Jawaharlal Nehru Technological University
Jawaharlal Nehru Technological University B.Tech I MATHEMATICS-I Previous Question Paper (December - 2017 - R16)
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.
Part A
Define exact differential equation. Give an example.
Answer
A differential equation of the form M(x, y)dx + N(x, y)dy = 0 is said to be exact if there exists a function f(x, y) such that df = Mdx + Ndy. This condition is satisfied if and only if the partial derivatives satisfy the equality ∂M/∂y = ∂N/∂x.
An example of an exact differential equation is:
Here, M = 2xy and N = x2 + 1. Since ∂M/∂y = 2x and ∂N/∂x = 2x, the equation is exact.
Find a particular integral of
Answer
The given differential equation is
Show that the matrix
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Find the values of a and b such that the system x + 2y + 3z = 5, 2x + 7y + 3z = 9, 3x + y + az = b has no solution.
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Find the sum and product of the Eigen values of the matrix
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Write the quadratic form corresponding to the matrix
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If u = f(x-y, y-z, z-x), find
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Expand f(x,y) = e^xy about origin up to 2nd degree terms.
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Form a partial differential equation by eliminating the arbitrary function f from z = f(x^2 + y^2).
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Solve p + q = 1.
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Part B
Solve (xy^2 - 2x^2y)dx + (3x^2y - x^3)dy = 0.
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Solve y'' + y = sin x.
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Apply the method of variation of parameters to solve y'' - y = x^2.
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If the temperature of the air is 30°C and the substance cools from 100°C to 70°C in 15 minutes, find when the temperature will be 40°C.
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Find the rank of the matrix by reducing to echelon form: [[0, 1, 3, 1], [0, 0, 1, 1], [3, 1, 0, 2], [1, 1, 2, 0]].
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Show that the system of equations 5x+3y+7z=4, 3x+26y+2z=9, 7x+2y+10z=5 is consistent and solve it.
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Solve the system of equations 2x-y+3z=4, x+y+z=1, x-3y-z=5 by LU – decomposition method.
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Find the Eigen values of A^5 - 2A^3 - A^2 - I if A = [[3, 7, 5], [2, 4, 3], [1, 2, 2]].
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Using Cayley-Hamilton theorem, find A^-1 and A^-2 if A = [[4, 6, 6], [1, 3, 2], [-1, -4, -3]].
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Reduce the quadratic form Q = 2x^2 + 2y^2 + 2z^2 + 8xy + 7xz + 3yz to canonical form and hence find its rank, nature, index and signature.
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If f(x,y) = x^4+y^4 / x+y, show that x(∂f/∂x) + y(∂f/∂y) = 3f.
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Determine whether the functions u = x+y/x-y, v = xy/x-y are dependent. If so, find the relation between them.
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Find the Taylor series expansion of f(x,y) = e^x cos y in powers of (x-1) and (y - π/4).
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Find the maximum and minimum values of the function f(x,y) = x^4 + y^4 - 2x^2 + 4xy - 2y^2 + 1.
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Find all possible second order partial differential equations by eliminating the arbitrary constants a, b, c from x^2/a^2 + y^2/b^2 + z^2/c^2 = 1.
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Solve (p-q)z = x + y.
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Reduce the equation p^2 + q^2 = z^2(px + qy) to F(p,q,z)=0 form and hence solve it.
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Solve p^2y + q^2x = 1.
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