JNTU
JNTU B.Tech I Year MATHEMATICS-I Previous Question Paper (March/April - 2023 - R13)
Note: i) Question paper consists of Part A, Part B.
ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions.
iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Define Hermitian, Skew-Hermitian and Unitary Matrices.
Answer
A square matrix A is Hermitian if
Define the rank and nature of a quadratic form.
Answer
The rank of a quadratic form is the number of non-zero terms in its canonical form. The nature is determined by the signs of the coefficients in the canonical form: positive definite (all positive), negative definite (all negative), positive semi-definite (non-negative with at least one zero), negative semi-definite (non-positive with at least one zero), or indefinite (mixed signs).
Write the geometrical interpretation of Lagrange’s mean value theorem.
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Explain in brief the method of Lagrange multipliers.
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Show that
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Evaluate
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Write the differential equation of the family of orthogonal trajectories of the curves
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Find the general solution of
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Write the linearity property in Laplace transforms.
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Define a periodic function. What is the Laplace transform of a periodic function?
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Part B
Find the rank of the matrix A = [[1, 2, 2, 4], [2, 3, 4, 6], [3, 5, 6, 10], [1, -1, 2, -2]].
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Solve completely, the system of equations x + y + z + w = 0, x - y + z - w = 0, x + y - z + w = 0, 4x + 5y + 8z + 2w = 0.
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Verify Cayley-Hamilton theorem for the matrix A = [[1, 2, 1], [2, -1, 2], [2, -2, 1]] and hence find its inverse.
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State Lagrange's mean value theorem, and using it prove that when 0 < a < b < 1, (b-a)/(1+b²) < tan⁻¹b - tan⁻¹a < (b-a)/(1+a²). Hence show that (π/4) + (3/46) < tan⁻¹(1/2) < (π/4) + (1/25).
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Find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid x²/a² + y²/b² + z²/c² = 1.
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Express the following integrals in terms of Gamma function: i) ∫₀^{π/2} tan θ dθ ii) ∫₀^{∞} e^{-ax} x^{b-1} dx
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Find Γ(-11/2).
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By changing into polar coordinates, evaluate ∫₀^{∞} ∫₀^{∞} e^{-(x²+y²)} dxdy.
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Using triple integral, find the volume of a sphere of radius ‘r’.
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Solve dy/dx + xy = xy².
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If the temperature of the air is 30°C, and the substance cools from 100°C to 80°C in 10 minutes, find the temperature of the substance after 20 minutes.
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Solve (D²-2D+2)y = e^x(x² + sin 2x).
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Find the Laplace transform of the function:
f(t) = e^{-t} (2 sin 3t + 3 sin 4t) + t³
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∫₀^{∞} (sin t / t) dt
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Solve by Laplace transform technique: d²y/dt² + 2dy/dt - 3y = sin t, where y(0)=0, y'(0)=0.
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