Jawaharlal Nehru Technological University Hyderabad
Jawaharlal Nehru Technological University Hyderabad B.Tech I Year MATHEMATICS I Previous Question Paper (December - 2017 - 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
Find the rank of the matrix:
Answer
Performing row operations on the matrix:
The matrix becomes:
Further operations R3 -> R3 + R2 yield:
There are 3 non-zero rows, so the rank is 3.
Find the nature of the quadratic form x2 + y2 + 2xy.
Answer
The quadratic form is f(x, y) = x2 + 2xy + y2 = (x + y)2. The matrix representation is:
The eigenvalues are found by det(A - λI) = (1-λ)2 - 1 = λ2 - 2λ = 0, giving λ = 0, 2. Since the eigenvalues are non-negative and one is zero, the quadratic form is positive semi-definite.
If x = r cos θ, y = r sin θ, then find the Jacobian J(x,y)/(r,θ).
Answer
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State Rolle’s mean theorem and explain its geometrical interpretation.
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Evaluate the integral of sin3θ cos5θ from 0 to π/2 using beta and gamma functions.
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Evaluate the triple integral of (x+y+z) dxdydz over the limits x:0 to 1, y:0 to 2, z:0 to 3.
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Find P.I of 1/(D2 + 16) sin x.
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Find the flow of the current in simple closed LR-circuit, initially the current is zero where L = 2H, R = 4Ω and source of the voltage E(t) = et, t > 0.
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Find L-1[1/(s2 - 2s + 5)].
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Let L{f(t)} = f(s), Prove that L{f(at)} = 1/a f(s/a).
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Part B
If a, b, c are distinct non-zero numbers, show that the homogeneous system with the provided coefficient matrix has no non-trivial solution. Find the Eigen values and Eigen vectors of the matrix.
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Reduce the quadratic form to canonical form and find the nature. Find the value of ‘k’ such that the rank of the matrix is 3.
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Examine the maxima and minima of the function and find the approximate value of 5th root of 245 using Lagrange’s mean value theorem.
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Find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid and verify Rolle’s theorem for log((x+a+b)/(x+ab)) on [a, b].
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Define Beta function, prove properties and evaluate the integral of x^2*sqrt(2-x) dx from 0 to 2.
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Evaluate the double integral of xy dxdy where R is the region bounded by x-axis, x = 2a and the curve x^2 = 4ay.
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Solve the given differential equations and find the orthogonal trajectory of the family of cardioids.
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Solve the differential equations using the method of undetermined coefficients or variation of parameters.
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Find the Laplace transform of the given function and solve the integral equation.
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Evaluate the given definite integral using Laplace transforms and solve the initial value problem for y'' + 4y = 0.
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