JNTU
JNTU B.Tech II MATHEMATICS-II Previous Question Paper (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
Define Divergence of a Vector.
Answer
The divergence of a continuously differentiable vector field F = F1i + F2j + F3k is a scalar function defined as the dot product of the del operator and the vector field. It is expressed as:
It represents the net rate of flow of the vector field out of an infinitesimal volume surrounding a given point.
Find a unit normal vector to the given surface x2+y2+2z2 = 26 at the point (2, 2, 3).
Answer
Let the given surface be defined by the scalar function f(x, y, z) = x2 + y2 + 2z2 - 26. The normal vector to the surface at any point is given by the gradient of the function, ∇f.
Evaluating the gradient at the point (2, 2, 3):
The magnitude of this gradient vector is:
The unit normal vector n is obtained by dividing the gradient vector by its magnitude:
Write the Dirichlet conditions for the existence of Fourier Series.
Answer
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State and prove Shifting property of Fourier Transforms.
Answer
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Derive the Normal Equations to fit a curve y = aebx, using method of least squares.
Answer
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Prove that 1 + μ2δ2 = (1 + 1/2 δ2)2.
Answer
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Explain Gauss Seidel iteration method to find the solution of linear system of equations.
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List out the merits and demerits of Newton Raphson method.
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Define Boundary value problem.
Answer
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Find the solution of dy/dx = -x/y, y(0)=1 at x=0.2 using Euler’s method.
Answer
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Part B
Prove that div(grad rm) = m(m+1)rm-2 where 𝑟̅= 𝑥𝑖̅+ 𝑦𝑗̅+ 𝑧𝑘̅ and r = |𝑟̅|.
Answer
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Find the directional derivative of the function xy2+yz2+zx2 along the tangent to the curve x=t, y=t2, z=t3 at the point (1,1,1).
Answer
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Verify the Gauss Divergence Theorem for 𝐹̅ = (𝑥3 − 𝑦𝑧)𝑖̅− (2𝑥2𝑦)𝑗̅+ (𝑧)𝑘̅ taken over the surface of the cube bounded by the planes x = y = z = a and coordinate planes.
Answer
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Find the Fourier Series for 𝑓(𝑥) = 𝑥2 in the interval [0,2π] and f(x+2π) = f(x).
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Obtain the Half range cosine series for 𝑓(𝑥) = 𝑥 in the interval (0, l).
Answer
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Find the Fourier Sine and Cosine Transform of 𝑓(𝑥) = e-ax/x and deduce that ∫ (e-ax-e-bx)/x sin(sx) dx = tan-1(s/a) - tan-1(s/b) from 0 to infinity.
Answer
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Using Gauss backward difference formula find y(8) from the following table: X: 0, 5, 10, 15, 20, 25; Y: 7, 11, 14, 18, 24, 32.
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Find the polynomial passing through the points (1,1), (3,27) and (4,64) using Lagrange’s interpolation formula.
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Fit a curve of the form y = abx for the following data by the method of least squares: X: 2, 3, 4, 5, 6; Y: 8.3, 15.4, 33.1, 64.2, 127.4.
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Fit a straight-line y=a+bx from the following data: X: 0, 1, 2, 3, 4; Y: 1, 1.8, 3.3, 4.5, 6.3.
Answer
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Find a real root of the equation 𝑥(log10 𝑥) = 1.2 by Regula False method.
Answer
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Using Jacobi Iteration method to solve the system: 10𝑥 + 2𝑦 + 𝑧 = 9; 𝑥 + 10𝑦 − 𝑧 = −22; −2𝑥 + 3𝑦 + 10𝑧 = 22.
Answer
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Find a Positive root of 3𝑥 = 𝑐𝑜𝑠𝑥 + 1 using Iteration Method.
Answer
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Solve the equations 𝑥 + 𝑦 + 𝑧 = 1; 3𝑥 + 𝑦 − 3𝑧 = 5; 𝑥 − 2𝑦 − 5𝑧 = 10 by LU Decomposition method.
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Find y(0.1) and y(0.2) using Euler’s modified formula given that dy/dx = -x/y, y(0)=1.
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Using R-k fourth order method, find y(0.2), given that dy/dx = y-x and y(0)=2.
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Evaluate ∫ 1/(1+x2) dx from 0 to 1 using i) Simpson’s 1/3 rule ii) Simpson’s 3/8 rule.
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Solve the boundary value problem 𝑦” − 64𝑦 + 10 = 0, 𝑦(0) = 𝑦(1) = 0 by Finite difference method [taking n=3].
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