JNTU
JNTU B.Tech II Year MATHEMATICS-II Previous Question Paper (December - 2018 - 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
State Divergence theorem.
Answer
The Divergence Theorem states that the outward flux of a vector field across a closed surface is equal to the volume integral of the divergence of the vector field over the region enclosed by the surface. Mathematically, if V is the region bounded by a closed surface S and F is a continuously differentiable vector field defined on a neighborhood of V, then ∫∫∫V (∇ ⋅ F) dV = ∫∫S F ⋅ n dS.
In what direction from the point (−1, 2, 2) is the direction derivative of 𝜙 = 𝑥𝑦2𝑧3 a maximum. Find the magnitude of this maximum.
Answer
The direction derivative of a scalar function 𝜙 in the direction of a unit vector u is given by ∇𝜙 ⋅ u. This direction derivative is maximum in the direction of the gradient vector ∇𝜙, and the magnitude of this maximum is |∇𝜙|.
Given 𝜙 = 𝑥 𝑦 2 𝑧 3, the gradient is:
∇𝜙 = (∂𝜙/∂x)i + (∂𝜙/∂y)j + (∂𝜙/∂z)k
∂𝜙/∂x = 𝑦 2 𝑧 3
∂𝜙/∂y = 2𝑥 𝑦 𝑧 3
∂𝜙/∂z = 3𝑥 𝑦 2 𝑧 2
So, ∇𝜙 = 𝑦 2 𝑧 3 i + 2𝑥 𝑦 𝑧 3 j + 3𝑥 𝑦 2 𝑧 2 k.
At the point (−1, 2, 2):
∇𝜙|(-1,2,2) = (2)^2 (2)^3 i + 2(-1)(2)(2)^3 j + 3(-1)(2)^2 (2)^2 k
∇𝜙|(-1,2,2) = (4)(8) i + (-4)(8) j + (-3)(4)(4) k
∇𝜙|(-1,2,2) = 32i - 32j - 48k
The direction of maximum direction derivative is the direction of the gradient vector 32i - 32j - 48k.
The magnitude of this maximum is:
|∇𝜙| = √(32^2 + (-32)^2 + (-48)^2)
|∇𝜙| = √(1024 + 1024 + 2304)
|∇𝜙| = √(4352)
|∇𝜙| = √(2048 * 2 + 256 * 9) = √(16 * 272) = 4√(272) = 4√(16 * 17) = 16√17
State the Dirichlet’s conditions of 𝑓(𝑥).
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Write the statement of Fourier integral theorem.
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With usual notations, show that 𝐸1/2 = 𝜇 + 1/2𝛿
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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.
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Explain bisection method.
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What is the condition for convergence of Gauss-seidel iteration method?
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Write the formula to evaluate ∫ 𝑦𝑑𝑥 between limits a and b by i) simpson’s 1/3rd rule ii) Simpson’s 3/8th rule.
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Write the merits and demerits of modified Euler’s method.
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Part B
If 𝐹 = (4𝑥𝑦 − 3𝑥2𝑧2) 𝑖 + 2𝑥2𝑗 − 2𝑥3𝑧 𝑘 , prove that ∫ 𝐹 ∙ 𝑑𝑟 is independent of the curve joining two points.
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Evaluate ∫ 𝐹 ∙ 𝑑𝑟 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).
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Verify Gauss Divergence theorem for 𝐹 = 2𝑥2𝑦 𝑖 − 𝑦2𝑗 + 4𝑥𝑧2𝑘 taken over the region of the first octant of the cylinder 𝑦2 + 𝑧2 = 9 and 𝑥 = 2.
Answer
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Obtain the Fourier series for the function 𝑓(𝑥) = 𝜋 − 𝑥2, 0 < 𝑥 < 2𝜋. Hence deduce that 1/12 + 1/22 + 1/32 + ... = 𝜋2/6.
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Find Fourier sine transform of 𝑥𝑒−𝑎𝑥.
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If 𝑓(𝑥) = 𝑘𝑥 (0 ≤ 𝑥 ≤ 𝜋/2), 𝑘(𝜋 − 𝑥) (𝜋/2 ≤ 𝑥 ≤ 𝜋), obtain the half-range sine series.
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Find the Fourier transform of 𝑓(𝑥) = 1 (𝑥 ≤ 𝑎), 0 (𝑥 > 𝑎). Hence prove that ∫ (sin2(𝑎𝑥)/𝑥2)𝑑𝑥 = 𝜋𝑎/2 from 0 to infinity.
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Construct the difference table and evaluate 𝑌(2.5) for x=1, 2, 3, 4, 5 and Y=4, 8, 32, 76, 102.
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Using Lagrange interpolation, fit a polynomial 𝑦(𝑥) for the data x= -1, 1, 4, 5 and y= -2, 5, 12, 20 and hence find 𝑓(1.5).
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Find the cubic polynomial that takes the following values and hence obtain 𝑦(4): 𝑦(0)=1, 𝑦(1)=0, 𝑦(2)=1, 𝑦(3)=10.
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Using least square method, fit an exponential curve of the form 𝑦 = 𝑎𝑒𝑏𝑥 to the data (x: 1, 1.1, 1.2, 1.3, 1.4, 1.5; Y: 0.23, 0.67, 1.56, 3.56, 5.66, 10.23) and hence evaluate Y(3.0).
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Solve the following system of equations using Jacobi iteration method: 𝑥 + 10𝑦 + 𝑧 = 6; 10𝑥 + 𝑦 + 𝑧 = 6; 𝑥 + 𝑦 + 10𝑧 = 6.
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Find a real root of the equation 𝑓(𝑥) = 𝑥 + log 𝑥 − 2 = 0 using Newton-Raphson method.
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Solve the system of equations 𝑥 + 𝑦 + 𝑧 = 1; 3𝑥 + 𝑦 − 3𝑧 = 5; 𝑥 − 2𝑦 − 5𝑧 = 10, using decomposition method.
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Evaluate ∫ sin 𝑥 𝑑𝑥 from 0 to 𝜋/2 using Simpson’s 1/3 rule with 6 sub intervals.
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Evaluate ∫ (1 + 𝑒2𝑥 + 𝑥2) 𝑑𝑥 from -1 to 1 using Gauss-Legendre 1-point, 2-point and 3-point formula.
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Use Taylor’s method to solve the differential equation 𝑑𝑦/𝑑𝑥 = 𝑥 + 𝑦, 𝑦(0) = 1 and evaluate 𝑦(0.2) correct to 4 decimal places. Taking h = 0.1.
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