JNTU
JNTU B.Tech I MATRICES AND CALCULUS Previous Question Paper (January/February - 2024 - R22)
Note: This question paper contains two parts A and B. Part-A is a compulsory question. Part-B consists of ten questions (numbered from 2 to 11) carrying 10 marks each. From each unit, there are two questions and the student should answer one of them. Hence, the student should answer five questions from Part-B.
Part A
Write the row elementary transformations on a matrix.
Answer
The row elementary transformations are: (i) Interchanging any two rows (Ri ↔ Rj), (ii) Multiplying a row by a non-zero scalar (Ri → kRi), and (iii) Adding a scalar multiple of one row to another row (Ri → Ri + kRj).
State the conditions for the existence of solution of AX=B.
Answer
The system AX=B is consistent if and only if the rank of the coefficient matrix A is equal to the rank of the augmented matrix [A|B], i.e., ρ(A) = ρ(A|B). If ρ(A) = ρ(A|B) = n (number of variables), the system has a unique solution; if ρ(A) = ρ(A|B) < n, it has infinitely many solutions.
Define the Eigenvalues and Eigenvectors of a matrix A.
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What is meant by index of a quadratic form?
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Is f(x) = |x| in [-1, 1] differentiable.
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State Rolle’s Mean value theorem.
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When we say the function is homogeneous?
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Write the relation between Gamma and Beta functions.
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Change the order of integral in 0 to infinity, 0 to infinity (e^-y/y) dy dx.
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Is integral of 0 to 1, 0 to 2, 0 to 3 dxdydz = 0 to 1, 0 to 2, 0 to 3 dzdydx?
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Part B
Reduce the matrix A = [[2, 1, 3, 4], [0, 3, 4, 1], [2, 3, 7, 5], [2, 3, 11, 6]] to the (a) Echelon form, (b) Normal form and hence find its rank.
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Using the Gauss-Seidel iteration method, solve the system of equations: 4x1 + x2 + x3 = 2, x1 + 5x2 + 2x3 = -6, x1 + 2x2 + 3x3 = -4.
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Verify Cayley-Hamilton theorem for matrix A = [[2, 1, 1], [0, 1, 0], [1, 1, 2]] and hence find A^-1 and A^8 - 5A^7 + 8A^6 - 3A^5 + 4A^4 - 11A^3 + 6A^2 - 11A + I.
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Reduce the quadratic form x^2 + 3y^2 + 3z^2 - 2yz to the canonical form. Hence find its rank, nature, index and signature.
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If f(x) = log x and g(x) = x^2 in [a, b] with 1 < a < b using Cauchy’s mean value theorem, prove or disprove that log(b/a) / (b^2 - a^2) = 1 / (2c^2) for a < c < b.
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Evaluate the Taylor series expansion of f(x) = tan^-1(x) about x = 0.
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Determine the integral 0 to 1 of x^3 * (ln(1/x))^4 dx.
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Evaluate 0 to pi/2 of (tan theta + sec theta)^2 d(theta).
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If xz + yz - x - y = 0, then show that (delta z / delta x)^2 + (delta z / delta y)^2 = (delta z / delta x - 1)^2 + (delta z / delta y - 1)^2.
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Given that x = x1(1-x2), y = x1*x2(1-x3), z = x1*x2*x3. Show that the Jacobian is x1^2 * x2.
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Discuss the maxima and minima of x^3 + y^3 - 3axy.
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Compute the double integral of (x^2 + y^2) dx dy, where D is bounded by y = x and y = x^2.
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Evaluate 0 to a, 0 to sqrt(a^2 - x^2) of dy dx / (x^2 + y^2) by transforming into polar coordinates.
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Obtain the value of 0 to 1, 0 to 1, 0 to 2 of x*y*z dx dy dz.
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Determine the triple integral of xyz dx dy dz taken through the positive octant of the sphere x^2 + y^2 + z^2 = a^2.
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