JNTU
JNTU B.Tech I MATRICES AND ORDINARY DIFFERENTIAL EQUATIONS Previous Question Paper (September - 2023 - R22)
Note: This question paper contains two parts A and B. Part-A is a compulsory question. Part-B consists of ten questions (numbered from 2 to 11) carrying 10 marks each. From each unit, there are two questions and the student should answer one of them.
Part A
Answer all sub-questions:
Define an Orthogonal matrix.
Answer
Define Normal form of a matrix.
Answer
If 1, 2, and 3 are the Eigen values of a Matrix A, then find the Trace of A and determinant of A.
Answer
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How do you find the inverse of a matrix by using Cayley-Hamilton theorem?
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State Cauchy’s mean value theorem.
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Find dy/dx from the implicit function f(x, y) = 0.
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State the linear differential equation of nth order.
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State Newton’s law of cooling.
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Find the general solution of (D + 1)(D - 2)y = 0.
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Find the PI of (D^2 + 4)y = sin 2x.
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Part B
Define a symmetric and skew symmetric matrices and give an example in each case.
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Discuss the consistency of the system of equations AX=B and hence solve the system: x1 - 2x2 + 3x3 + 4x4 = 2; x1 + x2 - 3x3 + 2x4 = -4; x1 + x2 + x3 + x4 = 4; x1 + x2 + x3 = 0.
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Find the rank of the following matrix by reducing it into Echelon form: A = [[1, 2, 1, 4], [2, 4, 3, 4], [1, 2, 3, 4], [1, 2, 6, 7]].
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Determine the values of λ for which the following system of equations may possess nontrivial solution and hence determine the solution for each permissible value of λ: x1 + x2 + 3x3 = 0; x1 + 2x2 + 3x3 = 0; x1 + 2x2 + λx3 = 0.
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Using Cayley-Hamilton theorem for the matrix A = [[1, 2, 3], [2, 4, 5], [3, 5, 6]], express B = A^8 - 11A^7 - 4A^6 + 11A^5 - 3A^4 - 2A^3 + 2A^2 + A + I as a quadratic polynomial in A. Also find the inverse of A.
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Find a matrix P which transforms the matrix A = [[1, 1, 3], [1, 5, 1], [3, 1, 1]] to the diagonal form.
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State Lagrange’s mean value theorem, and using it prove that for 0 < a < b, (b-a)/(1+b^2) < tan^-1(b) - tan^-1(a) < (b-a)/(1+a^2). Hence show that pi/4 + 3/25 < tan^-1(4/3) < pi/4 + 1/6.
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If u = (x+y)/(xy) and v = tan^-1(x) + tan^-1(y), find the Jacobian ∂(u, v)/∂(x, y). Are u and v functionally related? If so, find the relationship.
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Examine the function f(x, y) = x^3 + y^3 - 3axy for maxima and minima.
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Solve (1 + y^2)dx = (tan^-1 y - x)dy.
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The rate at which bacteria multiply is proportional to the instantaneous number present. If the original number doubles in 2 hours, in how many hours will it be triple?
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Solve dy/dx + (y/x) = y^3/x^6.
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If the temperature of the air is 30C, and the substance cools from 100C to 80C in 10 minutes, find the temperature of the substance after 20 minutes.
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Solve (D^2 - 2D + 2)y = e^x + sin 2x.
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Solve by method of variation of parameters d^2y/dx^2 + a^2y = tan(ax).
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