JNTU
JNTU B.Tech I Year MATHEMATICS - I Previous Question Paper (May - 2018 - R13)
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 and may have a, b, c as sub questions.
Part A
If A and B are square symmetric matrices of same order then prove that AB + BA is symmetric.
Answer
Given A and B are symmetric, 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, we get BA + AB = AB + BA. Since the transpose equals the original matrix, AB + BA is symmetric.
If one of Eigen vectors of
Answer
Let A = [[2, 2, 0], [2, 5, 0], [0, 0, 3]] and X = [[1], [2], [0]]. The eigenvalue λ satisfies AX = λX. Calculating AX: [[2, 2, 0], [2, 5, 0], [0, 0, 3]] * [[1], [2], [0]] = [[2(1)+2(2)+0(0)], [2(1)+5(2)+0(0)], [0(1)+0(2)+3(0)]] = [[6], [12], [0]]. This is equal to 6 * [[1], [2], [0]]. Thus, 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.
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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 x^2 + 6y^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].
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Find the maximum and minimum values of x^3 + y^3 - 3axy.
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If x = e^θ cos φ, y = e^θ sin φ, find J and verify that J * J' = 1, where J = ∂(x, y) / ∂(θ, φ).
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Evaluate ∫[0 to a] x^2 √(a^2 - x^2) dx.
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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^(m-1) (1-x)^(n-1) dx = Γ(m)Γ(n) / Γ(m+n).
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Evaluate by changing into polar coordinates: ∫[0 to a] ∫[0 to √(a^2-y^2)] (x^2+y^2) dx dy.
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Find the orthogonal Trajectory of the family of ax^2 = y^2.
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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?
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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.
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Find the inverse Laplace transform of log((s+1)/(s-1)).
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Solve the differential equation (D^2 + 1)y = t^2, using Laplace transform given that y(0) = 4, y'(0) = 2.
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