JNTU
JNTU B.Tech I MATRICES AND CALCULUS Previous Question Paper (March/April - 2023 - R22)
Part-A is a compulsory question which consists of ten sub-questions from all units carrying equal marks. 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
Answer the following ten sub-questions:
What is the value of ‘k’ if the rank of the matrix
Answer
For the rank to be 2, the determinant of the matrix must be 0. Calculating the determinant: 1(10k - 42) - 2(20 - 21) + 3(12 - 3k) = 10k - 42 + 2 - 12 + 9k = 0, which simplifies to 19k - 52 = 0. Thus, k = 52/19.
Find the value of ‘a’ for which the equations have infinite number of solutions:
Answer
The system has infinite solutions when the determinant of the coefficient matrix is 0. Calculating the determinant: 1(-a^2 + 9) - 1(a^2 - 15) + 1(-3a + 5a) = -a^2 + 9 - a^2 + 15 + 2a = -2a^2 + 2a + 24 = 0. Solving a^2 - a - 12 = 0 gives (a-4)(a+3) = 0, so a = 4 or a = -3.
If the eigenvalues of
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What is the nature of the quadratic form
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Find the value of the constant in Cauchy’s mean value theorem for
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Find the Taylor’s expansion of
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If
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Find the stationary points of the function
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Evaluate
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In the integral
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Part B
Find rank of matrix A by reducing it into normal form.
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Solve the system of equations: x+y+54z=110; 27x +6y –z =85; 6x+15y +2z =72 using Gauss-Seidel method.
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Solve the system of equations by Gauss -elimination method 5x+y+z+w=4, x+7y+z+w=12, x+y+6z+w=-5, x+y+z+4w=-6.
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Find the inverse of A using Gauss-Jordan method.
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Find the eigenvalues and the corresponding eigen vectors of the given 3x3 matrix.
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Using Cayley Hamilton theorem for the matrix A, find A^4.
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Reduce the quadratic form 3x^2 + 2y^2 + 3z^2 - 2xy - 2yz to the canonical form by orthogonal transformations and find rank, index, signature, nature of the quadratic form.
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Find the region in which f(x) = 1-4x-x^2 is increasing and the region in which it is decreasing using Mean Value Theorem.
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Find the volume of the solid generated by the revolution of the area bounded by y = x^2 and y = x about y-axis.
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Expand tan^-1(x) in powers of (x – 1) up to the term containing fourth degree.
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Evaluate the integral of log(x) from 0 to 1.
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If u = sin^-1((x+y)/(sqrt(x)+sqrt(y))), find the value of x(du/dx) + y(du/dy).
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If x + y + z = u, y + z = uv, z = uvw, then evaluate the Jacobian of (u, v, w) with respect to (x, y, z).
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If x = e^r sec(theta), y = e^r tan(theta), prove that the Jacobian product equals 1.
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The Temperature T at any point (x, y, z) in space is T = 400xyz^2. Find the highest temperature at the surface of the unit sphere x^2 + y^2 + z^2 = 1.
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By changing the order of integration, evaluate the double integral.
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Find the volume bounded by the cylinder x^2 + y^2 = 4 and the planes z = 0 and y + z = 4.
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Evaluate the integral by changing into polar coordinates.
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Using spherical polar coordinates, evaluate the triple integral of xyz / sqrt(x^2+y^2+z^2) over the first octant of the sphere x^2+y^2+z^2 = a^2.
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