JNTU B.Tech II MATHEMATICS - II Previous Question Paper (December - 2018 - R15)

State Divergence theorem.

In what direction from the point (−1, 2, 2) is the direction derivative of 𝜙 = 𝑥𝑦²𝑧³ a maximum. Find the magnitude of this maximum.

State the Dirichlet’s conditions of f(x).

Write the statement of Fourier integral theorem.

With usual notations, show that E^(1/2) = 𝜇 + (1/2)𝛿.

What is least square curve fitting? Write down the normal equations for fitting a straight line for the set of points (xᵢ, yᵢ), i = 1, 2, … n.

Explain bisection method.

What is the condition for convergence of Gauss-seidel iteration method?

Write the formula to evaluate the integral of y dx by i) simpson’s 1/3rd rule ii) Simpson’s 3/8th rule.

Write the merits and demerits of modified Euler’s method.

If F = (4xy − 3x²z²)i + 2x²j − 2x³z k, prove that the line integral of F is independent of the curve joining two points.

Evaluate the line integral of F by Stokes theorem if F = y²i + x²j + (x + z)k and C is the boundary of the region with vertices (0,0,0), (1,0,0) and (1,1,0).

Obtain the Fourier series for the function f(x) = (π − x)², 0 < x < 2π. Hence deduce that 1/1² + 1/2² + 1/3² + ⋯ = π²/6.

Find Fourier sine transform of x e^(-ax).

If f(x) = kx for 0 ≤ x ≤ π/2 and k(π − x) for π/2 ≤ x ≤ π, obtain the half-range sine series.

Find the Fourier transform of f(x) = 1 for |x| ≤ a, 0 otherwise. Hence prove that the integral of sin²(ax)/x² from 0 to infinity is πa/2.

Construct the difference table and evaluate Y(2.5) from the following table: x: 1, 2, 3, 4, 5; Y(x): 4, 8, 32, 76, 102.

Using Lagrange interpolation, fit a polynomial y(x) for the data x: -1, 1, 4, 5; f(x): -2, 5, 12, 20 and hence find f(1.5).

Find the cubic polynomial that takes the following values and hence obtain y(4): y(0)=1, y(1)=0, y(2)=1, and y(3)=10.

Using least square method, fit an exponential curve of the form y = ae^(bx) to the data x: 1, 1.1, 1.2, 1.3, 1.4, 1.5; Y(x): 0.23, 0.67, 1.56, 3.56, 5.66, 10.23 and hence evaluate Y(3.0).

Find a real root of the equation f(x) = x + log(x) − 2 = 0 using Newton-Raphson method.

Solve the system of equations x + y + z = 1; 3x + y − 3z = 5; x − 2y − 5z = 10, using decomposition method.

Evaluate the integral of sin(x) from 0 to π/2 using Simpson’s 1/3 rule with 6 sub intervals.

Evaluate the integral of (1 + e^(2x) + x²) from -1 to 1 using Gauss-Legendre 1-point, 2-point and 3-point formula.