JNTU
JNTU B.Tech I Year MATHEMATICS I Previous Question Paper (December - 2018 - R13)
Note: This question paper contains two parts A and B. 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
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:
Reducing to row echelon form, we find the rank of the coefficient matrix ρ(A) = 2 and the rank of the augmented matrix ρ(A|B) = 3. Since ρ(A) ≠ ρ(A|B), the system is inconsistent and has no solution.
Find A2 using Cayley-Hamilton theorem, given A = [3 1; -1 2].
Answer
The characteristic equation is |A - λI| = 0, which is λ2 - 5λ + 7 = 0. By Cayley-Hamilton theorem, A2 - 5A + 7I = 0, so A2 = 5A - 7I.
State Lagrange’s Mean Value theorem.
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Find the stationary points of x3y2(1 - x - y).
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Transform the integral of x2 + y2 dx dy over the region bounded by x=0, y=0, y=a, x=a to polar coordinates.
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Evaluate the integral from 0 to infinity of x5e-2x dx.
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Find the particular integral of (D2 + 4)y = 2ex sin x.
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Find L{ekt}.
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Find L{tet}.
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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 6x12 + 3x22 + 3x32 - 4x1x2 - 2x2x3 + 4x3x1 to canonical form.
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Verify Rolle’s theorem for the function f(x) = 2x3 + x2 - 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 = x2 + y2 + z2 - 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) = x2y3z4 subject to the condition x + y + z = 5.
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Change the order of integration in the integral and hence evaluate it integral xy dx dy.
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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 ex+y+z dz dy dx over the specified limits.
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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 x2/a2 + y2/(b2+λ) = 1, where λ is the parameter.
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Solve the differential equation (D2 - 5D + 6)y = xex.
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Find the Laplace transform of (sin t) / (t et).
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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 / (s2+a2)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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