JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (June - 2022 - R13)
Answer any five questions
All questions carry equal marks
Main
From the data find number of students who obtained marks between 40 and 45.
| Marks | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
|---|---|---|---|---|---|
| Number of students | 31 | 42 | 51 | 35 | 31 |
Answer
To find the number of students who obtained marks between 40 and 45, we use Newton's forward interpolation formula. Let x be the marks and y be the number of students. The class interval h = 10. We set the origin at x0 = 40, so y0 = 42.
The forward difference table is:
| x | y | Δy | Δ2y | Δ3y | Δ4y |
|---|---|---|---|---|---|
| 30 | 31 | 11 | -2 | -14 | 27 |
| 40 | 42 | 9 | -16 | 13 | |
| 50 | 51 | -7 | -3 | ||
| 60 | 35 | -10 | |||
| 70 | 31 |
Using Newton's forward interpolation formula:
Substituting the values y0=42, Δy0=9, Δ2y0=-16, Δ3y0=13, Δ4y0=27 and u=0.5:
The number of students who obtained marks between 40 and 45 is approximately 50 - 31 = 19 students (considering the range starts from 40).
Using Lagrange’s interpolation, find the interpolating polynomial for the data:
| X | 0 | 1 | 3 |
|---|---|---|---|
| F(X) | 1 | 3 | 55 |
Answer
Given data points are (x0, y0) = (0, 1), (x1, y1) = (1, 3), and (x2, y2) = (3, 55). Lagrange's interpolation formula is:
Calculating the Lagrange basis polynomials:
Substituting into the formula:
Using the least square method fit a straight line for the following data:
| X | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| Y(X) | 1 | 0 | 1 | 4 |
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Using the least square approximation fit a quadratic polynomial for the data:
| X | 0 | 1 | 2 |
|---|---|---|---|
| Y(X) | 1 | 6 | 17 |
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Using the Newton Raphson Method, find the cube root of 12.
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Solve the following system by Gauss-Seidel method:
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The velocity V(km/min) of a moped which starts from rest is given at fixed intervals of the time T(min) as follows:
| T | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|
| V | 10 | 18 | 25 | 29 | 32 | 20 | 11 | 5 | 2 | 0 |
Using the 1/3 Simpson’s rule estimate the distance covered in 20 minutes.
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Using 4th order Runge Kutta Method, with h = 0.1 find y(0.1) from y' = y - x with y(0) = 2
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Expand
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A tightly stretched string of length has its ends fastened at x=0 and x=L. The mid-point of the string is then taken to height h and then released from rest in that position. Find the expression for the displacement of the string at any point at any time.
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A bar of 10 cms long with insulated sides has its ends A and B maintained at temperature 50 °C and 100 °C respectively until the steady state conditions prevail. The temperature at A is suddenly raised to 90 °C and at the same time that at B is reduced to 60 °C. Find the temperature distribution in the bar at any point at any time.
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Verify the Stroke’s theorem for
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