JNTUH B.Tech - R25 - Matrices and Calculus - Important Questions

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JNTUH B.Tech - R25 - Matrices and Calculus - Important Questions

Unit 1 : Matrices

• Define Matrix Rank and explain how to find the rank of a matrix using echelon form.
• Determine the rank of a matrix using normal form.
• Explain the Gauss-Jordan Method to find the inverse of a non-singular matrix.
• Find the inverse of a non-singular matrix using the Gauss-Jordan method.
• Solve a system of Homogeneous Linear Equations.
• Solve a system of Non-Homogeneous Linear Equations.
• Discuss the consistency of a system of linear equations using matrix rank.
• State and explain the conditions for the existence of unique, infinite, and no solutions to a system of linear equations.
• Explain the Gauss-Seidel Iteration Method for solving simultaneous equations.
• Solve a system of linear equations using the Gauss-Seidel iteration method.

Unit 2 : Eigen values and Eigen vectors

• Define Linear Transformation and Orthogonal Transformation.
• Define Eigenvalue and Eigenvector and explain their properties.
• Find the eigenvalues and eigenvectors of a given matrix.
• Explain the process of Matrix Diagonalization.
• State and apply the Cayley-Hamilton Theorem.
• Find the inverse of a matrix using the Cayley-Hamilton theorem.
• Find higher powers of a matrix using the Cayley-Hamilton theorem.
• Define a Quadratic Form and determine its nature (positive definite, negative definite, indefinite).
• Reduce a quadratic form to canonical form using orthogonal transformation.
• Explain the significance and applications of eigenvalues, eigenvectors, and quadratic forms in engineering and data analysis.

Unit 3 : Single Variable Calculus

• Define Limit of a Function and Continuity of a Function and explain their properties.
• State and explain Rolle's Theorem with geometrical interpretation.
• State and explain Lagrange's Mean Value Theorem with geometrical interpretation and applications.
• State and explain Cauchy's Mean Value Theorem.
• Verify Rolle’s theorem for a given function.
• Verify Lagrange’s Mean Value Theorem for a given function and find the corresponding value of c.
• Apply Cauchy’s Mean Value Theorem to suitable functions.
• Expand a function using Taylor Series.
• Explain the steps involved in tracing curves in Cartesian coordinates.
• Trace curves in Cartesian coordinates by identifying symmetry, intercepts, tangents, and asymptotes.

Unit 4 : Multivariable Calculus (Partial Differentiation and applications)

• Define limits and continuity for functions of several variables and explain their significance.
• Define Partial Differentiation and solve basic problems.
• State and prove Euler's Theorem for Homogeneous Functions and explain its applications.
• Define and explain the Total Derivative.
• Define the Jacobian and discuss its properties.
• Explain Functional Dependence and Independence using Jacobians.
• Find the Jacobian of a given transformation.
• Determine whether functions are functionally dependent or independent.
• Find maxima and minima of functions of two variables and three variables.
• Explain the Method of Lagrange Multipliers and solve constrained maxima and minima problems.

Unit 5 : Multivariable Calculus (Integration)

• Define Double Integral and explain its evaluation in Cartesian coordinates.
• Evaluate double integrals in Polar Coordinate System.
• Explain the procedure for change of order of integration in Cartesian coordinates.
• Explain change of variables for double integrals from Cartesian to polar coordinates using the Jacobian.
• Define Triple Integral and explain its evaluation.
• Evaluate triple integrals in Cartesian coordinates.
• Explain change of variables for triple integrals from Cartesian to Cylindrical Coordinate System and Spherical Coordinate System.
• Find the area of a plane region using double integrals.
• Find the volume of a solid using triple integrals.
• Compare Cartesian, polar, cylindrical, and spherical coordinate systems and state when each is most convenient.