JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year MATRICES AND ORDINARY DIFFERENTIAL EQUATIONS Previous Question Paper (January/February - 2024 - 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. Hence, the student should answer five questions from Part-B.
Part A
Define a Skew-Symmetric matrix.
Answer
A square matrix A is called a skew-symmetric matrix if its transpose is equal to its negative, i.e., AT = -A. This implies that all diagonal elements are zero and aij = -aji for all i, j.
Define rank of a matrix.
Answer
The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix, or equivalently, the order of the largest non-zero minor of the matrix.
If λ is an Eigen value of a square matrix A, then find an Eigen value of A-1 and adj A.
Answer
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State Cayley-Hamilton theorem.
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State Rolle’s theorem.
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If u = sin(x/y), find ∂u/∂x.
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Write the condition for a differential equation Mdx+Ndy=0 to be exact.
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State the law of natural growth.
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If the roots of the auxiliary equation are 1, 2, and -2, then write the complementary function of a differential equation f(D)y = X.
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Find the PI of (D2 - 2D + 1)y = ex.
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Part B
Answer the following:
Find the rank of matrix A = [[1, 3, -6, 1], [1, 4, -5, 1], [1, 5, -4, 3]].
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Solve the system of homogeneous equations: x + 3y - 2z=0, 2x - y + 4z=0, x - 11y + 14z=0.
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Answer the following:
Define a Hermitian and a skew-Hermitian matrices and give an example in each case.
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Solve the equations 5x+3y+7z=4, 3x+26y+2z=9, 7x+2y+10z=5, using a matrix method.
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Find the Eigen values and Eigen vectors of
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Using Cayley-Hamilton theorem for the matrix A = [[2, 1, 1], [0, 1, 0], [1, 1, 2]], find the inverse of A. Also express B = A8 - 5A7 + 7A6 - 3A5 + A4 - 5A3 + 8A2 - 2A + I as a quadratic polynomial in A.
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Verify Rolle's theorem for f(x) = sin(x)/ex in [0, π].
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If u = log(x3+y3+3xyz), find du/dx.
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Verify Cauchy's mean value theorem for the functions log(x) and 1/x in [1, e].
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A rectangular box open at the top is to have volume of 32 cubic ft. Find the dimensions of the box requiring least material for its construction.
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Solve (1+x)dy = (1-y)dx.
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Uranium disintegrates at a rate proportional to the amount then present at any instant. If M1 and M2 grams of Uranium are present at times T1 and T2 respectively, find the half-life of Uranium.
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Solve (1+xy)y dx + (1-xy)x dy = 0.
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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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Solve y'' + 4y' + 4y = 3sin x + 4cos x, y(0)=1, y'(0)=0.
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Solve by method of variation of parameters d2y/dx2 + a2y = sec ax.
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