JNTU
JNTU B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (January/February - 2024 - R15)
Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Write the normal equations to fit the parabola y = a + bx + cx2 using least squares method.
Answer
To fit the parabola y = a + bx + cx2 to a set of n data points (xi, yi) using the method of least squares, we minimize the sum of the squares of the residuals. The resulting normal equations are:
Determine (Δ + E)2(x2+3x+2) by taking h=1.
Answer
Given the operator expression (Δ2 + 2E) acting on f(x) = x2 + 3 with h = 1, we apply the operators individually. Recall that Δf(x) = f(x+1) - f(x) and Ef(x) = f(x+1).
First, calculate Δ2(x2 + 3):
Next, calculate 2E(x2 + 3):
Summing the results:
Explain Trapezoidal rule to find the approximate value of the integral of f(x) from a to b.
Answer
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Derive the recurrence relation to evaluate √N using Newton Raphson’s method.
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Is f(x) = sin(x) + (1/3)sin(3x) + (1/5)sin(5x) + (1/7)sin(7x) periodic? If so, find its period.
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Obtain the Fourier sine integral of f(x) = e-kx, x > 0 & k ≥ 0.
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Using the method of separation of variables, find solution of (∂u/∂x) + (∂u/∂y) = 0.
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Form a partial differential equation by eliminating arbitrary constants a, b from z = (x-a)2 + (y-b)2.
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For what value(s) of p, the vector field F = (x+3y)i + (y-2z)j + (x+pz)k is solenoidal?
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Compute the work done in moving a particle in the force field F = 3i + 2j + zk along the straight line from (0,0,0) to (2,1,3).
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Part B
Prove or disprove: μ2 = 1 + (1/4)δ2.
Answer
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Use Lagrange's interpolation formula to fit a polynomial to the data: x={-1, 0, 2, 3}, y={-8, 3, 1, 2}. Hence find y(-2), y(1) and y(4).
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Explain the method of least squares.
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Determine the constants a and b by the method of least squares such that y = aebx for data x={2, 4, 6, 8, 10}, y={4.077, 11.084, 30.128, 81.897, 222.62}.
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Compute a positive real root of the equation x log10 x = 1.2 correct to three decimal places using Bisection method.
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Using the Gauss Seidel iterative method, solve: 5x-y=9; -x+5y-z=4; -y+5z=6.
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Use a) Taylor series method b) Runge-Kutta method of 4th order to solve dy/dx = x+y2, y(0)=1 for the interval 0 < x ≤ 0.2 with h=0.1.
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Find the Fourier series of periodic function f(x) of period 2, where f(x) = -1 for -1 < x < 0 and 1 for 0 < x < 1. Hence, prove that π/4 = 1 - 1/3 + 1/5 - 1/7 + ....
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State the Fourier integral theorem.
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Represent f(x) = sin(πx), 0 < x < 1, 0 otherwise as an exponential Fourier transform. Show that the result can be written as f(x) = (1/π) ∫ [ (1+cos(λπ))/(1-λ2) ] cos(λx) dλ.
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Solve (yz-2y)p + (xz-x)q = xy-xz.
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Solve p2+q2 = y2q/z.
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A string is stretched and fastened to two points l apart. Motion is started by displacing the string in the form y = a sin(πx/l) from which it is released at time t=0. Show that the displacement is y(x,t) = a sin(πx/l) cos(πct/l).
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Calculate the scalar potential φ such that F = ∇φ if the vector field F = (x2-y2)i - 2xyj + z2k is an irrotational vector.
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Evaluate the directional derivative of f = x2-y2+2z2 at P(1, 2, 3) in the direction of the line PQ where Q is (5, 0, 4).
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Verify the divergence theorem for F = (x-y2)i + (y-z2)j + (z-x2)k taken over the rectangular parallelepiped 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c.
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