JNTU
JNTU B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (May/June - 2019 - 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
(25 Marks)
If ∑x = 10, ∑x2 = 26, ∑y = 15, ∑xy = 18, n = 6, find the values a and b for best fit of a straight line y = a + bx.
Answer
The normal equations for the line y = a + bx are:
Substituting the given values:
Solving this system, we multiply the first by 5 and second by 3: 75 = 30a + 50b and 54 = 30a + 78b. Subtracting gives -21 = 28b, so b = -0.75. Substituting back, 15 = 6a - 7.5, so 6a = 22.5, yielding a = 3.75.
Find the missing value from the following data:
| x | 1 | 3 | 5 |
|---|---|---|---|
| y | - | 9 | 26 |
Answer
Using Lagrange interpolation for x = 1, 3, 5 and y = y1, 9, 26. Since the points are equally spaced, we use Newton's forward difference formula. Assuming a quadratic polynomial, the second order difference is constant. Δy1 = 9 - y1, Δy2 = 26 - 9 = 17. Δ2y1 = 17 - (9 - y1) = 8 + y1. Setting Δ2y1 = 0 for a linear fit, y1 = -8. If assuming a quadratic, more data is needed; however, for a simple missing value problem, we assume a linear progression: y = 8.5x - 16.5, giving y = -8 at x = 1.
How many number of subintervals are required for applying trapezoidal and Simpson rules to evaluate an integral.
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Derive an iterative formula for reciprocal of a non-zero number using Newton’s Raphson method.
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Find the Fourier sine series of f(x) = x2 in [0, π].
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Find the Fourier cosine Transform of f(x) = e-ax, a > 0.
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Solve the partial differential equation p + q = 1.
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Solve ∂u/∂x = 4 ∂u/∂t by method of separation of variables.
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Define solenoidal and irrotational vector.
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Find Curl r where r = xi + yj + zk.
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Part B
(50 Marks)
Prove that nth difference of a polynomial of degree n is constant and higher order differences are zero. Fit a straight line of the following data: x: 1, 3, 5, 7, 9; y: 1.5, 2.8, 4.0, 4.7, 6.0
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Interpolate by means of Gauss backward formula, the population of town for the year 1974. Also prove that (i) δμ = 1/2 Δ + 1/2 ∇ and (ii) hD = E, where h is step size, D is differentiable operator.
| Year | 1939 | 1949 | 1959 | 1969 | 1979 | 1989 |
|---|---|---|---|---|---|---|
| Population (000) | 12 | 15 | 20 | 27 | 39 | 52 |
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Solve by Jacobi iteration method for the system of equations: 8x - 3y + 2z = 20; 6x + 3y + 12z = 35 and 4x + 11y - z = 33.
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using Trapezoidal rule as well as Simpson’s rule, taking step size h=0.2.
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Find the Fourier series of f(x) = sin x for 0 ≤ x ≤ π, and f(x) = 0 for -π ≤ x ≤ 0; Hence show that 1/(1.3) + 1/(3.5) + 1/(5.7) + ... = 1/4(2 - π).
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Find the Fourier sine and cosine transform of f(x) = e-ax, and hence show that the integral of (e-ax - e-bx)/x sin sx dx = Tan-1(s/a) - Tan-1(s/b), a, b > 0.
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Solve p - q = x2 - y2 and Solve (y-z)p + (z-x)q = z(x-y).
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If a string of length l is initially at rest in equilibrium position and each of its points is given the velocity v = v0 sin(πx/l), find the displacement y(x,t).
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Prove that div(rn r) = (n + 3)rn. Calculate the work done by a force F = 3xyi - y2j in moving a particle in xy-plane from (0, 1) to (1, 2) along the parabola y = x2.
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Verify Stokes theorem for F = (x2 + y2)i - 2xyj taken around a rectangle bounded by the lines x = a, x = -a, y = 0 and y = b.
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