JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I MATHEMATICS-II Previous Question Paper (January - 2021 - R18)
Answer any five questions. All questions carry equal marks.
Main
Solve (3x y2 - y3) dx - (2x2y - xy2) dy = 0.
Answer
The given differential equation is (3xy2 - y3) dx - (2x2y - xy2) dy = 0. This is a homogeneous equation of degree 3. Let y = vx, then dy = v dx + x dv. Substituting these into the equation:
Simplifying the terms:
Dividing by x3:
Separating the variables:
Integrating both sides:
Substituting v = y/x:
Which simplifies to ln|x| = 2 ln|y| - 2 ln|x| - y/x + C, or 3 ln|x| - 2 ln|y| + y/x = C.
Radium decomposes at a rate proportional to the quantity present at time t. Suppose that it is found that in 25 years approximately 1.1% of certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose.
Answer
Let Q(t) be the quantity of radium at time t. The rate of decomposition is dQ/dt = -kQ. Solving this gives Q(t) = Q0e-kt. Given that 1.1% decomposes in 25 years, 98.9% remains:
To find the half-life (T), where Q(T) = 0.5 Q0:
It will take approximately 1567 years for one-half of the radium to decompose.
Solve y - 2px = tan-1(xp2); where p = dy/dx.
Answer
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A body of temperature 80oF is placed in a room of constant temperature 50oF at time t = 0. At the end of 5 minutes the body has cooled to a temperature of 70oF. After how many minutes will the temperature of the body be within 10oF of the constant 50oF temperature of the room?
Answer
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Using the method of variation of parameters solve d2y/dx2 + 4y = tan 2x.
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Solve (D2 - 4D + 4)y = x2 sinx + e2x + 3.
Answer
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Evaluate ∫∫R (x + y) dx dy; where R is the parallelogram in the xy-plane with vertices (1, 0), (3, 1), (2, 2), (0, 1) using the transformation u = x + y, v = x - 2y.
Answer
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Evaluate ∫0a ∫ya (x / (x2+y2)) dx dy by changing the order of integration.
Answer
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A vector field is given by A = (x2 + xy2) i + (y2 + x2y) j, show that the field is irrotational and find the scalar potential.
Answer
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Find the maximum value of the directional derivative of φ = x2yz at (1, 4, 1).
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Evaluate ∫∫S A . n ds where A = (x + y2) i - 2xj + 2yzk and S is a surface in the plane 2x + y + 2z = 6 in the first octant.
Answer
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If F = x3 i + x2yj + xyz2k, find Curl F.
Answer
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State and verify Stokes theorem for the function f = x2 i + xy j integrated round the square in the plane z = 0 whose sides are along the lines x = 0, y = 0, x = a, y = a.
Answer
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State and verify Gauss divergence theorem for f = (x3 - yz) i - 2x2yj + zk taken over the surface of the cube bounded by the planes x = y = z = a and coordinate planes.
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