | 6 | 9 | 11 |
| y | 12 | 13 | 14 | 16 |
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Using the method of least squares, fit a straight line of the form y = ax + b for the following data.
| x | 1 | 2 | 3 | 4 |
| y | 0 | 1 | 1 | 2 |
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Fit a curve of the form y = abx to the following data.
| x | 1 | 2 | 3 | 4 |
| y | 4 | 11 | 35 | 100 |
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Solve the following system of equations using L-U decomposition method.
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Find dy/dx at x = 0.1 from the following table.
| x | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |
| y | 0.0 | 0.12 | 0.48 | 1.10 | 2.0 | 3.20 |
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Find an approximate value of y(0.1) for dy/dx = (y-x)/(y+x), y(0)=1 by Euler’s method with h = 0.02.
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Obtain the Fourier series for f(x) = x2 in [-π, π] and hence show that 1/12 - 1/22 + 1/32 - 1/42 + ... = π2/12.
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Find the half range cosine series for the function f(x) = x in [0, π].
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Find the Fourier transform of f(x) = 1-x2 for |x|<1, and 0 for |x|>1, and hence evaluate: ∫0∞ ((x cos x - sin x)/x3) cos(x/2) dx.
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Solve p + q = z tan(x + 2y) + 5.
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Solve z = px + qy + p2 + q2 by Charpit’s method.
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A square plate is bounded by the lines x=0, y=0, x=20 and y=20 and its Faces are insulated. The temperature along the upper horizontal edge is given by u(x,20) = x(20-x), 0 < x < 20 while other three edges are kept at 0°C. Find the steady state temperature in the plate.
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Find the directional derivative of f(x,y,z) = x2y2z + 4xz at (1, -2, 1) in the direction of the vector 2i - j - 2k.
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If F = grad(x3+y3+z3-3xyz), find div F and curl F.
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Verify Green’s theorem for ∫c [(x-y)dx + (x+y)dy], where C is the rectangle with vertices (0,0), (π,0), (π,1) and (0,1).
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