Jawaharlal Nehru Technological University Hyderabad
Jawaharlal Nehru Technological University Hyderabad B.Tech I MATHEMATICS - I Previous Question Paper (May - 2018 - R15)
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
If A and B are square symmetric matrices of same order then prove that AB + BA is symmetric.
Answer
Given that A and B are symmetric, we have AT = A and BT = B. To prove AB + BA is symmetric, we take its transpose: (AB + BA)T = (AB)T + (BA)T = BTAT + ATBT. Substituting the symmetric properties, this becomes BA + AB, which is equal to AB + BA. Thus, the matrix is symmetric.
If one of Eigen vectors of
Answer
Let A = [[2, 2, 0], [2, 5, 0], [0, 0, 3]] and v = [1, 2, 0]T. The eigenvalue λ satisfies Av = λv. Calculating Av: [[2, 2, 0], [2, 5, 0], [0, 0, 3]] * [1, 2, 0]T = [2(1)+2(2)+0(0), 2(1)+5(2)+0(0), 0(1)+0(2)+3(0)]T = [6, 12, 0]T. Since [6, 12, 0]T = 6 * [1, 2, 0]T, the eigenvalue λ = 6.
Find the value of c in Roll’s theorem for
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Find the stationary points of the following functions
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Evaluate
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Evaluate
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Solve the differential equation
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Evaluate
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Find
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Find f(t), if
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Part B
Test for the consistency and hence solve the system: x + y + z = 6, x – y + 2z = 5, 3x + y + z = 8, 2x – 2y + 3z = 7
If λ1, λ2, ..., λn are the Eigen values of a non-singular matrix A of order n then prove that the Eigen values of A^-1 are 1/λ1, 1/λ2, ..., 1/λn.
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Reduce the quadratic form 6x^2 + 3y^2 + 3z^2 - 4xy - 2yz + 4xz by orthogonal reduction to the canonical form.
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Verify Lagranges mean value theorem for f(x) = log x in [1, e].
Find the maximum and minimum values of x^3 + y^3 - 3axy.
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If x = r cosθ and y = r sinθ, find J = ∂(x,y)/∂(r,θ) and verify that JJ' = 1.
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Evaluate ∫[0 to a] ∫[0 to √(a^2-x^2)] x^2 √(x^2+y^2) dy dx.
Change the order of integration and evaluate ∫[0 to a] ∫[x to a/x] xy dy dx.
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Prove that ∫[0 to 1] x^n(1-x)^m dx = Γ(n+1)Γ(m+1)/Γ(n+m+2).
Evaluate ∫∫[0 to a] ∫[0 to √(a^2-x^2)] (x^2+y^2) dy dx by changing into polar coordinates.
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Find the orthogonal trajectory of the family of ay^2 = x^3.
Solve the differential equation (D^2 + 9)y = cos 3x + sin 2x.
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If a population is increasing exponentially at the rate of 2% per year. What will be the percentage increase over a period of 10 years?
Solve by the method of variation of Parameters: d^2y/dx^2 + y = sec x.
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Evaluate ∫[0 to ∞] (sin t)/t dt.
Find the inverse Laplace transform of log((s+1)/(s-1)).
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Solve the differential equation (D^2 + 2D + 2)y = t^2, using Laplace transform given that y(0)=4, y'(0)=2.
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