JNTU
JNTU B.Tech I Year MATHEMATICS-I Previous Question Paper (December - 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.
Part A
Examine the consistency of the following system of equations: 12x - y - z = 2; x + 2y + z = 2; 4x - 7y - 5z = 2.
Answer
The augmented matrix [A|B] is:
Performing row operations: R1 ↔ R2 gives [[1, 2, 1, 2], [12, -1, -1, 2], [4, -7, -5, 2]]. R2 → R2 - 12R1 and R3 → R3 - 4R1 gives [[1, 2, 1, 2], [0, -25, -13, -22], [0, -15, -9, -6]]. Since rank(A) = rank(A|B) = 3, the system is consistent and has a unique solution.
Find A^2 using Cayley-Hamilton theorem, given A = [[3, 1], [-1, 2]].
Answer
The characteristic equation is |A - λI| = 0, which is (3-λ)(2-λ) + 1 = 0, leading to λ^2 - 5λ + 7 = 0. By Cayley-Hamilton theorem, A^2 - 5A + 7I = 0. Thus, A^2 = 5A - 7I = 5[[3, 1], [-1, 2]] - 7[[1, 0], [0, 1]] = [[15-7, 5-0], [-5-0, 10-7]] = [[8, 5], [-5, 3]].
State Lagrange’s Mean Value theorem.
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Find the stationary points of x^3y^2(1 - x - y).
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Transform the integral of (x^2 + y^2) over the region to polar coordinates.
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Evaluate integral from 0 to infinity of x^5 * e^(-2x) dx.
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Find the particular integral of (D^2 + 4)y = 2e^x * sin x.
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Find L{e^kt}.
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Find L{t * e^t}.
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Find L^-1{1 / (s(s+a))}.
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Part B
Find the matrix which transforms the matrix A = [[3, 1, 4], [2, 2, 4], [4, 1, 3]] to diagonal form.
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Reduce the quadratic form 6x1^2 + 3x2^2 + 3x3^2 - 4x1x2 - 2x2x3 + 4x3x1 to canonical form.
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Verify Rolle’s theorem for the function f(x) = 2x^3 + x^2 - 4x - 2 in [-2, 2]. Also find the value of C.
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Show that the functions u = x + y - z, v = x - y + z, w = x^2 + y^2 + z^2 - 2yz are not independent of one another. Also find the relation among them.
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Verify Cauchy Mean Value theorem for the functions f(x) = log x, g(x) = 1/x in [1, e].
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Find the stationary value of f(x, y, z) = x^2y^3z^4 subject to the condition x + y + z = 5.
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Change the order of integration in the integral and hence evaluate it.
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Find the area included between the curve r = a(sec θ + cos θ) and its asymptote.
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Derive the relationship between beta and gamma functions.
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Evaluate the triple integral of e^(x+y+z) dz dy dx.
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The rate of disintegration of a radioactive substance is proportional at any instant to the amount of substance present. It is found that 0.5% of the substance disappears in 12 years. What percentage will disappear in 1000 years?
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Solve the differential equation y'' + y = -cot x using the method of variation of parameters.
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Find the orthogonal trajectories of the confocal conics x^2/a^2 + y^2/(b^2+λ) = 1, where λ is the parameter.
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Solve the differential equation (D^2 - 5D + 6)y = xe^x.
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Find the Laplace transform of (sin t / t) * e^t.
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Find L^-1{log((s+a)/(s+b))}.
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State convolution theorem of Laplace transform and use it to find the inverse Laplace transform of s / (s^2 + a^2)^2.
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Solve the equation x'' + 2x' + 5x = e^-t sin t under the conditions x(0) = 0, x'(0) = 1.
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