Number Theory and Linear Algebra:
1. Show that the square of any odd integer is of the form 8k + 1.
2. Prove that every positive integer can be expressed uniquely as a sum of distinct powers of 2.
3. Find the general solution of the system of linear equations: 2x + y - z = 3, x + 2y + 3z = 4, 3x - y + 2z = 5.
4. Determine the rank and nullity of the matrix: A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
Calculus and Real Analysis:
5. Find the derivative of the function: f(x) = x^3 + 2x^2 - 5x + 1.
6. Evaluate the integral: ∫(x^2 + 3x - 1) dx.
7. Determine the convergence or divergence of the infinite series: ∑ (1/n^2).
8. State and prove the Mean Value Theorem for functions defined on a real interval.
Differential Equations:
9. Solve the first-order linear differential equation: dy/dx + y = e^x.
10. Find the general solution of the second-order homogeneous differential equation: y'' - 4y' + 3y = 0.
11. Use the Laplace transform to solve the initial value problem: y'' + 4y' + 3y = 0, y(0) = 1, y'(0) = 0.
12. Discuss the existence and uniqueness of solutions to first-order initial value problems.
Algebra:
13. Simplify and solve for x: (log2(x+2))(log4(2x-1)) = 3.
14. Find all solutions to the equation: x^3 - 2x^2 - x + 2 = 0.
15. Determine the eigenvalues and eigenvectors of the matrix: A = [[2, 1], [-1, 2]].
16. Prove that every finite integral domain is a field.
Statistics:
17. Calculate the mean, median, and mode of the following data: 10, 12, 15, 17, 20, 22, 25.
18. Construct a frequency distribution table and draw a histogram for the data: 90, 85, 92, 88, 95, 98, 94.
19. Test the hypothesis that the population mean is equal to 100, with a sample mean of 95 and a sample standard deviation of 5, using a two-tailed t-test at a significance level of 0.05.
20. Calculate the correlation coefficient between two variables: x = {10, 12, 15, 17, 20} and y = {20, 22, 25, 27, 28}.
Please note that this question bank is not exhaustive and can vary based on the specific syllabus and curriculum of the Madras University Institute of Distance Education.